[    {        "title": "1609.01330v1.Critical_Current_Oscillations_of_Josephson_Junctions_Containing_PdFe_Nanomagnets.pdf",        "content": "arXiv:1609.01330v1  [cond-mat.supr-con]  5 Sep 2016Critical Current Oscillations of Josephson Junctions\nContaining PdFe Nanomagnets\nJoseph A. Glick, Reza Loloee, W. P. Pratt, Jr. and Norman O. Bi rge\nDept. of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA\nEmail: birge@pa.msu.edu\nManuscript received September 5, 2016\nAbstract—Josephson junctions with ferromagnetic layers are\nvital elements in a new class of cryogenic memory devices. On e\nstyle of memory device contains a spin valve with one “hard”\nmagnetic layer and one “soft” layer. To achieve low switchin g\nfields, it is advantageous for the soft layer to have low magne tiza-\ntion and low magnetocrystalline anisotropy. A candidate cl ass of\nmaterials that fulfills these criteria is the Pd 1−xFexalloy system\nwith low Fe concentrations. We present studies of micron-sc ale\nelliptically-shapedJosephson junctionscontainingPd 97Fe3layers\nof varying thickness. By applying an external magnetic field , the\ncritical current of the junctions are found to follow charac teristic\nFraunhofer patterns. The maximum value of the critical curr ent,\nextracted from the Fraunhofer patterns, oscillates as a fun ction\nof the ferromagnetic barrier thickness, indicating transi tions in\nthe phase difference across the junction between values of z ero\nandπ.\nIndex Terms —Superconductivity, Josephson Junction, Ferro-\nmagnetism, Cryogenic Memory, Proximity Effect\nI. INTRODUCTION\nJosephson junctions containing ferromagnetic (F) layers\nare being studied by many researchers to create an energy-\nefficient, fast, non-volatile memory for superconducting c om-\nputing [1]–[7]. Recently our group demonstrated that a phas e-\ncontrollable memory element can be made from a Supercon-\nducting QUantum Interferance Device (SQUID) containing\ntwo Josephson junctionswith the structure S/F′/N/F′′/S, where\nS is a superconductor, F and F′are ferromagnetic materials,\nand N is a normal metal [7]. In this and other similar propos-\nals [1], [5], [6], one of the ferromagnetic layers (F′′, the “free\nlayer”) can be made to switch it’s magnetization direction t o\nbe parallel or anti-parallel to the other layer (F′, “hard layer”)\nby application of a small magnetic field. The thicknesses of\nthe F′and F′′layers are set so that when their magnetization\nvectors are parallel the junction is in the π-phase state, and\nwhen the two layers are anti-parallel the junction is in the\n0-phase state, as dictated by the superconducting proximit y\neffect.\nTo maximize energy-efficiency in a memory application, it\nis desirable to use a free layer whose magnetization directi on\ncan be controllably switched by a very low applied field. The\nmagnetic material used for the free layer should thus have lo w\nmagnetization and low magnetocrystalline anisotropy. Her e\nwe study the properties of the soft magnetic alloy Pd 97Fe3,\nwhich is underconsiderationfor the free layer material. Di lute\nPdFe alloys have been known for several decades to have verylow magnetocrystalline anisotropy [8], and our own previou s\nwork on the alloy with 1.3% Fe concentration found it to\nhave a spin diffusion length of 9.6 ±2 nm [9]. Josephson\njunctions containing PdFe with a lower Fe concentration of\n≈1% have already been studied by other groups [2]–[4] with\naneyetowardapplicationsincryogenicmemory.Wehavetrie d\nusing Pd 98.7Fe1.3as the free layer in controllable spin-triplet\nJosephson junctions [10], but the results were not satisfac tory.\nThat work provided the main motivation for studying PdFe\nalloys with somewhat higher Fe concentrations.\nII. SAMPLEFABRICATION AND CHARACTERIZATION\nWe first characterized the magnetic properties of unpat-\nterned continuous Pd 97Fe3films via SQUID magnetometry.\nThin films of Nb(5)/Cu(5)/PdFe(d F)/Cu(5)/Nb(5), with thick-\nnesses in nanometers, were deposited via dc sputtering in\nan Argon plasma with pressure 1.3 ×10−3Torr. Prior to\nsputtering the base pressure of the chamber was 2 ×10−8\nTorr. During the deposition the sample temperature was held\nbetween−30◦Cand−20◦C. The thicknesses of the various\ndeposited materials were controlled by measuring the depos i-\ntion rates (accurate to ±0.1˚A/s) using a crystal film thickness\nmonitor.\nThe samples were measured using a Quantum Design\nSQUID magnetometer at 5 K, with the applied magnetic\nfield parallel to the film plane. The hysteresis loops of films\nwithdPdFe= 8-16 nm are shown in Fig. 1. The saturation\nmagnetization per unit volume is nearly constant for the thr ee\nsamples. Plotting the saturation magnetic moment divided b y\nthe sample area versus dPdFeand fitting to a straight line gives\na slope which corresponds to a magnetization of M=90±9\nkA/m.Meanwhile,thex-interceptshowsamagneticdeadlaye r\nthickness of ddead=2.8±0.9nm. Note that these unpatterned\nfilms contain many magnetic domains so that the switching\nmechanism is governed by domain-wall motion; hence the\nfilm results should not be directly compared to the switching\nbehavior of the nanomagnets in our SFS junctions, discussed\nlater.\nIn a separate sputtering run we fabricated SFS\nJosephson junctions containing PdFe using the same\ntechniques described above. Prior to sputtering,\nphotolithography was used to define the geometry of\nour bottom wiring layer, which consists of the sequence\n[Nb(25)/Al(2.4)] 3/Nb(20)/Cu(5)/PdFe(d F)/Cu(5)/Nb(5)/Au(15),/s45/s56 /s45/s52 /s48 /s52 /s56/s45/s49/s48/s48/s48/s45/s53/s48/s48/s48/s53/s48/s48/s49/s48/s48/s48\n/s32/s56/s32/s110/s109\n/s32/s49/s50/s32/s110/s109\n/s32/s49/s54/s32/s110/s109\n/s32/s32/s109/s47/s65/s32/s40 /s74/s47/s84/s109/s50\n/s41\n/s72/s32/s40/s109/s84/s41\nFig. 1. Hysteresis loops of unpatterned films containing PdF e with thickness\ndPdFespanning 8-16 nm. Plotted is the magnetic moment divided by t he\nsample area versus the applied field, measured using SQUID ma gnetometry.\nFor the three samples the magnetization is approximately co nstant,M=90\n±9 kA/m. The data are slightly shifted along the field axis due t o a small\namount of trapped flux in the solenoid of the SQUID magnetomet er. From\nthe data we extract a magnetic dead layer thickness of ddead=2.8±0.9nm,\ndiscussed in the text.\nand was sputtered without breaking vacuum. A schematic of\nthe full sample structure is shown in Fig. 2.\nTo achieve sharp magnetic switching we grew the ferro-\nmagnets on a smooth [Nb/Al] multilayer used in previous\nworks [11]–[13]. Using atomic force microscopy (AFM),\nthe roughness of the [Nb/Al] multilayer we independently\nmeasured was ≈2.3˚A, which is smoother than our sputtered\nNb(100) films with roughness >5˚A. A 5 nm Cu spacer layer\nwas used on either side of the ferromagnet to improve it’s\nmagnetic properties and the samples were capped with a thin\nlayer of Nb and Au to prevent oxidation.\nThe elliptically-shaped junctions were patterned via\nelectron-beam lithography followed by ion milling in Argon ,\nwith the same process used our previous work [14], [15]. The\njunctions have an aspect ratio of 2.5 and area of 0.5 µm2,\nwhich is small enough to make some magnetic materials, such\nas NiFe and NiFeCo, single domain [15].\nOutside the mask region, ion milling was used to etch\nthrough the capping layer, the F layer, and half-way into the\nunderlying Cu spacer layer. After ion milling, we thermally\nevaporated a 50 nm thick SiO layer to electrically isolate th e\njunction and the bottom and top wiring layers. During the ion\nmilling and SiO deposition, to prevent the e-beam resist fro m\nover-heating, the back of the substrate was pressed against a\nCu heatsink coated with thin layer of silver paste to improve\nthermal contact.\nFinally, the top Nb wiring layer was patterned using similar\nphotolithography and lift-off processes as the bottom lead s.\nResidual photoresist was cleaned from the surface of the\nsamples with oxygen plasma etching followed by in-situion\nmilling in which 2 nm of the top Au surface was etched away\nprior to sputtering. The sputtered top electrode consists o f\nNb(150 nm)/Au(10 nm), ending with Au to prevent oxidation.\nFig. 2. A schematic showing the vertical cross-sectional st ructure of our SFS\nJosephson junctions. The Pd 97Fe3thickness dFranges from 9 to 36 nm. All\nthicknesses are given in nanometers.\nIII. MEASUREMENT AND ANALYSIS\nThe samples were wired to the leads of a dip-stick probe\nusing pressed indium solder and inserted into a liquid-He\ndewar outfitted with a Cryoperm magnetic shield. A super-\nconducting solenoid on the dipping probe is used to apply\nuniform magnetic fields along the long-axis of the elliptica l\njunctions over a range of -60 to 60 mT. The current-voltage\ncharacteristics of the junctions were measured in a standar d\nfour-terminal configuration at 4.2 K. The I-V curves were\nfoundto have the expectedbehaviorof overdampedJosephson\njunctions [16],\nV=RN/radicalbig\nI2−I2c, I≥Ic, (1)\nwhereIcisthe criticalcurrentand RNisthesample resistance\nin the normal state. RNis the slope of the linear region of the\nI-V curve when |I| ≫Ic, and was independently confirmed\nusing a lock-in amplifier. Measurements of the area-resista nce\nproduct in the normal state yielded consistent values of ARN\n= 11±1 fΩ-m2, an indicator of reproducible high quality\ninterfaces.\n“Fraunhofer” diffraction patterns, shown in Fig. 3, were\nobtained by plotting IcRNas a function of the applied\nmagneticfield.TheexpectedfunctionalformoftheFraunhof er\npattern for elliptical junctions is an Airy function [16],\nIc=Ic0|2J1(πΦ/Φ0)/(πΦ/Φ0)|, (2)\nwhereJ1is a Bessel function of the first kind, Ic0is the\nmaximumcritical current,and Φ0=h/2eis the flux quantum.\nThe magnetic flux through the junction is given by [17]1,\nΦ =µ0Hw(2λL+2dN+dF)+µ0MwdF,(3)\nwhereH,w,dNanddFare the applied field, the junction\nwidth, and the thicknesses of the normal metal and F layer,\nrespectively. λLis the London penetration depth of the Nb\nelectrodes, which we keep fixed at 85 nm, as determined by\ndata obtained in our group over many years [17]. The last\n1We correct a missing factor of µ0in the corresponding equation in\nRef. [17]/s48/s50/s53/s53/s48\n/s48/s53/s49/s48\n/s45/s54/s48 /s45/s51/s48 /s48 /s51/s48 /s54/s48/s48/s53/s49/s48\n/s32/s32/s32/s32/s100\n/s80/s100/s70/s101/s32/s61/s32/s50/s52/s32/s110/s109/s100\n/s80/s100/s70/s101/s32/s61/s32/s49/s53/s32/s110/s109\n/s32/s32\n/s100\n/s80/s100/s70/s101/s32/s61/s32/s57/s32/s110/s109\n/s32/s32/s73\n/s99/s82\n/s78/s32/s40 /s86/s41\n/s32/s32\n/s48/s72/s32/s40/s109/s84/s41/s97/s41\n/s98/s41\n/s99/s41\nFig. 3. PdFe Fraunhofer patterns: Critical current times th e normal state\nresistance, IcRN, is plotted versus the applied field H, for three samples with\ndPdFeequal to (a) 9 nm, (b) 15 nm, and (c) 24 nm. The data before Hswitch,\nthe field at which the PdFe magnetization vector reverses dir ection (solid\nmarkers), and the corresponding fits (lines) to Eqn.2 show go od agreement for\nboth the positive (red, dashed) and negative (blue) field swe ep directions. The\nhollow circles arethecorresponding datapoints after Hswitch.TheFraunhofer\npatterns display magnetic hysteresis and are increasingly shifted with larger\ndF.\nterm in Eqn. 3 describes the flux due to the magnetization\nMof the nanomagnet, which is valid if Mis uniform and\nis oriented in the same direction as the applied field H. In\nEqn. 3 we have omitted the much smaller flux terms from\nthe uniform demagnetizing field and any magnetic field from\nthe nanomagnet that returns between the top and bottom\nNb electrodes. The Fraunhofer pattern will be shifted by an\namountHshift=−MdF/(2λL+dF+2dCu)along the field\naxis due to Eqn. 3.\nThe Fraunhofer patterns in Fig. 3 were collected by the\nfollowing process: First we fully magnetized the nanomagne t\nwith an applied a field of -60 mT, then ramped the field to\n+60 mT in steps of 2.5 mT, measuring Icat each step.\nThe data follow the expected Airy function from the ini-\ntialization field up to the beginning of a small field range,\nHswitch>0, during which the ferromagnet switches the\ndirection of it’s magnetization vector. Beyond Hswitchthe\ndata jump to another Fraunhofer pattern that is shifted in\nthe opposite direction. To measure the magnetic hysteresis ,\nas done in previous works [6], [14], [15], we then swept the\napplied field in the opposite orientation.\nThe data prior to the magnetic switching event were fit to\nEqn. 2 with Ic0,w, andHshiftas fitting parameters. In Fig. 3,\nforboththepositive(red)andnegative(blue)sweepdirect ions,\nthe corresponding fits (lines) show excellent agreement wit h\nthe data (solid markers). The hollow markers denote the\ndata after Hswitch, and closely correspond to the Fraunhofer\npattern in which the field is swept in the opposite orientatio n.\nThe excellent nature of the Fraunhofer patterns allow us to\nextrapolate the maximum value of I c, albeit with a larger\nuncertainty, even when the value of Hshiftapproaches the first/s45/s54 /s45/s51 /s48 /s51 /s54/s54/s56\n/s32/s54/s48/s32/s116/s111/s32/s45/s54/s48/s32/s109/s84\n/s32/s53/s32/s116/s111/s32/s45/s53/s32/s109/s84\n/s32/s45/s54/s48/s32/s116/s111/s32/s54/s48/s32/s109/s84\n/s32/s45/s53/s32/s116/s111/s32/s53/s32/s109/s84\n/s32/s32/s73\n/s99/s82\n/s78/s32/s40 /s86/s41\n/s48/s72/s32/s40/s109/s84/s41\nFig. 4.IcRNis plotted versus the applied field Hfor the same Josephson\njunction shown in Fig. 3(b), zoomed-in on the central peak. S eparate measure-\nments using small initialization fields of ±5 mT and finer step size (green\nand orange data points) show the behavior of the magnetic swi tching. The\nreversal of the PdFe magnetization direction for the two swe ep directions\nbegins at Hswitch,1=1.0 mT (orange) and -0.5 mT (green) and ends at\nHswitch,2= 2.5 mT (orange) and -2.0 mT (green). During the switching\nevent the data deviate from the expected Fraunhofer pattern fit. As the field\napproaches Hswitch,2thedataconverge with thecorresponding measurements\nfrom Fig. 3(b) where much larger ±60 mTinitialization fields were used (blue\nand red points). Lines connect the adjacent finer spaced data for clarity.\nminimum in the Airy function. The nodes in the Fraunhofer\npattern nearly approach Ic= 0, indicating a robust SiO\nbarrier around the junction. The data typically follow the A iry\nfunctionthroughzerofieldbeforetherelativelysharpmagn etic\nswitching event, but for a few samples did not. Therefore, it\nis difficult to determine if the nanomagnets contain a single\nmagnetic domain near zero field.\nThe switching characteristics of the PdFe layer were main-\ntained even when smaller initialization fields were used. Af ter\nreturningthe field to zero, we measuredthe Fraunhoferpatte rn\nagain, sweeping the field from only ±5 mT in both directions\nat finer field steps of 0.5 mT, as shown in Fig. 4 (green\nand orange points), where we have zoomed-in on the central\npeak. It is clear that the junctions switch the direction of t heir\nmagnetization over a range of field values. To characterize\nthe magnetic switching we use two parameters: Hswitch,1,\ndenoting the beginning of the switching event, is the field\nat which Icbegins to deviate from the initial Airy function,\nandHswitch,2, denoting the end of the switching event, is\nthe field at which Icjoins the corresponding shifted Airy\nfunction. Across the range of thicknesses studied, on avera ge\nthe junctions began to switch at a very low field |Hswitch,1|\n= 0.4 mT with standard deviation 0.6 mT, and completed\nthe switching process at |Hswitch,2|= 2.4 mT with standard\ndeviation 0.9 mT. The value of |Hswitch,1|for PdFe is smaller\nthan foundin Ni 81Fe19-based junctionsof similar construction\nmeasured by our group [15], however |Hswitch,2|is compa-\nrable. The low Fe concentration in the Pd 97Fe3alloy may\ngive rise to this gradual switching behavior. Prior work on\nan alloy with lower Fe concentration, Pd 99Fe1, showed that\nthe ferromagnetic behavior of thin films are controlled by th e\npresence of weakly coupled ferromagnetic clusters [18].\nRepeating the measurement at even lower initialization\nfields (3 mT) sometimes caused irregular and irreproducible/s49/s49/s48/s49/s48/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48/s48/s53/s49/s48/s49/s53/s50/s48/s48\n/s32/s32/s73\n/s99/s82\n/s78/s32/s40 /s86/s41/s97/s41\n/s98/s41\n/s32/s32/s48/s72\n/s115/s104/s105/s102/s116/s32/s40/s109/s84/s41\n/s100\n/s80/s100/s70/s101/s32/s40/s110/s109/s41\nFig. 5. a) The maximal Ictimes R Nis plotted versus dPdFefor many\nsamples, with the error bars determined by the goodness of fit parameters\nof the individual Fraunhofer patterns. The minima indicate the critical PdFe\nthicknesses at which the junctions transition between the 0 andπ-phase states.\nThe solid red line is a fit to the data using Eqn. 4. b) The Fraunh ofer pattern\nfield shift Hshiftincreases with dPdFe. The blue line is the fit to Eqn. 5,\nwhich yields M= 72±16 kA/m and ddead= -4±5 nm.\nchanges to IcandHshift. We surmise that too low of an\ninitialization field allows domain walls to form within the\njunction, which disturb the magnetic switching. Hence, if\nPd97Fe3layers are used in cryogenicmemory,an initialization\nfield of at least 5 mT would be necessary to reproducibly\nmagnetize the nanomagnet.\nIn Fig. 5(a) we plot IcRNfor many samples of varying\nferromagnet thicknesses dF. The junctions transition from a 0\ntoπ-phase state at the value of dFwhere the first deep local\nminima occurs. In Fig. 5(a) Icdenotes the maximum critical\ncurrent obtained from the Fraunhofer pattern fits. The 0 to π-\nphase state transition occurs at thicknesses of about dF=16.5\nnm, followed by a π-to-0 phase transition near dF=38 nm.\nTheoretical predictions describe the behavior of IcRNver-\nsusdFas an oscillating function with either an exponential\ndecay for diffusive transport or an algebraic decay for ball istic\ntransport[19].Robinson etal.usedtheballisticformtofit data\nfrom junctions containing very thin elemental ferromagnet ic\nlayers like, Ni, Co, and Fe [20], [21], but when grown\nthicker, the data were better modeled by the diffusive limit .\nIn materials where the majority and minority spin bands have\nnearly identical properties, the diffusive limit is govern ed by\nthe Usadel equations [19]. We find that the diffusive limit\nagreesbest with our PdFe data in Fig. 5, after fitting the poin ts\nto the function,\nIcRN=V0∗e−dF/ξF1∗cos/parenleftbiggdF\nξF2−φ/parenrightbigg\n.(4)\nIn Eqn. 4 ξF1andξF2are length scales that control the\ndecay and oscillation period of IcwithdF, andφis an offset\nphase shift. ξF1,ξF2, andφare used as fitting parameters.\nIn diffusive systems, the simplest model of S/F/S Josephson\njunctions [19] predicts that ξF1=ξF2=/radicalbig\n¯hDF/Eexandφ=π/4, withDFandEexbeing the diffusion constant\nand exchange energy of F, respectively. However, in cases\nwith large spin-orbit or spin-flip scattering, one expects t o\nfindξF1< ξF2[22]. Heim et al.[23] have shown that the\nphase offset, φ, varies sensitively with the thickness and type\nof normal-metal spacer layers or insulating barriers withi n\nthe junction. The best-fit parameters are: V0= 85±13\nµV,ξF1= 13.6±1.3 nm, ξF2= 6.71±0.37nm, and\nφ= 0.88±0.14. The fits show that the junctions have\nξF1> ξF2, which was also the case for a PdNi alloy studied\npreviously [17]. Bergeret et al.have shown that ξF1> ξF2\nis a persistent feature in the semi-clean limit where ξF1=le,\nthe mean free path [24]. Our data suggest that Pd 97Fe3is also\nin the semi-clean limit.\nIn Fig. 5(b), we plot the average of Hshiftfrom the\nFraunhofer pattern fits for each sweep direction versus dF.\nWe find that Hshiftvs.dFincreases due to the magnetic flux\nin the junction contributed by the uniform magnetization of\nthe ferromagnet. Due to the fact that our λL≫dFthe trend\nis approximately linear. Despite the small magnetization o f\nPdFe, large field shifts ( >15mT for the thickest samples\nmeasured), commensurate with the width of the central peak\nof the Fraunhofer pattern are observed due to the large PdFe\nthickness. We fit these data to:\nHshift=M(dF−ddead)/(2λL+2dCu+dF),(5)\nwithMandddeadused as fitting parameters. The fit yields\nM= 72±16 kA/m and ddead= -4±5 nm. While the\nmagnetization value obtained from Fig. 5(b) lies within the\nuncertainty of that from Fig. 1, the value of ddeaddoes not,\neven with it’s large uncertainty. If we instead fix ddead=0 and\nre-fit the data in Fig. 5(b) we find M= 88±4 kA/m, which\nis closer to the result from Fig. 1.\nIV. CONCLUSION\nIn conclusion we have studied the magnetic and transport\nbehavior of micron-scale SFS Josephson junctions containi ng\nPd97Fe3. If used as a “free” magnetic layer in cryogenic\nmemory, Pd 97Fe3is advantageous in that its 0- πtransition\noccurs at a thickness of ≈16.5 nm, much greater than for\nNiFe, making Pd 97Fe3much less sensitive to small thickness\nvariations. Meanwhile, junctions with Pd 97Fe3maintain a\nrelatively low switching field |Hswitch,2|= 2.4 mT (with\nstandard deviation 0.9 mT). As a “free” layer Pd 97Fe3has\nsome disadvantages– the magnetic switching can occur over a\nrangeoffields,possiblyduetotheexistenceofweaklycoupl ed\nferromagnetic clusters. For reproducible magnetic switch ing,\nthe junctions had to be magnetized at an initialization field of\n5 mT or greater. In the future we plan to further increase the\nFe concentration, in the range of 5-7 %, to see if it is possible\nto improve the magnetic properties of this F-layer.\nACKNOWLEDGMENT\nWe thankB. Niedzielski, E. Gingrich,A. Herr,D. Miller,N.\nNewman, and N. Rizzo for helpful discussions, and B. Bi for\nhelp with fabricationusing the Keck MicrofabricationFaci lity.This research is supported by the Office of the Director of\nNationalIntelligence(ODNI),IntelligenceAdvancedRese arch\nProjects Activity (IARPA), via U.S. Army Research Office\ncontractW911NF-14-C-0115.Theviewsandconclusionscon-\ntained herein are those of the authors and should not be\ninterpreted as necessarily representing the official polic ies\nor endorsements, either expressed or implied, of the ODNI,\nIARPA, or the U.S. Government.\n%\nREFERENCES\n[1] C. Bell, G. Burnell, C. W. Leung, E. J. Tarte, D.-J. Kang, a nd M. G.\nBlamire, “Controllable Josephson current through a pseudo spin-valve\nstructure,” Appl. Phys. Lett. , vol. 84, pp. 1153–1155, 2004.\n[2] V. V. Ryazanov, V. V. Bolginov, D. S. Sobanin, I. V. Vernik , S. K.\nTolpygo, A. M. Kadin, and O. A. Mukhanov, “Magnetic josephso n\njunction technology for digital and memory applications,” Physics\nProcedia, vol. 36, pp. 35 – 41, 2012.\n[3] T. I. 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B , vol. 64, p. 134506, Sep 2001."    },    {        "title": "1609.02930v2.Effect_of_lithographically_induced_strain_relaxation_on_the_magnetic_domain_configuration_in_microfabricated_epitaxially_grown_Fe81Ga19.pdf",        "content": "1\nScientific  RepoRts  | 7:42107 | DOI: 10.1038/srep42107www.nature.com/scientificreportsEffect of lithographically-\ninduced strain relaxation on the \nmagnetic domain configuration in \nmicrofabricated epitaxially grown \nFe81Ga19\nR. P . Beardsley1, D. E. Parkes1, J. Zemen2, S. Bowe1,3, K. W. Edmonds1, C. Reardon4, \nF. Maccherozzi3, I. Isakov2,5, P . A. Warburton5, R. P . Campion1, B. L. Gallagher1, S. A. Cavill3,4 & \nA. W. Rushforth1\nWe investigate the role of lithographically-induced strain relaxation in a micron-scaled device \nfabricated from epitaxial thin films of the magnetostrictive alloy Fe81Ga19. The strain relaxation due \nto lithographic patterning induces a magnetic anisotropy that competes with the magnetocrystalline and shape induced anisotropies to play a crucial role in stabilising a flux-closing domain pattern. We \nuse magnetic imaging, micromagnetic calculations and linear elastic modelling to investigate a region \nclose to the edges of an etched structure. This highly-strained edge region has a significant influence on the magnetic domain configuration due to an induced magnetic anisotropy resulting from the inverse magnetostriction effect. We investigate the competition between the strain-induced and shape-\ninduced anisotropy energies, and the resultant stable domain configurations, as the width of the bar is \nreduced to the nanoscale range. Understanding this behaviour will be important when designing hybrid magneto-electric spintronic devices based on highly magnetostrictive materials.\nMany existing and proposed spintronic device concepts make use of magnetic domains and domain walls to store \nand process data. Examples include magnetoresistive random access memory\n1,2, racetrack memory3,4 and domain \nwall logic architectures5. The drive to develop these technologies has led to a large and growing body of work on \nthe behaviour and structure of magnetic domain configurations and domain walls6.\nThe majority of recent studies have used electrical currents to manipulate magnetization, typically by spin \ntransfer torque7 or by magnetic field8. Whilst these methods have received significant attention they have not \nyet adequately addressed the problem of Joule heating, which presents an increasing problem as logic and mem-ory devices are reduced in size. It has been shown that using electric fields to manipulate magnetization can be many times more efficient than electrical current, due to the achieved reduction in power dissipation within the device\n9,10. Hybrid ferromagnet/piezoelectric devices, in which magnetic anisotropy is controlled by voltage \ninduced strain, are increasingly being seen as a viable and pragmatic solution to the problem of electrical control of magnetization for spintronic applications\n11–14. One of the key elements of such hybrid devices is a magneto-\nstrictive ferromagnetic material, of which Fe81Ga19 has the highest magnetostriction coefficient for non-rare-earth \ncontaining materials15, and displays highly magnetostrictive behaviour in the form of thin films16, making it an \nexcellent candidate for integration into the hybrid structures.\nThe configuration of ferromagnetic domains and domain walls in a lithographically patterned structure \nis determined by the balance of the anisotropy energies, including magnetocrystalline, magnetoelastic and \n1School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK. 2Department of Physics, \nBlackett Laboratory, Imperial College, Prince Consort Road, London SW7 2AZ, UK. 3Diamond Light Source Chilton, \nDidcot, Oxfordshire OX11 0DE UK. 4Department of Physics, University of York, Heslington, York, YO10 5DD, UK. \n5London Centre of Nanotechnology, University College London, London, WC1H 0AH, UK. Correspondence and \nrequests for materials should be addressed to S.A.C. (email:stuart.cavill@york.ac.uk) or A.W.R. (email: Andrew.\nrushforth@nottingham.ac.uk)Received: 16 September 2016\naccepted: 05 January 2017\nPublished: 10 February 2017OPENwww.nature.com/scientificreports/2\nScientific  RepoRts  | 7:42107 | DOI: 10.1038/srep42107shape-induced anisotropy terms. The relative magnitude of these different anisotropies is dependent on device \nsize and aspect ratio. In earlier studies we have shown that the magnetization in large scale (~50 μ m) epitax-\nial Fe81Ga19 structures is dominated by magnetoelastic and cubic magnetocrystalline anisotropy terms, and the \ndomains appear disordered with domain walls forming at nucleation sites determined by imperfections in the device structure\n12. In narrower devices, with width ~15 μ m and length ~90 μ m, shape-induced anisotropy plays \na more important role and magnetic domains form a pattern which minimises the stray field from the device17. \nIn this letter we discuss an additional contribution to the anisotropy energy which can result from the relaxation of growth strain at the edges of lithographically patterned bars when the width is reduced to the order of 1 μ m \nor less. This strain-relaxation originates from a lattice mismatch between the epitaxially grown magnetic layer and the substrate. The lattice mismatch imposes a built-in compressive strain in the magnetic layer, which can be \nrelaxed by etching of the continuous film into patterned devices. The effect of strain-relaxation on magnetic ani-\nsotropy was studied extensively in the diluted magnetic semiconductor (Ga,Mn)As\n18–21. There, the low magnetic \nmoment prevented the formation of regular domain patterns and the observations were interpreted using a single domain model. In our high moment Fe\n81Ga19 devices this additional contribution to the magnetic anisotropy \nenergy results in the stabilisation of a flux closure magnetic domain pattern, which is distorted compared to the pattern observed in wider bars\n17 where the effects of the lattice relaxation are less significant.\nThe sample consisted of a 14.3 nm Fe81Ga19 epitaxial thin film grown by molecular beam epitaxy on a 500 nm \nSi-doped buffer on a GaAs (001) substrate. A 1.5 nm amorphous GaAs capping layer was grown to protect the \nmetallic layer from oxidation. The layer structure is shown in Fig.1(a). X-ray diffraction measurements using a Phillips X-Pert materials research diffractometer on material grown under similar conditions\n16 show a single peak \ncorresponding to the Fe81Ga19 layer, indicating that the layer is a single crystal phase with a vertical lattice param-\neter of =.⊥a 0296nmFeGa. The lattice constant of the A2 phase of single crystal bulk Fe81Ga19 is known to be \n=. a 0287 nmoFeGa 22. Taking the in plane lattice constant of a fully strained film to be half the GaAs substrate lat-\ntice parameter ( =. =. aa 05 0283nmFeGa GaAs\n0) and assuming a Poisson ratio of 0.4523 this would imply that the \nout of plane lattice constant of a fully strained Fe81Ga19 film on GaAs would be 0.296  nm, consistent with the \nmeasured value. Superconducting quantum interference device (SQUID) magnetometry shows that the cubic magnetocrystalline anisotropy favouring the [100]/[010] directions has a magnitude K\nC =  18.9 ×  103 J/m3 for the \nunpatterned film, and the weaker uniaxial anisotropy favouring the [110] direction is KU =  12.4 ×  103 J/m3. The \nlatter contribution is typically observed for ferromagnetic films grown on GaAs substrates and is induced by the substrate/film interface\n24,25. The saturation magnetisation of the film is 1.4 ×  106 A/m. An L-shaped structure with \narms of width 1.2 μ m and length 10  μ m along the [100]/[010] directions was fabricated using electron beam \nlithography and Ar ion milling. The milled depth was greater than the film thickness which resulted in a 100 nm \nGaAs mesa, measured by atomic force microscopy, with the Fe81Ga19 and capping layers on top (Fig.1(a)). In this \nletter we focus on the behaviour in one 10 μ m-long arm of the L-shaped structure.\nFinite element calculations were used to gain insight into the structural behaviour of a bar of infinite length \nand the same cross-sectional dimensions as the experimental device. A fixed structural constraint was set at the \nsubstrate boundaries and an initial in-plane compressive strain of −= . aa a () /1 4%FeGa FeGa FeGa\n00 was included \nin the Fe81Ga19 layer. The strain profiles in the wire cross section were calculated using a partial differential equa-\ntion solver (the COMSOL software package) implementing the theory of an anisotropic elastic medium. From \nhere on we define positive strain as the difference between the in-plane lattice spacing along the directions per -\npendicular and parallel to the stripe such that the zero strain state corresponds to the unrelaxed film.\nFigure1(a,b) show the calculated strain relaxation across the wire cross-section, defined as εxx −  εyy where εxx \nand εyy are the in-plane components of strain in directions perpendicular and parallel to the wire. The strain pro-\nfile at the Fe81Ga19/cap interface, shown in Fig.1(c), reveals that there is a nonzero strain relaxation in the centre \nof the cross-section, which increases in amplitude away from the bar centre. The relaxation in the edge-region was also seen in the previous studies on (Ga,Mn)As-based devices\n18–21. An interesting feature of this profile is the \nabrupt decrease in amplitude in the regions near to the edges of the bar. This discontinuity is only observed with the inclusion of the GaAs capping layer, which tends to suppress the relaxation of the in-built strain by partially clamping the top surface as shown in Fig.1(b).\nMagnetic domains were imaged using photoemission electron microscopy (PEEM) on beamline I06 of the \nDiamond Light Source\n26. Illuminating the sample at oblique incidence and making use of X-ray magnetic circu-\nlar dichroism at the Fe L3 edge as the contrast mechanism allowed sensitivity to in-plane moments with a spatial \nresolution of approximately 50 nm. Figure2(a) shows an image of the domain configuration in a 1.2 μ m ×  6 μ m \nsection of the bar. The flux-closure domain configuration observed is different to that seen in previous studies of wires with width 15 μ m\n17 in that the regions with magnetisation perpendicular to the length of the bar are \nbroadened at the edges of the bar. In this study there is no externally induced strain and we attribute the observed domain behaviour to the relatively large effect of non-linear strain relaxation at the bar edges in our narrower device.\nTo understand the experimentally observed domain configuration we performed micromagnetic calcu-\nlations carried out using the Object Oriented Micromagnetic Framework (OOMMF)\n27. The simulation used \nmagnetocrystalline anisotropy coefficients determined by the SQUID magnetometry measurements of the unpat-terned Fe\n81Ga19 film, a cell size of 1  nm ×  1 nm ×  10 nm and critical damping. The OOMMF simulation was ini -\ntialised in a flux-closing state, with flux-closing units having an aspect ratio of 1:1.3, which is the average aspect ratio present in the experimental image.\nThe magneto-elastic coupling present in the Fe\n81Ga19 film leads to a strain-induced uniaxial anisotropy across \nthe width of the bar, with a profile determined by the strain profile shown in Fig.1(c). The relation between strain and magneto-elastic anisotropy energy is, Δ  K\nme =  Bmeε, where Δ  Kme represents the change in the magnetic \nanisotropy energy and ε  is the position-dependent uniaxial strain perpendicular to the bar length. We set the www.nature.com/scientificreports/3\nScientific  RepoRts  | 7:42107 | DOI: 10.1038/srep42107magneto-elastic constant, Bme =  1.56 ×  107 J/m3, as determined previously for an epitaxial thin Fe81Ga19 film16. \nWe have approximated the anisotropy energy as a function of position by fitting a first order polynomial to the \nedge region, and an exponential function to the central region of the calculated strain profile. This strain-induced anisotropy energy was incorporated into the OOMMF calculation as an additional uniaxial magnetocrystalline anisotropy term, with the anisotropy axis perpendicular to the length of the bar. The results of micromagnetic \ncalculations based on the approximated anisotropy profile are shown in Fig.2(b). Similar to the experimental \ndata, the ground state is a flux-closure pattern with regions magnetised perpendicular to the length of the bar. To observe broadening of the domain boundaries at the edges of the bar to an extent similar to that observed in the experimental data, we scale the magnitude of the strain-induced anisotropy by a factor of 0.4 in the simulations, otherwise the calculated broadening is too large. A possible explanation for the need to scale the anisotropy might \nFigure 1. The calculated strain relaxation across the micro-bar. (a) Cross-sectional view of the layer structure \nof the experimental device and simulated colour scale map showing the relaxation of the growth strain as a function of depth in the bar. (b) A zoomed section of the colour scale map showing the relaxation of the growth strain as a function of depth in the edge region of the bar. (c) The simulated strain profile across the Fe\n81Ga19 bar \nat the Fe81Ga19/cap interface.www.nature.com/scientificreports/4\nScientific  RepoRts  | 7:42107 | DOI: 10.1038/srep42107arise from damage to the Fe81Ga19 layer at the edges of the bar during device fabrication, which would degrade \nthe magnetism in the region where the strain relaxation is largest. If the strain-induced anisotropy is not included \nin the calculation we find that the flux closure domain configuration is not the lowest energy state of the system. Calculations initialised with a single domain state, where magnetisation is aligned uniformly along the length of the bar, evolve into the S-shaped domain pattern shown in Fig.2(c). In the absence of the strain-induced \nanisotropy the S-shaped pattern represents a lower total energy state than the flux closure state. The situation is \nreversed when the strain-induced anisotropy is included in the calculation. Calculations on bars with the same 1.2 μ m width, but using periodic boundary conditions to simulate infinite length reveal that a very similar flux \nclosure pattern represents the lowest energy ground state when the strain-induced anisotropy is included, but that without this anisotropy term a single domain configuration with magnetisation pointing along the length of the bar is the lowest energy configuration.\nTo investigate the competition between the shape- and strain-induced anisotropies and to determine the limit \nin which shape-induced anisotropy will overcome the strain-induced anisotropy, we carried out calculations for bars of different widths, but with the same length to width ratio, thickness and etch depth as in the calculations \ndescribed above. Figure3(a) shows the calculated strain profile as a function of the normalised position across \nthe bar. It can be observed that the maximum strain at the edges of the bar, where the relaxation is largest, remains roughly the same for each bar width. The strain relaxation at the centre of the bar, and therefore also the average strain relaxation across the bar, becomes larger as the bar width decreases. Figure3(b) shows the difference,  \nΔ E, between the total energies of the flux closure and S-shaped states as a function of bar width. For widths in the \nrange 100 nm to 2000 nm (300 nm or greater and less than 2000 nm for the infinitely long bar), the flux closure \nstate is the energetically favourable state when the strain-induced anisotropy is included in the calculation. For widths greater than 500 nm (300 nm for the infinitely long bar) the magnitude of Δ  E decreases as width increases, \nrepresenting the reducing significance of the strain-induced anisotropy which acts mainly at the edges of the \nbar. Below 500 nm (300 nm for the infinitely long bar) the magnitude of Δ E decreases as the bar width decreases \nuntil eventually the S-shaped state becomes energetically more favourable. This transition occurs below a width \nof 100 nm (300 nm for the infinitely long bar) and represents the increasing significance of the demagnetising \nfield with respect to the strain-induced anisotropy as the width of the bar is reduced to these dimensions. For the 100 nm wide bar the flux closure domain pattern is the energetically favoured state. In this case the strain \nrelaxation is significant across the whole width of the bar. However, Fig.2(d) reveals that the broadening of the transverse domains at the edges of the bar is reduced compared to the case of the 1200 nm bar (Fig.2(b)) \ndue to the increased significance of the demagnetising field which competes with the strain-induced anisotropy. We note that, although not considered in the present study, it would be important to include the effects of the \nstrain-induced anisotropy in the length direction of the bar for devices with dimensions in the sub-micron limit. \nIn the absence of strain-induced anisotropy the S-shaped state is energetically favourable for both the finite and infinite length bars over the whole range of widths considered.\nFigure 2. The magnetic domain configuration. (a) Experimental top down PEEM image of 1.2 μ m ×  6 μ m \nregion of a bar with arrows indicating the magnetization direction. (b) Micromagnetic simulation for the  1.2 μ m ×  6 μ m bar with an anisotropy profile that includes the calculated strain relaxation profile scaled \nby a factor of 0.4. (c) Micromagnetic simulation for the 1.2 μ m ×  6 μ m bar initialised in a single domain \nconfiguration without the inclusion of a strain-induced anisotropy. (d) As in (b), but for a 100 nm ×  500 nm bar.www.nature.com/scientificreports/5\nScientific  RepoRts  | 7:42107 | DOI: 10.1038/srep42107In our Fe81Ga19 structures the cubic magnetocrystalline anisotropy supports magnetic easy axes along the \n[100] and [010] directions. The small intrinsic uniaxial anisotropy along the [110] direction acts to distort the \nshape of the magnetic domains and leads to a canting of the magnetic moments towards the [110] direction. This feature is present in the experimental data (Fig.2(a)) and is also revealed in the calculations. Furthermore, the magnetostriction constant in Fe\n81Ga19 is largest along the [100]/[010] directions, hence magneto-elastic effects \nwill be maximised by strain relaxation along these directions23, as is the case with our device.\nMagnetic contrast imaging by Kerr microscopy on patterned and unpatterned films reveals no evidence of the \nformation of magneto-statically and magneto-elastically self-sufficient domains, which were reported by Chopra et al. for Fe\nxGa1−x single crystals after high-temperature thermal processing28. This may be due to the different \ngrowth method used in our study or the fact that our epitaxial films are clamped to a thick substrate. In our case, a micromagnetic model incorporating strain relaxation induced anisotropy energy is sufficient to understand the experimental observations.\nIn conclusion we have demonstrated that relaxation of growth strain is an important factor in determining \nthe magnetic domain configuration of micron and sub-micron sized devices based on epitaxial Fe\n81Ga19. In the \n1.2 μ m wide bars investigated experimentally, the strain-induced anisotropy stabilises the formation of a regular \nflux-closure domain configuration and distorts the features near the edges of the bar. The competition between strain- and shape-induced anisotropy energies determines the stable domain configuration over a range of device dimensions. Growth strain is an additional degree of freedom to be considered and manipulated in the design of \nmicro- and nano-scale magnetic devices.\nReferences\n1. Fukami, S. et al. 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Origin of in-plane uniaxial magnetic \nanisotropy in CoFeB amorphous ferromagnetic thin films. Phys. Rev. B 83, 212404 (2011).\n26. Dhesi, S. S. et al. The Nanoscience Beamline (I06) at Diamond Light Source. AIP Conference Proceedings 1234, 311–314 (2010).\n27. Donahue, M. J. & Porter, D. G. OOMMF User’s Guide, Version 1, Interagency Report NISTIR 6376 (1999).\n28. Chopra, H. D. & Wuttig, M. Non-Joulian magnetostriction. Nature , 521, 340 (2015).\nAcknowledgements\nThe authors would like to acknowledge Diamond Light Source for the provision of beamtime under SI-8560 and 7601. \nThis work was supported by the Engineering and Physical Sciences Research Council [grant number EP/H003487/1].  \nWe are grateful for access to the University of Nottingham High Performance Computing Facility.\nAuthor Contributions\nR.B., D.E.P ., S.B., K.W .E., F.M., S.A.C. and A.W .R. conducted the PEEM measurements. R.B., D.E.P . and C.R. performed electron beam lithography. II carried out ion milling. J.Z. performed structural calculations of the strain profile using the COMSOL package. D.E.P . and A.W .R. carried out the micromagnetic simulations. R.P .C. was responsible for the growth of the semiconductor and metallic layers. All authors contributed to writing the manuscript.\nAdditional Information\nCompeting financial interests: The authors declare no competing financial interests.\nHow to cite this article: Beardsley, R. P . et al. Effect of lithographically-induced strain relaxation on the \nmagnetic domain configuration in microfabricated epitaxially grown Fe81Ga19. Sci. Rep. 7, 42107; doi: 10.1038/\nsrep42107 (2017).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and \ninstitutional affiliations.\nThis work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, \nunless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/\n \n© The Author(s) 2017"    },    {        "title": "1609.03383v1.Spin_structures_of_textured_and_isotropic_Nd_Fe_B_based_nanocomposites__Evidence_for_correlated_crystallographic_and_spin_texture.pdf",        "content": "arXiv:1609.03383v1  [cond-mat.mes-hall]  12 Sep 2016Spin structure of textured and isotropic Nd-Fe-B-based nan ocomposites: evidence for\ncorrelated crystallographic and spin texture\nA. Michels,1,∗R. Weber,1I. Titov,1D. Mettus,1´E.A. P´ erigo,1,2I. Peral,1,3\nO. Vallcorba,4J. Kohlbrecher,5K. Suzuki,6M. Ito,7A. Kato,7and M. Yano7\n1Physics and Materials Science Research Unit, University of Luxembourg,\n162a avenue de la Faencerie, L-1511 Luxembourg, Luxembourg\n2ABB Corporate Research Center, 940 Main Campus Drive, 27606 Raleigh, North Carolina\n3Materials Research and Technology Department,\nLuxembourg Institute of Science and Technology, 41 rue du Br ill, L-4422 Belvaux, Luxembourg\n4Alba Synchrotron, BP 1413, km 3.3, Cerdanyola del Vall `es, Spain\n5Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerlan d\n6Department of Materials Science and Engineering,\nMonash University, Clayton, Victoria 3800, Australia\n7Advanced Material Engineering Division, Toyota Motor Corp oration, Susono 410-1193, Japan\n(Dated: November 8, 2021)\nWe report the results of a comparative study of the magnetic m icrostructure of textured and\nisotropic Nd 2Fe14B/α-Fe nanocomposites using magnetometry, transmission elec tron microscopy,\nsynchrotron x-ray diffraction, and, in particular, magneti c small-angle neutron scattering (SANS).\nAnalysis of the magnetic neutron data of the textured specim en and computation of the correlation\nfunction of the spin misalignment SANS cross section sugges ts the existence of inhomogeneously\nmagnetized regions on an intraparticle nanometer length sc ale, about 40 −50nm in the remanent\nstate. Possible origins for this spin disorder are discusse d: it may originate in thin grain-boundary\nlayers (where the materials parameters are different than in the Nd 2Fe14B grains), or it may reflect\nthe presence of crystal defects (introduced via hot pressin g), or the dispersion in the orientation\ndistribution ofthe magnetocrystalline anisotropy axes of the Nd 2Fe14B grains. X-raypowder diffrac-\ntion data reveal a crystallographic texture in the directio n perpendicular to the pressing direction\n– a finding which might be related to the presence of a texture i n the magnetization distribution,\nas inferred from the magnetic SANS data.\nPACS numbers: 75.40.-s; 75.50.Tt; 75.75.-c\nI. INTRODUCTION\nNd-Fe-B-based nanocomposite permanent magnets,\nwhich consist of exchange-coupled nanocrystalline hard\n(Nd2Fe14B) and soft ( α-Fe orFe 3B) magnetic phases, are\nof potential interest for electronic devices due to their\npreeminent magnetic properties such as high remanence\nand magnetic energy product [1, 2]. The major chal-\nlenge remains the understanding of how the details of\nthe microstructure (e.g., average particle size and shape,\nvolume fraction of soft phase, texture, interfacial chem-\nistry) correlate with their magnetic properties. In or-\nder to tackle this issue a multiscale characterization ap-\nproach is adopted, which comprises a suite of both ex-\nperimental and theoretical state-of-the-art methods such\nashigh-resolutionelectronmicroscopy,electronbackscat-\ntering diffraction, three-dimensional atom-probe analy-\nsis, Lorentz and Kerr microscopy, or atomistic and con-\ntinuum micromagnetic simulations.\nRecent investigations by Liu et al.[3] and Sepehri-\nAminet al.[4] demonstrate that the properties of\nthe interface regions between the Nd 2Fe14B grains deci-\nsively determine the coercivity of the sample. The grain-\n∗andreas.michels@uni.luboundarylayers(andtriple-junctionsbetweenthegrains)\nhave a thickness between about 1 −15nm and can be\nboth in a crystalline or amorphous state. Moreover, as\nfar as their magnetism is concerned, the intergranular\nregions are characterized by different magnetic interac-\ntions (exchange, magnetocrystalline anisotropy, satura-\ntion magnetization) as compared to the Nd 2Fe14B crys-\ntallites and, hence, they represent potential sources for\nthe nucleation of inhomogeneous spin textures during\nmagnetization reversal. Indeed, the micromagnetic sim-\nulation results reported in [4] suggest that the existence\nof a thin ( <5nm) ferromagnetic grain-boundary phase\nwith reduced magnetocrystalline anisotropy, exchange-\nstiffness constant, and saturation magnetization causes\nthe magnetization reversal to occur from the soft inter-\ngranular phase into the hard Nd 2Fe14B phase at a field\nof−2.5T. When the intergranularphase is nonmagnetic,\nthen the nucleation of reversed domains starts from the\ntriple junctions of the Nd 2Fe14B grains at a higher field\nof−3.2T.\nIn this context it also worth noting that first-\nprinciples density functional theory calculations on an\nexchange-spring multilayer system [5] predict a de-\npendency of the exchange coupling on the crystallo-\ngraphic orientation at the interface between Nd 2Fe14B\nandα-Fe; specifically, ferromagnetic coupling is pre-\ndicted for the Nd 2Fe14B(001)/α-Fe(001) interface model,2\nwhereas antiferromagnetic interactions are obtained for\nNd2Fe14B(100)/α-Fe(110). If this prediction were true,\nthen it may negatively influence the magnetic proper-\nties of this class of materials (e.g., the maximum energy\nproduct). Indeed, a recent experimental study using fer-\nromagnetic resonanceand Kerr microscopy[6] reports on\nthe predicted negative exchange coupling.\nThe above discussed examples ultimately demonstrate\nthat the magnetic microstructure of nanocrystalline Nd-\nFe-B-based magnets is characterized by inhomogeneous\nmagnetization structures and that interfacial regions are\na major cause for the nanoscale spin disorder. In addi-\ntion to the grain boundaries, there exist, however, other\nsources of spin disorder in such materials: ultrafine-\ngrained textured nanocomposites are produced from\nmelt-spun ribbons via hot compaction [3, 4, 7, 8]; this\nprocess may introduce crystal defects which locally act\nas nucleation centers for nonuniform magnetization tex-\ntures. Furthermore, one has to invoke a magnetiza-\ntion inhomogeneity which is due to the nonideal align-\nment (dispersion) of the crystallographic c-axes (of the\nNd2Fe14B grains) along the pressing direction during hot\ndeformation; the spins haveto undergorotations in order\nto accommodate to the changes in the easy-axis mag-\nnetization direction from grain to grain. Last but not\nleast, there is the magnetic shape anisotropy of the usu-\nally platelet-shaped Nd 2Fe14B particles, which may re-\nsult in a small spin canting towards the plane perpendic-\nular to the c-axis. It is certainly true that the magnetic\nanisotropy field of the Nd 2Fe14B phase (about 8T at\n300K [9]) is much larger than any shape-anisotropy field\n(assuming, e.g., 0 .5T for strongly anisotropic grains),\nbut neverthelessweak spin canting (tan−1(0.5/8)∼=3.6◦)\nmight be produced by the competition between shape\nand magnetocrystalline anisotropy.\nIn order to scrutinize the above-sketched issue, we\nhave carried out a comparative study of the magnetic\nmicrostructure of textured (hot-deformed) and isotropic\nnanocrystalline Nd 2Fe14B/α-Fe by means of magnetic\nsmall-angle neutron scattering (SANS). Specifically, the\ncentral aim of our investigation is to detect and quan-\ntify the presumed nanoscale spin disorder, which is com-\nmonly only indirectly inferred by combining results from\nelectron microscopy, magnetization, and micromagnetic\nsimulations.\nThe SANS technique (see Ref. [10] for a review) pro-\nvides information on variations of both the magnitude\nand orientation of the magnetization on a nanometer\nlength scale ( ∼1−300 nm). SANS is extremely sensi-\ntive to long-wavelengthmagnetizationfluctuations and it\nhas only recently been employed for characterizing Nd-\nFe-B-based permanent magnets: for example, the field\ndependence of characteristic magnetic length scales dur-\ning the magnetization-reversal process in isotropic Nd-\nFe-B-based nanocomposites [11] and in isotropic sintered\nNd-Fe-B[12] wasstudied, the exchange-stiffnessconstant\nhas been determined [13], the observationof the so-called\nspike anisotropy in the magnetic SANS cross section hasbeen explained with the formation of flux-closure pat-\nterns[14], magneticmultiplescatteringhasbeendetected\n[15], texturedNd-Fe-Bhasbeeninvestigated[16], andthe\neffect of grain-boundary diffusion on the magnetization-\nreversal process of isotropic [17] and hot-deformed tex-\ntured [18–20] nanocrystalline Nd-Fe-B magnets has been\nstudied.\nII. EXPERIMENTAL\nTwo Nd 2Fe14B/α-Fe nanocomposites containing, re-\nspectively, 5wt% of Fe were investigated in this study.\nBoth samples were prepared by means of the melt-\nspinning technique. One sample was subsequently hot-\ndeformed in order to obtain a textured magnet. For this\npurpose, the melt-spun ribbons were crushed into pow-\nders of a few hundred micrometers in size and then sin-\ntered at 973Kunder a pressure of400MPa. The sintered\nbulk was hot-deformed with a height reduction of about\n75% to develop the [001] texture of the Nd 2Fe14B phase\nalong the pressing direction [4, 8]. This results in the\nformation of platelet-shaped Nd 2Fe14B grains with an\naverage thickness of ∼110nm and an average diameter\nof∼140nm. The Nd 2Fe14B platelets are stacked along\nthe nominal c-axis, which wedefine as the [001]direction,\nwith some degrees of misorientation. The isotropic sam-\nple had an average grain size of about 20nm. We have\nalso investigated composites with 0wt% and 10wt% of\nFe, which, as far as the neutron results are concerned,\nshow qualitatively the same behavior as the 5wt% sam-\nple. For further details, see Refs. [18–20].\nThe neutron experiment has been carried out at 300K\nat the instrument SANS-I at the Paul Scherrer Institute,\nSwitzerland, using unpolarized neutrons with a mean\nwavelength of λ= 4.5˚A and ∆ λ/λ= 10% (FWHM)\n[21, 22]. The external magnetic field H0(provided by a\ncryomagnet; µ0Hmax= 9.5T) was applied perpendicular\nand parallel to the wave vector k0of the incoming neu-\ntron beam (compare Fig. 1); this corresponds to the sit-\nuation that H0is parallel ( k0⊥H0) and perpendicular\n(k0∝bardblH0)tothenominal c-axis(pressingdirection)ofthe\ntextured sample. Neutron data were corrected for back-\nground scattering (empty sample holder), transmission,\nand detector efficiency using the GRASP software pack-\nage. The measured transmission was larger than 90%\nfor both samples at all fields investigated. Further sam-\nple characterizationwas done by means ofvibrating sam-\nplemagnetometry, transmissionelectronmicroscopy,and\nsynchrotron x-ray diffraction (at beamline BL04-MSPD\nat the Alba synchrotron, Barcelona, Spain [23]).\nIII. UNPOLARIZED SANS CROSS SECTIONS\nAND CORRELATION FUNCTION\nThe elastic unpolarized SANS cross section dΣ/dΩ at\nmomentum-transfer vector qtakes on different forms de-3\nFIG. 1. Sketch of the perpendicular (a) and parallel (b) scat -\ntering geometry, which, respectively, have the applied mag -\nnetic field H0perpendicular and parallel to the wave vec-\ntork0of the incident neutron beam; q=|q|= 4πλ−1sinψ,\nwhere 2ψdenotes the scattering angle and λis the mean neu-\ntron wavelength. Note that H0/bardblezinbothgeometries and\nthatq∼=(0,qy,qz) =q(0,sinθ,cosθ) fork0⊥H0(a) and\nq∼=(qx,qy,0) =q(cosθ,sinθ,0) fork0/bardblH0(b). The press-\ning direction is horizontal, which is along ezin (a) and along\nexin (b).\npending onthe relativeorientationbetweenthe wavevec-\ntork0of the incident neutron beam and the externally\napplied magnetic field H0[10]; for the perpendicular ge-\nometry ( k0⊥H0), we obtain\ndΣ\ndΩ(q) =8π3\nV/parenleftBig\n|/tildewideN|2+b2\nH|/tildewiderMx|2+b2\nH|/tildewiderMy|2cos2θ\n+b2\nH|/tildewiderMz|2sin2θ−b2\nH(/tildewiderMy/tildewiderM∗\nz+/tildewiderM∗\ny/tildewiderMz)sinθcosθ/parenrightBig\n,\n(1)\nwhereas for the parallel case ( k0∝bardblH0)\ndΣ\ndΩ(q) =8π3\nV/parenleftBig\n|/tildewideN|2+b2\nH|/tildewiderMx|2sin2θ+b2\nH|/tildewiderMy|2cos2θ\n+b2\nH|/tildewiderMz|2−b2\nH(/tildewiderMx/tildewiderM∗\ny+/tildewiderM∗\nx/tildewiderMy)sinθcosθ/parenrightBig\n;\n(2)\nVdenotes the scattering volume, bH= 2.91×\n108A−1m−1,/tildewideN(q) is the nuclear scattering ampli-\ntude, and /tildewiderM(q) ={/tildewiderMx(q),/tildewiderMy(q),/tildewiderMz(q)}represents\nthe Fourier transform of the magnetization M(r) =\n{Mx(r),My(r),Mz(r)};c∗is a quantity complex-\nconjugated to c. We would like to emphasize that the\nmagnetization vector field of a bulk ferromagnet is a\nfunction of the position r={x,y,z}inside the ma-\nterial, i.e., M=M(x,y,z), and that, consequently,\n/tildewiderM=/tildewiderM(qx,qy,qz). However, the Fourier components\nwhich appear in the above SANS cross sections represent\nprojections into the qy-qz-plane for k0⊥H0(qx∼=0)\nand into the qx-qy-plane for k0∝bardblH0(qz∼=0) (compareFig. 1). In polar coordinates, the /tildewiderMx,y,zthen depend (in\naddition to the applied field and the magnetic interac-\ntions) on both the magnitude qand the orientation θof\nthe scattering vector q[24].\nIn our neutron data analysis below, we subtract the\nrespective SANS signal at the largest available field of\n9.5T (approach-to-saturation regime, compare Fig. 2)\nfrom the measured data at lower fields. This subtrac-\ntion procedure eliminates the nuclear SANS contribution\n(∝ |/tildewideN|2), which is field independent, and it yields the so-\ncalled spin-misalignment SANS cross section dΣM/dΩ,\nwhich we display here for simplicity only for the parallel\nscattering geometry:\ndΣM\ndΩ=8π3\nVb2\nH/parenleftBig\n∆|/tildewiderMx|2sin2θ+∆|/tildewiderMy|2cos2θ\n+∆|/tildewiderMz|2+∆CTsinθcosθ/parenrightBig\n,(3)\nwhere ∆|/tildewiderMx|2:=|/tildewiderMx|2(H)−|/tildewiderMx|2(9.5T) (and so on for\nthe other Fourier coefficients) represents the difference\nbetween the value of |/tildewiderMx|2at the actual field Hand the\nmeasurement at 9 .5T [CT:=−(/tildewiderMx/tildewiderM∗\ny+/tildewiderM∗\nx/tildewiderMy)]. If\nit would be possible to fully saturate the sample (i.e.,\nM(r) ={0,0,Mz=Ms(r)}) and if one restricts the\nconsiderations (subtraction procedure) to the approach-\nto-saturation regime, where the field dependence of the\nlongitudinal Fourier component can be neglected (i.e.,\n|/tildewiderMz|2(H)− |/tildewiderMs|2→0), then dΣM/dΩ (fork0∝bardblH0)\nreduces to\ndΣM\ndΩ=8π3\nVb2\nH/parenleftBig\n|/tildewiderMx|2sin2θ+|/tildewiderMy|2cos2θ\n+CTsinθcosθ),(4)\nand likewise for the k0⊥H0geometry.\nFurthermore, it is decisive for the later discussion\nto note that for a ferromagnet with a statistically-\nisotropic microstructure the parallel total (nuclear and\nmagnetic) dΣ/dΩ anddΣM/dΩ [Eqs. (2) and (3)] are\ngenerally isotropic, i.e.,θ-independent (see, e.g., Fig. 21\nin Ref. [10], Fig. 4 in Ref. [25], or Figs. 5(b), 7(c),\nand 7(d) below). In other words, although the indi-\nvidual contributions to the parallel SANS cross section\nare highly anisotropic (e.g., |/tildewiderMx|2sin2θ), their corre-\nsponding sums in Eqs. (2) and (3) are isotropic for a\nstatistically-isotropic ferromagnet; this is not true for\nthe perpendicular geometry[Eq. (1)], which generallyex-\nhibits a pronounced angular anisotropy.\nThe (normalized) correlation function c(r) of the\nspin misalignment can be computed from azimuthally-\naveraged data via [26]\nc(r) =/integraltext∞\n0dΣM\ndΩ(q)J0(qr)qdq/integraltext∞\n0dΣM\ndΩ(q)qdq, (5)\nwhereJ0(qr) denotes the zeroth-order Bessel function.\nAnalysis of c(r) provides information on the characteris-\ntic magnetic length scales [11, 12, 17].4\nFIG. 2. Room-temperature magnetization curves of textured (a) and isotropic (b) Nd 2Fe14B/α-Fe (5wt% Fe). Measurements\nhave been carried out for the magnetic field applied parallel and perpendicular to the texture axis (pressing direction) in (a),\nand for two different in-plane directions in (b) (“in-plane 2 ” direction is rotated by 90◦with respect to “in-plane 1” direction).\nMagnetization data (on the rectangular-shaped samples) ha ve been corrected for demagnetizing effects using the magnet ometric\ndemagnetizing factor [27].\nFIG. 3. Bright-field transmission electron microscopy imag es of the textured [(a) and (b)] and isotropic (c) Nd 2Fe14B/α-Fe\nnanocomposites (5wt% Fe). The average sizes of the anisotro pic grains of the textured sample have been estimated [from\n(a) and (b)] as, respectively, ∼110nm (parallel to the pressing direction “ p”) and∼140nm (perpendicular to the pressing\ndirection “p”), while the average grain diameter of the isotropic sample has been found to be ∼20nm.\nIV. RESULTS AND DISCUSSION\nFigure 2 displays the room-temperaturemagnetization\ncurves of textured [Fig. 2(a)] and isotropic [Fig. 2(b)]\nNd2Fe14B/α-Fe. The coercive fields are µ0Hc= 0.57T\n(textured) and µ0Hc= 0.61T (isotropic). The satu-\nration polarization was estimated by extrapolating the\ndata to infinite field: we find Js= 1.57T (textured)\nandJs= 1.53T (isotropic) with ensuing remanence-\nto-saturation ratios of about 0 .64 (textured easy), 0 .36\n(textured hard), and 0 .53 (isotropic). Consistent with\nthe magnetization data, the transmission electron mi-\ncroscopy images of the textured sample [Fig. 3(a) and3(b)] reveal a weakly anisotropic microstructure, while\nthe isotropic sample [Fig. 3(c)] exhibits equiaxed grains.\nX-ray diffraction measurements carried out (in trans-\nmission mode) at the Alba synchrotron (Fig. 4) unam-\nbiguously prove the presence of a weak texture along the\n(horizontal) pressing direction; specifically, diffraction\npeaks of the type (00 l) do present two maxima around\nθ= 0◦andθ= 180◦in the Debye Scherrer rings. Ad-\nditionally, we find evidence for the presence of texture\nalong other crystallographicdirections; peaks of the type\n(hk0) exhibit two maxima around 90◦and 270◦. This\nlatter observation will be of relevance when discussing\nthe results of the magnetic neutron data analysis (see5\nFIG. 4. Synchrotron x-ray diffraction data of the isotropic ( a) and textured (b) Nd 2Fe14B/α-Fe nanocomposites (5wt% Fe).\nThe pressing direction of the hot-deformed sample is horizo ntal (same scattering geometry as in the neutron experiment ,\ncompare Fig. 1). (left images) Integrated intensity as a fun ction of azimuthal angle θand scattering angle 2 ψ; (right images)\ncorresponding Debye-Scherrer diffraction rings. Radial in tegration of synchrotron data has been performed with the Fi t2D\nsoftware [28].\nbelow).\nFigure 5 depicts (for k0∝bardblH0) the two-dimensional\nunpolarized total scattering cross sections dΣ/dΩ of the\ntextured and isotropicNd-Fe-B-basednanocomposites at\nselected applied magnetic fields (9 .5T, remanence, co-\nercive field). The isotropic sample [Fig. 5(b)] exhibits\nan isotropic scattering pattern at all fields investigated,\nwhereasthetexturedsample[Fig.5(a)]showsanisotropic\nscattering with an elongation along the horizontal direc-\ntion. The corresponding (over 2 π) azimuthally-averaged\ndata sets are displayed in Fig. 6; between the coer-\ncive field and the largest available field of 9 .5T, the\ncrosssection ofthe isotropicsample changes(roughly) by\nabout an order of magnitude at the smallest momentum-\ntransfers q(and about half an order of magnitude for the\ntextured sample).\nWhile the textured nanocomposite revealsa power-law\ntype scattering over most of the q-range, the isotropic\nsample exhibits a more structured dΣ/dΩ with signif-\nicant curvature at lower and medium q. This differ-\nence indΣ/dΩ is most likely related to the difference\nin the average grain sizes and the ensuing magnetization\nfluctuations on a nanometer length scale: the isotropic\nsample has an average grain size of ∼20nm, while the\ntextured Nd-Fe-B possesses a larger particle size of the\norder of 100nm (compare the TEM images in Fig. 3).We also note that the dΣ/dΩ of both samples (data not\nshown) as well as the spin-misalignment SANS cross sec-\ntiondΣM/dΩ [Fig. 6(c)] are characterized by power-law\nexponents nthat are larger than 4. This is in agree-\nment with the notion of spin-misalignment scattering,\ni.e., scattering due to canted spins with a characteris-\ntic magnetic-field-dependent wavelength [10]. It is also\nquite obvious from this observation that the correspond-\ning magnetization fluctuations in real space are not ex-\nponentially correlated (see Fig. 8 below).\nAs discussed previously, for k0∝bardblH0, any anisotropy\nofdΣ/dΩ (or of dΣM/dΩ) is indicative of an anisotropic\nmicrostructure. At magnetic saturation , thetotalSANS\nsignalarisesfromnanoscalespatialfluctuationsinthenu-\nclear density and in the saturation magnetization Ms(r),\npresumably at internal Nd 2Fe14B/α-Fe interfaces. The\nnuclear scattering-length density contrast between the\nNd2Fe14B phase and the α-Fe phase amounts to ∆ ρnuc∼=\n1.63×1014m−2, whereas – at saturation – the mag-\nnetic contrast can be estimated as ∆ ρmag=bH∆M∼=\n1.37×1014m−2, where ∆ Mdenotes the difference in\nsaturation magnetization between α-Fe (Js= 2.2T) and\nNd2Fe14B (Js= 1.61T). By assuming that the elements\nof the microstructure which give rise to nuclear scatter-\ning|/tildewideN|2are identical to those which give rise to lon-\ngitudinal magnetic scattering b2\nH|/tildewiderMz|2, one finds for a6\nFIG. 5. Color-coded two-dimensional intensity maps of the t otal unpolarized dΣ/dΩ in the plane perpendicular to the incoming\nneutron beam at selected applied magnetic fields (see insets ) (logarithmic color scale) ( k0/bardblH0).dΣ/dΩ of the textured (a)\nand isotropic (b) Nd 2Fe14B/α-Fe nanocomposite. H0is normal to the detector plane.\nFIG. 6. Azimuthally-averaged total unpolarized SANS cross sectionsdΣ/dΩ at selected applied magnetic fields (see insets)\n(log-log scale) ( k0/bardblH0).dΣ/dΩ of the textured (a) and isotropic (b) Nd 2Fe14B/α-Fe nanocomposite. (c) Applied-field\ndependence of the power-law exponent nindΣM/dΩ =K/qnfor the textured and isotropic Nd 2Fe14B/α-Fe nanocomposite.\ndΣM/dΩ has been obtained by subtracting, respectively, the total dΣ/dΩ at 9.5T; the fits were restricted to the interval\n0.4nm−1/lessorsimilarq/lessorsimilar0.6nm−1. Dotted horizontal line ( n= 4) corresponds to scattering due to sharp interfaces (Poro d) or to\nexponentially correlated magnetization fluctuations.\nsaturated sample that the ratio of nuclear to magnetic\nSANS equals |/tildewideN|2/(b2\nH|/tildewiderMz|2)∼=1.42. With reference to\nthe electron-microscopy results (Fig. 3), which reveal a\n(weakly) anisotropic grain shape (aspect ratio ∼1.3),\nit is then obvious that a (weakly) horizontally-elongated\nSANS pattern can already be generated at saturation by\nthe combined action ofthe nuclearand longitudinal mag-\nnetic form factors.\nSubtracting the total dΣ/dΩ at 9.5T from the total\ndΣ/dΩ at lower fields, we obtain the spin-misalignmentSANS cross section dΣM/dΩ [Eq. (3)], which is free of\nnuclear SANS. The results for dΣM/dΩ for the textured\nnanocomposite [Fig. 7(a) and 7(b)] still reveal an angu-\nlar anisotropy with maxima parallel and antiparallel to\nthe horizontal texture axis. Inspection of Eq. (3) then\nsuggests that this observation may be due to (i) spin\ncomponents which are directed along the ±ey-direction\n[cf. the term ∆ |/tildewiderMy|2cos2θin Eq. (3)] and/or due to (ii)\nthe particle form factor anisotropy (cf. terms ∝∆|/tildewiderMz|2).\nHowever, measurements in the k0⊥H0geometry [com-7\nFIG. 7. Selected results for thespin-misalignment SANScro ss\nsectiondΣM/dΩ of the textured and isotropic Nd 2Fe14B/α-\nFe nanocomposite for k0/bardblH0[(a)−(d)] and for k0⊥H0\n[(e)−(f)] (logarithmic color scale). The respective data set\nat the maximum applied field of 9 .5T has been subtracted.\nIn (a)−(d),H0is normal to the detector plane, whereas in\n(e)−(f)H0is horizontal in the plane.\npare Fig. 1(a)] suggest that longitudinal magnetization\nfluctuations play only a minor role: if the dΣM/dΩ (for\nk0⊥H0) were dominated by ∆ |/tildewiderMz|2, a sin2θ-type\nanisotropy with intensity maxima along the vertical di-\nrection would result [compare Eq. (1)]. This is, how-\never, not visible in the experimental data [Fig. 7(e) and\n7(f)], which exhibit a horizontal elongation [cf. the term\n|/tildewiderMy|2cos2θin Eq. (1)]. In other words, the anisotropyof\nthe scattering pattern for k0∝bardblH0[Fig. 7(a) and 7(b)] is\ndue to an anisotropy in the magnetic microstructure, not\nto the form-factor anisotropy ∆ |/tildewiderMz|2of the particles.\nWe emphasize that, although the mean magnetization\nin the remanent state is directed along the c-axis [ez-\ndirection in Fig. 1(a) and ex-direction in Fig. 1(b)], the\nmagnetic neutron scattering cross section is in both ge-\nometries dominated by the respective |/tildewiderMy|2cos2θterm,\nwhich (in real space) is related to small misaligned spin\ncomponents varying along the ±ey-direction [29]. This\nanisotropy in the spin microstructure may be related\nto the finding of a crystallographic texture: as shown\nin Fig. 4, diffraction peaks of the type ( hk0) exhibit\ntwo maxima along the vertical direction ( θ= 90◦andθ= 270◦). The investigation of the relation between this\ncrystallographic texture and the spin texture is of inter-\nest in its own right and beyond the scope of this paper.\nHowever,wewouldliketoemphasizethatrecentelectron-\nmicroscopy and three-dimensional atom-probe tomogra-\nphy work by Liu et al.[30] also reports anisotropic\nproperties of the grain-boundary phase in hot-deformed\nnanocrystalline Nd-Fe-B magnets; namely, these authors\nfound that the concentration of rare-earth elements is\nhigher for intergranularphases parallel to the flat surface\nof the platelet-shaped Nd 2Fe14B grains as compared to\nintergranular phases along the short side of the platelets.\nThe characteristic size of the spin inhomogeneities in\nthe remanent state along the vertical and horizontal di-\nrection has been estimated by computing [using Eq. (5)]\nthe correlation function c(r) of the spin misalignment\n(Fig. 8). The exp( −1)-lengths are lC∼=53nm along the\nvertical direction, and lC∼=42nm along the horizontal\ndirection; lC∼=28nm for the isotropic nanocomposite.\nNotethattakingtheexp( −1)-lengthsdoesnotimplythat\nthe correlations decay exponentially. For the textured\nspecimen, both lCvalues are smaller than the average\nparticle size, which suggests the existence of intraparti-\ncle spin disorder, whereas lC∼=Dfor the isotropic sam-\nple. Compatible with [30], these results indicate that the\nmicroscopic nature of the microstructural defects (e.g.,\nthe Nd 2Fe14B/α-Feinterfaces) alongthese two directions\nare different (as is manifest by the different correlation\nlengths). In this respect field-dependent SANS measure-\nments are helpful, since they allow one to determine the\nfield evolution of lC, from which the size of the defect\n(causing the spin perturbation) and the exchange corre-\nlation length may be obtained [11, 12].\nV. CONCLUSION\nUsingmagneticsmall-angleneutronscattering(SANS)\nwe have provided a comparative study of the magnetic\nmicrostructure of textured and isotropic Nd 2Fe14B/α-\nFe nanocomposites. Our neutron-data analysis suggests\nthat the spin-misalignment scattering of the textured\nsampleis dominated byspin components alongone direc-\ntion perpendicular to the easy c-axis (pressing direction)\nof the Nd 2Fe14B grains. This anisotropy in the magne-\ntization distribution is accompanied by the presence of\na crystallographic texture along these directions. Possi-\nble originsfor the spin canting(on an intraparticlelength\nscale)havebeen discussedandarerelatedtothe presence\nof perturbed interface regions, crystalline imperfections,\nand/or a dispersion in the orientation distribution of the\neasyc-axes. In agreement with the x-ray synchrotron\nand neutron data, we find anisotropic real-space correla-\ntions, with a correlation length that has been estimated\nat about 40 −50nm in the remanent state. The results\ndemonstrate the power of magnetic SANS for analyzing\nanisotropic magnetic structures on a nanometer length\nscale; in particular, the complimentary use of the per-8\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s32/s32/s116/s101/s120/s116/s117/s114/s101/s100/s44/s32/s118/s101/s114/s116/s105/s99/s97/s108\n/s32/s32/s116/s101/s120/s116/s117/s114/s101/s100/s44/s32/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108\n/s32/s32/s116/s101/s120/s116/s117/s114/s101/s100/s44/s32/s50 /s32/s97/s118/s101/s114/s97/s103/s101\n/s32/s32/s105/s115/s111/s116/s114/s111/s112/s105/s99/s44/s32/s50 /s32/s97/s118/s101/s114/s97/s103/s101/s99 /s40 /s114 /s41\n/s114 /s32/s32/s40/s110/s109/s41\nFIG. 8. Normalized correlation function c(r) of the\nspin misalignment [Eq. (5)] for the textured and isotropic\nNd2Fe14B/α-Fe nanocomposite in the remanent state. c(r)\nof the textured sample has been computed using dΣM/dΩ\naveraged along the vertical and horizontal directions ( ±7.5◦\nsector averages) as well as using the full circular (2 π) aver-\nage ofdΣM/dΩ; thec(r) of the isotropic sample was com-\nputed using the corresponding 2 π-averageddΣM/dΩ (see in-\nset). Solid horizontal line: C(r) = exp( −1). The physically\nrelevant information content of c(r) is restricted to the inter-\nval [rmin,rmax] with approximately rmin∼=2π/qmax= 2nm\nandrmax∼=2π/qmin= 130nm.pendicular and parallel scattering geometrieshas (for the\ntextured sample) provided results that were otherwise\nnot accessible with only one geometry.\nACKNOWLEDGEMENTS\nDenis Mettus acknowledges financial support from\nthe National Research Fund of Luxembourg (IN-\nTER/DFG/12/07). This paper is based on results ob-\ntainedfromthefuture pioneeringprogram“Development\nof magnetic material technology for high-efficiency mo-\ntors” commissioned by the New Energy and Industrial\nTechnology Development Organization (NEDO). The\nneutron experiments were performed at the Swiss spalla-\ntion neutron source SINQ, Paul Scherrer Institute, Vil-\nligen, Switzerland. ALBA synchrotron is acknowledged\nfor the provision of beamtime. We thank Birgit Heiland\n(INM, Saarbr¨ ucken)and J¨ orgSchmauch (Universit¨ at des\nSaarlandes) for the electron-microscopy work.\n[1] O. Gutfleisch, M. A. Willard, E. Br¨ uck, C. H.Chen, S. G.\nSankar, and J. P. Liu, Adv. Mater. 23, 821 (2011).\n[2] J. P. Liu, in Lect. Notes Phys. 678 Nanoscale magnetic\nmaterials and applications , edited by J. P. Liu, E. Fuller-\nton, O. Gutfleisch, and D. J. Sellmyer (Springer, New\nYork, 2009) pp. 309–335.\n[3] J. Liu, H. Sepehri-Amin, T. Ohkubo, K. Hioki, A. Hat-\ntori, T. Schrefl, and K. 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P´ erigo, E. P. Gilbert, and A. Michels, Acta Mater.\n87, 142 (2015).\n[13] J.-P. Bick, K. Suzuki, E. P. Gilbert, E. M. Forgan,\nR.Schweins, P.Lindner, C. K¨ ubel, andA.Michels, Appl.\nPhys. Lett. 103, 122402 (2013).\n[14] E. A.P´ erigo, E.P.Gilbert, K.L.Metlov, andA.Michels ,\nNew. J. Phys. 16, 123031 (2014).\n[15] T. Ueno, K. Saito, M. Yano, M. Ito, T. Shoji, N. Sakuma,\nA. Kato, A. Manabe, A. Hashimoto, E. P. Gilbert,\nU. Keiderling, and K. Ono, Sci. Rep. 6, 28167 (2016).\n[16] E. A. P´ erigo, D. Mettus, E. P. Gilbert, P. Hautle,\nN. Niketic, B. vandenBrandt, J. Kohlbrecher, P. McGui-\nness, Z. Fu, and A. Michels, J. Alloys Comp. 661, 110\n(2016).\n[17] E. A. P´ erigo, I. Titov, R. Weber, D. Honecker, E. P.\nGilbert, M. F. De Campos, and A. Michels, J. Alloys\nComp.677, 139 (2016).\n[18] M. Yano, K. Ono, A. Manabe, N. Miyamoto, T. Shoji,\nA. Kato, Y. Kaneko, M. Harada, H. Nozaki, and\nJ. Kohlbrecher, IEEE Trans. Magn. 48, 2804 (2012).\n[19] M. Yano, K. Ono, M. Harada, A. Manabe, T. Shoji,\nA.Kato, andJ.Kohlbrecher,J.Appl.Phys. 115, 17A730\n(2014).\n[20] K. Saito, T. Ueno, M. Yano, M. Harada, T. Shoji,\nN. Sakuma, A. Manabe, A. Kato, U. Keiderling, and\nK. Ono, J. Appl. Phys. 117, 17B302 (2015).\n[21] J. Kohlbrecher and W. Wagner, J. Appl. Cryst. 33, 804\n(2000).9\n[22] N. Niketic, B. van den Brandt, W. Th. Wenckebach,\nJ. Kohlbrecher, and P. Hautle, J. Appl. Cryst. 48, 1514\n(2015).\n[23] F. Fauth, R. Boer, F. Gil-Ortiz, C. Popescu,\nO. Vallcorba, I. Peral, D. Full` a, J. Benach, and J. Juan-\nhuix, The European Physical Journal Plus 130, 1 (2015).\n[24] S. Erokhin, D. Berkov, and A. Michels, Phys. Rev. B\n92, 014427 (2015).\n[25] A. Michels, S. Erokhin, D. Berkov, and N. Gorn, J.\nMagn. Magn. Mater. 350, 55 (2014).\n[26] D. Mettus and A. Michels, J. Appl. Cryst. 48, 1437\n(2015).[27] A. Aharoni, J. Appl. Phys. 83, 3432 (1998).\n[28] A. P. Hammersley, S. O. Svensson, M. Hanfland, A. N.\nFitch, and D. Hausermann, High Pressure Research 14,\n235 (1996).\n[29] A. Michels, R. N. Viswanath, and J. Weissm¨ uller, Euro-\nphys. Lett. 64, 43 (2003).\n[30] J. Liu, H. Sepehri-Amin, T. Ohkubo, K. Hioki, A. Hat-\ntori, and K. Hono, Journal of Applied Physics 115, 17\n(2014)."    },    {        "title": "1610.00365v1.Intrinsic_magnetic_properties_of___R__Fe___1_x__Co___x______11__Ti_Z_____R____Y_and_Ce___Z____H__C__and_N_.pdf",        "content": "Intrinsic magnetic properties of R(Fe 1\u0000xCo x)11TiZ(R= Y and Ce; Z= H, C, and N)\nLiqin Ke1,\u0003and Duane D. Johnson1, 2\n1Ames Laboratory US Department of Energy, Ames, Iowa 50011\n2Materials Science &Engineering, Iowa State University, Ames, Iowa 50011-2300\n(Dated: September 16, 2018)\nAbstract\nTo guide improved properties coincident with reduction of critical materials in permanent magnets, we investigate via\ndensity functional theory (DFT) the intrinsic magnetic properties of a promising system, R(Fe1\u0000xCox)11TiZwith R=Y, Ce\nand interstitial doping ( Z=H, C, N). The magnetization M, Curie temperature TC, and magnetocrystalline anisotropy energy\nKcalculated in local density approximation to DFT agree well with measurements. Site-resolved contributions to Kreveal that\nall three Fe sublattices promote uniaxial anisotropy in YFe 11Ti, while competing anisotropy contributions exist in YCo 11Ti.\nAs observed in experiments on R(Fe1\u0000xCox)11Ti, we \fnd a complex nonmonotonic dependence of Kon Co content, and\nshow that anisotropy variations are a collective e\u000bect of MAE contributions from all sites and cannot be solely explained by\npreferential site occupancy. With interstitial doping, calculated TCenhancements are in the sequence of N >C>H, with volume\nand chemical e\u000bects contributing to the enhancement. The uniaxial anisotropy of R(Fe1\u0000xCox)11TiZgenerally decreases with\nC and N; although, for R=Ce, C doping is found to greatly enhance it for a small range of 0.7 <x< 0.9.\nI. INTRODUCTION\nThe search for new permanent magnets without crit-\nical materials has generated great interest in the mag-\nnetism community.[1, 2] Developing CeFe 12-based rare-\nearth(R)-transition-metal( TM) intermetallics[3{7] is an\nimportant approach, considering the relative abundance\nof Ce among Relements and the large content of inexpen-\nsive Fe. To improve CeFe 11Ti as a permanent magnet,\nit is desired to modify the compound to achieve the best\npossible intrinsic magnetic properties, such as magneti-\nzationM, Curie temperature TC, and magnetocrystalline\nanisotropy energy (MAE) K. Both substitutional dop-\ning with Co[8] and interstitial doping with small elements\nof H, C or N can strongly a\u000bect its magnetic properties.\nA theoretical understanding of intrinsic magnetic proper-\nties in this system and the e\u000bect of doping will help guide\nthe experiments and help ascertain the best achievable\npermanent magnet properties.\nBinary iron compounds of RFe12do not form for\nanyRelements unless a small amount of stabilizer el-\nements are added, such as T=Ti, Si, V, Cr, Mo, or W.[9]\nSuchRFe12\u0000zTzcompounds are generally regarded as\nternaries rather than pseudobinaries because the third\nelement,T, atoms often have a very strong site prefer-\nence and exclusively sit at one of three nonequivalent Fe\nsites.[10] Magnetization often decreases quickly with the\nincrease of Tcomposition and a minimum amount of Ti\n(z=0.7) is needed to stabilize the structure, resulting in\nTi compounds having better magnetic properties than\nothers.[11] Prototype yttrium compounds are often stud-\nied to focus on the properties of the TM sublattices in\nthe corresponding R-TM systems because yttrium can\nbe regarded as a nonmagnetic, rare-earth element.\nIn comparison to other R-Fe systems,[11, 12] such as\n\u0003Corresponding author: liqinke@ameslab.govY2Fe17and Y 2Fe14B, Fe sublattices in 1-12 compounds\nhave relative low magnetization due to a more compact\nstructure, but at low temperatures a very high uniaxial\nMAE, e.g., K=2 MJm\u00003in YFe 11Ti.[8, 13] Curie temper-\natures are relatively low; MandKquickly decrease with\nincreasing temperature.[14{16] CeFe 11Ti hasTC\u0019485 K,\nand a low-temperature magnetization within a range of\n17:4\u000020:2\u0016B/f.u., while YFe 11Ti has a slightly larger M\nandTC. At room temperature, CeFe 11Ti has a larger K\n(1.3 MJm\u00003) than YFe 11Ti (0.89 MJm\u00003). This may in-\ndicate that the Ce sublattice has a positive contribution\nto the uniaxial anisotropy.[15]\nThe substitutional doping with Co is a common\napproach to improve TCinR-Fe compounds.[8] Pure\nphaseR(Fe1\u0000xCox)11Ti exists over the whole compo-\nsition range for both R=Y and Ce.[17] The largest\nmagnetization in Y(Fe 1\u0000xCox)11Ti occurs at YFe 8Co3Ti\nwhile theTCincreases continuously with Co composi-\ntion until it reaches the maximum in YCo 11Ti.[18, 19]\nFor Ce compounds, the maximum TCis obtained in\nCeFe 2Co9Ti.[17] The dependence of MAE on the Co com-\nposition in R(Fe1\u0000xCox)11Ti is more intriguing and not\nunderstood. Although early experiments[8, 14] suggested\nthat YCo 11Ti has a planar anisotropy, later experiments\nagreed that YCo 11Ti has uniaxial anisotropy,[17{23] but\nwith a magnitude smaller than those of YFe 11Ti. For the\nintermediate Co composition, anisotropy changes from\nuniaxial to planar and then back to uniaxial with the\nincrease of Co composition in both of Y(Fe 1\u0000xCox)11Ti\nand Ce(Fe 1\u0000xCox)11Ti.[17{23]\nThe interstitial doping with H,[15] N [5, 24, 25] and C\n[26{28] can increase MandTC, and provide control of the\nmagnitude and sign of the MAE constants in RFe11Ti.\nHydrogenation simultaneously increases all three intrin-\nsic magnetic properties in YFe 11Ti, and enhancements\nare \u0001M=1\u0016B/f.u. at 4.2 K, \u0001 TC=60 K,[16, 29] and\n\u0001K= 6.5%,[16] respectively. Insertion of larger C and\nN atoms has a much stronger e\u000bect on the enhance-\nments ofMandTC.[27, 30] Unfortunately, it is achieved\nat the expense of uniaxial anisotropy. In compari-\n1arXiv:1610.00365v1  [cond-mat.mtrl-sci]  2 Oct 2016son with YFe 11Ti, enhancements of \u0001 M=2.6\u0016B/f.u.\nand \u0001TC=154 K were observed in YFe 11TiC 0:9, and\n\u0001M=2.7\u0016B/f.u. and \u0001 TC=218 K in YFe 11TiN 0:8. The\nMAE decreases by \u0001 K=0.6\u00180.7 MJm\u00003in both com-\npounds. Doping in\ruences the MAE contributions from\nbothTM sublattice and rare-earth atoms. It had been\nargued that the proximity of doping atoms to the rare-\nearth atoms in RTiFe 11Nxmay lead to drastic changes\nin the rare-earth sublattice anisotropy,[25] and N doping\noften has an opposite e\u000bect on MAE as H doping.[30]\nFor Ce compounds, a similar amount of TCenhancement\nwas obtained upon nitriding,[5] and the e\u000bect of H doping\nis much smaller. Isnard et al. [15] found that not much\nchange is observed upon H insertion either in the room\ntemperature anisotropy or in saturation magnetization.\nOther possible interstitial doping elements such as B,\nSi or P atoms are much less favored to occupy the inter-\nstitial sites due to chemical or structural reasons.[26] In\nfact, interestingly, it has been found that the B atoms\nprefer to substitute for some of the Ti atoms and drive\nthe Ti into the interstitial.[31]\nThe nature of the Ce 4 fstate di\u000berent Ce- TM com-\npounds is often a controversial subject.[32] The anomalies\nin the lattice constants as well as the magnetic moment\nand Curie temperature have been interpreted as evidence\nof the mixed-valence (between Ce3+and Ce4+) behavior\nof the cerium ion. It is further complicated by the dop-\ning. Controversy remains on how Ce valence states are af-\nfected upon hydrogenation.[15, 33] It also has been shown\nthat Ce 4fstates are itinerant and, as such, the standard\nlocalized 4fpicture is not appropriate for systems such\nas CeCo 5.[34, 35] Moreover, in the (Nd-Ce) 2Fe14B sys-\ntem, the mixed valency of Ce has been shown to be due\nto local site volume and site chemistry e\u000bects.[36] In this\npaper the 4 fstates in Ce are treated as itinerant and\nincluded as valence states, and we found that magnetic\nproperties calculated are in good agreement with exper-\niments.\nII. CALCULATION DETAILS\nA. Crystal structure\nRFe11Ti has a body-center-tetragonal ThMn 12-type\n(I4=mmm space group, no.139) structure, which is\nclosely related to the 1-5 and 2-17 R-TM structures.[11]\nThe primitive unit cell contains one formula unit (f.u.).\nAs shown in Fig. 1, Ratoms occupy the 2 a(4=mmm ) site,\nwhile transition metal atoms are divided into three sub-\nlattices, 8i(m2m), 8j(m2m) and 8f(2=m), each of which\nhas fourfold multiplicities. The 8 jand 8fsites bear a\ngreat similarity in their local environments with respect\nto the distribution of coordinated atoms,[37] whereas the\n8isites, often referred to as dumbbell sites, form -Fe-Fe-\nR- chains with Ratoms along the basal axes, instead of\nthecaxis, as in the 2-17 structure.[38] Ti atoms occupy\nnearly exclusively on the 8 isites, however, the distribu-\n8j8f\n2a8i\nFIG. 1: (Color online) Schematic representation of the crys-\ntal structures of RFe11Ti.R(2a) atoms are indicated with\nlarger spheres in yellow color. Three transition metal sublat-\ntices 8 i, 8j, and 8 fare in red, blue and green, respectively.\nEach Ratom has four nearest 8 i, eight nearest 8 j, and 8 f\nneighbor atoms. Among three TM sites, the 8 isite has the\nshortest distance from the Ratom. Interstitial sites 2 b(not\nshown) are halfway between two Ratoms along the caxis and\ncoordinated by an octahedron of two Rand four Fe(8 j) sites.\ntion of Fe and Ti atoms within the 8 isites is disordered.\nTo calculate RFe11Ti, we replace one of four Fe(8 i)\natoms with Ti in the primitive cell of RFe12and neglect\nthe e\u000bect of the arti\fcial Ti ordering introduced by using\nthis unit cell. Although the I4=mmm symmetry is low-\nered by Ti substitution or the spin-orbit coupling (SOC)\nin the anisotropy calculation, we still use the notations\nof 8j, 8i, and 8fsites for simplicity.\nFor Co doping, M ossbauer spectroscopy found that\nCo atoms preferentially occupy the 8 fsites in\nY(Fe 1\u0000xCox)11Ti,[39] while the high-resolution neutron\npowder di\u000braction experiments concluded that Co atoms\npreferentially occupy sites in the sequence of 8 j >8f >\n8i.[40] For interstitial H, C, and N doping, neutron scat-\ntering has shown that dopants prefer to occupy the larger\noctahedral 2 binterstitial sites,[15, 25] which have the\nshortest distance from the rare-earth sites among all\nempty interstitial sites. In all our calculations, we also\nassume that H, C, or N atom occupies the 2 bsites.\nB. Computational methods\nMost magnetic properties were calculated using a\nstandard linear mu\u000en-tin orbital (LMTO) basis set[41]\ngeneralized to full potentials.[42] This scheme employs\ngeneralized Hankel functions as the envelope functions.\nFor MAE calculation, the SOC was included through\nthe force theorem.[43] The MAE is de\fned below as\nK=E110\u0000E001, whereE001andE110are the summa-\ntion of band energies for the magnetization being oriented\nalong the [001] and [110] directions, respectively. Positive\n2(negative)Kcorresponds to uniaxial (planar) anisotropy.\nIt should be noted that, due to the presence of Ti in the\nprimitive cell, the two basal axes become inequivalent,\nwith -Ti-Fe- R- chains along the [100] direction and -Fe-\nFe-R- chains along the [010] direction. E100andE010\nbecome di\u000berent, which is an artifact introduced by us-\ning the small primitive cell and arti\fcial ordering of Ti\nwithin the 8 isublattice.\nWe found that the [100] direction is harder than the\n[010] in YFe 11Ti, and vice versa in YCo 11Ti.E110is\nusually about the average of E100andE010. Thus, we\nuse [110] as the reference direction for the basal plane.\nA 16\u000216\u000216k-point mesh is used for MAE calcula-\ntions to ensure su\u000ecient convergence; MAE in YFe 11Ti\nchanged by less than 3% when a denser 32 \u000232\u000232\nmesh was employed. To decompose the MAE, we eval-\nuate the on-site SOC matrix element hVsoiand the cor-\nresponding anisotropy Kso=1\n2hVsoi110\u00001\n2hVsoi001. Unlike\nMAE,Ksocan be easily decomposed into sites, spins,\nand orbital pairs. According to second-order perturba-\ntion theory,[44, 45]\nK\u0019X\niKso(i); (1)\nwhereiindicates atomic sites. Equation (1) holds true for\nall compounds that we investigated in this paper. Hence,\nwe useKso(i) to represent the site-resolved MAE. For\nsimplicity, we write it as K(i).\nExchange coupling parameters Jijare calculated us-\ning a static linear-response approach implemented in a\nGreen's function (GF) LMTO method, simpli\fed using\nthe atomic sphere approximation (ASA) to the potential\nand density.[46, 47] The scalar-relativistic Hamiltonian\nwas used so SOC is not included, although it is a small\nperturbation on Jij's. In the basis set, s;p;d;f orbitals\nare included for Ce, Y, Fe, and Co atoms, and s;por-\nbitals are included for H, C, and N atoms. Exchange pa-\nrametersJij(q) are calculated using a 163k-point mesh,\nandJij(R) can be obtained by a subsequent Fourier-\ntransforming. TCis estimated in the mean-\feld approx-\nimation (MFA) or random-phase approximation (RPA).\nSee Ref. 46 for details of the methods to calculate TC.\nFor all magnetic property calculations, the e\u000bec-\ntive one-electron potential was obtained within the lo-\ncal density approximation (LDA) to DFT using the\nparametrization of von Barth and Hedin.[48] However,\nwith the functional of Perdew, Becke, and Ernzerhof\n(PBE) being better at structural relaxation for most of\nthe solids containing 3 delements,[49] we use it to fully\nrelax the lattice constants and internal atomic positions\nin a fast plane-wave method, as implemented within the\nVienna ab initio simulation package (VASP).[50, 51] The\nnuclei and core electrons were described by the projec-\ntor augmented wave (PAW) potential[52] and the wave\nfunctions of valence electrons were expanded in a plane-\nwave basis set with a cuto\u000b energy of up to 520 eV. All\nrelaxed structures are then veri\fed in FP-LMTO before\nthe magnetic property calculations are performed.TABLE I: Calculated and measured (Exp.) values for the lat-\ntice parameters and volume are listed for various compounds.\nCompounds aa(\u0017A) c( \u0017A) V( \u0017A3) \u0001V=V Ref.\nYFe 11Ti (Exp.) 8.480 4.771 343.08 [53]\nYFe 11Ti 8.472 4.720 338.78 0\nYFe 12 8.447 4.695 334.94 -1.1\nYFe 11TiH 8.457 4.732 338.43 -0.10\nYFe 11TiC 8.517 4.834 350.67 3.51\nYFe 11TiN 8.563 4.791 351.31 3.70\nYCo 11Ti (Exp.) 8.367 4.712 329.87 [54]\nYCo 11Ti 8.328 4.673 324.08 0\nYCo 12 8.268 4.655 318.21 -1.81\nYCo 11TiH 8.343 4.688 326.30 0.68\nYCo 11TiC 8.396 4.767 336.08 3.70\nYCo 11TiN 8.436 4.716 335.59 3.55\nCeFe 11Ti (Exp.) 8.539 4.780 348.53 [15]\nCeFe 11Ti 8.524 4.670 339.35 0\nCeFe 12 8.504 4.648 336.12 -0.95\nCeFe 11TiH 8.498 4.738 342.13 0.82\nCeFe 11TiC 8.501 4.891 353.45 4.16\nCeFe 11TiN 8.570 4.809 353.17 4.07\nCeCo 11Ti (Exp.) 8.380 4.724 331.74 [17]\nCeCo 11Ti 8.360 4.657 325.46 0\nCeCo 12 8.291 4.648 319.51 -1.82\nCeCo 11TiH 8.359 4.694 327.94 0.76\nCeCo 11TiC 8.383 4.811 338.07 3.87\nCeCo 11TiN 8.442 4.735 337.44 3.68\naExcept for the hypothetical 1-12 compounds, Ti substitution in\nthe 13-atom cell breaks the symmetry of CeFe 12, and lattice pa-\nrameters aand bbecome nonequivalent. The listed calculated ais\nan average of aand bof the unit cell used in the calculation.\nIII. RESULTS AND DISCUSSION\nA. structure\nLattice constants and volumes are listed in Table I,\nthe calculated lattice constants are in good agreement\nwith experiments. The strong Ti site preference on the\n8isite[3, 15, 54] had been interpreted in terms of atomic\nvolume, coordination number, and enthalpy. It had been\nargued that enthalpy associated with Rand Ti, V, or\nMo atoms are positive and 8 isites have the smallest con-\ntact area with Ratoms. To identify quantitatively the\nsite-preference e\u000bect, we calculated the total energy of\nCeFe 11Ti with one Ti atom occupying at the 8 i, 8j, or\n8fsites, respectively, in the 13-atom primitive cell. The\nlowest-energy structure is the one with Ti atoms on the\n8isite. Energies are higher by 42 meV =atom and 60\nmeV=atom with Ti atom being on the 8 jand 8fsites,\nrespectively. Hence, Ti atom should have a strong pref-\nerence to occupy the 8 isites, as observed in the experi-\nments.\n3In comparison to the hypothetical 1-12 compounds, the\nreplacement of Fe or Co atoms with Ti increases volume\nby 1% or 2%, respectively. Experimentally, H doping\nslightly increases the volume by 1% in YFe 11TiH, which is\nnot observed in our calculation. The calculated volume of\nCeFe 11TiH is 0.82% larger than CeFe 11Ti. Calculations\nshow that carbonizing and nitriding have a larger e\u000bect\non volume expansion than hydrogenation and volume ex-\npansion is larger in Ce compounds than in Y compounds,\nboth of which agree with experiments.\nThe total density of states of YFe 11Ti and YFe 11TiN\ncompares reasonably well with previously reported\nLMTO-ASA calculations.[6] Figure. 2 shows the scalar-\nrelativistic partial density of states (PDOS) projected on\nindividual elements in YFe 11Ti, YFe 11TiH, YFe 11TiC,\nand YFe 11TiN. The Fe PDOS are averaged over 11 Fe\natoms. The interstitial doping elements on 2 bsites hy-\nbridizes with neighboring Rand Fe(8j) atoms. H- s\nstates hybridizes with neighboring Y and Fe(8 j) atoms at\naround -7 eV below the Fermi level in YFe 11TiH. The C- p\nand N-pstates have larger energy dispersion in YFe 11TiC\nand YFe 11TiN, respectively. The Fe states hybridized\nwith interstitial elements, as shown in Fig. 2, are mostly\nfrom four (8 j) out of 11 Fe sites. Fe(8 f) sites are the\nfurthest away from the interstitial 2 bsites and their hy-\nbridization with doping elements are negligible. The Ce\ncompounds have a large f-states above the Fermi level\nand share lots of similar PDOS features with the corre-\nsponding Y compounds below the Fermi level.\nB. Magnetization, Exchange Couplings and TC\nIntrinsic magnetic properties of each compound are\nlisted in Table II. Experimental magnetization and\nanisotropy values vary. The calculated magnetizations\nin YFe 11Ti, YCo 11Ti, and CeFe 11Ti compare well with\nexperiments. For CeCo 11Ti, only a limited number of\nstudies had been reported, and the calculated magne-\ntization is larger than experimental ones. Ti spin mo-\nments couple antiparallel to those of Fe and Co sublat-\ntices, which is typical for the light 3 dand 4delements.[56]\nIn CeFe 11Ti, the Ce spin moments antiferromagneti-\ncally couple with the TM sublattice as expected.[57] Ce\nhas a spin moment ms\u0019-0.7\u0016Band an orbital moment\nml\u00190.3\u0016Bwith the opposite sign, which re\rects Hund's\nthird rule. The calculated Fe spin moments on the in-\ndividual sublattice have the magnitude in the sequence\nofms(8i)>m s(8j)>m s(8f), which agrees with previous\nexperiments and calculations.[58] The dumbbell 8 isites\nhave larger spin magnetic moments because of the rela-\ntive larger surrounding empty volume and smaller atomic\ncoordination number. The orbital magnetic moments\ncalculated are larger in the Co-rich compounds than the\nFe-rich compounds. MFA overestimated TCby about\n200 K in Fe compounds and about 50 \u0000100 K in Co com-\npounds, respectively. RPA gives lower TCvalues, e.g.,\n−101\n  \nYFe11Ti (a)\nY\nTi\nFe\nFe(8i)\nFe(8j)\nFe(8f)\n−101DOS ( states ( eV spin atom )−1)\n  \nYFe11TiH (b)\nY\nTi\nFe\nH\n−101\n  \nYFe11TiC (c)\nY\nTi\nFe\nC\n−8 −6 −4 −2 0 2 4−101\nE(eV)  \nYFe11TiN (d)\nY\nTi\nFe\nNFIG. 2: (Color online) Atom- and spin-projected, partial\ndensities of states (DOS) in ( a) YFe 11Ti, (b) YFe 11TiH, ( c)\nYFe 11TiC, and ( d) YFe 11TiN within the LDA and no SOC.\nFor YFe 11Ti, the of Fe DOS are further resolved by aver-\naging states projected on 8 i, 8j, and 8 fsites. Majority spin\n(positive values) and minority spin (negative values) DOS are\nshown separately. Fermi energy EFis at 0 eV.\n489 K in YFe 11Ti, and 461 K in CeFe 11Ti, respectively.\nThe experimental TCfalls between the MFA and RPA\nvalues, and is much closer to the latter.\nTi additions decrease the magnetization by 20% in\nRFe11Ti andRCo11Ti relative to their 1-12 hypothetical\ncounterparts. The magnetization reduction is not only\ndue to the replacement of ferromagnetic Fe by antiferro-\nmagnetic Ti atoms (spin moment \u00000.54\u0016B), but also the\n4TABLE II: Calculated spin Ms, orbital Ml, and total Mtmagnetization, exchanges J0, Curie temperature TCestimated in\nthe mean-\feld approximation, and magnetocrystalline anisotropy Kin various compounds. Unless speci\fed, experimental\nmagnetization and anisotropy Kvalues from previous studies were measured or evaluated for low temperature ( <5 K).\nCompoundMsMl Mt J0(meV) TC \u0001TC KRef.\n(\u0016B\nf.u.) 8 i(Ti) 8 i 8j 8f Y K (meV\nf.u.) (MJ\nm3)\nYFe 12 24.20 0.61 24.81 7.57 4.91 5.10 1.34 689 -38 1.40 1.34\nYFe 11Ti (Exp.) 19 \u000020.6 524 \u0000538 2.0 [8, 11, 16, 19, 27]\nYFe 11Ti 19.75 0.60 20.35 5.29 7.17 6.70 6.61 1.43 727 0 1.93 1.83\nYFe 11TiH 19.92 0.54 20.46 4.99 7.63 7.46 7.57 1.40 778 51 2.07 1.96\nYFe 11TiC 20.64 0.55 21.19 5.51 8.58 8.83 8.66 1.67 884 157 0.95 0.87\nYFe 11TiN 22.11 0.57 22.68 5.44 9.36 8.91 9.29 1.30 938 211 1.80 1.65\nYCo 11Ti (Exp.) 14.2 \u000015.7 1020 \u00001050 0.75a[18, 19]\nYCo 11Ti 14.42 0.82 15.24 3.93 10.50 10.13 11.13 1.45 1091 364 0.94 0.93\nYCo 12 18.42 0.90 19.32 12.52 13.12 13.84 1.66 1374 647 0.48 0.48\nCeFe 12 24.02 0.78 24.80 8.69 7.33 6.12 1.75 806 131 1.77 1.69\nCeFe 11Ti (Exp.) 17.4 \u000020.2 482 \u0000487 1.3a\u00002.0a[5, 15, 17, 18, 55]\nCeFe 11Ti 19.19 0.72 19.91 4.69 6.26 7.04 5.95 2.16 675 0 2.09 1.98\nCeFe 11TiH 20.24 0.77 21.01 4.67 6.87 7.42 7.04 2.30 736 61 2.03 1.90\nCeFe 11TiC 19.84 0.73 20.57 5.45 9.86 8.62 8.62 3.44 908 233 1.09 0.99\nCeFe 11TiN 21.48 0.67 22.15 5.51 9.09 8.53 8.99 1.09 905 230 1.78 1.62\nCeCo 11Ti (Exp.) 10.9 \u000012.53a920\u0000937 Axial [17, 18]\nCeCo 11Ti 13.77 1.32 15.09 4.07 10.40 9.38 10.94 3.76 1044 369 1.29 1.23\nCeCo 12 17.35 1.36 18.71 12.03 12.29 12.97 3.80 1286 611 1.24 1.24\naMeasured at room temperature.\nsuppression of the ferromagnetism on the neighboring Fe\nsublattices. This is a common e\u000bect of doping early 3 d\nor 4delements on the Fe or Co sublattice.[56] On the\nother hand, the addition of the Ti atom barely a\u000bects\nthe Ce moment. Interestingly, although magnetization\ndecreased by 20% upon the Ti addition, the calculated\nTCis even slightly higher in YFe 11Ti than in YFe 12. This\nis somewhat re\rected in the experiments, in which no ob-\nviousTCdependence on Ti composition was observed in\nYFe 11\u0000zTizover the homogeneous 1-12 phase composi-\ntion range, 0.7\u0014z\u00141.25.[11]\nTo understand this phenomenon, we investigated the\ne\u000bective exchange coupling parameters J0(i)=P0\njJijand\ncompareJ0values in YFe 12and YFe 11Ti. With Ti re-\nplacing one Fe atom, J0values increase for all sites ex-\ncept the pair of Ti-Fe dumbbell sites. The overall J0and\nthe mean-\feld TCincrease. The site-resolved e\u000bective ex-\nchange parameters J0(i) for various compounds are listed\nin Table II.\nFigure 3 shows the magnetization as a function of the\nCo composition in YFe 11Ti, with similar behavior to the\nSlater-Pauling curve. The maximum magnetization oc-\ncurs atx=0.2, while in experiments it is at x=0.3.[19]\nSimilarly, for Ce(Fe 1\u0000xCox)11Ti, the experimental maxi-\nmum magnetization occurs at x=0.1\u00000.15.[55] As shown\nin Table II, the RCo11Ti compounds have much larger\n0 0.2 0.4 0.6 0.8 11516171819202122\nxM (µB/f.u.)\n  \nTheory\nExp. at 5K (Wang et al.)\nExp. at 1.5K (Yang et al.)FIG. 3: (Color online) Comparison of measured and cal-\nculated (squares) Mversus Co content in Y(Fe 1\u0000xCox)11Ti.\nExperimental data are from Wang et al. [19] at 5 K (circles)\nand Yang et al. [8] at 1.5 K (triangles).\nTCthan the corresponding RFe11Ti compounds, which\nagrees with experiments.[17]\nAll interstitial doping increases MandTCin YFe 11Ti\nand CeFe 11Ti, and nitriding has the strongest e\u000bect.\nWith H, C, and N doping, the calculated Curie tem-\nperature in YFe 11Ti increases by 51, 157, and 211 K,\n5respectively, which is consistent with experiments. J0\nvalues on all three TM sublattices increase with inter-\nstitial doping. Although DFT underestimats the volume\nexpansion with H doping, the calculated \u0001 TCis only\nslightly smaller than the experimental value. The calcu-\nlated \u0001TCis larger with N doping than C doping, while\ntheir calculated volume expansions are similar. This in-\ndicates that both volume and chemical e\u000bects are im-\nportant for the TCenhancement. To estimate qualita-\ntively the relative magnitudes of the two e\u000bects, we cal-\nculate theTCof several hypothetical compounds related\nto YFe 11TiN by removing the N atom in the unit cell or\nreplacing it with H or C atoms, respectively. The cal-\nculated \u0001TCof those structures relative to YFe 11Ti are\n53, 80, and 169 K, respectively. Obviously, both volume\nand chemical e\u000bects contribute to the TCenhancement\nand the chemical e\u000bects of interstitial elements are in the\nsequence of N >C>H.\nC. MAE in R(Fe1\u0000xCox)11Ti\nAs listed in Table II, both YFe 11Ti and YCo 11Ti have\nuniaxial anisotropy. Calculated MAE in YFe 11Ti is in\ngood agreement with the experimental value. CeFe 11Ti\nhas a slightly larger MAE than YFe 11Ti as found in\nexperiments.[15, 32] The PBE functional (not shown)\ngives a smaller MAE than LDA in YFe 11Ti and CeFe 11Ti.\nThe Fe sublattice anisotropy may have a strong de-\npendence on the composition of stabilizer atoms.[14] To\nunderstand how Ti a\u000bects the magnetic anisotropy and\nthe origin of the nonmonotonic dependence of MAE on\nCo composition, we resolved MAE into sites by evaluat-\ning the matrix element of the on-site SOC energy.[44, 45]\nFor intermediate Co composition, we investigate the\nMAE in YFe 7Co4Ti and YFe 3Co8Ti. We calculated\nthe formation energy relative to YFe 11Ti and YCo 11Ti\nand found that YFe 7Co4Ti has a formation energy\nEfmn=\u000034 meV=atom with four Co atoms on the 8 jsites\nandEfmn=\u000028 meV=atom with four Co atoms on the 8 f\nsites. Both values are lower than Efmn=\u000010 meV=atom,\nthe formation energy of YFe 8Co3Ti with all three Co\natoms being on the 8 isites. Hence, the site preference\nof Co atoms is 8 j>8f>8i, which agrees with the neutron\nscattering experiments.[40] For YFe 3Co8Ti, we occupy\nanother four Co atoms on the 8 fsites and the corre-\nsponding formation energy is \u000031 meV=atom.\nFigure 4 shows the total MAE values and their\nsublattice-resolved components, in YFe 12, YFe 7Co4Ti,\nYFe 3Co8Ti and YCo 11Ti. Obviously, Eq. (1) is well\nsatis\fed in all compounds and Ksopresents well the\nsite-resolved MAE. The Y sublattice has a negligible\ncontribution to anisotropy, as expected for a weakly\nmagnetic atom, because the spin-parallel components\nof MAE contribution cancel out the spin-\rip ones.[45]\nSublattice-resolved MAE contributions in YFe 12shows\nK(8j)>K(8i)>0>K(8f), which agrees with the pre-\n−2−1012Anisotropy (meV/sublattice)\nYFe12YFe11Ti YFe 7Co4Ti YFe 3Co8Ti YCo 11Ti  \nKso(8j)\nKso(8i)\nKso(8f)\nKso(Y)\nKso\nK\nKexptFIG. 4: (Color online) Total and sublattice-resolved Ksoin\nYFe 11Ti, YFe 7Co4Ti, YFe 3Co8Ti and YCo 11Ti. Calculated\nKand measured Kexptvalues are also compared. Experimen-\ntal values were from Refs. 29 and 18, measured at 4.2K for\nYFe 11Ti and 293K for YCo 11Ti, respectively. In calculations,\nwe assume that all four Co occupy the 8 jsites in YFe 7Co4Ti\nwhile all eight Co occupy the 8 jand 8 fsites in YFe 3Co8Ti.\nvious estimation in sign but di\u000bers in the order.[11]\nConsidering Fe(8 i) sites have positive contributions to\nthe uniaxial anisotropy in YFe 12, one may expect\nthat replacing Fe atoms by the Ti atoms on the 8 i\nsite would decrease MAE. Interestingly, we found that\nYFe 11Ti has even larger uniaxial anisotropy than YFe 12.\nAnisotropies of all three sublattices become more uniax-\nial andK(8j)>K(8i)>K(8f)>0 in YFe 11Ti, which indi-\ncates that the introduction of Ti atoms modi\fes the elec-\ntronic structure of neighboring sites and enhances their\ncontribution to uniaxial anisotropy. Similarly, other com-\npounds, such as YCo 11Ti, CeFe 11Ti, and CeCo 11Ti, are\nalso found to have MAE values larger than or similar to\ntheir corresponding hypothetical 1-12 counterparts.\nThe dependence of MAE on the Co composi-\ntion is nonmonotonic and also found in other R-TM\nsystems.[59] As shown in Fig. 4, the calculated MAE re-\nproduce the trend observed in experiment. For interme-\ndiate Co compositions, YFe 7Co4Ti compound has pla-\nnar anisotropy while YFe 3Co8Ti compound has a very\nsmall uniaxial anisotropy. The 8 jsublattice is the ma-\njor contributor to the uniaxial anisotropy in YFe 11Ti.\nWith all four 8 jFe atoms being replaced by Co atoms\nin YFe 7Co4Ti,K(8j) becomes very negative. Moreover,\nK(8i) andK(8f) are also strongly a\u000bected and become\nnegative. Further Co doping on 8 fsites changes K(8i)\nandK(8f) back to positive in YFe 3Co8Ti. Finally, in\nYCo 11Ti bothK(8i) andK(8f) increase and K(8j) be-\ncomes less planar and we have K(8i)>K(8f)>0>K(8j).\nThe nonmonotonic composition dependence is often\ninterpreted by preferential site occupancy,[59] however,\nsuch an explanation is an oversimpli\fcation for a metal-\nlic system, such as Y(Fe 1\u0000xCox)11Ti. The MAE con-\n6tributions from each TM sublattice may depend on the\ndetailed band structure around the Fermi energy. The\ndoping of Co on particular sites unavoidably a\u000bects the\nelectronic structure of neighboring TM sublattices due\nto the hybridization between them, which changes the\nMAE contribution from neighboring sites. Obviously, as\nshown in Fig. 4, with a sizable amount of Co doping, the\nvariation of anisotropy is a collective e\u000bect instead of a\nsole contribution from the doping sites.\nAmong three TM sublattices, the dumbbell 8 isites\nhave the largest contribution to the uniaxial anisotropy\nin YCo 11Ti, which we found also true in CeCo 11Ti,\nand hypothetical YCo 12and CeCo 12. It is interest-\ning to compare the MAE contributions from Co sub-\nlattices in RCo12andR2Co17, in which the dumbbell\nCo sites have the most negative contribution to the uni-\naxial anisotropy.[38] In both cases, the moments of the\ndumbbell sites prefer to be perpendicular to the dumbbell\nbonds, which are along di\u000berent directions in two struc-\ntures, i.e., basal axes in the 1-12 structure and caxis in\nthe 2-17 structure. As a result, dumbbell Co sites have\nMAE contributions of opposite sign in two structures.\nIn a real sample, Co likely also partially occupies the\n8jand 8fsites instead of exclusively only the 8 jsite. We\ninvestigate the scenario at the other extreme by assuming\nCo occupies the three TM sublattices with equal prob-\nability and calculate composition dependence of MAE\nusing the virtual crystal approximation (VCA). Inter-\nestingly, the nonmonotonic behavior is also observed as\nshown in Fig. 5. The easy direction changes from uni-\naxial to in-plane and then back to uniaxial. The varia-\ntions of each individual TM sublattice share a similarity\nwith the trend shown in Fig. 4. With increasing of x\nin Y(Fe 1\u0000xCox)11Ti,K(8j) decreases and becomes neg-\native while K(8i) andK(8f) become negative for the\nintermediate Co composition and then change back to\npositive at the Co-rich end. Thus, the nonmonotonic\nbehavior is con\frmed with or without considering pref-\nerential occupancy. The spin-reorientation transition[21]\nfrom axis to in-plane occurs in Y(Fe 1\u0000xCox)11Ti but not\npure YFe 11Ti,[21] which may relate to the fact that the\ncompeting anisotropies between three TMsublattices ex-\nist in Y(Fe 1\u0000xCox)11Ti while all three TM sublattices\nsupport the uniaxial anisotropy in YFe 11Ti. As shown in\nFig. 5(Top), MAE in Y(Fe 1\u0000xCox)11Ti barely changes or\neven slightly increases with a very small Co composition.\nA similar feature had been observed experimentally.[60]\nIt is caused by the partial occupation of Co on 8 fsites in\nYFe 11Ti. We found that replacing Fe atoms in YFe 11Ti\nwith Co atoms on the 8 fsites increases the MAE.\nIt is commonly assumed that the MAE contribu-\ntions from the TMsublattices are similar in R-TMcom-\npounds with di\u000berent R, and such contributions are often\nestimated experimentally from measurements on corre-\nsponding yttrium compounds.[12] As shown in Fig. 5,\nMAE contributions from TM sublattices in YFe 11Ti and\nCeFe 11Ti are similar but not identical. All three TM\nsublattices have positive contributions to the uniaxial\n0 0.2 0.4 0.6 0.8 1−1012\nxAnisotropy (meV/sublattice)\n  \nKso(8j)\nKso(8i)\nKso(8f)\nKso(Y)\nK\n0 0.2 0.4 0.6 0.8 1−2−1012\nxAnisotropy (meV/sublattice)\n  \nKso(8j)\nKso(8i)\nKso(8f)\nKso(Ce)\nKFIG. 5: (Color online) Kand sublattice-resolved Ksoin Y-\nbased (top) and Ce-based (bottom) R(Fe1\u0000xCox)11Ti.\nanisotropy and K(8j)>K(8i)>K(8f)>0. However, mag-\nnitudes of each sublattice di\u000ber in two compounds, which\nsuggest that the hybridization TM sites have with dif-\nferentRatoms a\u000bects their contributions to the MAE.\nUnlike the Y sublattice in YFe 11Ti, Ce provides a posi-\ntive contribution to the uniaxial anisotropy in CeFe 11Ti.\nD. E\u000bect of interstitial doping\nInterstitial doping with N, C, and H a\u000bects the MAE\nfrom both of the Fe and Rsublattices.[30] As shown in\nTable II, H doping barely changes or slightly increases\nthe uniaxial anisotropy in YFe 11Ti and CeFe 11Ti while\ncarbonizing and nitriding weaken the uniaxial anisotropy,\nwhich agrees with experiments.[5, 27] Simultaneous sub-\nstitutional Co doping and interstitial doping with H, C,\nor N is of interest. Although the uniaxial anisotropy may\nnot improve that much at the low temperature, the e\u000bect\ncould be more signi\fcant at room temperature. For ex-\nample, upon hydrogenation, a signi\fcant increase of K1\n70 0.2 0.4 0.6 0.8 1−2−1012\nxK(MJ/m3)\n  \nYFe1−xCoxTi\nYFe1−xCoxTiH\nYFe1−xCoxTiN\nYFe1−xCoxTiC\n0 0.2 0.4 0.6 0.8 1−2−1012\nxK(MJ/m3)\n  \nCeFe 1−xCoxTi\nCeFe 1−xCoxTiH\nCeFe 1−xCoxTiN\nCeFe 1−xCoxTiCFIG. 6: (Color online) Kversus Co content in\nR(Fe1\u0000xCox)11TiZwith R=Y (top) and R=Ce (bottom),\nwith and without Z=H, C, and N.\nwith a factor 1.8 was observed in YFe 9Co2Ti at room\ntemperature.[60]\nTo our knowledge, simultaneous doping of Co and\ninterstitial elements C and N atoms is not well stud-\nied. We calculated the MAE dependence on Co com-\npositions in Ce(Fe 1\u0000xCox)11TiZwithZ=H, C, and N,\nand results are shown in Fig. 6. The site preference\nof Co is not considered and VCA is used. The maxi-\nmum of uniaxial anisotropy in Y(Fe 1\u0000xCox)11TiH is ob-\ntained atx=0.1 while experiments found the maximum\nat YFe 9Co2TiH.[60] For the Fe-rich CeCo 11TiZ, only H\ndoping slightly increases the MAE, while C and N quickly\ndecrease uniaxial anisotropy. For Y(Fe 1\u0000xCox)11TiZ, it\nis unlikely we can have better uniaxial anisotropy (at\nleast at low temperature) over the whole range of Co com-\nposition. Interestingly, for Co-rich Ce(Fe 1\u0000xCox)11TiZ,interstitial C doping signi\fcantly improves the uniaxial\nanisotropy in Ce(Fe 1\u0000xCox)11TiZfor 0.7<x< 0.9. Con-\nsidering the relative high Curie temperature on the Co-\nrich end, it has an attractive combination of all three\nintrinsic magnetic properties, M,J, andK, for perma-\nnent magnet application.\nIV. CONCLUSION\nUsing DFT methods, the intrinsic magnetic proper-\nties ofRFe11Ti-related systems were investigated for the\ne\u000bects of substitutional alloying with Co and intersti-\ntial doping with H, C, and N. All properties and trends\nwere well described within the local density approxima-\ntion to DFT. In comparison to the hypothetical YFe 12,\nTi quickly decreases the magnetization and increases the\nuniaxial magnetic anisotropy in YFe 11Ti. The calculated\nCo site preference is 8 j >8f > 8iin Y(Fe 1\u0000xCox)11Ti\nwithx < 0:4, in agreement with neutron experiments.\nThe enhancement of MandTCdue to Co doping and in-\nterstitial doping are in good agreement with experiments.\nCompared with YFe 11Ti, the calculated TCincreases\nby 51, 157 and 211 K in YFe 11TiZwithZ=H, C, and N,\nrespectively, with both volume and chemical e\u000bects con-\ntributing to the enhancement. We found that all three Fe\nsublattices promote uniaxial anisotropy in the sequence\nofK(8j)> K(8i)> K(8f)>0 in YFe 11Ti, while com-\npeting contributions give K(8i)> K (8f)>0> K (8j)\nin YCo 11Ti. For intermediate Co composition, we con-\n\frm that the easy direction changes with increasing Co\ncontent from uniaxial to in-plane and then back to uni-\naxial. Substitutional doping a\u000bects the MAE contri-\nbutions from neighboring sites and the nonmonotonic\ncomposition dependence of anisotropy is a collective ef-\nfect, which can not be solely explained by preferential\noccupancy. The Ce sublattice promotes the uniaxial\nanisotropy in CeFe 11Ti and CeCo 11Ti. Interstitial C\ndoping signi\fcantly increases the uniaxial anisotropy in\nCe(Fe 1\u0000xCox)11Ti for 0:7< x < 0:9, which may pro-\nvide the best combination of all three intrinsic magnetic\nproperties for permanent applications.\nAcknowledgments\nWe thank B. Harmon, A. Alam, C. Zhou, and R. 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Johnson1, 3\n1Ames Laboratory, U.S. Department of Energy, Ames, Iowa 50011, USA\n2Institute for Metal Physics, 36 Vernadsky Street, 03142 Kiev, Ukraine\n3Departments of Materials Science &Engineering and Physics, Iowa State University, Ames, Iowa 50011-2300\n(Dated: September 4, 2018)\nMagnetocrystalline anisotropy (MCA) in doped Ce 2Co17and other competing structures was in-\nvestigated using density functional theory. We con\frmed that the MCA contribution from dumbbell\nCo sites is very negative. Replacing Co dumbbell atoms with a pair of Fe or Mn atoms greatly en-\nhance the uniaxial anisotropy, which agrees quantitatively with experiment, and this enhancement\narises from electronic-structure features near the Fermi level, mostly associated with dumbbell sites.\nWith Co dumbbell atoms replaced by other elements, the variation of anisotropy is generally a col-\nlective e\u000bect and contributions from other sublattices may change signi\fcantly. Moreover, we found\nthat Zr doping promotes the formation of 1-5 structure that exhibits a large uniaxial anisotropy,\nsuch that Zr is the most e\u000bective element to enhance MCA in this system.\nI. INTRODUCTION\nThe quest for novel high energy permanent magnet\nwithout critical elements continues to generate great in-\nterest [1]. While a rare-earth-free permanent magnet is\nappealing, developing a Ce-based permanent magnet is\nalso very attractive, because among rare-earth elements\nCe is most abundant and relatively cheap. Among Ce-Co\nsystems, Ce 2Co17has always attracted much attention\ndue to its large Curie temperature TCand magnetization\nM. The weak point of Ce 2Co17is its rather small easy-\naxis magnetocrystalline anisotropy (MCA), which must\nbe improved to use as an applicable permanent magnet.\nThe anisotropy in Ce 2Co17, in fact, can be improved\nsigni\fcantly through doping with various elements. Ex-\nperimental anisotropy \feld HAmeasurements by dopant\nand stoichiometry are shown in Fig. 1. This anisotropy\nenhancement has been attributed to the preferential sub-\nstitution e\u000bects of doping atoms [2, 3]: (i) The four\nnon-equivalent Co sites contribute di\u000berently [4] to the\nmagnetic anisotropy in Ce 2Co17. Two out of the 17 Co\natoms occupy the so-called dumbbell sites and have a\nvery negative contribution to uniaxial anisotropy, lead-\ning to the small overall uniaxial anisotropy; (ii) Dop-\ning atoms preferentially replace the dumbbell sites \frst,\neliminating their negative contribution and increasing the\noverall uniaxial anisotropy. The above explanation is\nsupported by the observation that with many di\u000berent\ndopants, the anisotropy \feld in Ce 2TxCo17\u0000xshows a\nmaximum around x= 2. This corresponds to the num-\nber of dumbbell sites in one formula unit [5].\nNumerous experimental e\u000borts have explored the pref-\nerential substitution e\u000bect and site-resolved anisotropy.\nStreever [12] studied the site contribution to the MCA in\nCe2Co17using nuclear magnetic resonance and concluded\nthat the dumbbell sites in Ce 2Co17have a very nega-\ntive contribution to uniaxial anisotropy. Neutron scat-\n\u0003Corresponding author: liqinke@ameslab.gov\n 0 10 20 30 40 50 60 70\n 0 1 2 3 4 5 6 7 8HA (Koe)\nxAl\nSi\nGa\nZr\nHf\nTi\nV\nCr\nMn\nFe\nCuFIG. 1. Experimental anisotropy \felds HAin Ce 2TxCo17\u0000x\nwithT=Al [6, 7], Si [6, 8], Ga [6, 9], Zr [10], Hf [10], V [5],\nCr [5], Mn [5, 11], Fe [5], and Cu [5].\ntering or M ossbauer studies have suggested that Fe [12{\n15], Mn [16], and Al [7, 17, 18] atoms prefer to substitute\nat dumbbell sites.\nHowever, it is not clear whether only the preferential\nsubstitution e\u000bect plays a role in HAenhancement for all\ndoping elements. For elements such as Zr, Ti, and Hf, the\nsubstitution preference is not well understood. Replacing\nthe dumbbell Co atoms with a pair of large atoms may\nnot always be the only energetically favorable con\fgu-\nration. For Mn and Fe, known to substitute at dumb-\nbell sites, the elimination of negative contributions at\nthose sites may explain the increase of magnetocrystalline\nanisotropy energy (MAE). It is yet unclear why di\u000ber-\nent elements give a di\u000berent amplitude of MAE enhance-\nment or what mechanism provides this enhancement. For\npermanent magnet application, Fe and Mn are partic-\nularly interesting because they improve the anisotropy\nwhile preserving the magnetization with x < 2. Other\ndopants quickly reduce the magnetization and Curie tem-\nperature. Further tuning of magnetic properties for com-\npounds based on Fe-or-Mn-doped Ce 2Co17would bene\ftarXiv:1610.02767v2  [cond-mat.mtrl-sci]  10 Apr 20182\nfrom this understanding.\nIn this work, we use density functional theory (DFT) to\ninvestigate the origin of the MAE enhancement in doped\nCe2Co17. By evaluating the on-site spin-orbit coupling\n(SOC) energy [19, 20], we resolved anisotropy into contri-\nbutions from atomic sites, spins, and orbital pairs. Fur-\nthermore, we explained the electronic-structure origin of\nMAE enhancement.\nII. CALCULATION DETAILS\nA. Crystal structure\nCe2Co17crystallizes in the hexagonal Th 2Ni17-type\n(P63=mmc , space group no. 194) structure or the\nrhombohedral Zn 17Th2-type (R3mh, space group no.\n166) structure, depending on growth condition and dop-\ning [10]. As shown in Fig. 2, both 2-17 structures can\nbe derived from the hexagonal CaCu 5-type (P6=mmm\nspace group 191) structure with every third Ce atom be-\ning replaced by a pair of Co atoms (referred to as dumb-\nbell sites). The two 2-17 structures di\u000ber only in the\nspatial ordering of the replacement sites. In the CeCo 5\ncell, a Ce atom occupies the 1 a(6=mmm ) site and two Co\natoms occupy the 2 c(\u00006m2) site, together forming a Ce-\nCo basal plane. Three Co atoms occupy the 3 g(mmm )\nsites and form a pure Co basal plane. The primitive\ncell of hexagonal Ce 2Co17(H-Ce2Co17) contains two for-\nmula units while the rhombohedral Ce 2Co17(R-Ce2Co17)\ncontains one. The Co atoms are divided into four sub-\nlattices, denoted by Wycko\u000b sites 18 h, 18f, 9d, and 6c\nin the rhombohedral structure, and 12 k, 12j, 6g, and\n4fin the hexagonal structure. The 6 cand 4fsites are\nthe dumbbell sites. In the R-structure, Ce atoms form\n-Ce-Ce-Co-Co- chains with Co atoms along the zaxis.\nTheH-structure has two inequivalent Ce sites, denoted\nas 2cand 2b, respectively. Along the zdirection, Ce 2b\nform pure -Ce- atoms chains and Ce 2cform -Ce 2c-Co-\nCo- chains with Co dumbbell sites.\nB. Computational methods\nWe carried out \frst principles DFT calculations using\nthe Vienna ab initio simulation package (VASP) [21, 22]\nand a variant of the full-potential linear mu\u000en-tin or-\nbital (LMTO) method [23]. We fully relaxed the atomic\npositions and lattice parameters, while preserving the\nsymmetry using VASP. The nuclei and core electrons\nwere described by the projector augmented-wave poten-\ntial [24] and the wave functions of valence electrons were\nexpanded in a plane-wave basis set with a cuto\u000b energy\nof 520 eV. The generalized gradient approximation of\nPerdew, Burke, and Ernzerhof was used for the correla-\ntion and exchange potentials.\nThe MAE is calculated below as K=E100\u0000E001, where\nE001andE100are the total energies for the magnetizationoriented along the [001] and [100] directions, respectively.\nPositive (negative) Kcorresponds to uniaxial (planar)\nanisotropy. The spin-orbit coupling is included using the\nsecond-variation procedure [25, 26]. The k-point integra-\ntion was performed using a modi\fed tetrahedron method\nwith Bl ochl corrections. To ensure the convergence of the\ncalculated MAE, dense kmeshes were used. For example,\nwe used a 163k-point mesh for the calculation of MAE\ninR-Ce2Co17. We also calculated the MAE by carry-\ning out all-electron calculations using the full-potential\nLMTO (FP-LMTO) method to check anisotropy results.\nTo decompose the MAE, we evaluate the anisotropy of\nthe scaled on-site SOC energy Kso=1\n2hVsoi100\u00001\n2hVsoi001.\nAccording to second-order perturbation theory [19, 20],\nK\u0019P\niKso(i), whereiindicates the atomic sites. Un-\nlikeK, which is calculated from the total energy di\u000ber-\nence,Ksois localized and can be decomposed into sites,\nspins, and subband pairs [19, 20].\nIII. RESULTS AND DISCUSSION\nA. Ce 2Co17\nTABLE I. Atomic spin msand orbital mlmagnetic mo-\nments (\u0016B/atom) in CeCo 5,R-Ce2Co17andH-Ce2Co17.\nAtomic sites are grouped to re\rect how the 2-17 structure\narises from the 1-5 structure. Calculated interstitial spin mo-\nments are around \u00001:1\u0016B=f:u:in Ce 2Co17and\u00000:4\u0016B=f:u:in\nCeCo 5. Measured magnetization is 26 :5\u0016B=f:u:inH-Ce2Co17\nat 5K[6], and 7:12\u0016B=f:u:in CeCo 5[27]. Dumbbell sites are\ndenoted as 6 cand 4finR-Ce2Co17andH-Ce2Co17, respec-\ntively.\nCeCo 5 2c 3g 1a(Ce) Total\nms 1.33 1.44 -0.76 6.22\nml 0.14 0.12 0.30 0.92\nR-Ce2Co1718f18h9d 6c 6c(Ce) Total\nms 1.53 1.43 1.52 1.65 -0.85 23.94\nml 0.10 0.09 0.07 0.07 0.35 2.17\nH-Ce2Co1712j12k6g 4f2c(Ce) 2b(Ce) Total\nms 1.56 1.51 1.51 1.65 -0.84 -0.90 24.50\nml 0.11 0.10 0.08 0.07 0.38 0.42 2.43\nAtomic spin and orbital magnetic moments in Ce 2Co17\nand CeCo 5are summarized in Table I. The calcu-\nlated magnetization are 25.2 and 25 :8\u0016B=f:u:inR-\nCe2Co17andH-Ce2Co17, respectively, and 6 :75\u0016B=f:u:\nin CeCo 5, which agree well with experiments [6]. Ce\nspin couples antiferromagneticlly with the Co spin.\nThe orbital magnetic moment of Ce is antiparallel to\nits spin, which re\rects the Hunds' third rule. In\nthe Ce-Co plane of Ce 2Co17the Ce atoms are par-\ntially replaced by dumbbell Co atoms and this leads\nto an increased moment for the Co atoms (in that\nplane) as compared to CeCo 5, The dumbbell sites3\n(a) (b)(c)\n2c 1a3g\n4f2b 12j2c12k 6g\n6c6c 18f18h 9dDumbbell site\nFIG. 2. Schematic crystal structures of (a) CeCo 5, (b) hexagonal H-Ce2Co17, and (c) rhombohedral R-Ce2Co17. Ce atoms\nare indicated with large (yellow or magenta colored) spheres. Co atoms are denoted by Wycko\u000b sites. Dumbbell (red) sites are\ndenoted in H-Ce2Co17(4fsites) and in R-Ce2Co17(6csites), and indicated further by arrows and label. We use larger cells\nfor CeCo 5andR-Ce2Co17to compare with H-Ce2Co17.\nhave the largest magnetic moment due to its rela-\ntively large volume. Calculation shows Ce 2Co17has a\nsmall uniaxial anisotropy, 0 :13 meV=f:u:(0.09 MJm\u00003)\nand 0:47 meV=f:u:(0.30 MJm\u00003) forR-Ce2Co17andH-\nCe2Co17, respectively. The experimental values fall\nslightly above the calculated ones, see Fig. 3.\nTo understand the low uniaxial anisotropy in Ce 2Co17,\nwe resolve the anisotropy into atomic sites by evaluat-\ningKso. The anisotropy contributions in Ce 2Co17can\nbe divided into three groups: the pure Co plane (3 g\nin CeCo 5, 12k+ 6ginH-Ce2Co17, or 18h+ 9dinR-\nCe2Co17), the Ce-Co plane, and the Co dumbbell pairs.\nWe found that the MAE contributions from these three\ngroups in the two 2-17 structures are very similar: the\ndumbbell Co sites have a very negative contribution to\nuniaxial anisotropy; the pure-Co basal plane has a negli-\ngible or even slightly negative contribution to the uniaxial\nanisotropy; only the Ce-Co basal plane provides uniax-\nial anisotropy in Ce 2Co17. The two inequivalent Ce sites\ncontribute di\u000berently to the uniaxial anisotropy in H-\nCe2Co17structure. Ce(2 b) supports uniaxial anisotropy\nwhile Ce(2c) moment prefer to be in-plane. However, the\ntotal contribution from the two Ce sites is positive, as in\ntheR-structure.\nIntrinsic magnetic properties and the e\u000bect of dop-\ning on them are very similar in the two 2-17 struc-\ntures. We only discuss the results calculated using the\nR-structure because it has a smaller primitive cell than\ntheH-structure, and the most interesting substituents,\nFe and Mn, promote its formation [5].\nB. MAE in Ce 2T2Co15\nWe \frst calculate the MAE in Ce 2T2Co15with a va-\nriety of doping elements T, by assuming the pair of Co\n012345K(MJ/m3)\nHf Zr Ti V Cr Mn Fe Co Ni Cu Zn  \nR-Ce 2T2Co15\nH-Ce 2Co17(Ce 0.67T0.33)Co 5\nExperiments1-5\nR-2-17FIG. 3. Magnetic anisotropy in Ce 2T2Co15and\nCe0:66T0:33Co5withT=Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zr,\nand Hf. In Ce 2T2Co15,Tatoms occupy the dumbbell sublat-\ntice. The Ce 0:66T0:33Co5structure was obtained by replacing\nthe pair of dumbbell Co atoms in the original Ce 2Co17with\na singleTatom.Kvalues derived from experimental HA\nmeasurements [5, 11] by using K=1\n2\u00160MsHAare also shown.\ndumbbell atoms is replaced by a pair of doping atoms.\nThe calculated MAE as a function of doping elements\nforT=Zr and 3delements is shown in Fig. 3. Fe and\nMn doping increase the MAE, aligning with with exper-\nimental results. However, the MAE calculated for light\ndelementsT=Ti, V, and Zr are rather small while ex-\nperiments show that large enhancements of MAE can\nbe achieved with a small amount of doping of those ele-\nments. Interestingly, large MAE values are obtained in\nCe2T2Co15withT=Cu or Zn. In fact, a small amount\nof Cu are often added to the alloy to improve the co-\nercivity and the enhancement had been interpreted as\nprecipitation hardening by Cu. It may not be unex-4\npected that the enhancement of coercivity may also par-\ntially arise from the increase of MAE, although Cu atoms\nhad been reported to randomly occupy all Co sites [17].\nMoreover, the trend of MAE in Ce 2T2Co15, as shown\nin Fig. 3, is rather generic. We also found the simi-\nlar trend in Y 2T2Co15and La 2T2Co15, MAE increases\nwithT=Mn, or late 3 delements. Calculations using FP-\nLMTO method also shows similar trends of MAE.\n−101234Kso(meV/sublattice)\nTi V Cr Mn Fe Co Ni Cu Zn  \n6c\nOthers\nTotal\nFIG. 4. Anisotropy of the scaled on-site SOC energy Ksoin\nCe2T2Co15and its contributions from the dumbbell sublattice\nT(6c) and the rest sublattices.\nThe total Kso, its contribution from the dumbbell\nsite, and the other sublattices' contributions are shown\nin Fig. 4. Total Ksoclosely follows Kfor all dop-\ning elements, thus validating our use of Ksoto re-\nsolve the MAE and understand its origin. As shown\nin Fig. 4, the Co dumbbell sublattice in R-Ce2Co17has\na very negative contribution to the uniaxial anisotropy\nKso(6c)=1 meV=f:u:(0:5 meV=atom). Replacing Co\nwith other 3 delements decreases or eliminates this neg-\native contribution, or even make it positive, as with\nT=Mn. For the dumbbell site contributions, only four\nelements with large magnetic moments (all ferromagnet-\niclly couple to Co sublattice), Mn, Fe, Co, and Ni, have\nnon-trivial contributions. Atoms on both ends of the\n3delements have negligible contributions to the uniax-\nial anisotropy as expected. Although Cu and Zn have\nthe largest SOC constants among 3 d, they are nearly\nnon-magnetic, hence, they barely contribute to the MAE\nitself [20]. The light elements Ti, V, and Cr have small\nspin moments between 0.36 and 0.55 \u0016B(antiparallel to\nthe Co sublattice) and smaller SOC constants, together\nresulting in a small Kso(T).\nAlthough the dumbbell site contribution dominates\nthe MAE enhancement for T=Fe and Mn, it is obvious\nthat the variation of MAE is a collective e\u000bect, espe-\ncially forT=Cu, or Zn. While the \u00001 meV=f:u:nega-\ntive contribution from the dumbbell sublattice is elim-\ninated with T=Cu and Zn, the contributions from the\nrest sublattices increase by about 2 and 3 meV =f:u:, re-\nspectively. Similarly, for the doping of non-magnetic Al\natoms, the calculated MAE in Ce 2Al2Co15has a largevalue ofK= 3:8 meV=f:u:. Experimentally, Al atoms\nhad been found to prefer to occupy the dumbbell site and\nalso increase the uniaxial anisotropy [7, 17]. MAE often\ndepends on subtle features of the bandstructure near the\nFermi level; therefore, the collective e\u000bect of MAE vari-\nation should be expected for a metallic system [28]. The\nmodi\fcation of one site, such as doping, unavoidably af-\nfects the electronic con\fguration of other sites and their\ncontribution to MAE.\nC. Origin of MAE in Ce 2T2Co15withT=Fe and Mn\n−1.2−0.8−0.400.40.81.2Kso(meV/sublattice)\n18f 18h 9d T Ce  \n(a)\nMn\nFe\nCo\n−0.4−0.200.20.4\n|−2⟩↔| 2⟩| −1⟩↔| 1⟩| ±1⟩↔| 0⟩| ±1⟩↔| ±2⟩Kso(meV/atom)\n  \n(b)\nMn\nFe\nCo\nFIG. 5. (a) Site-resolved anisotropy of the on-site SOC\nenergyKsoand (b) orbital-resolved Kso(6c) in Ce 2Co17\u0000xTx\nwithT=Co, Fe, and Mn.\nWe found that all dopings except Fe and Mn de-\ncrease the magnetization, which is consistent with the\nexperiments by Fujji et al. [5], and Schaller et al. [29].\nCe2Fe2Co15and Ce 2Mn2Co15have slightly larger mag-\nnetization than Ce 2Co17by 5% and 8%, respectively. It\nis worth noting that experimental result on Mn doping is\nrather inconclusive. A slight decrease of magnetization\nwith Mn doping has also been reported [11].\nSublattice-resolved Ksoin Ce 2T2Co15forT=Co, Fe,\nand Mn are shown in Fig. 5(a). The dominant enhance-\nment of MAE are from the dumbbell site, although con-\ntributions from other sublattices also vary with T. To5\nunderstand this enhancement of Ksofrom the dumbbell\nsites, we further resolved Ksointo contributions from al-\nlowed transitions between all pairs of subbands. The\ndumbbell sites have 3 msymmetry. Without considering\nSOC, \fvedorbitals onTsites split into three groups: dz2\nstate, degenerate ( dyz,dxz) states, and degenerate ( dxy,\ndx2\u0000y2) states. Equivalently, they can be labeled as m=0,\nm=\u00061, andm=\u00062 using cubic harmonics. Kso(T) can\nbe written as [20]\nKso(T) =\u00182\n4(4\u001f\u000f\n22+\u001f\u000f\n11\u00003\u001f\u000f\n01\u00002\u001f\u000f\n12); (1)\nwhere\u0018is the SOC constant and \u001f\u000f\nmm0is the di\u000berence\nbetween the spin-parallel and spin-\rip components of or-\nbital pair susceptibility. It can be written as\n\u001f\u000f\nmm0=\u001f\"\"\nmm0+\u001f##\nmm0\u0000\u001f\"#\nmm0\u0000\u001f#\"\nmm0: (2)\nContributions to Kso(T) resolved into transitions be-\ntween pairs of subbands are shown in Fig. 5(b). The\nfour groups of transitions correspond to the four terms in\nEq. (1). The dominant e\u000bect is from j0i$j\u0006 1i, namely\nthe transitions between dz2and (dyzjdxz) orbitals. This\ncontribution is negative for T=Co, nearly disappears for\nT=Fe, and even becomes positive and large for T=Mn.\nThe interesting dependence of j0i$j\u0006 1icontribution\nonTcan be understood by investigating how the elec-\ntronic structure changes with di\u000berent Telements. The\nsign of the MAE contribution from transitions between a\npair of subbandsjm;\u001biandjm0;\u001b0iis determined by the\nspin and orbital character of the involved orbitals [20, 30].\nInter-jmjtransitionsj0i $ j\u0006 1ipromote easy-plane\nanisotropy within the same spin channel and easy-axis\nanisotropy when between di\u000berent spin channels.\nThe scalar-relativistic partial densities of states\n(PDOS) projected on the dumbbell site are shown in\nFig. 6. For T=Co, the majority spin channel is nearly\nfully occupied and has very small DOS around the Fermi\nlevel, while the minority spin channel has a larger DOS.\nThe transitions between dz2and (dyz;dxz) states across\nthe Fermi level and within the minority spin chan-\nnel, namelyj0;#i $ j\u0006 1;#i, promote the easy-plan\nanisotropy. For T=Fe, the PDOS of dz2and (dyz;dxz)\nare rather small near the Fermi level in both spin chan-\nnels and the net contribution from j0i $ j\u0006 1ibe-\ncomes negligible. For T=Mn, the Fermi level inter-\nsects a large peak of the dz2state at the Fermi level\nin the minority spin channel. The spin-\rip transitions\nj0;#i$j\u0006 1;\"igive rise to a large positive contribution\nto uniaxial anisotropy.\nD. Zr, Ti, and Hf doping in Ce 2Co17\nThe failure to reproduce high anisotropy introduced\nby other dopants, such as Zr, Ti, and V, is likely due\n−0.8−0.400.40.8Co(a)\n−0.8−0.400.40.8DOS ( states ( eV spin atom )−1)\n  \nFe(b)m=±2\nm=±1\nm=0\n−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.8−0.400.40.8\nE(eV)Mn(c)FIG. 6. The scalar-relativistic partial density of states pro-\njected on the 3 dstates ofTsites inR-Ce2T2Co15withT=Co,\nFe, and Mn. Tatoms occupy the dumbbell (6 c) sites.\nto our oversimpli\fed assumption that a pair of Tatoms\nalways replaces a pair of Co dumbbell atoms. Unlike Fe\nand Mn, the site occupancy preference for those dopants\nis not well understood [31]. Considering Zr doping most\ne\u000bectively enhanced HAin experiments, here we focus on\nZr doping.\nBoth volume and chemical e\u000bects likely play important\nroles in substitution site preference. To have a better un-\nderstanding of the Zr site preference, we calculated the\nformation energy of Ce 2ZrCo 16with the Zr atom occu-\npying one of the four non-equivalent Co sites and found\nthat Zr also prefers to occupy the dumbbell sites { likely\ndue to the relatively large volume around the dumbbell\nsites. The formation energies are higher by 39, 58, and\n81 meV=atom when Zr occupies the 18 f, 18h, or 9dsites,\nrespectively. Considering Zr atoms are relatively large,\nwe investigated another scenario by replacing the pair of\nCo dumbbell atoms with a single Zr atom, as suggested\nby Larson and Mazin [31]. Indeed, this latter con\fg-\nuration of Ce 2ZrCo 15has the lowest formation energy,\nwhich is 3 meV =atom lower than that of Ce 2Zr2Co15and\n1 meV=atom lower than Ce 2Co16Zr (with Zr replacing\none of the two dumbbell Co atoms in Ce 2Co17). That is,\nwith Zr additions the CeCo 5structure is preferred over\nthe Ce 2Co17-based structure. The resulting Ce 2ZrCo 156\nhas a 1-5 structure (Ce 0:67Zr0:33)Co5, with one-third of\nthe Ce in the CeCo 5structure, shown in Fig. 2(a), re-\nplaced by Zr atoms. Hence, the formation energy cal-\nculation indicate that the realized structure is likely a\nmix of 2-17 and 1-5 structures. Interestingly, this may\nbe related to experimental observations that successful\n2-17 magnets usually have one common microstructure,\ni.e., separated cells of 2-17 phase surrounded by a thin\nshell of a 1-5 boundary phase, and Zr, Hf, or Ti additions\npromote the formation of such structure [3].\nThe calculated anisotropy in Ce 2ZrCo 15, or equiva-\nlently (Ce 0:67Zr0:33)Co5, is about 4 MJ m\u00003and much\nlarger than that of Ce 2Zr2Co15. Analysis of Ksoreveals\nthat not only is the negative contribution from the pre-\nvious dumbbell sites eliminated, but more importantly,\nthe pure Co plane becomes very uniaxial. For T=V and\nTi, the calculated MAE in this con\fguration is also much\nlarger than that of Ce 2T2Co15, as shown in Fig. 3. Sim-\nilarly, a large MAE of 2 :41 meV=f:u:was obtained for\n(Ce0:67Hf0:33)Co5.\nIV. CONCLUSION\nUsing density functional theory, we investigated the\norigin of anisotropy in doped Ce 2Co17. We con\frmed\nthat the dumbbell sites have a very negative contri-\nbution to the MAE in Ce 2Co17with a value about\n0:5 meV=atom. The enhancement of MAE due to Fe and\nMn doping agrees well with experiments, which can be\nexplained by the preferential substitution e\u000bect becausethe enhancement is dominated by dumbbell sites. The\ntransitions between the dz2and (dyzjdxz) subbands on\ndumbbell sites are responsible for the MAE variation, and\nthese transitions can be explained by the PDOS around\nthe Fermi level, which in turn depends on the element\nToccupying on the dumbbell site. For Zr doping, the\ncalculated formation energy suggests that the real struc-\nture is likely a mix of 2-17 and 1-5 structures, and the\nresulted 1-5 structure has a large anisotropy, which may\nexplain the large MAE enhancement observed in experi-\nments. The variation of MAE due to doping is generally\na collective e\u000bect. Doping on dumbbell sites may signif-\nicantly change the contributions from other sublattices\nand then the overall anisotropy. It is worth investigating\nother non-magnetic elements with a strong dumbbell site\nsubstitution preference because it may increase the total\nanisotropy in this system by increasing the contributions\nfrom other sublattices.\nV. ACKNOWLEDGMENTS\nWe thank B. Harmon, T. Ho\u000bmann, M. K. Kashyap,\nR. W. McCallum, and V. Antropov for helpful discus-\nsions. Work at Ames Laboratory was supported by the\nU.S. Department of Energy, ARPA-E (REACT Grant\nNo. 0472-1526). The relative stability and formation en-\nergy investigation were supported by O\u000ece of Energy Ef-\n\fciency and Renewable Energy (EERE) under its Vehicle\nTechnologies Program. Ames Laboratory is operated for\nthe U.S. Department of Energy by Iowa State University\nunder Contract No. DE-AC02-07CH11358.\n[1] R. McCallum, L. Lewis, R. Skomski, M. Kramer, and\nI. Anderson, Annual Review of Materials Research 44,\n451 (2014).\n[2] K. H. J. Buschow, Reports on Progress in Physics 40,\n1179 (1977).\n[3] K. Strnat (Elsevier, Amsterdam, 1988), vol. 4 of Hand-\nbook of Ferromagnetic Materials , pp. 131 { 209.\n[4] S. Yajima, M. Hamano, and H. Umebayashi, Journal of\nthe Physical Society of Japan 32, 861 (1972).\n[5] H. Fujii, M. V. Satyanarayana, and W. E. Wallace, Jour-\nnal of Applied Physics 53, 2371 (1982).\n[6] S. Hu, X. Wei, D. Zeng, X. Kou, Z. Liu, E. Brck,\nJ. Klaasse, F. de Boer, and K. Buschow, Journal of Alloys\nand Compounds 283, 83 (1999).\n[7] B. Shen, Z. Cheng, S. Zhang, J. Wang, B. Liang,\nH. Zhang, and W. Zhan, Journal of Applied Physics 85,\n2787 (1999).\n[8] X. Wei, S. Hu, D. Zeng, X. Kou, Z. Liu, E. Br uck,\nJ. Klaasse, F. de Boer, and K. Buschow, Physica B: Con-\ndensed Matter 262, 306 (1999).\n[9] X. Wei, S. Hu, D. Zeng, X. Kou, Z. Liu, E. Brck,\nJ. Klaasse, F. de Boer, and K. Buschow, Journal of Alloys\nand Compounds 279, 301 (1998).[10] H. Fujii, M. Satyanarayana, and W. Wallace, Solid State\nCommunications 41, 445 (1982).\n[11] Z. Sun, S. Zhang, H. Zhang, J. Wang, and B. Shen, Jour-\nnal of Physics: Condensed Matter 12, 2495 (2000).\n[12] R. Streever, Phys. Rev. B 19, 2704 (1979).\n[13] J. Deportes, D. Givord, R. Lemaire, H. Nagai, and\nY. Yang, Journal of the Less Common Metals 44, 273\n(1976).\n[14] R. Perkins and P. Fischer, Solid State Communications\n20, 1013 (1976).\n[15] P. Gubbens and K. Buschow, physica status solidi (a)\n34, 729 (1976).\n[16] A. Kuchin, A. Pirogov, V. Khrabrov, A. Teplykh, A. Er-\nmolenko, and E. Belozerov, Journal of Alloys and Com-\npounds 313, 7 (2000).\n[17] K. Inomata, Phys. Rev. B 23, 2076 (1981).\n[18] C. de Groot, F. de Boer, K. Buschow, Z. Hu, and\nW. Yelon, Journal of Alloys and Compounds 233, 188\n(1996).\n[19] V. Antropov, L. Ke, and D. \u0017Aberg, Solid State Commu-\nnications 194, 35 (2014).\n[20] L. Ke and M. van Schilfgaarde, Phys. Rev. B 92, 014423\n(2015).\n[21] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).7\n[22] G. Kresse and J. Furthm uller, Phys. Rev. B 54, 11169\n(1996).\n[23] M. Methfessel, M. van Schilfgaarde, and R. Casali, in\nLecture Notes in Physics , edited by H. Dreysse (Springer-\nVerlag, Berlin, 2000), vol. 535.\n[24] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).\n[25] D. Koelling and B. Harmon, Journal of Physics C: Solid\nState Physics 10, 3107 (1977).\n[26] A. Shick, D. Novikov, and A. Freeman, Phys. Rev. B 56,\nR14259 (1997).[27] M. Bartashevich, T. Goto, A. Korolyov, and A. Er-\nmolenko, Journal of Magnetism and Magnetic Materials\n163, 199 (1996).\n[28] L. Ke and D. D. Johnson, Phys. Rev. B 94, 024423\n(2016).\n[29] H. Schaller, R. Craig, and W. Wallace, Journal of Applied\nPhysics 43, 3161 (1972).\n[30] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuur-\nmans, Phys. Rev. B 50, 9989 (1994).\n[31] P. Larson and I. I. Mazin, Phys. Rev. B 69, 012404\n(2004)."    },    {        "title": "1610.03544v1.Large_magnetic_anisotropy_predicted_for_rare_earth_free_Fe16_xCoxN2_alloys.pdf",        "content": " 1 Large magn etic anisotropy predicted for r are-earth free Fe 16-xCoxN2 \nalloys  \nXin Zhao*, Cai -Zhuang Wang§, Yongxin Yao, and Kai -Ming Ho  \nAmes Laboratory, US DOE and Department of Physics and Astronomy, Iowa State University, \nAmes, Iowa 50011, USA  \n*E-mail: xzhao@iastate.edu ; §E-mail: wangcz@ameslab.gov   \nABSTRACT  \nStructures and magnetic properties of Fe 16-xCoxN2 are studied using adapt ive genetic algorithm \nand first -principles calculations. We show  that substitut ing Fe by Co in Fe 16N2 with Co/Fe ratio  ≤ \n1 can greatly improve the magnetic anisotropy of the material. The magnetocrystalline \nanisotropy energy from first -principles calculations reach es 3.18 MJ/m3 (245.6 μe V per metal \natom) for Fe 12Co4N2, much larger than that of Fe 16N2 and is one of the largest among the \nreported rare -earth free magnets. From our systematic crystal structure search es, we show that \nthere is a structure transition from tetragonal Fe 16N2 to cubic Co 16N2 in Fe16-xCoxN2 as the Co \nconcentration  increases , which can be well explained by electron counting analysis. Different \nmagnetic properties between the Fe -rich (x ≤ 8) and Co -rich (x > 8) Fe 16-xCoxN2 is closely \nrelated to the structural tra nsition.  \nPACS Numbers: 75.30.Gw , 75.50.Ww, 75.50.Bb , 61.50. -f  2 Permanent -magnets (PM) are important energy and information storage/conversion materials, \nand play a crucial role in new energy ec onomies. Currently, the most widely used PM are \nNd2Fe14B and SmCo 5, both containing rare -earth (RE) elements. Because of the concerns on \nlimited RE mineral resources and RE supplies, there have been increasing interests in \ndiscovering strong RE-free PM materials.1  \nIn addition to the microstructures, the performance of PM relies on the intrinsic magnetic \nproperties of their crystal structures, such as saturation magnetization, Curie temperature, and \nmagnetocrystalline anisotropy energy  (MAE).  Among the R E-free magnets, the metastable \ntetragonal 𝛼′′-Fe16N2 phase of iron nitrides have attracted considerable experimental and \ntheoretical attentions due to the low cost of Fe and high magnetization in 𝛼′′-Fe16N2 thin films.2 \nIn experiments, a very large value  (~ 3 μ B) of average Fe magnetic moment in Fe 16N2 thin film \nwas reported,2,3 which is much higher than that of pure bulk Fe (2.2 μ B). The MAE of the \ntetragonal Fe 16N2 phase has been measured by several experiments,3-7 but the results are not \nconclusive, ra nging from 0.44 to 2.0 MJ/m3. Because of the promising magnetic properties \nobserved in thin films, efforts  to synthesize bulk samples of the tetragonal 𝛼′′-Fe16N2 phase  have \nalso been made .8-10 In the 𝛼′′-Fe16N2 structure, the body -centered cubic ( bcc) Fe lattice is \nexpanded into a distorted body -centered tetragonal ( bct) lattice due to the presence  of N. Close to \n500 K, the tetragonal 𝛼′′-Fe16N2 phase  decomposes into α -Fe and Fe 4N phases ,11 making the  \nthermal stability one of the major issues for its synthesis and application. It has been shown that \nadding a small amount of third elements, such as Co, Mn, or Ti, can stabilize the 𝛼′′-Fe16N2 \nphase  in thin film samples .12,13 Whether such doping  approach can stabilize the  𝛼′′-Fe16N2 in the \nsynthesis of  bulk samples  is still under investigation .14   3 Theoretical ly, many studies have also been devoted to exploring the origin of the magnetic \nmoment and MAE enhancement in 𝛼′′-Fe16N2.15-17 Using density functional theory (DFT) and \nquasiparticle self -consistent GW calculations, Ke et al.  studied the intrinsic magnetic properties \nof Fe 16N2 and reported a magnetic moment of ~2.5 μ B and Curie temperature of ~1300 K.17 For \nMAE, the y obtained a  uniaxial magnetocrystalline anisotropy of 1.03 MJ/m3 by local density \napproximation (LDA) and 0.65 MJ/m3 by generalized gradient approximation  (GGA).  \nCalculations by Ke et al.  also indicate possible improvement of magnetocrystalline anisotropy by \nsubstituti ng a small amount (less than 15%) of Fe by Co or Ti in the 𝛼′′-Fe16N2 phase .17 \nNevertheless, the calculations were done using the 𝛼′′-Fe16N2  structure and the effect of \ntransition metals (TM) substitution on the crystal structures and phase stabilities has not been \ninvestigated.  \nIn this paper, we systematically stud y the structural evolution and magnetic properties of Fe 16-\nxCoxN2 with x  rangin g from  0 to 16 .  We show that substituting minority of Fe by Co results in \nbetter magnetic properties, especially a large value of MAE (~ 3.18 MJ/m3) is achieved at the \ncomposition of Fe 12Co4N2 (i.e. x = 4). We demonstrate  that there is a tetragonal  (Fe16N2) to cubic \n(Co16N2) structur al transition in Fe16-xCoxN2 as the Co concentration  is increased. The changes in \nthe magnetic properties with the Co concentration  are strongly correlated with the structural \nchanges.   \nOur crystal structure searches were perf ormed using adaptive genetic algorithm (AGA)18,19 with \nreal space cut -and-paste operations for generating offspring structures20. No constrains were \napplied on the structure symmetries during the AGA search. We explored the crystal structures \nof Fe 16-xCoxN2 (18 atoms per unit cell) with x ranging from 0 to 16. During the AGA searches, \nauxiliary classical potential based on embedded -atom method21 was used. The first -principles  4 calculations were carried out using spin -polarized density functional theory (DFT) . Generalized -\ngradient approximation (GGA) in the form of PBE22 implemented in the VASP code23 is used. \nKinetic energy cutoff was set to be 520 eV. The Monkhorst -Pack’s scheme24 was used for \nBrillouin zone sampling with a k -point grid resolution of 2π x 0. 05 Å-1 during the structure \nsearches. In the final structure refinements, a denser grid of 2π x 0.03 Å-1 was used and the ionic \nrelaxations stop when the force on each atom is smaller than 0.01 eV/Å. Intrinsic magnetic \nproperties, such as magnetic moment and MAE, were also calculated by VASP based on the \ntheoretically optimized structures. All symmetry operations are switched off completely when \nthe spin -orbit coupling is turn on. Meanwhile, a much denser k -point grid (2π x 0.016 Å-1) is \nused in the MAE ca lculations to achieve better k -point convergence.  \nBased on the structures obtained from our AGA searches, we found that mixing Co with Fe can \nsignificantly enhance the magnetic anisotropy as compared to that of  Fe16N2 when the Co \nconcentration is smaller than that of Fe. A large value of MAE (~ 3.18 MJ/m3) is found in one of \nthe metastable Fe 12Co4N2 structures, which is the highest among the rare -earth free magnets \nreported  so far . In Fig. 1 (a) – (c), the crystal structures of Fe 16N2, Fe12Co4N2 (the one w ith \nlargest MAE) and Co 16N2 are plotted.  The Fe 12Co4N2 structure can be considered as a substituted \nFe16N2 structure with Fe at the 4d sites being replaced by Co. For Fe 16N2, our calculation gives \nan MAE of 50.1 μeV/Fe atom (or 0.64 MJ/m3), consistent with Ref. 1 7 where a value of 52 \nμeV/Fe atom (or 0.65 MJ/m3) was reported from GGA calculations using  the theoretically \noptimized structure. The Co 16N2 structure has nearly zer o MAE due to its cubic symmetry.  \nTo demonstrate that these structures  are dynamically stable, in Fig. 1(d), the phonon density -of-\nstates of these three structures  are plotted . The phonon calculation was performed using a \nsupercell approach by the Phonopy  code,25 where supercells with sizes of 324 atoms for Fe 16N2  5 and Fe 12Co4N2 and 288 atoms for Co 16N2 were used. The results show that there are no negative \nfrequencies in all three structures, indicating these structures are dynamically stable.  \n \nFIG. 1 Crystal structures of (a) Fe 16N2, space group I4/mmm  with a=5.68 Å, c=6.22  Å and N 2a \n(0.0, 0.0, 0.0), Fe 4d (0.0, 0.5, 0.25) ; (b) Fe 12Co4N2, space group I4/mmm  with a=5.64 Å, c=6.24 \nÅ and N 2b (0.0, 0.0, 0.5), Co 4d (0.0, 0.5, 0.25) , Fe 8h (0.255, 0.255, 0.0), Fe 4e (0.0, 0.0, \n0.793) ; (c) Co 16N2, space group Fm-3m with a=7.19 Å  and N 4a (0.0, 0.0, 0.0), Co 24e (0.260, \n0.260, 0.260), Co 8c (0.25, 0.25, 0.25) . (d) Phonon density -of-states of the structures plotted in \n(a) – (c). \nIn addition to the Fe 12Co4N2 structure, other metastable Fe -Co-N structures from our AGA \nsearches also exhibit large MAE. The compositions, energies and MAE of these structures are \npresented in Fig. 2. Different symbols used in Fig. 2 indicate structures with different MAE \nvalues: 1. 5 MJ/m3 < MAE < 2.0 MJ/m3 (green squares), 2.0 MJ/m3 < MAE < 3.0 MJ/m3 (black \ncircles) and MAE > 3.0 MJ/m3 (red stars), respectively. From Fig. 2, it is clear that no structure \nin the Co -rich side has MAE larger than 1.5 MJ/m3. On the contrary, many struct ures in the Fe -\n 6 rich side have high MAE. We will show that t he trend of MAE is correlated with a structural \ntransition as the function of Co concentration.  \nBased on the results from our AGA search, the lowest-energy  structure of Co 16N2 is cubic rather \nthan tetragonal. In Fig. 2, Co 16N2 is plotted in a different view so that its relationship to the \ntetragonal Fe 16N2 structure can be seen more clearly. The structure of Fe 16N2 has larger distortion \nthan Co 16N2, causing the symmetry degraded to a tetragonal spac e group. In fact, Fe atoms in \nFe16N2 have been considered to form distorted bct structure in previous studies .14,17 In the Fe 16N2 \nstructure the bond length between each Fe and its 12 nearest neighbors var ies gradually  from \n2.43 to 3.11 Å.  While in the Co 16N2 structure, the Co atoms are in a face-centered cubic ( fcc) \nlattice, thus each Co bonds to 12 Co neighbors  with almost the same bond length . \nIn order to build the connection between these two structures, we notice that the fcc structure of \nCo16N2 (a = 7.19 Å) can also be represented by a bct unit cell with  𝑎′=𝑎/√2 = 5.08 Å and  𝑐=\n𝑎 = 7.19 Å, i.e. the c/a ratio of the tetragonal cell equals to  √2. The Fe 16N2 structure, on the other \nhand, as being squeezed along c axis, has a larger a (= 5.68 Å) and smal ler c (= 6.22 Å). The c/a \nratio in the Fe 16N2 structure is about 1.09, much smaller than that of Co 16N2. Despite the \ndifference in the c/a ratio, these two structures can be classified as the same type of structure, \nwhich will be referred to as the TM 16N2-type in the following discussions. As shown in Fig. 2, \nthe lowest -energy structures at different compositions all  belong to  the TM 16N2-type structures \n(represented by the blue squares and connected by the blue line), except the lowest -energy \nstructure  of Fe8Co8N2 which has a different prototype structure as shown in Fig.  2.   7  \nFIG. 2 Formation energies and the structural evolution of Fe 16-xCoxN2 as the function of Co \nconcertation x. The formation energy is calculated using Fe 16N2 and Co 16N2 as references: \nEf(Fe 16-xCoxN2) = [E(Fe 16-xCoxN2) – (1-x/16)*E(Fe 16N2) – x/16*E(Co 16N2)]/18. The solid blue \nline connects the TM 16N2-type structures (represented by the blue squares, see also text). The \nconvex hull is shown by the black dash line. Calculated structures with high MAE’s are indicated \nby different symbols: green squares (1.5 < MAE < 2.0 MJ/m3), black circles (2.0 < MAE < 3.0 \nMJ/m3) and red stars (MAE > 3.0 MJ/m3). Crystal structures are plotted for a, b … f as labeled \nin the formation energy figure.  \n \nWe not e that there is a structural transition as the Co concentration  increases. To obtain the \ntransition point, we plot the c/a ratio and volume of the TM 16N2-type structures (represented by \nthe blue squares in Fig. 2) as a function of Co conce ntration as shown  in Fig. 3(a). For cubic \n 8 structures such as Co 16N2 and Co 12Fe4N2, the c/a value is √2 as explained above. The results in \nFig. 3(a) clearly show a sudden increase in c/a at the composition of Fe 8Co8N2. For the Co -rich \nside, all the c/a ratios are close to  √2, while on  the Fe -rich si de including Co:Fe  = 1, the c/a \ndrops to around 1.1. Note that the bct structures with c/a ratios of 1 and  √2 are both cubic \nstructures as discussed above.  In order to measure the tetragonality of the Fe 16-xCoxN2 structures, \nwe defin e a parameter  𝛾=√|(𝑐/𝑎−1)×(√2−𝑐/𝑎)|, which is the geometric mean of the \ndistance s between  the c/a ratio of a given  structure from that of the bcc ( c/a=1) and fcc ( c/a=√2) \nstructure s. We found that γ is around 0.18 for x ≤ 8, while drops  to ~ 0.0 for x >  8, thus serving \nas a clear indicator  of the tetragonal -like to cubic -like structure transition.  In addition to c/a, at \nthe transition point from Fe -rich side to Co -rich side, there is nearly 4 percent sudden drop in \nvolume, which can also be consider as the signal of the structure transformation.  \nThe sudden changes in c/a ratio (as well as γ) and volume at the transition can be attributed to the \nmagnetic effects.  It is interesting to note that without spin polarization, the above mentioned  \nstructure transition disappears as shown by the dashed lines in Fig. 3(a). All structures with \ndifferent compositions have the c/a ratio close to √2 if spin -polarization is turn off in the \ncalculations. At the same time, the volume of the structures is muc h smaller and increases very \nslowly as the Co concentration increases. These results indicate that magnetism plays a crucial \nrole for the structural transition in this system.   9  \nFIG. 3 (a) The c/a ratio and v olume per formula unit (8 TM atoms, 1 N atom) of  the Fe 16-xCoxN2 \nstructures as a function of x from DFT calculations with spin-polariz ation  (solid  blue and \nredlines) and without spin polarization  (dash blue and red lines). Black dash line represents c/a \n= √2, which is the value for an ideal fcc  structure. (b) Electronic density -of-state of the Fe 16N2 in \nthe cubic (Fe 8N-c) and tetragonal (Fe 8N-t) structures. Fermi level is indicated by the vertical \ndash lines (two lines are very close). (c) Energy difference between the tetragonal (E t) and cubic \n(Ec) phases evaluated using rigid -band analysis as a function of the electron doping. (d) The \nlargest MAE values obtained at each composition and  magnetic moment of the transition metal \natoms calculated  for the TM 16N2-type structures . In plots (a) and (d), the calculated volume and \nmagnetic moment for the lowest energy Fe 8Co8N2 structure are represented by green open \nsquares.  \n \n 10 We further carried out a rigid band perturbation analysis to better understand the mechanism of \nthe structural  transition upon TM substitution  in Fe16N2. Figure 3(b) shows the electronic \ndensity -of-states of Fe 16N2 in cubic and tetragonal phases. The electrostatic potential is aligned \nsuch that the band energy is equal to the total energy in each system, which was  previously used \nin the development of the tight -binding potentials .26 By valence electron counting, substitution of \none Fe atom with a Mn, Co, or Ni atom is corresponding to doping of 1 hole, 1 electron or 2 \nelectrons, respectively. If all Fe atoms are re placed, the structures will be Mn 16N2, Co 16N2 and \nNi16N2, respectively. The difference of band energy between the tetragonal and cubic phases as \nthe function of electron doping (i.e., the number of valence electron per TM atom relative to that \nof Fe) in the system is shown by the solid line in Fig. 3(c). The trend predicted by the rigid band \nanalysis is remarkabl y consisten t with the actual total energy calculation  results shown by  the \nstars where the atomic relaxations  are also included . Similar analyses  based on the Co 16N2 cubic \nand tetragonal structures give the same trend, as shown by the dashed line in Fig. 3(c). \nTherefore  the band structure effect  through electron or hole doping by substitution of other TM \natoms with similar atomic radius in Fe16N2 plays a dominant role in the structural transition of \nthis series of compounds.  \nThe structural transition from tetragonal Fe 16N2 to cubic Co 16N2 is strongly correlated with the \nmagnetic properties of the system. The magnetic moments of the TM atoms calculat ed for the \nlowest -energy TM 16N2-type structures are plotted in Fig. 3(d), together with that of the Fe 8Co8N2 \nground -state structure. It can be seen that the behavior of magnetic moment variation is almost \nthe same as that of volume variation. A sudden decr ease in magnetic moment near the structural \ntransition point is also observed, which can be attributed to the well -known fact that larger \nvolume gives larger moment .17,27 For the magnetic anisotropy, as mentioned earlier, after the  11 structure transition, i.e. on the Co -rich side , no structure is found to have MAE larger than 1.5 \nMJ/m3, while before the transition, structures are found to have much larger MAE’s , which can \nalso be seen from Fig. 3(d) . This can now be explained from the perspective of crystal structures. \nThe Co -rich structures prefer cubic -like structures, thus unlikely to have strong anisotropy. On \nthe other hand, the shape anisotropy in the Fe -rich tetragonal structures can enhance the MAE.  \nIt should be noted that the magnetic properties of the TM 16N2-type structures presented in Fig. \n3(d) are for stoichiometric compounds. Since the energies of Fe -Co-N alloys with different \nFe/Co site occupations are very close to each other, it is likely that at finite temperature \ndisordered site occupation between Fe and Co would occur, thus alloy of Fe 16-xCoxN2 will form \ninstead of stoichiometric compounds. The effects of Fe and Co occupation disorder on the \nmagnetic properties should be considered in material design and processing. A s discussed above, \nthere is a sharp structural phase transition at x=8 between Fe -rich tetragonal and Co -rich cubic \nstructures. Therefore, it is a good approximation to adopt the lattice parameters of tetragonal \nFe16N2 for the Fe -rich (x≤8) and those of cu bic Co 16N2 for the Co -rich (x≥8) Fe 16-xCoxN2 alloys. \nWe enumerated  all the possible Fe/Co occupations in the tetragonal -Fe16N2/cubic -Co16N2 \nstructures using a unit cell size of 18 atoms. The atomic positions in the unit cells are fully \nrelaxed by DFT calcu lations while the lattice parameters of the structures are kept fixed. The \nresults are shown in Fig. 4. As one can see from Fig. 4(a), the energy spread due to the site \noccupation disorder can be as large as 50 -60 meV/atoms. However, the magnetic moment as  the \nfunction of composition averaged over all the structures at each composition as shown in Fig. \n4(b) is very similar to that of the lowest -energy structure plotted in Fig. 3(d). Since the \ncalculation of MAE is very costly, in Fig. 4(b), we plotted the M AE results averaged over 10 \nlowest -energy configurations for different Co concentrations. Although the averaged MAE’s are  12 slightly smaller than the results calculated for stoichiometric compounds, the maximum value \nappears at the same composition i.e. Fe 12Co4N2. Between the MAE and magnetic moment, it is \neasy to notice that the error bar in MAE is much larger, indicating that magnetic anisotropy is \nmore sensitive to the change of structures. We have also performed calculations in which both \natomic position and lattice parameters are allowed to relax. The behavior of the magnetic \nmoment and MAE as the function of Co concentration is essentially the same as Fig. 4 (b) except \nthe transition at x(Co)=8 is smoothed out when the lattice parameters are allowed to re lax. \n \nFIG. 4 (a) Formation energies and (b) magnetic properties  of the structures obtained from \nenumerating all the possible Fe/Co occupations in the tetragonal -Fe16N2 (for x(Co) ≤ 8) and \ncubic -Co16N2 (for x(Co) ≥ 8) structure s using a unit cell size of 1 8 atoms . The energies of \ntetragonal -Fe16N2 and cubic -Co16N2 structures  were used as reference in the formation energy \n 13 calculation . “< >” in (b) represents the averaged value s: the magnetic moments were averaged \nover all the calculated structures, while t he MAE numbers were averaged over 10 lowest -energy \nstructures  at each composition. Error bar represents  one standard deviation.  \n \nIn conclusion, our study on Fe 16-xCoxN2 reveals a structure transition as the Co concentration \nincreases, which can be well exp lained by rigid band perturbation analysis. From the cubic \nCo16N2 structure to the tetragonal Fe 16N2 structure, magnetic moments of the system increase \nwhile magnetocrystalline anisotropy is maximized  at the composition near Fe 12Co4N2. The \nsubstantial improvement in the magnetic properties predicted for both ordered stoichiometric \ncompounds and  alloys with disordered Fe/Co occupations  makes it possible for Fe 16-xCoxN2 to \nfind applications as a rare -earth free magnet. The stability of t his system is still an important \nissue to consider. Although the structures  discussed here  are proven to be dynamically stable, we \nsee that using Fe 16N2 and Co 16N2 (which are metastable already) as references, only Fe 4Co12N2 is \nstable against decomposition . But it is also noticed that energies of the other structures are within \n20 meV/atom above the convex hull, which may be attainable by synthesis techniques at far -\nfrom -equilibrium conditions.   \n \nACKNOWLEDGMENT  \nThis work was supported by the National Scienc e Foundation (NSF), Division of Materials \nResearch (DMR) under Award DMREF: SusChEM 1436386. The development of adaptive \ngenetic algorithm (AGA) and the method for rigid band perturbation analysis was supported by \nthe US Department of Energy, Basic Energy Sciences, Division of Materials Science and  14 Engineering, under Contract No. DE -AC02 -07CH11358, including a grant of computer time at \nthe National Energy Research Scientific Computing Center (NERSC) in Berkeley, CA . \n \nREFERENCES  \n1. R. W. McCallum,  L. H. Lewis,  R. Skomski, M. J. Kramer,  I. E. Anderson,  Annu. Rev. \nMater. Res.  44, 451 -477 (2014) . \n2. T. K. Kim and M.  Takahashi , Appl. Phys. Lett.  20, 492  (1972 ). \n3. Y. Sugita, K. Mitsuoka, M. Komuro,  H. Hoshiya, Y. Kozono, and M. Hanazono,  J. Appl. \nPhys.  70, 5977 (1991) . \n4. H. Takahashi, M. Igarashi, A. Kaneho,  H. Miyajima, and Y. Sugita,  IEEE Trans. Magn.  35, \n2982  (1999 ). \n5. S. Uchida, T. Kawakatsu, A. Sekine, and T. Ukai, J. Magn. Magn. Mater.  310, 1796  (2007) . \n6. E. Kita, K. Shibata,  H. Yanagihara, Y. Sasaki  and M.  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Here we show\nthat the N\u0013 eel order dynamics a\u000bects the mechanical motion of a rigid body by modifying its inertia\ntensor in the presence of strong magnetocrystalline anisotropy. This e\u000bect depends on temperature\nwhen magnon excitations are considered. Such a spin-mechanical inertia can produce measurable\nconsequences at small scales.\nI. INTRODUCTION\nSpin-mechanics, also known as magnetomechanics, is\na venerable branch of modern physics that has attracted\ncontinuous attention. It explores the coupled dynamics of\nquantum spins and mechanical motions of a crystal back-\nground,1,2where the conservation of angular momentum\nserves as the governing principle. For example, when\na paramagnet is placed in a magnetic \feld to polarize\nthe atomic spins inside, it undergoes a spontaneous ro-\ntation to balance the angular momentum acquired from\nthe magnetic \feld, known as the Einstein-de Haas e\u000bect.3\nThe reverse process, i.e., a crystal rotation generating\nspin polarization, has also been discovered around the\nsame period by Barnett.4\nDi\u000berent from paramagnets, spins in a ferromagnetic\nmaterial order collectively into a magnetization even in\nthe absence of an external magnetic \feld. The magneti-\nzation serves as an order parameter and carries an intrin-\nsic angular momentum. As a consequence of the angu-\nlar momentum conservation, a reorientation of the order\nparameter is necessarily accompanied by a mechanical\nrotation, and vice versa . Following the seminal discov-\nery of the Einstein-de Haas e\u000bect, the idea that angular\nmomentum can transfer between magnetic and mechani-\ncal degrees of freedom has fertilized a broad spectrum of\napplications such as magnetic force microscopy5{7, me-\nchanical manipulations of spins8{10,etc.\nHowever, this simple picture seems to break down in\nantiferromagnets (AFs) with vanishing magnetization.\nIn this case, the ground state is characterized by the N\u0013 eel\norder parameter which does not carry an angular mo-\nmentum. Only when the N\u0013 eel order is driven into motion\ndoes a small magnetization develop;11,12it is this induced\nmagnetization that is subjected to angular momentum\nconservation. In other words, unlike its ferromagnetic\ncounterparts, the order parameter dynamics in AFs is\nnot directly dictated by any conservation law. There-\nfore, to study how spin-mechanical e\u000bects can manifest\nthrough the N\u0013 eel order parameter instead of the small\nmagnetization, one must seek new physics beyond angu-\nlar momentum conservation.\nIn this regard, a well-established phenomenon provides\na critical hint: The coordinated motion of antiparallel\nmagnetic moments in an AF creates a \fctitious inertiain the e\u000bective N\u0013 eel order dynamics13{16, in sharp con-\ntrast to the non-inertial behavior of the magnetization\ndynamics in ferromagnets: The N\u0013 eel order behaves more\nlike a massive particle that can be accelerated via exter-\nnal forces rather than an angular momentum regulated\ndirectly by the conservation law. Regarding this unique\nfeature, it is tempting to ask whether the \fctitious inertia\nof the N\u0013 eel order can lead to any mechanical consequence.\nIn this paper, we demonstrate that the \fctitious inertia\nof the N\u0013 eel order modi\fes the inertia tensor that char-\nacterizes the rigid background rotation, giving rise to a\nmechanically measurable e\u000bect. We term this e\u000bect spin-\nmechanical inertia, which re\rects a quantum correction\nto the otherwise classically de\fned inertia. Our claim is\njusti\fed by modeling a collinear AF as a hybrid system\nconsisting of antiferromagnetic spins and a rigid mechani-\ncal rotation; they couple through an easy-axis anisotropy.\nWhen the two subsystems operate at vastly di\u000berent time\nscales, their coupled motion can be solved by the adia-\nbatic approximation. The spin-mechanical inertia is then\nderived as a result of the adiabatic approximation. Fur-\nthermore, by considering magnon excitations, we \fnd\nthat the spin-mechanical inertia is subject to a reduc-\ntion in two aspects: zero-point quantum \ructuation and\nthermal \ructuations. The former is independent of tem-\nperatureT, whereas the latter results in an appreciable\ntemperature dependence when the thermal energy kBT\nis comparable to the magnon gap. Finally, we invoke\nthe path integral formalism to derive a criterion for the\nadiabatic approximation by inquiring into non-adiabatic\ncorrections, which turns out to be well suppressed in typ-\nical situations. Our result establishes spin-mechanical in-\nertia as an essential ingredient of spin-mechanics in the\ncontext of AFs.\nThis paper is organized as follows. In Sec. II, we study\nthe simplest case of a uniform AF to illustrate the es-\nsential physics, supplemented by a discussion on possi-\nble detection schemes. In Sec. III, we consider the tem-\nperature dependence of spin-mechanical inertia arising\nfrom magnon excitations. In Sec. IV, non-adiabatic cor-\nrections are derived using the path integral formalism.\nIn Sec. V, we discuss the underlying physics of spin-\nmechanical inertia from several fundamental aspects and\npotential issues that may complicate out result. Mathe-\nmatical details are presented in the Appendices.arXiv:1611.00100v3  [cond-mat.mes-hall]  22 Aug 20172\nII. MACROSPIN MODEL\nTo capture the essential physics, we \frst consider the\nsimplest case where the antiferromagnetic ordering is spa-\ntially homogeneous and described by two macrospins SA\nandSBwith equal magnitude jSAj=jSBj=S. The\ntwo macrospins couple through the Heisenberg exchange\ninteraction H=JSA\u0001SB. De\fning the N\u0013 eel order as\nN= (SA\u0000SB)=2S, we follow the standard procedure\nto eliminate the small magnetization m=SA+SB17{19,\nwhich yields an e\u000bective action of N:\nSN=~2V\n2ZJa3Z\ndtj@tNj2; (1)\nwhereVis the system volume, ais the lattice constant,\nandZis the coordination number. For simplicity, we\nhave assumed a cubic lattice. In fact, SNis equivalent\nto the action of a rigid rod with supporting point on its\ncenter of mass. This property indicates that the N\u0013 eel\norder acquires an e\u000bective inertia from the exchange in-\nteraction between SAandSB.\nNext we picture the crystal background as a rigid body.\nWhen it rotates about a \fxed-axis, its kinetic energy is\nT=1\n2I_'2withIthe moment of inertia and 'the angle of\nrotation around the axis. More generally, when the rigid\nbody rotates about a \fxed point (a spinning top), its in-\nstantaneous orientation is characterized by the principal\naxes of the body-frame, which are speci\fed by three Euler\nangles\u0015(t)\u0011f\u0012(t);\u001e(t); (t)gas shown in Fig. 1(a). We\nnow consider a symmetric top in which the e1ande2axes\nare equivalent and the system preserves cylindrical sym-\nmetry with respect to the e3axis. Accordingly, the ki-\nnetic energy is T=1\n2I?(_\u00122+sin2\u0012_\u001e2)+1\n2Ik(_ +cos\u0012_\u001e)2,\nwhereI?andIkare the moments of inertia with respect\nto the e1(ore2) and e3axes, respectively20. In the ab-\nsence of gravitational torques (known as the Euler top),\nthe system action only has the kinetic energy, thus its\naction can be written as21\nSR=1\n2Z\ndtGij(\u0015)_\u0015i_\u0015j; (2)\nwhere repeated indices are summed. Gij(\u0015) made up by\nI?,Ik, and\u0015; it plays the role of an e\u000bective metric in\nthe parameter space spanned by the Euler angles.\nWe assume that the spin subsystem couple to the rigid\ncrystal background (an Euler top) through an easy-axis\nanisotropy, described by the action\nSK=KV\n2a3Z\ndt\f\fN\u0001ek(\u0015)\f\f2; (3)\nwhereK > 0 in our convention22. The easy-axis ekis\na function of \u0015since its direction depends on the ori-\nentation of the rigid body. It is this \u0015-dependence that\nconnects the two subsystems. If the anisotropy Kis su\u000e-\nciently strong such that the N\u0013 eel order is able to adjust to\nthe easy-axis at any instant of time, then the entire sys-\ntem moves as a whole as if a rigid rod is \frmly attached\n✓\u0000 e3e2e1xyz(a)(b)\neke3e2e1\u0000\u0000+↵\u00000LFIG. 1. (a) Euler angles \u0012,\u001e, and specify the relative\norientation of the body frame labeled by the principal axes\ne1-e2-e3with respect to the laboratory frame x-y-z. (b) The\nekandLvectors in the body frame.\nto the Euler top, which de\fnes an adiabatic motion of\nthe hybrid system.\nWe now check the behavior of the hybrid system in\nthe adiabatic limit ~!rb=K!0, where!rbis the rota-\ntional frequency of the rigid body. Deviations from the\nadiabatic limit ( i.e. non-adiabatic corrections) will be\ndiscussed in Sec. IV. In this limit, SKis a constant, so\nthe e\u000bective action becomes Se\u000b=SN+SR. As illus-\ntrated in Fig. 1(b), we \fx the body frame by choosing the\ne1axis coplanar with e3andek:ek= cos\re3+ sin\re1.\nSuch a choice is always possible for a symmetric Euler\ntop. The e\u000bective action Se\u000bthen becomes\nSe\u000b[\u0015] =1\n2Z\ndt\u0014\nGij+~2V\nZJa3gij\u0015\n_\u0015i_\u0015j; (4)\nwheregijis a correction of the parameter-space metric\ntensor originating from the N\u0013 eel order dynamics; it is a\nfunction of the Euler angles and \r:\ng\u0012\u0012=\u0000\n3 + cos 2\r\u00002 sin2\rcos 2 \u0001\n=4;\ng\u001e\u001e= sin2\rcos2 + (cos\rsin\u0012\u0000sin\rcos\u0012sin )2;\ng  = sin2\r;\ng\u0012\u001e= sin\rcos (cos\rcos\u0012+ sin\rsin\u0012sin );\ng\u001e = sin\r(sin\rcos\u0012\u0000cos\rsin\u0012sin );\ng\u0012 = sin\rcos\rcos : (5)\nIn the presence of gij, the inertia tensor is no longer di-\nagonal in the body-frame labeled by the principal axes.\nOr equivalently, we can say that the principal axes them-\nselves are changed by the spin-mechanical coupling. We\nterm this e\u000bect spin-mechanical inertia . In the limit that\n\r!0,i.e., when the easy-axis ekis parallel to the prin-\ncipal axise3, only two components survive: g\u0012\u0012= 1 and\ng\u001e\u001e= sin2\u0012. In this case, gijreduces to a spherical met-\nric and the principal axes do not change. Nevertheless,\nmoments of inertia associated with the principal axes, I?\nandIk, are still modi\fed.\nIn Eq. (4), the strength of spin-mechanical inertia\nseems to be proportional to the system volume V. How-\never, sinceGijscales asVd2withdthe body dimension\ntransverse to the instantaneous rotation axis, the relative3\n\u0000\n\u0000II+\u0000I~2VZJa3zero-point fluctuation2kBT~!000.511.52\u0000I\nFIG. 2. Left: Schematics of the moment of spin-mechanical\ninertia \u0001Iin a sphere with uniform mass distribution. Right:\nWhen averaged over magnon excitations, \u0001 Idecreases with\nan increasing temperature, supplemented by a residual zero-\ntemperature reduction due to quantum \ructuation. Parame-\nters for the plot: V= 103a3andJ= 102~!0.\nstrength of spin-mechanical inertia scales as d\u00002instead\nofV. Therefore, we expect a pronounced e\u000bect only in\nsmall systems.\nA. Rotation about a \fxed-axis\nTo demonstrate the physical consequences of the spin-\nmechanical inertia, we now consider a rotation about a\n\fxed axis, where the inertia tensor reduces to a moment\nof inertiaI. For instance, a sphere with uniform mass dis-\ntribution has a constant Iregardless of how the sphere is\nsuspended. By contrast, the moment of spin-mechanical\ninertia depends on the relative orientation of Nwith\nrespect to the rotation axis ^z. According to Eq. (5),\n\u0001I=~2V=(ZJa3)j^z\u0002Nj2. \u0001Ireaches maximum for\nN?^z, and thus the period of oscillation measured by a\ntorsion balance, as illustrated in Fig. 2, reaches a max-\nimum forN?^z. The relative correction \u0001 I=I, which\nscales asd\u00002as mentioned above, gets larger when the\nsphere gets smaller.\nAdmittedly, the torsion balance shown in Fig. 2 is\nprobably not a realistic setup for detection at micro-\nscopic scales. For example, if the sphere considered above\nrefers to an AF molecule, then both the spin dynamics\nand the molecular rotation should be treated quantum\nmechanically. Therefore, a possible way to observe the\nspin-mechanical inertia is to measure the change of the\nrotational spectrum when a N\u0013 eel ordering is introduced\n(e.g., by lowering the temperature). For example, if we\nregard the AF molecule as a quantum rotor with moment\nof inertiaI, the energy is quantized as E=~2n2=2Iwith\nn= 0;1;2\u0001\u0001\u0001. Since the spin-mechanical inertia changes\nIintoI+\u0001I, the energy splitting is slightly reduced. By\nmonitoring the shift of spectral lines stemming from tran-\nsitions between the ground state and states with large n\n(so that the change is magni\fed by n2), one should be\nable to identify the existence of spin-mechanical inertia.\nWe estimate the e\u000bect in an AF molecule consisting\nof thousands of atoms. Suppose the magnetic moments\nClassical Euler Top\nEuler Top with Spin-mechanical EffectRicci ScalarFIG. 3. Ricci curvature (colored) and geodesic curves (solid\nblack) in the parameter space of the Euler angles. The Euler\ntop hasI?=Ik=2 = 10 ~2V=(ZJa3) and ek?e3. Plot\nrange:\u00122[0;\u0019),\u001e2[0;2\u0019), and 2[0;2\u0019). In the absence\n(presence) of spin-mechanical inertia, the Ricci curvature is\na constant R0= 1 (a periodic function in all Euler angles).\nRdiverges at \u0012= 0;\u0019and =\u0006\u0019=2;\u0006\u0019, where the Euler\nangles are ill-de\fned. In the \u0012\u0000 subspace, we cut o\u000b the\ncolor bar at R=R0\u00065:5.\noriginate from transition metal elements, such as iron\nand nickel, and take Jto be tens of meV (similar to\nthe superexchange interaction in bulk antiferromagnetic\ncrystals), then \u0001 I=Ifalls somewhere between 10\u00002and\n10\u00003. If we further consider that the value of Jin mag-\nnetic molecules is smaller than that in magnetic crystals,\nthen the spin-mechanical inertia should be more signi\f-\ncant than the above estimation.\nHere we emphasize one point: The validity of the rigid\nbody action Eq. (2) does not require that the mechanical\nmotion is classical. In fact, Eq. (2) can describe a fully\nquantum rotation in the path integral formalism to be\nexploited below. The spin-mechanical correction of the\ninertia tensor holds, whether the mechanical motion is\nclassical or quantum.\nB. Rotation about a \fxed-point\nFor rotations about a \fxed point, the tensorial nature\nof inertia is reinstated. Although this general case is not\nnecessary for practical measurements, it entails a beauti-\nful geometrical interpretation of spin-mechanical inertia.\nTo this end, it is adequate to consider a classical Euler\ntop, the motion of which can be obtained by minimizing\nthe e\u000bective action Eq. (4) with respect to the Euler an-\ngles\u0015. The result is a geodesic equation in the parameter\nspace:\n\u0015k+ \u0000k\nij_\u0015i_\u0015j= 0; (6)\nwhere \u0000k\nij=1\n2Gk`(@iGj`+@jGi`\u0000@`Gij) is the connection\nwithGij=Gij+~2V\nZJa3gijthe total metric and Gijsatis-4\nfyingGi`G`j=\u000ei\nj. The metric tensor Gijfully determines\nthe geometry of the parameter space.\nIn the absence of spin-mechanical e\u000bect, the geodesic\ncurve solved by Eq. (6) corresponds to a trivial great arc\nin the\u0012\u0000 subspace as shown in Fig. 3. Here \u0012is a\nconstant of motion while \u001eand precess uniformly. We\nhave chosen parameters to make the  \u0000\u001ephase portrait\ncommensurate. To further understand this feature from a\ngeometrical perspective, we calculate the Ricci curvature\nof the parameter space de\fned as\nR=Gij \n@\u0000k\nij\n@\u0015k\u0000@\u0000k\nik\n@\u0015j+ \u0000`\nij\u0000k\nk`\u0000\u0000`\nik\u0000k\nj`!\n;(7)\nwhich is analogous to the inverse radius of a sphere. It\nturns out that Ris a constant throughout the parame-\nter space if spin-mechanical inertia is disregarded, which\nexplains why the geodesic curve is trivial. However, the\nspin-mechanical inertia introduces an induced metric gij\non top ofGij, which changes the geometry and distorts\nthe parameter space so that the Ricci curvature is no\nlonger a constant. Consequently, the geodesic curve char-\nacterizing the rigid body rotation de\rects from its origi-\nnal path, as if a \fctitious gravity appears in the parame-\nter space. As depicted in Fig. 3, a free Euler top has no\nnutation and the geodesic curve projected onto the  \u0000\u001e\nsubspace is commensurate with our chosen parameter.\nThe spin-mechanical inertia breaks these properties: It\nnot only generates a small nutation, but also decommen-\nsurates the orbit in the  \u0000\u001esubspace.\nIII. MAGNON EXCITATIONS\nSo far the N\u0013 eel order Nhas been regarded as a uni-\nform vector, which is a reasonable approximation at low\ntemperatures. To investigate how the spin-mechanical\ninertia is a\u000bected by thermal \ructuations embedded in\nthe spin dynamics, we need to go beyond the macrospin\nmodel and introduce magnon excitations. Since magnons\nare inhomogeneous deviations from the uniform ground\nstate, we need to generalize the N\u0013 eel vector into a stag-\ngered \feldN=N(t;r), and promote Eq. (1) into the\nnonlinear sigma model13\nSN=~2\n2ZJa3Z\nd4r(\u0011\u0016\u0017@\u0016N\u0001@\u0017N); (8)\nwherec=Ja=~is the spin wave velocity, r\u0016=ft;rg\nis the joint spacetime coordinate, d4r= dtd3r, and\n\u0011\u0016\u0017= diag[1;\u0000c2;\u0000c2;\u0000c2] is the spacetime metric of\nthe laboratory frame. The anisotropy term is now\nSK=K\n2a3Z\nd4rjN(t;r)\u0001ek(\u0015)j2: (9)\nNext we employ the standard procedure19to decompose\nthe staggered \feld into\nN(t;r) =L(t)p\n1\u0000j\u0019(t;r)j2+\u0019(t;r); (10)whereL(t) is a time-dependent unit vector and \u0019(t;r)\nis the magnon \feld (a transverse \ructuation) that satis-\n\fesL\u0001\u0019= 0. We restrict our discussion to the low-\ntemperature regime where j\u0019j \u001c 1. Our goal is to\neliminate\u0019and derive an e\u000bective action of Lwith a\ntemperature-dependent coe\u000ecient, which is supposed to\nreplace the original action Eq. (1).\nTo this end, we insert Eq. (10) into Eq. (8) and Eq. (9).\nAfter some tedious algebra, as detailed in Appendix A,\nwe obtain\nSN=~2\n2ZJa3Z\nd4rn\n(1\u0000j\u0019j2)@tL\u0001@tL\n+\u0011\u0016\u0017\u0014\n@\u0016\u0019\u0001@\u0017\u0019+(\u0019\u0001@\u0016\u0019)(\u0019\u0001@\u0017\u0019)\n1\u0000j\u0019j2\u0015\n+\u0011\u0016\u0017\u0019a\u0019b@\u0016ea\u0001@\u0017ebo\n; (11)\nSK=K\n2a3Z\nd4rh\n(1\u0000j\u0019j2)\f\fL\u0001ek\f\f2+\f\f\u0019\u0001ek\f\f2i\n;(12)\nwherefeagforms a set of local coordinates labeling the\ntransverse plane normal to the instantaneous L(t); the\nmagnon \feld \u0019(t;r) resides in this plane. In Eqs. (11)\nand (12), terms that will not survive the thermal aver-\naging operation below have been omitted; they are listed\nin Appendix A. The sum SN+SKde\fnes a coupled \feld\ntheory consisting of Land\u0019.\nIntegrating out the \u0019\feld, again, requires the adi-\nabatic approximation. But at \fnite temperatures, the\nmeaning of the adiabatic approximation changes. It now\nmeans that the rigid body rotates su\u000eciently slow such\nthat the staggered \feld remains in thermal equilibrium\nwith respect to the instantaneous crystal orientation at\nall times. This allows us to take a thermal average over\nthe\u0019\feld by freezing L(t), which \fnally leads to an\ne\u000bective description of L(t) with temperature-dependent\nparameters. The result comes in the form of a thermally-\naveraged action \u0016SL\u0011hSN+SKith. As derived in Ap-\npendix A, \u0016SLreads\n\u0016SL=V\n2Z\ndt\u0014~2\nZJa3\u0002(T)_L2+K\na3\u0000\nL\u0001ek\u00012\u0015\n;(13)\nwhere the temperature-dependent factor is\n\u0002(T) = 1\u0000Za3\nVX\nkJ\n~!kcoth~!k\n2kBT; (14)\nwithkBthe Boltzmann constant and !kthe dispersion.\nIt is clear that the net e\u000bect of magnon excitations is\nto replace the original action Eq. (1) with Eq. (13), in\nwhich the unit vector L(t) becomes an e\u000bective order\nparameter, and the coe\u000ecient acquires a temperature de-\npendence through \u0002( T). This change, in turn, yields a\ntemperature-dependent spin-mechanical inertia re\rected\nin a thermally averaged action\n\u0016Se\u000b[\u0015] =1\n2Z\ndt\u0014\nGij+ \u0002(T)~2V\nZJa3gij\u0015\n_\u0015i_\u0015j;(15)5\nwhich replaces our previous result Eq. (4).\nTo assess the signi\fcance of magnon excitations in the\nspin-mechanical inertia, we consider an AF nanomag-\nnet with quantized magnon modes subjected to rotations\nabout a \fxed-axis (as illustrated in Fig. 2). The moment\nof spin-mechanical inertia is simply\n\u0001I(T) =~2V\nZJa3\u0002(T) (16)\nwith \u0002(T) given by Eq. (14). Because of the prominent\nenergy splitting at the nanometer scale, the lowest mode\n!0is well separated from all other modes. Since coth x\nconverges to unity rapidly with x, the dominant contri-\nbution to the temperature dependence originates from\nthe lowest mode, which scales as (J\n~!0)(a3\nV) coth~!0\n2kBTac-\ncording to Eq. (14).\nWithout loss of essential physics, we will only keep\nthe lowest mode, which occurs at k= 0 with a gap ~!0\nbeing few Kelvins. In typical antiferromagnets such as\nMnF 2, the ratioK=J is around 10\u00003to 10\u00004. Since the\nN\u0013 eel temperature is in a loose sense proportional to J\nthat far exceeds the gap ~!0\u0018p\nZJK\u001810\u00002J, it is\nstill within the low temperature regime even when kBT\nis comparable to ~!0. In MnF 2, for example, ~!0\u00182 K\nwhile the N\u0013 eel temperature is around 60 to 80 K, thus\nour theory remains valid up to few Kelvins.\nWith these considerations, we plot the spin-mechanical\ninertia Eq. (16) as a function of temperature in Fig. 1,\nassumingV=a3= 103andJ=~!0= 102. There are two\nnoticeable features in Fig. 1: (i) \u0001 I(T) starts to bend\ndown at around kBT\u0018~!0, which marks the onset of\nsubstantial thermal \ructuations. (ii) There is a residual\nreduction of spin-mechanical inertia even at zero tem-\nperature, \u0001 I(T!0)<~2V\nZJa3. This is attributed to the\nzero-point quantum \ructuation of Naround the easy-\naxis. According to Eq. (14), ~!0\u0018p\nZJK , this zero-\ntemperature correction vanishes in the limit K!1 .\nIV. NON-ADIABATIC EFFECT\nWe \fnally derive a criterion for the adiabatic assump-\ntion employed in previous sections. Since the in\ruence\nof magnon excitations has been resolved by the tempera-\nture dependence of the spin-mechanical inertia during the\nthermal averaging operation, we can now treat L(t) as\nthe real order parameter23. In the body frame depicted\nby Fig. 1(b), Lcan be decomposed as\nL= cos(\r+\u000b)e3+ sin(\r+\u000b) cos\fe1\n+ sin(\r+\u000b) sin\fe2; (17)\nwhere\u000band\fare two independent variables character-\nizing the deviation of Lfrom the easy-axis ek. For large\nbut \fnite anisotropy K, misalignment between Landek\nshould be small, so we assume that \u000b\u001c1 and\f\u001c1.\nAs detailed in Appendix B, by adopting the path integralformalism18and expanding the action \u0016SLup to second\norder in\u000band\f, we can analytically integrate out the\nfast variableLas\nZ=Z\nD\u0015DL\u000e3(L2\u00001) exp\u0014i\n~\u0000\nSR+\u0016SL\u0001\u0015\n=Z\nD\u0015exp\u0014i\n~\u0000\u0016Se\u000b+ \u0001S\u0001\u0015\n; (18)\nwhere \u0016Se\u000bis given by Eq. (15). The \u0001 Sterm includes\nall non-adiabatic corrections, which can be expressed as\na series summation\n\u0001S=~2V\n2ZJa3\u0002(T)1X\nn=1\u0002(T)n(\u00001)n\n\u0002Z\ndtXT(\u0015)\u0014~2\nZJK@2\nt\u0015n\nX(\u0015); (19)\nwhere the vector XT(\u0015) =fX1(\u0015);X2(\u0015)grepresents\na particular combination of the Euler angles: X1(\u0015) =\n\u0000sin _\u0012+ cos sin\u0012_\u001eandX2(\u0015) = sin\r(cos\u0012_\u001e+_ )\u0000\ncos\r(cos _\u0012+ sin\u0012sin _\u001e). In Eq. (19), the small quan-\ntity of expansion is~2\nZJK@2\nt, which is proportional to\n(!rb=!0)2with!rbthe frequency of rigid body rota-\ntion and ~!0the anisotropy gap used earlier. In typical\nAFs such as transition metal oxides or \ruorides, !0is in\nthe Terahertz regime, which coincides with the frequency\nscale of vibrational modes in a magnetic molecule. On\nthe other hand, !rbcorresponds to the frequency of ro-\ntational modes that is typically far below the vibrational\nfrequency. Therefore, the adiabatic condition is likely to\nbe well respected.\nV. DISCUSSIONS\nIt is worthwhile to distinguish the spin-mechanical ef-\nfect explored in this paper from the well-established mag-\nnetoelastic phenomena. The latter scenario primarily fo-\ncuses on the hybridization of magnetic and mechanical\nexcitations. For example, when spin dynamics is driven\nby a current, mechanical vibrations are agitated24. By\ncontrast, our attention is paid on the ground state where\nthe e\u000bect is maximum at zero temperature; elementary\nexcitations reduce the strength of the e\u000bect. The spin-\nmechanical inertia we predict is a conceptual progress\nthat poses a serious challenge to the common belief that\nmoment of inertia is a classical quantity.\nWe also mention that the spin-mechanical inertia does\nnot modify the inertial mass of the crystal. It only makes\nsense when a rigid body undergoes rotations instead of\nlinear motions. By de\fnition, the moment of inertia is\nthe response coe\u000ecient of the angular acceleration versus\nan external torque. In classical mechanics, this coe\u000ecient\nturns out to be, but is not de\fned as, a quantity that is\nmerely determined by the mass distribution of the body.\nWhat we have shown in this paper is that this response6\ncoe\u000ecient also depends on the spin degree of freedom of\nthe constituent atoms in AFs, which cannot be described\nby classical mechanics.\nBesides magnons, thermal excitations also come in the\nform of phonons. Phonons play a signi\fcant role in spin-\nmechanical e\u000bects of ferromagnets because local distor-\ntions of the lattice background directly couple to the\nmagnetization in the form of _m\u0001(r\u0002u), whereu(t;r) is\nthe local displacement \feld of the lattice and m(t;r) is\nthe local magnetization vector. This form of coupling can\neither be justi\fed by angular momentum conservation\nor derived from a simple model including the easy-axis\nanisotropy25. In an AF, the latter approach is apparently\nmore reasonable, since angular momentum conservation\ndoes not explicitly rule the N\u0013 eel order dynamics. (Cau-\ntion: Angular momentum is always conserved, but the\nN\u0013 eel order does not carry one.)\nIt is straightforward to check that the local lattice dis-\ntortionr\u0002ualways couple to m(t;r) instead ofN(t;r).\nHowever, in a collinear AF, local magnetization develops\nonly when the staggered \feld is driven into motion11,12:\nm\u0018N\u0002_N=J. Therefore, the magnitude of mscales asp\nK=J, which is typically few percents. This implies that\nphonons are far less important in collinear AFs than in\nferromagnets regarding spin-mechanical e\u000bects. Never-\ntheless, our discussions refer to transverse phonons only.\nThere might be a strong e\u000bect from the longitudinal\nphonons as they would a\u000bect the exchange interaction\nJby modulating distances between neighboring spins.\nACKNOWLEDGMENTS\nWe are grateful to A. H. MacDonald, J. Zhu, and\nS. Okamoto for inspiring discussions. X.W. also thanks\nN. P. Ong for insightful comments. This work was sup-\nported by the Department of Energy, Basic Energy Sci-\nences, Grant No. DE-SC0012509. D.X. also acknowl-\nedges support from a Research Corporation for Science\nAdvancement Cottrell Scholar Award.\nR.C. and X.W. contributed equally to this work.\nAppendix A\nThe time dependence of L(t) originates from the rigid\nbody rotation, while that of \u0019(t;r) stems from thermal\nagitations. Equation (10) can be further written as\nN(t;r) =L(t)p\n1\u0000j\u0019j2+\u0019a(t;r)ea(t); (A1)\nwherefea(t)gfora= 1;2 and e0\u0011Ltogether forms a\nlocal orthonormal base with respect to the instantaneous\nL(t), satisfyingLL+P2\na=1eaea=I. To improve visual\nclarity, hereafter we will omit the spacetime argument\nunless necessary. We de\fne Aa0\u0011ea\u0001@tL,A0a\u0011L\u0001@tea,\nandAab\u0011ea\u0001@tebas temporal connections of the local\nbase. They obey @teA=eBABA,AAB+ABA= 0, andACAACB=@teA\u0001@teB, whereA;B take 0;1;2 (c.f.,a;b\nonly take 1 ;2) and repeated indices are summed. With\nthese notations, we have\n@\u0016N=\u000e\u00160(@tL)p\n1\u0000j\u0019j2+\u0019\u0001@\u0016\u0019p\n1\u0000j\u0019j2L\n+ (@\u0016\u0019a)ea+\u000e\u00160eaAab\u0019b; (A2)\nwhere\u0016=f0;1;2;3g \u0011 ft;x;y;zg. Now we insert\nEq. (A2) into the actions Eq. (11) and (12), and reor-\nganize the terms of the total action into three parts\nSAF\u0011SN+SA=Z\nd4r(LL+L\u0019+Lodd);(A3)\nwhere the \frst two terms are, respectively,\nLL=~2\n2ZJa3\u0002\n(1\u0000j\u0019j2)@tL\u0001@tL+@tea\u0001@teb\u0019a\u0019b\u0003\n+K\n2a3\u0000\nL\u0001ek\u00012; (A4)\nL\u0019=~2\n2ZJa3\u0011\u0016\u0017Gab@\u0016\u0019a@\u0017\u0019b\n+K\n2a3h\u0000\n\u0019\u0001ek\u00012\u0000j\u0019j2\u0000\nL\u0001ek\u00012i\n; (A5)\nwhere Gabis the metric in the local base19de\fned as\nGab(\u0019) =\u000eac\u0019c\u0019d\n1\u0000j\u0019j2\u000edb+\u000eab; (A6)\nherea;b;c;d take 1;2. The third term of Eq. (A3) reads\nLodd=~2\n2ZJa3h\nAab\u0019b@t\u0019a+Aa0\u0019a\u0019\u0001@t\u0019p\n1\u0000j\u0019j2\n+p\n1\u0000j\u0019j2Aa0\u0000\n@t\u0019a+Aab\u0019b\u0001i\n+K\na3\u0000\nL\u0001ek\u0001\u0000\nea\u0001ek\u0001\n\u0019ap\n1\u0000j\u0019j2; (A7)\nwhich, to be shown below, will vanish identically under\nthermal averaging.\nNext we integrate out the \u0019\feld and derive an e\u000bective\nLagrangian for L. However, since \u0019represents magnon\nexcitations driven by thermal \ructuations, the integra-\ntion should be performed in the Euclidean space where\ntemperature plays the role of time. In other words, we\nare dealing with an adiabatic process in which \u0019stays\nin thermal equilibrium with respect to the instantaneous\nL. Retaining to the non-interacting order in the small \u0019\n\feld, the expected Lagrangian density for Lbecomes\nLL=~2\n2ZJa3\u0002\n(1\u0000h\u0019a\u0019aith)@tL\u0001@tL\n+AcaAcbh\u0019a\u0019bith\u0003\n+K\n2a3\u0000\nL\u0001ek\u00012:(A8)\nThe key issue boils down to the calculation of the thermal\ncorrelation function\n\u0019a\u0019b\u000bth. To ful\fll this task, we7\nperform a Wick rotation for the \u0019\feld and freeze the\ntime variable of L. Since deviations of Lfromekare\nsmall and\u0019is virtually perpendicular to ek, we have\u0000\n\u0019\u0001ek\u00012\u001cj\u0019j2\u0000\nL\u0001ek\u00012. Consequently, we can ignore\nthe\u0000\n\u0019\u0001ek\u00012term in Eq. (A5). In the non-interacting\norder,\u0019is just a Klein-Gordon \feld with dispersion !k= p\nc2k2+ZJK=~2. The Matsubara propagator is\n\n\u0019a(\u001c;r)\u0019b(0;0)\u000bth\n0\n=ZJa3\n\f~2\u000eabX\nnZd3k\n(2\u0019)3ei(k\u0001r+!k\u001c) 1\n!2n=~2+!2\nk\n=ZJa3\n2\u000eabZd3k\n(2\u0019)3eik\u0001r\n~!k[fB(!k)e!k\u001c\n+ (1 +fB(!k))e\u0000!k\u001c\u0003\n; (A9)\nwhere\u001cis the imaginary time, fB(!k) =1\ne\f!k\u00001with\n\f= 1=kBT, and!n=2\u0019n\n\f(n2Z) is the Matsubara\nfrequency. A relevant quantity that can be constructed\nfrom the propagator is the one-loop integral\n\n\u0019a(\u001c;r)\u0019b(\u001c;r)\u000bth\n=Za3\n2\u000eabZd3k\n(2\u0019)3J\n~!kcoth\u0012\f~!k\n2\u0013\n:(A10)\nFollowing the same spirit, terms in Eq. (A7) that areodd in the power of the \u0019\feld should vanish identically:\nj\u0019j2n\u0019a\u000bth=\nj\u0019j2n\u0019a\u0019\u0001@t\u0019\u000bth=\nj\u0019j2n@t\u0019a\u000bth= 0.\nMoreover,\n\u0019b@t\u0019a\u000bth\u0018R\nd!k!kR\nd3k\n\u0019b\u0019a\u000bth, which\nvanishes as well. Therefore, there is no term in Loddthat\ncan survive the thermal averaging process.\nFinally, inserting Eq. (A10) into Eq. (A8) and noticing\nthatAcaAcb\u000eab=_L2, we arrive at\nLL=~2\n2ZJa3\u0002(T)_L2+K\n2a3\u0000\nL\u0001ek\u00012; (A11)\n\u0002(T) = 1\u0000Za3\n~Zd3k\n(2\u0019)3J\n!kcoth~!k\n2kBT: (A12)\nIfkis quantized due to geometric con\fnement, thenRd3k\n(2\u0019)3should be understood as1\nVP\nk. Equations (A11)\nand (A12) prove our central result, Eqs. (13) and (14).\nAppendix B\nAs expressed by Eq. (17), \u000b(t) and\f(t) parametrize\nthe deviation of L(t) fromek(t) in the body frame, while\n\ris a constant. Since the coordinates in the body frame\nare time dependent, we \frst need to relate them to the\nlaboratory frame coordinates\n0\n@e1(t)\ne2(t)\ne3(t)1\nA=0\n@cos\u001ecos \u0000cos\u0012sin\u001esin sin\u001ecos + cos\u0012cos\u001esin sin\u0012sin \n\u0000cos\u001esin \u0000cos\u0012sin\u001ecos \u0000sin\u001esin + cos\u0012cos\u001ecos sin\u0012cos \nsin\u0012sin\u001e \u0000sin\u0012cos\u001e cos\u00121\nA0\n@ex\ney\nez1\nA; (B1)\nwhere all Euler angles depend on time. Next we insert\nthe above expression into Eq. (17), followed by insertingthus-obtained L(t) into Eq. (A11). Expanding Eq. (A11)\nto quadratic orders in \u000band\f, we have\nLL=~2\n2ZJa3\u0002(T)gij_\u0015i_\u0015j+~2\n2ZJa3\u0002(T)\u0010\n_\u000b2+ sin2\r_\f2\u0011\n+K\na3\u0000\n1\u0000\u000b2\u0000\f2sin2\r\u0001\n+~2\nZJa3\u0002(T)n\n_\u000b\u0010\n\u0000sin _\u0012+ cos sin\u0012_\u001e\u0011\n+_\fsin\rh\nsin\r\u0010\ncos\u0012_\u001e+_ \u0011\n\u0000cos\r\u0010\ncos _\u0012+ sin\u0012sin _\u001e\u0011io\n; (B2)\nwhere the leading term gijis the spin-mechanical inertia\nthat does not depend on \u000band\f.\nAs the adiabatic approximation has frozen the time\nfor the Euler angles, the integral over Lconverts into\nthat over\u000band\f, which is Gaussian type according toEq. (B2). The measure of the integral is\nd3L\u000e\u0000\nL2\u00001\u0001\n=\f\f\f\f\f@L3\n@\u000b@L3\n@\f\n@\n@\u000barctanL2\nL1@\n@\farctanL2\nL1\f\f\f\f\fd\u000bd\f\n= sin\rd\u000bd\f: (B3)8\nTo perform the integral, we set\nx1=\u000b; (B4)\nx2=\fsin\r; (B5)\nX1=\u0000sin _\u0012+ cos sin\u0012_\u001e; (B6)\nX2= sin\r(cos\u0012_\u001e+_ )\n\u0000cos\r(cos _\u0012+ sin\u0012sin _\u001e): (B7)Then we \fnally obtain\nZAF=Z\nDx1Dx2exp\u001a\ni~V\u0002(T)\nZJa3Z\ndt\u00141\n2\u0000\n_x2\n1+ _x2\n2\u0001\n\u0000ZJK\n~2\u0002(T)\u0000\nx2\n1+x2\n2\u0001\n+ ( _x1X1+ _x2X2)\u0015\u001b\n=Z\nDx1Dx2exp\u001a\ni~V\u0002(T)\n2ZJa3Z\ndt\u0014\nx1\u0012\n@2\nt\u00002ZJK\n~2\u0002(T)\u0013\nx1+x2\u0012\n@2\nt\u00002ZJK\n~2\u0002(T)\u0013\nx2\u00002\u0010\nx1_X1+x2_X2\u0011\u0015\u001b\n=(V=a3)p\n2K=(ZJ)\u0002(T)\n2\u0019isinhq\n2ZJK\n~2\u0002(T)(tf\u0000ti)iexp(\ni\n~1X\nn=1~2V\u0002(T)\n2ZJa3Z\ndtXT\u0014\n\u0000~2\u0002(T)\nZJK@2\nt\u0015n\nX)\n; (B8)\nwhereX=fX1;X2g, andtf(ti) is the upper (lower) limit of the time integral.\n1E. M. Chudnovsky, Phys. Rev. Lett. 72, 3433 (1994).\n2J. E. Losby and M. R. Freeman, Spin Mechanics , Physics\nin Canada 72(2), 2016.\n3A. Einstein, W. J. de Haas, Royal Neth. Acad. Arts Sci-\nences (KNAW) 18, 696 (1915).\n4S. J. Barnett, Phys. Rev. 6, 239 (1915).\n5H. Hopster and H. P. Oepen, Magnetic Microscopy of\nNanostructures , Springer, 2005.\n6D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui,\nNature 430, 329 (2004).\n7J. E. Losby, et al. , Science 350, 798 (2015).\n8E. M. Chudnovsky and D. A. Garanin, Phys. Rev. Lett.\n93, 257205 (2004).\n9A. A. Kovalev, G. E. W. Bauer, and A. Brataas, Phys.\nRev. Lett. 94, 167201 (2005).\n10M. Matsuo, J. Ieda, K. Harii, E. Saitoh, and S. Maekawa,\nPhys. Rev. B 87, 180402(R) (2013).\n11F. Ke\u000ber and C. Kittel, Phys. Rev. 85, 329 (1952).\n12R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Phys. Rev.\nLett. 113, 057601 (2014).\n13F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).\n14N. Papanicolaou, Phys. Rev. B 51, 15062 (1995).\n15A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev,\nA. Kirilyuk, and Th. Rasing, Nat. Phys. 5, 727 (2009).\n16H. V. Gomonay, R. V. Kunitsyn, and V. M. Loktev, Phys.Rev. B 85, 134446 (2012).\n17V. G. Baryakhtar, B. A. Ivanov, M. V. Chetkin, and S.\nN. Gadetskii, Dynamics of Topological Magnetic Solitons\n(Springer, Berlin, 1994).\n18E. Fradkin, Field Theories of Condensed Matter Physics ,\n(Cambridge University Press, Cambridge, 2013).\n19A. Auerbach, Interacting Electrons and Quantum Mag-\nnetism , (Springer, Berlin, 1994).\n20W. Greiner, Classical Mechanics , (Springer-Verlag, New\nYork, 2003).\n21H. Kleinert, Path Integrals in Quantum Mechanics, Statis-\ntics, Polymer physics, and Financial Markets , (World Sci-\nenti\fc, Singapore, 2009).\n22The full form of the anisotropy should be\f\fM\u0001ek\f\f2+\f\fN\u0001ek\f\f2where Mis the magnetization vector. But in\nderiving the e\u000bective action of N,e.g., see Ref. 18, the\n\frst term is negligible since K\u001cJ.\n23The temperature-dependent factor \u0002( T) is derived in the\nadiabatic limit where the staggered \feld is in thermal equi-\nlibrium, so using it here for the non-adiabatic correction is\nnot exact, but rather an approximation.\n24H. V. Gomonay, S. V. Kondovych, and V. M. Loktev, L.\nTemp. Phys. 38, 633 (2012).\n25D. A. Garanin and E. M. Chudnovsky, Phys. Rev. B 92,\n024421 (2015)."    },    {        "title": "1611.06599v3.The_effect_of_dynamical_compressive_and_shear_strain_on_magnetic_anisotropy_in_low_symmetry_ferromagnetic_film.pdf",        "content": "arXiv:1611.06599v3  [cond-mat.mes-hall]  3 May 2017The effect of dynamical compressive and shear\nstrain on magnetic anisotropy in low symmetry\nferromagnetic film\nT. L. Linnik,1V. N. Kats,2J. J¨ ager,3A. S. Salasyuk,2\nD. R. Yakovlev,2,3A. W. Rushforth,4A. V. Akimov,4\nA. M. Kalashnikova,2M. Bayer,2,3and A. V. Scherbakov2\n1Department of Theoretical Physics, V. E. Lashkaryov Instit ute of\nSemiconductor Physics, 03028 Kyiv, Ukraine\n2Ioffe Institute, Russian Academy of Sciences, 194021 St. Pet ersburg, Russia\n3Experimentelle Physik 2, Technische Universit¨ at Dortmun d, 44221 Dortmund,\nGermany\n4School of Physics and Astronomy, University of Nottingham, Nottingham NG7\n2RD, United Kingdom\nE-mail:scherbakov@mail.ioffe.ru\nMarch 6, 2017\nAbstract. Dynamical strain generated upon excitation of a metallic fil m by\na femtosecond laser pulse may become a versatile tool enabli ng control of\nmagnetic state of thin films and nanostructures via inverse m agnetostriction\non a picosecond time scale. Here we explore two alternative a pproaches to\nmanipulate magnetocrystalline anisotropy and excite magn etization precession\nin a low-symmetry film of a magnetic metallic alloy galfenol ( Fe,Ga) either\nby injecting picosecond strain pulse into it from a substrat e or by generating\ndynamical strain of complex temporal profile in the film direc tly. In the former\ncase we realize ultrafast excitation of magnetization dyna mics solely by strain\npulses. In the latter case optically-generated strain emer ged abruptly in the film\nmodifies its magnetocrystalline anisotropy, competing wit h heat-induced change\nof anisotropy parameters. We demonstrate that the opticall y-generated strain\nremains efficient for launching magnetization precession, w hen the heat-induced\nchanges of anisotropy parameters do not trigger the precess ion anymore. We\nemphasize that in both approaches the ultrafast change of ma gnetic anisotropy\nmediating the precession excitation relies on mixed, compr essive and shear,\ncharacter of the dynamical strain, which emerges due to low- symmetry of the\nmetallic film under study.\nPACS numbers: 75.78.Jp, 75.30.Gw, 75.80.+q, 75.50.Bb\nKeywords : ultrafast laser-induced magnetization dynamics, picosecond mag neto-\nacoustics, magnetic anisotropy, inverse magnetostriction, optic ally-induced strainThe effect of dynamical compressive and shear strain on magne tic anisotropy in low symmetry ferromagnetic film 2\n1. Introduction\nThe lattice symmetry sets up the common and im-\nportant feature of all crystalline magnetically-ordered\nmaterials: the magnetocrystalline anisotropy (MCA).\nThe MCA determines such parameters of a magnetic\nmedium as magnetization direction, magnetic reso-\nnances frequencies, coercive fields etc. Since the MCA\nrelates directly to the lattice, applying stress to a mag-\nnetic medium allows modifying static and dynamical\nmagnetic properties of the latter. This effect known as\ninverse magnetostriction or Villary effect was discov-\nered at the end of 19th century and is widely used in\nboth fundamental research and applications. Inverse\nmagnetostriction plays a tremendous role in scaling\nmagnetic devices down, e.g. in tailoring the MCA pa-\nrameters of nanometer ferromagnetic films by properly\nchosen lattice mismatch with substrate and providing\nsensing mechanism in microelectromechanical systems\n(MEMS). Recently, the importance of inverse magne-\ntostriction was also recognized in the emerging field of\nultrafast magnetism focused at manipulating magnetic\nstate of matter on the (sub-) picosecond time scale [1].\nMagnetization control via the inverse magne-\ntostriction at ultrashort timescale is based on the tech-\nniques of generating picosecond strain pulses in solids\ndeveloped in picosecond ultrasonics [2]. When an\nopaque medium is subjected to a pico- or femtosecond\nlaserpulse, the lightabsorptionandthe followingrapid\nincrease of lattice temperature induces thermal stress\nin a surface region. This results in generation of a pi-\ncosecond strain pulse with spatial size down to 10nm\nand broad acoustic spectrum (up to 100 GHz), which\npropagates from excited surface as coherent acoustic\nwavepacket. It has been demonstrated experimentally,\nthat injection of such a strain pulse into a thin film of\nferromagnet can modify the MCA and trigger the pre-\ncessionalmotion ofmagnetization [3]. This experiment\nhas initiated intense experimental and theoretical re-\nsearch activities [3, 4, 5, 6, 7, 8, 9, 10, 11] in what is\nnow referred to as ultrafast magnetoacoustics.\nHigh interest to ultrafast magnetoacoustics is\ndriven by a number of features, which are specific\nto interaction between coherent acoustic excitation\nand magnetization, and do not occur when other\nultrafast stimuli are employed. The wide range of\ngenerated acoustic frequencies overlaps with the range\nof magnetic resonances in the magnetically-ordered\nmedia. Furthermore, thin films and nanostructures\npossess specific magnetic and acoustic modes, and\nmatching their frequencies and wavevectors may\ndrastically increase their coupling efficiency [4].\nFinally, there is a well-developed theoretical and\ncomputational apparatus for high-precision modeling\nof spatial-temporal evolution of the strain pulse and\nrespective modulation of the MCA [12, 13, 14].These advantages, however, may be exploited only if\nthe strain-induced effects are not obscured by other\nprocesses triggered by direct ultrafast laser excitation.\nGenerally, there are two main approaches to\nsingle-out the strain-induced impact on magnetization.\nThe first one is the spatial separation, when the\nresponse of magnetization to the strain pulses is\nmonitored at the sample surface opposite to the one\nexcited by a laser pulse. It has been used in a number\nofexperiments with variousferromagneticmaterials[3,\n4, 5, 6]. The irrefutable advantage of such an approach\nis that the laser-induced heating of a magnetic\nmedium is eliminated due to spatial separation of the\nlaser-impact area and the magnetic specimen. An\nalternative approach employs the spectral selection\ninstead, when the initially generated strain with\nbroad spectrum is converted into monochromatic\nacoustic excitation. In this case the efficiency of its\ninteraction with ferromagnetic material is controlled\nby external magnetic field, which shifts the magnetic\nresonance frequency. This approach was realized in\nthe experiments with ferromagnetic layer embedded\ninto acoustic Fabry-Perot resonator [9] and by means\nof lateral patterning of ferromagnetic film or optical\nexcitation resulting in excitation of surface acoustic\nwaves [7, 10, 11].\nVery recently we have demonstrated that the\nstrain-induced impact on the MCA can be reliably\ntraced even in a ferromagnetic film excited directly\nby a femtosecond laser pulse, despite the complexity\nof the laser-induced electronic, lattice and spin\ndynamics emerging in this case [15]. Here we present\noverview of our recent experimental and theoretical\nstudies of the ultrafast strain-induced effects in\nferromagnetic galfenol films, where the dynamical\nstrain serves as a versatile tool to control MCA.\nMagnetization precession serves in this experiments\nas the macroscopic manifestation of ultrafast changes\nof MCA. We demonstrate the modulation of the\nMCAand the correspondingresponseofmagnetization\nunder two different experimental approaches, when\nthestrain pulses are injected into the film from\nthe substrate, and when the strain with a step-like\ntemporal profile is optically generated directly in a\nferromagnetic film. In the case of direct optical\nexcitation we also compare the strain-induced change\nof MCA to the conventional change of anisotropy via\noptically-induced heating emerging, and demonstrate\nthat these two contributions can be unambiguously\ndistinguished and suggest the regimes at which either\nof them dominates. In these studies we have utilized\nthe specific MCA of low-symmetry magnetostrictive\ngalfenol film grown on a (311)-GaAs substrate, which\nenables generation of dynamical strain of mixed,\ncompressive and shear character, as compared to theThe effect of dynamical compressive and shear strain on magne tic anisotropy in low symmetry ferromagnetic film 3\npure compressive strain in high-symmetry structures.\nThe paper is organized as follows. In Sec.2\nwe describe the sample under study and three\nexperimental geometries which enable us to investigate\nultrafast changes of magnetic anisotropy. In Sec.3\nwe describe phenomenologically magnetocrystalline\nanisotropyofthe(311)galfenolfilmandconsiderhowit\ncan be altered on an ultrafast timescale. The following\nSec.4 is devoted to generation of dynamical strain in\nmetallic films of low symmetry. In Secs.5,6 we present\nexperimental results and analysis of the magnetization\nprecession triggered by purely acoustical pump and by\ndirect optical excitation and demonstrate that even\nin the latter case optically-generated strain may be\na dominant impact allowing ultrafast manipulation of\nthe MCA.\n2. Experimental\n2.1. Sample\nFilm of a galfenol alloy Fe 0.81Ga0.19(thickness\ndFeGa=100nm) was grown on the (311)-oriented GaAs\nsubstrate (dGaAs=100µm) (Fig.1(a)). As was shown\nin our previous works [3, 6], the magnetic film of\nthis content and thickness of 100nm facilitates a\nstrong response of the magnetization to picosecond\nstrain pulses. The film was deposited by DC\nmagnetron sputtering at a power of 22W in an Ar\npressure of 1.6mTorr. The GaAs substrate was\nfirst prepared by etching in dilute hydrochloric acid\nbefore baking at 773K in vacuum. The substrate\nwas cooled down to 298 K prior to deposition.\nDetailed x-ray diffraction studies [16] revealed that\nthe film is polycrystalline, and the misorientation of\ncrystallographic axes of crystallites, average size of\nwhich was of a few nanometers, was not exceeding a\nfew degrees. Therefore, the studied film can be treated\nas the single crystalline one. The equilibrium value\nof the saturation magnetization is Ms=1.59T [17].\nThe SQUID measurements confirmed that the easy\nmagnetization axis is oriented in the film plane along\nthe [0¯11] crystallographic direction ( y-axis). In our\nexperiments external DC magnetic field Bwas applied\nin the sample plane along the magnetization hard axis,\nwhich lies along [ ¯233] crystallographic direction ( x-\naxis). In this geometry magnetization Morients along\nthe applied field if the strength of the latter exceeds\nB=150mT. At lower field strengths magnetization is\nalong an intermediate direction between the x- andy-\naxes.\n2.2. Experimental techniques\nThree experimental geometries were used in order\nto explore the impact of dynamical strain on theGaAs FeGa\nGaAs FeGaGaAs FeGa Al\noptical \npumpacoustical \npump\nprobeprobe\nprobeoptical \npump\noptical \npump(b)\n(c)\n(d)\n(a)\nzxB\nBB\nzxzx\nB\nFigure 1. (Color online) (a) Schematic presentation of the\ngalfenol film grown on the (311) GaAs substrate. x′-,y′-\nandz′-axes are directed along the crystallographic [100], [010]\nand [001] axes, respectively. DC magnetic field Bis applied\nalong the [ ¯233] crystallographic direction, which is the hard\nmagnetization axis. (b-d) Experimental geometries. (b) Th e\noptical pump pulses excite 100nm thick Al film on the back of\nthe GaAs substrate, thus generating strain pulses injected into\nthe substrate. They act as the acoustical pump triggering th e\nmagnetization precession in the galfenol film. The precessi on is\ndetected by monitoring the rotation of polarization plane o f the\nprobe pulses reflected from the galfenol film. (c) The optical\npump pulses excite the galfenol film directly. The propagati ng\nstrain pulses are detected by monitoring polarization rota tion\nfor the probe pulses, which penetrate into the GaAs substrat e.\n(d) The optical pump pulses excite the galfenol film directly ,\nincreasing the lattice temperature and generating the dyna mical\nstrain in the film. Excited magnetization precession is dete cted\nby monitoring the rotation of polarization of the probe puls es\nreflected from the galfenol film. Experiment (b) was performe d\natT=20K, experiments (c,d) were performed at room\ntemperature.\nMCA of the galfenol film. First, the experiments\nwere performed with the dynamical strain being the\nonly stimulus acting on the galfenol film (Fig.1(b)).\nA 100-nm thick Al film was deposited on the back\nside of the GaAs substrate and was utilized as an\noptoacoustic transducer to inject picosecond strain\npulses into the substrate [18]. The 100-fs optical\npump pulses with the central wavelength of 800nm,\ngeneratedby a Ti:sapphire regenerativeamplifier, were\nincident on the Al film inducing rapid increase of its\ntemperature. As a result, as discussed in detail in\nSec.4, the picosecond strain pulses were injected into\nthe GaAs substrate. These pulses propagated through\nthe substrate, reached the film (Fe,Ga) film, modified\nits MCA and triggered the magnetization precession.\nThe probe pulses split from the same beam\nwere incident on the (Fe,Ga) film at the angle close\nto 0, and the time-resolved polar magneto-optical\nKerr effect (TRMOKE) was measured. In this\nexperimental geometry, TRMOKE rotation angle βK\nis directly proportional to the out-of-plane deviation of\nmagnetization ∆ Mzinduced by a pump:\n∆βK(t) = [√ε0(ε0−1)]−1χxyz∆Mz(t), (1)The effect of dynamical compressive and shear strain on magne tic anisotropy in low symmetry ferromagnetic film 4\nwhereε0is the diagonal dielectric permittivity tensor\ncomponent of (Fe,Ga) at the probe wavelength,\nχxyzis the magneto-optical susceptibility at the\nsame wavelength, which enters off-diagonal dielectric\npermittivity component as iεxy=iχxyzMz[21]. By\nnormalizing TRMOKE rotation by the static one at\nsaturation ( βs\nK∼Ms), one gets the measure of\ndeviation ofthe magnetizationout ofthe sampleplane,\n∆Mz(t)/Ms= ∆βK(t)/βs\nK. These experiments were\nperformed at T=20K. The choice of low temperature\nin this experiment was dictated by the fact, that this\nprevents attenuation of higher frequency components\nof the strain pulses in the GaAs substrate [22], thus\nallowing excitation of precession with high frequency\nin relatively high applied magnetic fields.\nSecond and the third types of experiments\n(Fig.1(c,d))wereconductedinthegeometry,wherethe\n(Fe,Ga) film was directly excited by the optical pump\npulses. In this geometrythere weretwocontributionto\nthe changeofMCA: (i) direct modification ofthe MCA\ndue to heating [23] and (ii) inverse magnetostrictive\neffects (See Sec.3 for details). In these experiments\nwe used 170-fs pump and probe pulses of the 1030-\nnm wavelength generated by the Yb:KGd(WO 4)2\nregenerative amplifier. These experiments were\nperformed at room temperature.\nIn the second geometry the probe pulses were\nincident onto the back side of the GaAs substrate\n(Fig.1(c)). Since the probe pulses wavelength is well\nbelow the GaAs absorption edge, it penetrated the\nsubstrate and reached the magnetic film. Thus, here\nwe were able to probe optically excited dynamics of\nthe magnetization of the (Fe,Ga) film. Additionally,\nthisexperimentalgeometryenablesonetodetectstrain\npulses injected into the substrate from the film with\nthe velocity sj, wherejdenotes the particular strain\npulses polarization. Upon propagation through GaAs\nthese pulses modified its dielectric permittivity via\nphotoelastic effect. The intensity and the polarization\nof the probe pulses were therefore modified in the\noscillating manner [18], with the frequency\nνj= 2sj√ε0λ−1\npr, (2)\nwhereλpris the probe wavelength, ε0is the dielectric\npermittivity of GaAs, and the angle of incidence for\nthe probe pulses is taken to be 0. These oscillations\nareoften referredto asBrillouin oscillations. The main\npurpose of this experiment was to confirm generation\nof dynamical strain upon excitation of the (Fe,Ga) film\nby optical pump pulses.\nThird type of experiments was performed in the\nconventional optical pump-probe geometry, when both\noptical pump and probe pulses were incident directly\non the galfenol film (Fig.1(d)). This is the main\nexperiment in our study, which demonstrates howvarious contributions to the optically-induced MCA\nchange can be distinguished and separated.\n3. Thermal and strain-induced control of the\nmagnetic anisotropy in (311) galfenol film\nThe magnetic part of the normalized free energy\ndensity of the single crystalline galfenol film FM=\nF/Msgrown on the (311)-GaAs substrate (Fig.1(a))\ncan be expressed as\nFM(m) =−m·B+Bdm2\nz (3)\n+K1/parenleftbig\nm2\nx′m2\ny′+m2\nz′m2\ny′+m2\nx′m2\nz′/parenrightbig\n−Kum2\ny\n+b1(ǫx′x′m2\nx′+ǫy′y′m2\ny′+ǫz′z′m2\nz′)\n+b2(ǫx′y′mx′my′+ǫx′z′mx′mz′+ǫy′z′my′mz′),\nwherem=M/Ms. Here for a sake of convenience\nZeeman, shape, and uniaxial anisotropy terms are\nwritten in the coordinate frame associated with the\nfilm, i.e the z-axis is directed along the sample normal.\nCubic anisotropy term and the magneto-elastic terms\nare written in the frame given by the crystallographic\naxesx′y′z′(Fig.1(a)). Strain components ǫijare\nconsidered to be zero at equilibrium. Corresponding\nequilibrium orientation of magnetization is given by\nthe direction of an effective magnetic field expressed as\nBeff=−∂FM(m)\n∂m. (4)\nRapid change of any of the terms in Eq.(3) under\nan external stimulus may result in reorientation of the\neffective field (4). This and thus trigger magnetization\nprecession, which can be described by the Landau-\nLifshitz equation [24, 25]:\ndm\ndt=−γ·m×Beff(t), (5)\nwhereγis the gyromagnetic ratio. This precession\nplays a two-fold role. On the one hand, magnetization\nprecession triggered by an ultrafast stimulus is in\nitself an important result attracting a lot of attention\nnowadays. On the other hand, magnetization\nprecession is the macroscopical phenomenon, which\ncan be easily observed in conventional pump-probe\nexperiments and, at the same time, allows getting\na insight into the complex microscopical processes\ntriggered by various ultrafast stimuli.\nWe exclude from the further discussion the\nultrafast laser-induced demagnetization [26], which\nmay trigger the magnetization precession [27] due\nto the decrease of the demagnetizing field µ0Ms/2.\nThis contribution to the change of the effective field\norientation is proportional to z-component of M\nat equilibrium, which is zero in the experimental\ngeometry discussed here, with magnetic field applied\nin the film plane. Thus, we focus on the effects relatedThe effect of dynamical compressive and shear strain on magne tic anisotropy in low symmetry ferromagnetic film 5\nto the change of the MCA solely and consider two\nmechanisms.\nFirst mechanism allowing ultrafast change of the\nMCA relies on heat-induced changes of the parameters\nK1andKuin Eq.(3). This phenomenon is inherent to\nvarious magnetic metals [23], semiconductors [29], and\ndielectrics [30]. In metallic films absorption of laser\npulse results in subpicosecond increase of electronic\ntemperature Te. Subsequent thermalization between\nelectrons and lattice takes place on a time scale\nof several picoseconds and yields an increase of the\nlattice temperature Tl. Magnetocrystalline anisotropy\nof a metallic film, and galfenol in particular, is\ntemperature-dependent [28]. Therefore, laser-induced\nlattice heating results in decrease of MCA parameters.\nImportantly, this mechanism is expected to be efficient\nif magnetization is not aligned along the magnetic\nfield [31, 32]. This can be realized by applying\nmagnetic field of moderate strength along the hard\nmagnetization axis. Otherwise, decrease of K1,uwould\nnot tiltBeffalready aligned along B.\nSecond mechanism relies on inverse magnetostric-\ntion. As it follows from Eq.3, dynamical strain ˆ ǫin-\nduced in a magnetic film can effectively change the\nMCA. Such dynamical strain can be created in a film\neither upon injection from the substrate [3], or due\nto the thermal stress induced by rapid increase of the\nlattice temperature by optical pulse. It is important\nto emphasise, that, in contrast to the heat-induced\nchange of the magnetocrystallineanisotropyconstants,\nthestrain-induced mechanism can be efficient even if\ntheBeffis aligned along B, if the symmetry of the\nfilm and the polarization of the dynamical strain are\nproperly chosen.\n4. Optical generation of the dynamical\ncompressive and shear strain in a metallic film\non a low-symmetry substrate\n4.1. Optical generation of the dynamical strain\nIncrease of the electronic Teand lattice Tlof a metallic\nfilm excited by a femtosecond laser pulse is described\nby the coupled differential equations:\nCe∂Te\n∂t=κ∂2Te\n∂z2−G(Te−Tl)+P(z,t);\nCl∂Tl\n∂t=−G(Tl−Te), (6)\nwhereP(z,t) =I(t)(1−R)αexp(−αz) is the absorbed\noptical pump pulse power density, with I(t) describing\nthe Gaussian temporal profile, αis the absorption\ncoefficient, Ris the reflection Fresnel coefficient. Ce=\nAeTeandClare the specific electronic and lattice heat\ncapacities, respectively; κ- the thermal conductivity,\nG- the electron-phonon coupling constant, consideredto be temperature independent. Tlstands for the\nlattice temperature. Heat conduction to the substrate\nis usually much less than the one within the film\nand, thus, is neglected. The boundary conditions are\n∂Te/∂z= 0 atz=0, andTe=Tl=Tatz=∞, whereT\nis the initial temperature.\nLattice temperature increase sets up the thermal\nstress, which in turn leads to generation of dynamical\nstrain [18, 13, 14]. Details of this process are\ndetermined by the properties of the metallic film\nand of the interface between the metallic film and\nthe substrate. As a generalization, we consider the\nstrain mode with the polarization vector ejand the\namplitude u0,j. Following the procedure described\nin [14] for a high-symmetry film we express the\ndisplacement amplitude in the frequency domain as\nδTe(z,ω) =α(1−R)\nκI(ω)\nα2−p2\nT/bracketleftbigg\ne−αz+α\npTe−pTz/bracketrightbigg\n; (7)\nδTl(z,ω) =δTe(z,ω)\n1−iωClG−1; (8)\nu0,j(z,ω) =σj/parenleftBigg\ne−αz\nα2+k2\nj−e−pTz\np2\nT+k2\nj\n+ekjz\n2ikj/bracketleftbigg1\nα+ikj−1\npT+ikj/bracketrightbigg\n+e−kjz\n2ikj/bracketleftbigg1\nα−ikj+1\npT−ikj/bracketrightbigg/parenrightbigg\n+Ajeikjz+Bje−ikjz, (9)\nwhere we introduced the parameters\nσj=ej,z\nρs2\njβCl\n1−iωClG−1α2(1−R)I(ω)\nκ(α2−p2\nT),\npT=/radicalBigg\n−iωCe\nκ/parenleftbigg\n1+ClC−1e\n1−iωClG−1/parenrightbigg\n, (10)\nkj=ωs−1\nj,Re(pT)>0.\nHereβis Gruneisen parameter, ρis the galfenol\ndensity. The constants AjandBjare determined from\nthe boundary condition at the free surface z= 0 and\nat the (311)-(FeGa)/GaAs interface.\nFrom Eq.(9) it can be seen that thermal stress\ninduces two contributions to the strain in the\nmetallic film. First one is maximal at the film\nsurface and decays exponentially along z, which\nis shown schematically in Figs.1(b-d). In fact,\nit closely follows spatial evolution of the lattice\ntemperature Tlin Eq.(8). In the time domain,\nthis contribution emerges on a picosecond time scale\nfollowing lattice temperature increase and decays\nslowly towards equilibrium due to the heat transfer to\nthe substrate. Therefore, on the typical time scale of\nexperiment on ultrafast change of the MCA, i.e ∼1ns,\nthis contribution can be considered as the step-like\nstrain emergence . Second contribution describes theThe effect of dynamical compressive and shear strain on magne tic anisotropy in low symmetry ferromagnetic film 6\npicosecond strain pulse propagatingawayfromthe film\nsurface along z[20].\n4.2. Injection of compressive and shear dynamical\nstrain pulses into (311)-galfenol film\nFirst we consider the scenario illustrated in Fig.1(b).\nThe Al film serving as the optoacoustic transducer, is\npolycrystalline and, thus, acoustically isotropic. Thus,\nlongitudinal (LA) strain is generated due to optically-\ninducedthermalstress. Itspolarizationvectoris eLA=\n(0,0,1) and the amplitude is u0,LA. Corresponding\nstraincomponentis ǫLA\nzz=eLA,z∂u0,LA/∂z. Thisstrain\nis purely compressive/tensile. Due to mode conversion\nat the interface shear strain may be also generated,\nbut the efficiency of this process is low [33]. After\ntransmission of the strain pulse through the interface\nbetween elastically isotropic Al film and anisotropic\nlow-symmetry single crystalline (311)-GaAs substrate,\ntwo strain pulses emerge, quasi-longitudinal (QLA)\nand quasi-transversal (QTA), with the polarization\nvectorseQLA=(0.165,0,0.986) and eQTA=(0.986,0,-\n0.165), propagating further to the substrate [20].\nImportantly, both QLA and QTA strain pulses have\nsignificant shear components. Expressions for the\ncorresponding amplitudes are found in [20] by taking\ninto account interference between LA and TA modes\nwithin the film and multiple reflections and mode\nconversion at the interface. QLA and QTA pulses\ninjected thus into GaAs substrate propagatewith their\nrespective sound velocities.\nUpon reaching magnetic (Fe,Ga) film these strain\npulses can trigger the magnetization precession [3,\n6], by modifying magneto-elastic terms in Eq.(3).\nSince the QTA and QLA pulse velocities in the\n100µm (311)-GaAs substrate are sQTA=2.9km·s−1\nandsQLA=5.1km·s−1[19, 20], they reach(Fe,Ga) film\nafter 35 and 20ns, respectively, and thus, their impact\non the magnetic film can be separated in time. Strictly\nspeaking, polarization vectors of QL(T)A in (Fe,Ga)\nand in GaAs differ, and transformation of the strain\npulses upon crossing GaAs/(Fe,Ga) interface should\nbe taken into account. However, since the mismatch\nis rather small and both QL(T)A strain pulses remain\npolarized in the xzplane, we neglect it in the analysis.\nTherefore, in the experimental geometry, shown in\nFig.1(b), propagating strain pulses (9) are employed\nto control magnetization.\n4.3. Generation of compressive and shear dynamical\nstrain pulses in (311)-galfenol film\nBy contrast to polycrystalline Al film, in the single\ncrystalline (Fe,Ga) film on the (311)-GaAs substrate\nthe elasticanisotropyplaysessentialrolealreadyat the\nstage of the strain generation [34]. Two strain compo-nentsǫxzandǫzzariseduetocouplingofthermalstress\nto QLA and QTA acoustic waves. Their polarizations\nareeQLA=(0.286,0,0.958) and eQTA=(0.958,0,-0.286)\n[15] in the film coordinate frame xyz(Fig.1(a)). Cor-\nresponding strain components can be found as\nǫQL(T)A\nxz= 0.5eQL(T)A,x∂u0,QL(T)A\n∂z\nǫQL(T)A\nzz=eQL(T)A,z∂u0,QL(T)A\n∂z, (11)\ni.e. the generated strain is of mixed, compressive\nand shear, character. Both step-like emergence of the\nstrain and propagating strain pulses can modify MCA.\nPossible contribution from this step-like emergence\nof the strain to the change of MCA was pointed\nout in [35], however, no detailed consideration was\nperformed allowing to confirm feasibility of this\nprocess. Importantly, since the step-like emergence\nof the strain closely follows temporal and spatial\nevolution of the lattice temperature, distinguishing\ntheir effect on the magnetic anisotropy can be\nambiguous. We note that in the case of optically\nexcited (Fe,Ga) film the QLA and QTA strain pulses\nwill be also injected into GaAs, and can be detected\nemploying the scheme shown in Fig.1(c).\n5. Magnetization dynamics in the (311)\ngalfenol film induced by picosecond strain\npulses\nFirst we examine excitation of the magnetization\nprecession by dynamical strain only, which is realized\nin the experimental geometry shown in Fig.1(b). In\nFig.2(a) we present changes of the probe polarization\nrotation measured as a function of pump-probe time\ndelaytafter QLA or QTA strain pulse arrives to the\ngalfenol film. Time moment t=0 for each shown trace\ncorresponds to the time required for either QLA or\nQTA pulse to travel through the 100 µm thick GaAs\nsubstrate, and was verified by monitoring reflectivity\nchange [6]. As one can see, both the QLA and QTA\npulses excite oscillations of the probe polarization.\nTwolinesareclearlyseenintheFastFourierTransform\n(FFT) spectra of the time traces (Fig.2(b)) separated\nby few GHz. Frequencies of both lines change with\nthe applied field (Fig.2(c)), thus confirming that the\nobservedoscillationsoftheprobepolarizationoriginate\nfrom the magnetization precession triggered by QLA\nand QTA strain pulses. The character of the field\ndependence of ν(Fig.2(c)) corresponds to the one\nexpected for the geometry, when the external magnetic\nfield is applied along the magnetization hard axis.\nPresence of two field dependent frequencies in the\nFFT spectra can be attributed to the excitation of\ntwo spin wave modes, which is one of the signatures\nof the magnetization precession excited by picosecondThe effect of dynamical compressive and shear strain on magne tic anisotropy in low symmetry ferromagnetic film 7\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48\n/s48 /s52/s48/s48 /s56/s48/s48 /s49/s50/s48/s48/s48/s50/s48/s52/s48/s40/s99/s41/s40/s98/s41\n/s81/s76/s65 /s51/s48/s48/s32/s109/s84/s53/s48/s48/s32/s109/s84/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s100/s101/s118/s105/s97/s116/s105/s111/s110/s32/s77\n/s122/s40/s116/s41/s47/s77\n/s115/s32/s40/s97/s114/s98/s46/s117/s110/s46/s41\n/s84/s105/s109/s101/s32/s100/s101/s108/s97/s121/s32/s116/s32/s40/s112/s115/s41/s66/s61/s56/s48/s48/s32/s109/s84\n/s81/s84/s65/s81/s76/s65\n/s81/s84/s65/s81/s76/s65\n/s81/s84/s65/s40/s97/s41\n/s84/s61/s50/s48/s32/s75\n/s112/s117/s109/s112/s58\n/s66/s61/s53/s48/s48/s32/s109/s84/s70/s70/s84/s32/s65/s109/s112/s108/s105/s116/s117/s100/s101/s32/s40/s97/s114/s98/s46/s32/s117/s110/s46/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32 /s32/s40/s71/s72/s122 /s41/s81/s76/s65/s45/s112/s117/s109/s112\n/s81/s84/s65/s45/s112/s117/s109/s112/s80/s114/s101/s99/s101/s115/s115/s105/s111/s110/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s32/s40/s71/s72/s122/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s66/s32/s40/s109/s84/s41/s32/s32/s81/s76/s65/s45/s112/s117/s109/s112/s32/s32 /s32/s81/s84/s65/s45/s112/s117/s109/s112\n/s32\n/s49 /s32/s32/s32/s32/s32/s32/s32/s32 /s32\n/s49 \n/s32\n/s50 /s32/s32/s32/s32/s32/s32/s32/s32 /s32\n/s50 \nFigure 2. (Color online) (a) Probe polarization rotation vs.\ntime delay tmeasured in the geometry shown in Fig.1(b)\nat various values of the magnetic field. t=0 is the moment\nof arrival of the QLA or QTA pulse to the galfenol film,\nand corresponds to 20 and 35ns after the excitation with the\noptical pump pulse, respectively. (b) FFT spectra of the\ntime delay dependence measured at B=500mT. The two lines\nseen in each spectrum correspond to two spin-wave modes\n(see text for details). (c) Frequency of the probe polarizat ion\noscillations caused magnetization precession excited by Q LA\n(closed symbols) and QTA (open symbols) pulses. Optical pum p\nfluence was of P=40mJ·cm−2. Note, that in (a) the curves are\nshifted along the vertical axis for a sake of clarity.\nacoustic pulses. As discussed in details in [4] excitation\nof several spin waves is enabled by the broad spectrum\nof the strain pulses and is governed by the boundary\nconditions in the thin film.\nBoth the QLA and QTA pulses contain com-\nponentsǫxzandǫzz(11). The QLA (QTA)\nstrain pulse enters the magnetic film and propagates\nthough it with the sound velocity of sQLA=6.0km·s−1\n(sQTA=2.8km·s−1). Upon propagation it contributes\nto the change of the magneto-elastic term in the free\nenergy in Eq.(3), modifying the MCA of the film, and\ncausing effective magnetic field Beffto deviate from\nits equilibrium. As a result, magnetization starts to\nmove away from its equilibrium orientation following\ncomplex trajectory [3]. QL(T)A strain pulse leaves the\nfilmafter2dFeGas−1\nQL(T)A,i.e. 33and70ps,respectively,\nandBeffreturns to its equilibrium value, while mag-\nnetizations relaxes towards Beffprecessionally on the\nmuch longer nanosecond time scale.\nAs seen from Fig.2(a), amplitude of the magneti-\nzation precession excited by QLA phonons is higher\nthan of that excited by QTA phonons. This is in\nagreement with experimental and theoretical results\non propagationof QLA and QTA phonons through the\n(311)-GaAs substrate [20], which showed that the am-\nplitude of the displacement associated with the QTA\npulses is smaller by a factor of ∼5 than that of QLA,\nwhile the magnetoelastic coefficients for the shear andcompressive strain are the same in galfenol.\nThus, the experiment on excitation of the (311)-\n(Fe,Ga) film by the picosecond strain pulses clearly\ndemonstrates that dynamical strain effectively excites\nthe magnetization precession in the film in the fields\nupto 1.2T, i.e. when the equilibrium magnetization is\nalready along the applied magnetic field. We note that\nhere we reported the magnetization excitation in the\nparticular geometry, when the magnetic field is applied\nalong the magnetization hard axis. Previously some of\nthe authors also demonstrated analogous excitation in\n(Fe,Ga) with the field applied in the (311) plane at\n45oto [¯233] direction, as well as in the field applied\nalong the [311] axis [6]. It has been also shown\nthat all the features of the excitation observed at low\ntemperature remain valid at room temperature as well.\nThus, reported here results obtained at T=20K can\nbe reliably extrapolated to the room temperature, at\nwhich the direct optical excitation of the precession in\nthe galfenol film was studied.\n6. Magnetization dynamics in (311) galfenol\nfilm induced by direct optical excitation\nWhileintheexperimentsdescribedinSec.5picosecond\nstrain pulses are the only stimulus driving the\nmagnetization precession, the processes triggered by\ndirect optical excitation of a metallic magnetic film\nare more diverse, and may contribute to both strain-\nrelated and other driving forces (see Sec.3). First,\nin order to confirm generation of dynamical strain in\nthe optically-excited galfenol film we have detected\npropagatingQLA and QTA strain pulses by measuring\nthe polarization rotation for the probe pulses incident\nonto the back side of the (311)-(Fe,Ga)/GaAs sample\n(Fig.1(c)). Fig.3(a) shows the time traces obtained at\nvarious magnetic fields. There are several oscillating\ncomponents clearly present, as can be seen from the\nFourier spectra in Fig.3(b). The field dependences\nof these frequencies are shown in Fig.3(c). The\nlines atνQTA=20GHz and νQLA=35GHz are field-\nindependent and are attributed to the Brillouin\noscillations caused by the QTA and QLA strain pulses\n(2), respectively, propagating away from the galfenol\nfilm towards the back side of the GaAs substrate with\nthe velocities sQTA<sQLA.\nA line in the FFT spectra marked in Fig.3(b)\nasMzand possessing the field dependent frequency ν\ncorresponds to the optically triggered precession of the\nmagnetization in the galfenol film. This experiment,\ntherefore, confirms concomitant generation of the\ndynamical strain and excitation of the magnetization\nprecession in the optically excited galfenol film.\nThe mechanism behind the precession excitation is,\nhowever, more intricate than in the case of injectionThe effect of dynamical compressive and shear strain on magne tic anisotropy in low symmetry ferromagnetic film 8\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48 /s49/s50/s48/s48 /s49/s52/s48/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48/s48/s49/s48/s50/s48/s51/s48/s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53\n/s40/s99/s41/s40/s98/s41\n/s66/s61/s53/s48/s48/s32/s109/s84/s81/s76/s65\n/s32/s32/s70/s70/s84/s32/s65/s109/s112/s108/s105/s116/s117/s100/s101/s32/s40/s97/s114/s98/s46/s32/s117/s110/s46/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32 /s32/s40/s71/s72/s122/s41/s81/s84/s65/s77\n/s122 /s40/s97/s41\n/s66/s61/s53/s48/s48/s32/s109/s84\n/s52/s48/s48/s32/s109/s84\n/s51/s48/s48/s32/s109/s84\n/s50/s48/s48/s32/s109/s84/s80/s114/s111/s98/s101/s32/s112/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s114/s111/s116/s97/s116/s105/s111/s110/s32\n/s75/s32/s40/s109/s100/s101/s103/s41\n/s84/s105/s109/s101/s32/s100/s101/s108/s97/s121/s32/s116/s32/s40/s112/s115/s41/s49/s48/s48/s32/s109/s84/s84/s61/s51/s48/s48/s75\n/s66/s114/s105/s108/s108/s111/s117/s105/s110/s32/s111/s115/s99/s105/s108/s108/s97/s116/s105/s111/s110/s115\n/s81 /s84 /s65 \n/s81 /s76 /s65 \n/s80/s114/s101/s99/s101/s115/s115/s105/s111/s110\n/s32\n/s49\n/s32\n/s50/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s32/s40/s71/s72/s122/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s66/s32/s40/s109/s84/s41\nFigure 3. (Color online) (a) Probe polarization rotation vs.\ntime delay tmeasured in the geometry shown in Fig.1(c)\nat various magnetic fields. (b) FFT spectrum of the time\ndelay dependence measured at B=500mT.νQLA,νQTA, and\nMzdenote the lines corresponding to the Brillouin oscillatio ns\nrelated to the QLT and QTA strain pulses, and to the\nmagnetization precession, respectively. (c) Brillouin fr equencies\nin the probe polarization oscillations related to the QTA (c losed\ncircles) and QLA (open circles) pulses, and the frequencies\nrelated to the optically excited magnetization precession\n(triangles). Optical pump fluence was of P=10mJ·cm−2.\n/s115/s105/s110/s50 /s116\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48/s48/s49/s50/s51/s52/s53/s84/s61/s51/s48/s48/s32/s75\n/s49/s53/s48/s32/s109/s84/s51/s48/s48/s32/s109/s84/s66/s61/s53/s48/s48/s32/s109/s84\n/s32/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s100/s101/s118/s105/s97/s116/s105/s111/s110/s32 /s77\n/s122/s47/s77\n/s115/s32/s40/s37/s41\n/s84/s105/s109/s101/s32/s100/s101/s108/s97/s121 /s32/s116/s32/s40/s112/s115/s41/s55/s53/s32/s109/s84/s40/s97/s41\n/s48 /s50/s53 /s53/s48 /s55/s53/s99/s111/s115/s50 /s116/s40/s98/s41/s32 /s32/s32\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s40/s99/s41/s32\n/s32/s70/s70/s84/s32/s65/s109/s112/s108/s105/s116/s117/s100/s101/s32/s40/s97/s114/s98/s46/s32/s117/s110/s46/s41/s32\n/s32/s40/s71/s72/s122/s41\nFigure 4. (Color online) (a) Probe polarization rotation vs.\ntime delay tmeasured in the geometry shown in Fig.1(d) at\nvarious values of the magnetic field. (b) The same traces\nmeasured with higher temporal resolution. (c) FFT spectra\nof the time delay dependence. Optical pump fluence was of\nP=10mJ·cm−2.\nof strain pulses in the film.\nIn order to get an insight into the problem\nof direct optical excitation of (311)-(Fe,Ga) film we\nhave performed experiment in the geometry shown in\nFig.1(d), with both pump and probe pulses incident\ndirectly on the galfenol film. Figs.4(a,b) show the\ntemporal evolution of the TRMOKE signal following\nexcitation of the sample by femtosecond laser pulses.\nFFT spectra (Fig.4(c)) contain one line with fielddependent frequency (Fig.5(a)). Thus, we observe\nexcitation of the magnetization precession. Oscillatory\ncomponent in the observedsignal can be approximated\nby the function\n∆Mz(t)\nMs=∆Mmax\nz\nMse−t/τsin(2πνt+ψ0). (12)\nAs can be seen from Figs.5(a,b) the frequency νis\nminimal and the amplitude ∆ Mmax\nz/Msis maximal\natB=150mT, i.e. when the magnetization becomes\nparallel to the external field. This is a conventional\nbehaviour, if the magnetic field is applied along\nthe magnetization hard axis. We also note that\nthe frequency vs. applied field dependences in the\ncase of strain-induced and direct optical excitation\nresembleeachother(seeFigs.2(c)and5(a)), withsome\ndeviation observed at low fields, which could be due to\na fact that the measurements were performed at T=20\nand 293K, respectively.\nWe have studied the evolution of the magnetiza-\ntion right after the direct optical excitation in more\ndetail(Fig.4(b)) and, inparticular, determinedtheini-\ntial phases ψ0of precession (12). The initial evolution\nof the magnetization suggest that Beffdemonstrates\na step-like jump from its equilibrium orientation upon\nthe optical excitation and remains in this orientation\nfor the time much longer than the precession decay.\nThis is opposite to the case of injected strain pulse [3],\nwhen the magnetization takes a complex path before\nthe harmonic oscillations start, which reflects the fact\nthatBefffollows the strain while it propagatesthrough\nthe film and returns back to equilibrium orientation\nonce the strain pulse has left the film.\nThe most striking result is that the initial\nphaseψ0of the oscillations possesses non-monotonous\nfield dependence. In particular Mz(t) demonstrates\npuresine−like behaviour when the magnetic field\nis ofB=150mT, and pure cosine−like behavior at\nB=500mT. Detailed fielddependence ofthe precession\ninitial phase is shown in Fig.5(c). Keeping in mind\nthat att= 0 at any strength of the in-plane magnetic\nfield the magnetization is oriented in the film plane,\none concludes that the sine-like (ψ0= 0) temporal\nevolution of Mzat the applied field of B=150mT\ncorresponds to the magnetization precessing around\nthe transient effective field Beff(t), which lies in the\nsample plane. By contrast, cosine-like (ψ0=π/2)\nbehavior of the Mzcorresponds to the precession\naroundBeff(t), having finite out-of-plane component.\nObserved change of the initial phase of the\nmagnetization precession gives a hint that there are,\nin fact, two competing mechanism of the precession\nexcitation, which relative and absolute efficiencies\nchange with increase of the applied magnetic field.\nIn experiment with optical excitation (Fig.1(d)) two\nmechanisms, heat- and strain-induced ones, consideredThe effect of dynamical compressive and shear strain on magne tic anisotropy in low symmetry ferromagnetic film 9\nBeff\u00010=0\u00000=π/2 \n\u0002 \u0003=0; Δφ=0\nΔθ=0; Δφ=0\nFigure 5. (Color online)Field dependencies of (a) the frequency\nν, (b) amplitude Mz0/Msand (c) initial phase ψ0of the\nmagnetization precession excited optically and detected i n the\ngeometry shown in Fig.1(d). Lines show the results of the\ncalculations (see text): solidlinesshow the results obtai ned when\nboth heat- and strain-induced contribution to the anisotro py\nchange are taken into account; blue and red dashed lines show\nthe results obtained when only the heat- (red) or strain-ind uced\n(blue) change of the MCA is considered. Open symbols in (a)\nshow the frequency of the magnetization excited acoustical ly by\nQTA strain pulse (see also Fig.2(b). (d) Calculated equilib rium\nin-plane components of the effective field Beff. (e) Field\ndependences of the Befftilt angles ∆ θ(dashed lines) and ∆ φ\n(solid lines) under the optical excitation resulting in the increase\nof temperature by 120K from equilibrium one (RT). The red\nand blue lines show the results when only heat-related or str ain-\nrelated mechanisms is taken into account, respectively. Sh ort-\ndashed black line shows the net tilt of Beff, induced by both\nmechanisms. The upper and lower insets show two cases of\npurely in-plane and out-of-plane tilts of Beffrespectively.\nin Sec.3 are expected to affect the magnetic anisotropy\nof the galfenol film. The heat-induced mechanism,\nbased on the rapid increase of the temperature and\ndecrease of the MCA constants, is expected to trigger\nthe precession at relatively low fields. The strain-\ninduced mechanism, resulting from the thermally-\ninduced stress, in turn, can be efficient at high fields as\nwell. The latter is demonstrated in our experiments\nwith purely acoustic excitation of the studied film,\nwhere the precession is observed in the applied fields\nupto at least 1.2T (Fig.2(a,c))).\nIn order to test this model we calculated the\nchangesoftheMCAparameters K1,uandthemagneto-\nelastic part of the free energy (3) of the optically-\nexcited (Fe,Ga) film, using the routine described\nbriefly in Secs.3 and 4 and in more detail in\n[15]. Since some required parameters are unknown\nfor galfenol, for calculations we used those of Fe:\nAe=672J·m−3K−2[36],Cl=3.8·106J·m−3K−1,κ=80.4\nW·m−1K−1[37]. The electron-phonon coupling\nconstantG=8·1017W·m−3K−1was obtained from [23]\nwhere the electron-phonon relaxation time equal to\n∼CeG−1was found to be of 250fs. The values\nofRandαwere determined experimentally. Thegalfenol density ρfor Fe0.81Ga0.19is estimated to be of\n7.95·103kg·m−3. The equilibrium magnetic anisotropy\nparameters K1=30mT,Ku=45mT and the magneto-\nelastic coefficients b1=-6T,b2=2T were found using\nliterature data [17, 38, 39] as well as from the fit\nof the field dependence of the precession frequency\n(Fig.5(a)). Fig.5(d) shows calculated equilibrium in-\nplane orientation of the effective field Beff, confirming\nthat it aligns with the external field when the latter\nexceedsB=150mT, in agreement with the SQUID\ndata (not shown).\nThe calculations of the laser-induced magnetiza-\ntion dynamics were performed for the optical exci-\ntation density of P=10mJ·cm−2. For this laser flu-\nence the lattice temperature increase calculated with\nEq.(8) was found to be of ∆ Tl=120K. Correspond-\ning change of the MCA parameters was found to be\nof ∆K1=-4.75mT and ∆ Ku=-2.2mT, and the persis-\ntent components of the compressive and shear dynam-\nical strain were found to be ǫzz= ∆ǫzz=1.2·10−3and\nǫxz= ∆ǫxz=-4·10−4. From these values the optically-\ntriggered out-of-plane ∆ θand in-plane ∆ φdeviations\nof the effective field Beff(Fig.1(a)) were found, as\nshown in Fig.5(e). As expected, the heat-induced\nchange of the MCA affects the orientation of the ef-\nfective field predominantly in the range of the applied\nfields below and close to B=150mT. By contrast, the\nstrain-induced deviation of the effective field remains\nsignificant even when the applied field is as high as\n500mT.Atlowerfieldsthiscontributioncompeteswith\nthe heat-induced one. The calculations also confirm\nthat the propagating strain pulses also generated by\nthe optical excitation contribute much weakly to the\nMCA change.\nImportantly, at high fields optically-generated\nstrain results in the out-of-plane deviation ∆ θofBeff.\nAt intermediate and low field the combined effect\nof the heat- and strain-induced anisotropy change\nresults in both ∆ θand ∆φto be non-zero. At\nB=150mT, i.e. when equilibrium magnetization is\naligned along the external field, the heat-induced\nchange of magnetocrystalline constants dominates and\nBeffdeviates mostly in plane. These two limiting\nsituations are illustrated in the insets of Fig.5(e).\nFinally, the amplitude and the initial phase\nof the magnetization precession triggered via heat-\nand strain-induced change of magnetic anisotropy\nof galfenol film were calculated (Fig.5(b,c)). Good\nagreement between the experimental data and the\ncalculated one confirms that the MCA in the optically-\nexcited(311)-(Fe,Ga)filmisindeedmodifiedand, thus,\ntriggers the precession, via two distinct mechanisms.\nAt relatively low fields the heat-induced change\nof anisotropy parameters is efficient, as was also\nshown in a number of previous works [23, 32]. InThe effect of dynamical compressive and shear strain on magne tic anisotropy in low symmetry ferromagnetic film 10\naddition, optically-generated strain modifies MCA\nvia inverse magnetostriction. Furthermore, as the\napplied field increases, the heat-induced contribution\nto the magnetic anisotropy change decreases more\nrapidly than the strain-induced one. As a result,\nin the relatively high fields magnetization precession\nis excited mostly due to the optically-generated\npersistent strain.\nIt is instructive to note, that the physical origin\nof the MCA change - inverse magnetostriction - is\nthe same when either the step-like strain is induced\noptically in the magnetic film, or the strain pulse is\ninjected into the film. However, the magnetization\nprecession trajectories may appear to be very distinct.\nIn the caseofthe direct optical excitation the temporal\nprofile of the Beffmodified due to abruptly emerged\nstrain can be seen as the step-like jump. This sets\nthe amplitude and initial phase of the magnetization\nprecession, which are uniquely linked to the angle\nbetween the equilibrium and modified directions of\nBeff[15]. This situation becomes much more intricate\nwhen the strain pulse drives the MCA change. As\ndiscussed in [3], strain pulse alternates the direction\nofBeffupon propagation through the film triggering\nthe magnetization precession, which then proceeds\naround the equilibrium Beffones the strain pulse left\nthe film. Thus, the amplitude and the initial phase\nof the excited precession would be dependent on the\nparticular spatial and temporal profile of the strain\npulse.\n7. Conclusions\nIn conclusion, we have demonstrated two alternative\napproaches allowing one to modify magnetocrystalline\nanisotropy of a metallic magnetic film at ultrafast time\nscale by dynamical strain, with inverse magnetostric-\ntion being the underlying mechanism. Using 100-nm\nfilm of a ferromagnetic metallic alloy (Fe,Ga) grown on\na low symmetry (311)-GaAs substrate, we were able\nto trigger the magnetization precession by dynamical\nstrain of a mixed, compressive and shear, character.\nPicosecondquasi-longitudinalandquasi-transversal\nstrain pulses can be injected into the galfenol film from\nthe substrate, which leads to efficient excitation of the\nmagnetization precession. In this case, owing to the\ndistinct propagationvelocities ofQLA and QTA pulses\nin the substrate, their impact on MCA can be easily\ndistinguished and analyzed. Importantly, dynamical\nstrainremainstheefficientstimulustriggeringthemag-\nnetization precession in the applied magnetic fields up\nto 1.2T.\nAlternatively, one can directly excite the galfenol\nfilm by a femtosecond laser pulse. In this case\nthere are two competing mechanism mediating theultrafast change of MCA. Rapid increase of the\nlattice temperature results in the decrease of the\nMCA parameters. The lattice temperature increase\nalso sets up the thermal stress which results in\noptically generated strain. We demonstrate that\nthe heat-induced decrease of the MCA parameters\nand the change of MCA mediated by the inverse\nmagnetostriction compete and both can trigger the\nmagnetization precession. Despite the fact that\nthe temporal and the spatial profiles of the lattice\ntemperature increase and optically-generated strain\nclosely resemble each other, their impact on MCA\ncan be distinguished. This is possible owing to\ndistinct response of magnetization to the heat-\ninduced decrease of anisotropy parameters and to\nthe strain-induced change and reorientation of the\neffective field describing the MCA. In the former\ncase the magnetization precession is triggered only\nif the external magnetic field applied along the\nmagnetization hard axis is of moderate strength. In\nthelattercasethisconstrainislifted andtheprecession\nexcitation was observed in the applied fields as strong\nas at least 0.5T. The experiments with strain pulses\ninjected into the film suggest that strain-induced\nprecession excitation would remain efficient at higher\napplied field values as well.\nWe would like to emphasise, that in order\nto realize the strain-induced control of magnetic\nanisotropy in the optically excited metallic film, low\nsymmetry and elastic anisotropy of the latter is of\nprimary importance. As we have shown in our recent\nstudy [15] in the galfenol film of high symmetry grown\non the (001) GaAs substrate magnetic anisotropy\nchange is dominated by the lattice heating and related\ndecrease of MCA parameters. Thus, low-symmetry\nfilms and structures, where the dynamical strain can\nefficiently modify MCA at relatively high magnetic\nfields, may be promising objects, when one aims at\nthe excitation of magnetization precession of high\nfrequency. This finding highlights further importance\nof low-symmetry ferromagnetic structures for ultrafast\nmagneto-acoustic studies [40].\nFinally, we note that direct detection of the\noptically-generatedquasi-persistentstrainin ametallic\nfilm is a challenging task. This strain component is\nintrinsically accompanied by the laser-pulse induced\nlattice heating. Therefore, detection of this strain\ncomponent by optical means is naturally obscured by\nthe change of optical properties of the medium [18].\nRealized in our experiments at the high magnetic\nfield limit observation of the magnetization precession\naround new MCA direction, which was set mostly by\nthe emerged quasi-persistent strain, can be considered\nas an indirect way to probe this constituent of the\nultrafast lattice dynamics.The effect of dynamical compressive and shear strain on magne tic anisotropy in low symmetry ferromagnetic film 11\n8. Acknowledgements\nThis work was supported by the Russian Scientific\nFoundation [grant number 16-12-10485] through fund-\ning the experimental studies at the Ioffe Institute;\nand by the Engineering and Physical Sciences Re-\nsearch Council [grant number EP/H003487/1]through\nfunding the growth and characterization of Galfenol\nfilms. The experimental work at TU Dortmund was\nsupported by the Deutsche Forschungsgemeinschaft in\nthe frame of Collaborative Research Center TRR 160\n(project B6). The Volkswagen Foundation [Grant\nnumber 90418] supported the theoretical work at the\nLashkaryov Institute. A.V.A. acknowledges the sup-\nport from Alexander von Humboldt Foundation.\n[1] A. Kirilyuk, A. V. Kimel, and T. Rasing, Ultrafast optica l\nmanipulation of magnetic order, Rev. Mod. Phys. 82,\n2731 (2010).\n[2] C.Thomsen, J. Strait, Z. Vardeny, H. J. Maris, J. Tauc, an d\nJ. J. Hauser, Coherent phonon generation and detection\nby picosecond light pulses, Phys. Rev. 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Appl. 5,\n031002 (2016)."    },    {        "title": "1611.07335v1.Uranium_ferromagnet_with_negligible_magnetocrystalline_anisotropy_____mathrm_U__4_Ru__7_Ge__6___.pdf",        "content": "Uranium ferromagnet with negligible magnetocrystalline anisotropy - U4Ru7Ge6\nMichal Vališka,1,\u0003Martin Diviš,1and Vladimír Sechovský1\n1Faculty of Mathematics and Physics, Charles University,\nDCMP, Ke Karlovu 5, CZ-12116 Praha 2, Czech Republic\nStrong magnetocrystalline anisotropy is a well-known property of uranium compounds. The\nalmost isotropic ferromagnetism in U4Ru7Ge6reported in this paper represents a striking excep-\ntion. We present results of magnetization, AC susceptibility, thermal expansion, specific heat and\nelectrical resistivity measurements performed on a U4Ru7Ge6single crystal at various tempera-\ntures and magnetic fields and discuss them in conjunction with results of first-principles electronic-\nstructure calculations. U4Ru7Ge6behaves as an itinerant 5 f-electron ferromagnet ( TC= 10:7 K,\n\u0016S= 0:85\u0016B=f:u:at1:9 K. The ground-state easy-magnetization direction is along the [111] axis of\nthe cubic lattice. The anisotropy field \u00160Haalong the [001] direction is only of \u00180:3 T, which is\nat least 3 orders of magnitude smaller value than in other U ferromagnets. At Tr= 5:9 Kthe easy\nmagnetization direction changes for [001] which holds at temperatures up to TC. This transition\ncausing a change of magnetic symmetry is significantly projected in low-field magnetization, AC\nsusceptibility and thermal-expansion data whereas only a weak anomaly is observed at Trin the\ntemperature dependence of specific heat and electrical resistivity, respectively. The magnetoelastic\ninteraction induces a rhombohedral (tetragonal) distortion of the paramagnetic cubic crystal lattice\nin case of the [111]([001]) easy-magnetization direction. The rhombohedral distortion is connected\nwith two crystallographically inequivalent U sites. The ab initio calculated ground-state magnetic\nmoment of 1:01\u0016B=f:u:is oriented along [111]. The two crystallographically inequivalent U sites are\na consequence of spin-orbit coupling of the U 5 f-electrons. In the excited state which is only 0:9 meV\nabove the ground state the moment points to the [001] direction in agreement with experiment. A\nscenario of the origin of the very weak magnetic anisotropy of U4Ru7Ge6is discussed considering\ninteractions of the U-ion 5 f-electron orbitals with the nearest neighbor ions.\nPACS numbers: 75.30.Gw, 75.50.Cc, 75.40.Cx, 71.15.Mb\nKeywords: Itinerant 5 f-electron ferromagnetism, low anisotropy, magnetostriction\nI. INTRODUCTION\nMagnetocrystalline anisotropy (MA) is manifested by\nlocking the magnetic moments in a specific orientation\n(usually the easy magnetization direction) with respect\nto crystal axes. A quantitative measure of MA, the\nanisotropy field Ha, is the magnetic field needed to be\napplied in the hard magnetization direction for reach-\ning the easy-axis magnetization value. The key prereq-\nuisites of MA are the orbital moment of a magnetic ion,\nthe spin-orbit (s-o) interaction coupling the orbital and\nspin moment, and interactions with neighboring ions1,2.\nThe s-o interaction is a relativistic effect which becomes\nstronger in heavier atoms. Consequently MA dominates\nmagnetisminmaterialswithlanthanideandactinideions\nbearing magnetic moments of the 4 f- and 5f-electrons,\nrespectively.\nThe widely accepted scenario of the origin of mag-\nnetocrystalline anisotropy involves the crystal electric\nfield (CEF) interaction, the single-ion mechanism born\nin the electrostatic interaction of the anisotropic crys-\ntallineelectricfield(thepotentialcreatedatthemagnetic\nion site by the electric charge distribution in rest of the\ncrystal) with the aspherical charge cloud of the magnetic\nelectrons. The electron orbital adopts the direction that\nminimizes the CEF interaction energy. The single-ion\nanisotropy is most often encountered in compounds with\nlanthanides having the well-localized 4 f-electrons3,4.\nContrary to the 4 f-orbitals deeply buried in the coreelectron density of lanthanide ions the spatially extended\nuranium 5f-electron wave functions interact with the\noverlapping 5 f-orbitals of the nearest neighbor U ions\n(5f-5foverlap) and with valence electron orbitals of lig-\nands (5f-ligand hybridization5). Consequently the 5 f-\nelectron wave functions loose the atomic character and\nsimultaneously the U magnetic moments melt down. De-\nspite that the strong spin-orbit coupling induces a pre-\ndominant orbital magnetic moment antiparallel to the\nspin moment in the spin-polarized 5 f-electron energy\nbands. This effect was first demonstrated for the itin-\nerant 5f-electron magnetism in UN6. In some cases very\nsmall or eventually zero total magnetic moment of a U\nion are observed as a result of mutual compensation of\nthe antiparallel spin and orbital component, respectively.\nThe itinerant 5 f-electron ferromagnet UNi 27with the U\nmagnetic moment of few hundredths \u0016Bserves as an ex-\ncellent example as documented by results of polarized\nneutron measurements8and first principles electronic\nstructurecalculations9. Despitetheitinerantcharacterof\nmagnetism UNi 2exhibits very strong magnetocrystalline\nanisotropy with \u00160Ha\u001d35 Tat4:2 K10.\nThe very strong MA seems to be inherent to the ura-\nnium magnetism. The typical values of the MA field of\nmost uranium intermetallic compounds are of the order\nof hundreds T11. The strong anisotropy is reported also\nfor the cubic U pnictides and chalcogenides12,13.\nThe strong interaction of the spatially extended U\n5forbitals with surrounding ligands in the crystal andarXiv:1611.07335v1  [cond-mat.str-el]  22 Nov 20162\nparticipation of 5 felectrons in bonding14,15imply an\nessentially different mechanism of magnetocrystalline\nanisotropy based on a two ion (U-U) interaction. The\nanisotropy of the bonding and 5 f-ligand hybridization\nassisted by the strong spin-orbit interaction are the key\ningredients of the two-ion anisotropy in 5 f-electron mag-\nnets.\nThe systematic occurrence of particular types of\nanisotropy related to the layout of the U ions in a crys-\ntal lattice suggests that in materials, in which the U-U\nco-ordination is clearly defined in the crystal structure,\nthe easy-magnetization direction is perpendicular to the\nnearest U-U links11,16. Cooper and co-workers17,18have\nformulated a relatively simple model of the two-ion in-\nteraction. This model, leading to qualitatively realistic\nresults, is based on Coqblin-Schrieffer approach to the\nmixing of ionic f-states and conduction-electron states,\nin which the mixing term of the Hamiltonian of Ander-\nson type is treated as a perturbation, and the hybridiza-\ntion interaction is replaced by an effective f-electron-\nband electron resonant exchange scattering. The the-\nory has been further extended so that each partially\ndelocalized f-electron ion is coupled by the hybridiza-\ntion to the band electron sea; and these both lead to\na hybridization-mediated anisotropic two-ion interaction\ngiving an anisotropic magnetic ordering.\nIn this paper we focus on magnetism in the U4Ru7Ge6\ncompound. Although the first U4Ru7Ge6single crys-\ntals were grown already in late Eighties only a vague\nreport on ferromagnetism ( TC\u00187 K,0:2\u0016B=Uion in\n5 Tat4:3 K)19and no information on anisotropy can\nbe found in literature. Therefore we have grown a high\nquality single crystal of this compound and measured on\nit the magnetization, AC susceptibility, thermal expan-\nsion, specific heat and electrical resistivity with respect\nto temperature and magnetic field. A weakly anisotropic\nferromagnetism has been observed below TC= 10:7 K.\nThe ferromagnetic ground state is characterized by the\neasy-magnetization axis along the [111] crystallographic\ndirection. A magnetic phase transition at which the easy\nmagnetizationaxischangesfrom[111]to[001]isobserved\natTr= 5:9 K. The [111] ([001]) phase is associated with\na tiny rhombohedral (tetragonal) distortion of the para-\nmagnetic cubic crystal structure which hardly can be ob-\nserved by the X-ray diffraction but we demonstrate that\nit can be indicated by thermal-expansion data. To shed\nmorelightonmicroscopicmechanismsresponsibleforthe\nU4Ru7Ge6magnetisminsomeaspectsveryunusualforU\ncompounds we also performed first-principles electronic\nstructure calculations.\nII. EXPERIMENTAL DETAILS\nThe U4Ru7Ge6single crystal has been grown by\nCzochralski method in a tri-arc furnace from constituent\nelements (purity of Ru 3N5 and Ge 6N). The uranium\nmetal was purified using Solid State Electrotransporttechnique (SSE). Half of the crystal was wrapped in the\nTa foil (purity 4N), sealed in a quartz tube under the\nvacuum of 1\u000110\u00006mbarand subsequently annealed at\n1000 °C for 7 days. The high quality of both crystals was\nverified by Laue diffraction using a Photonic Science X-\nRay Laue system with a CCD camera and single crystal\nX-ray diffraction on a Rigaku R-Axis Rapid II diffrac-\ntometer with a Mo lamp. A part of each, the as-cast and\nannealed single crystal, respectively, was pulverized and\ncharacterized by X-ray powder diffraction (XRPD) at\nroom temperature on a Bruker D8 Advance diffractome-\nter with a Cu lamp. The obtained data were evaluated\nbyRietveldtechnique20usingFULLPROF/WINPLOTR\nsoftware21,22. The chemical composition of each single\ncrystal was verified by a scanning electron microscope\n(SEM) Tescan Mira I LMH equipped with an energy dis-\npersive X-ray (EDX) detector Bruker AXS. Samples for\nindividual experiments were afterward cut from the an-\nnealed crystal with a fine wire saw to prevent induction\nof additional stresses and lattice defects.\nThe magnetization (in a magnetic field up to 7 Tap-\nplied along the [001] and [111] directions, respectively)\nand AC susceptibility (the AC magnetic field with the\namplitude of 300\u0016Tapplied along [001]) were measured\nby a SQUID magnetometer in MPMS 7T in the tem-\nperature range from 1:9 Kto300 K. The magnetization\nalong[111]wasmeasuredupto 14 TbyaVSMdeviceata\nPPMS 14T. Specific heat data were collected by the ther-\nmal relaxation technique in the temperature range from\n0:4to20 Kin the magnetic fields up to 9 Tby a PPMS\n9T. The electrical resistivity measurements were realized\nwith the ACT option of the PPMS instruments with AC\ncurrent applied along [100] and [111], respectively, in the\ntemperature range from 1:9 Kto300 K. The MPMS 7T\nand both the PPMS apparatuses are of Quantum Design\nInc. production. The thermal expansion measurements\nalong the [100], [001] and [111] directions, respectively,\nin the temperature range from 1:9 Kto30 Kusing a\nminiature capacitance dilatometer23implemented in the\nPPMS 9T. All the instrumentation mentioned above is a\npart of the Magnetism and Low Temperature Laborato-\nries – MLTL ( http://mltl.eu/ ).\nIII. RESULTS\nA. Crystal structure and chemical composition\nThe Rietveld refinement of XRPD data collected on\nthe pulverized as-cast (see the powder pattern in Figure\n1) and annealed single crystal, respectively, confirmed\npreviousreports19,24,25thatU4Ru7Ge6possessesatroom\ntemperature a cubic crystal structure of the Im\u00163mspace\ngroup. The corresponding lattice parameters (see Table\n1) determined for the as cast and annealed crystal, re-\nspectively, are nearly identical.\nElemental mapping by EDX confirmed homogeneity\nof all the studied samples. The average of multiple point3\nFigure 1. X-ray powder diffraction pattern of a pulverized\nU4Ru7Ge6as cast single crystal.\nSpace group Im\u00163m as-cast annealed\na 8:2934(2)Å 8:2933(3)Å\nU(x;y;z );8c(0.25, 0.25, 0.25) (0.25, 0.25, 0.25)\nRu1(x;y;z );12d(0.25, 0, 0.5) (0.25, 0, 0.5)\nRu2(x;y;z );2a (0, 0, 0) (0, 0, 0)\nGe(x;y;z );12e(0.31375(13), 0, 0) (0.31360(11), 0, 0)\nTable I. Results of structure analysis.\nscans from the different part of the sample gives resulting\nstoichiometry4.4(4):6.9(2):5.7(1). ItpointstoaslightGe\ndeficiency.\nB. Low temperature magnetization\nThe magnetization curves M(\u00160H)measured at 1:9 K\nin a magnetic field applied along the [001] and [111] di-\nrection, respectively, document that U4Ru7Ge6is at this\ntemperature ferromagnetic with the easy magnetization\ndirection along the [111] axis (see Figure 2). The spon-\ntaneous magnetization MS[111] = 0:88 (1)\u0016B(obtained\nas an extrapolation of the M[111](\u00160H)dependence to\n\u00160H= 0 T). The spontaneous magnetization value along\nthe [001] direction is lower, MS[001] = 0:51 (1)\u0016B, re-\nspectively. This value is in very good agreement with\nNéel phase law26for a cubic system. It proposes that\nMS[001]can be obtained from the MS[111]value multi-\nplied by the appropriate direction cosine – i.e. MS[001] =q\n1\n3MS[111]. For fields higher than 0:3 TtheM[111](\u00160H)\nandM[001](\u00160H)curves merge (we estimate the \u00160Ha\nvalue 300 mT), however, do not saturate in the field up\nto the 14 Twhere the magnetization reaches the value of\n1:4\u0016B=f:u:. The poor saturation of the magnetization in\nhigh magnetic fields is typical for itinerant electron ferro-\nFigure 2. Field dependence of magnetization isotherms of\nU4Ru7Ge6at1:9 Kfor [111], and [001] directions of applied\nmagnetic field. Inset shows a low field detail.\nmagnets. The increasing magnetic moment with increas-\ningthemagneticfieldisthenreflectingthemagnetic-field\ninduced change of electronic structure (additional split-\nting of the majority and minority subbands).\nThe observed very weak magnetization anisotropy\nmakes U4Ru7Ge6a unique exception among the U fer-\nromagnets which are by rule strongly anisotropic with\nanisotropy fields of several hundred T11. In the next\nchapterweapproachthisissuebythefirst-principleselec-\ntronic structure calculations focused on the microscopic\norigin of U magnetic moments and magnetocrystalline\nanisotropy.\nC. First-principles calculations\nTo obtain microscopic information about magnetism\ninU4Ru7Ge6we applied the first-principles methods\nbasedonthedensityfunctionaltheory(DFT).TheKohn-\nSham-Dirac 4-components equations have been solved by\nusing the latest version of the full-potential-local-orbitals\n(FPLO) computer code27. Severalk-meshes in the Bril-\nlouin zone were involved to ensure the convergence of\ncharge densities, total energy and magnetic moments.\nFor the sake of simplicity we assumed a collinear ferro-\nmagnetic structure. In U4Ru7Ge6the total ground-state\nmagneticmomenthasbeenfoundpointingalongthe[111]\ndirection. Due to the s-o interaction the symmetry is re-\nduced from 48 to 12 symmetry operations. Instead of\nfour symmetrically equivalent U ion sites in the scalar\nrelativistic treatment with a spin-only magnetic moment\nwehavethespinandorbitalangularmomentacoupledby\nthe relativistic s-o interaction which divides the U ions in\ntwo subgroups consistently with the expected rhombohe-\ndral distortion induced by magnetoelastic interaction in\ncaseofthe[111]easymagnetizationdirection. TheUions4\nFigure 3. Structure of U4Ru7Ge6together with magnetic mo-\nments from theoretical calculations. Arrows for magnetic mo-\nments are in proper relative scale but in arbitrary units. U1\nions are blue, U2are green, Ru3andRu4are gray and Ge5\nare yellow.\nin the first subgroup at the U1positions (0.25, 0.25, 0.25)\nand(0.75, 0.75, 0.75), respectivelyhaveaverysmalltotal\nmagnetic moment of 0:01\u0016Bdue to cancellation of the al-\nmost equal-size antiparallel spin and orbital moment (see\nFigure 3). The second subgroup includes the U ions at\nthe remaining six U2positions bearing the spin magnetic\nmoment of \u00000:56\u0016Band the orbital magnetic moment\nof0:79\u0016B. There are also some hybridization-induced\nRu magnetic moments, which summed up give 0:29\u0016B,\nwhereas the calculated Ge magnetic moment is negligi-\nble. The summation over the seventeen ions (one formula\nunit) in the primitive crystallographic cell gives the total\nmagnetic moment of 1:01\u0016Bwhich is somewhat larger\nthan the spontaneous magnetic moment determined by\nexperiment.\nWe have also performed relativistic calculation with\nthe moment along [001]. In this case the cubic symme-\ntry is reduced to tetragonal and all U moments have the\nsame value of spin ( \u00000:529\u0016B) and orbital ( 0:706\u0016B)\ncomponent.\nWhen calculating the magnetocrystalline anisotropy\nenergy between the configurations of magnetic moments\naligned along the [111] and [001] axis the values of the\ntotal energy were used. In agreement with experiment\nwe have found the ground state with the total moment\npointing to the [111] direction. The excited state with\nthe moment pointing to the [001] direction is 0:9 meV\nabove the ground state. This value is as to order of mag-\nFigure 4. Temperature dependence of the magnetization of\nU4Ru7Ge6measured in the magnetic field of 10 mT(in the\nZFC regime) applied along the [111] and [100] direction, re-\nspectively (a) and temperature dependence of the AC suscep-\ntibility in the AC field applied along the [111] (b). Inset in\nthe upper panel shows field dependence of magnetization at\n9 K.\nnitude in agreement with the energy corresponding to\nTr, the temperature of [111] to [100] spin reorientation\ntransition.\nD. Magnetization near the phase transitions\nThe Curie temperature, TC, of a ferromagnet is fre-\nquently estimated as the temperature of the inflection\npoint of the Mvs.Tcurve measured in a low magnetic\nfield and/or of the temperature dependence of the AC-\nsusceptibility. In Figure 4 (a) one can see that the tem-\nperature dependences of magnetization M(T)measured\nin the external magnetic field of 10 mTapplied along the\n[111] and [100] direction, respectively, have one inflection\npoint at the same temperature of 10:6\u00060:1 K(\u0018TC) and\nare crossing at Tr= 5:9\u00060:1 K. Two sharp anomalies at\ncorresponding temperatures can be seen in the \u001fAC(T)\ndependenceseenintheFigure4(b). Theseresultsclearly\ndocument that U4Ru7Ge6orders atTCferromagnetically\nwith the easy magnetization axis [100] (which is demon-\nstrated by the 9 Kmagnetization isotherms in the inset\nof Figure 4 (a)). At Trit undergoes a spin reorientation\ntransition to the ground state characterized by the easy\nmagnetization axis along [111].\nA rigorous way of determining Curie temperature of5\nFigure 5. Arrott plots for U4Ru7Ge6in the magnetic field\napplied along the [001] direction.\na ferromagnet from magnetization data is based on the\nanalysis of Arrott plots ( M2vs.\u00160H=M)28. Linear Ar-\nrott plots are in fact a graphical representation of the\nGinsburg-Landau mean field theory of magnetism in the\nvicinity of the ferromagnetic to paramagnetic second or-\nder phase transition. In Figure 5 one can see that the\nArrott plots for U4Ru7Ge6in the magnetic field applied\nalong the easy magnetization direction [001] are almost\nlinear with varying slope for magnetic fields between 1\nand7 Twhereas for lower fields they became slightly con-\ncave. The linear extrapolations of high-field data to the\nvertical axis are marking the values of M2, which are\nconsidered as estimates of the square spontaneous mag-\nnetizationM2\nS. The spontaneous magnetization as the\norder parameter of a ferromagnetic phase vanishes at TC.\nNote that the use of the linear extrapolations from high\nfields in case of the concave curvature of U4Ru7Ge6Ar-\nrott plots in low fields leads to a certain overestimation\nofMSvalues and consequently to a higher estimated TC\nvalue.\nA more precise TCvalue may be expected from the\ngeneralizedapproachusingtheArrott-Noakesequationof\nstate (\u00160H=M )1=\r= (T\u0000TC)=T1+ (M=M 1)1=\f, where\nM1andT1are material constants29. We re-analyzed our\ndata by plotting them as M1=\fvs.(\u00160H=M )1=\rwhile\f\nand\rvalueswerechosentogetthebestpossiblelinearity\nof these plots keeping them parallel with constant slope.\nThe values \f= 0:31\u00060:03and\r= 0:81\u00060:04lead to\na linear dependence in the broad field range except the\nvery low fields. This construction is plotted in Figure 6\nfor all measured isotherms. This approach leads to TC=\n10:7\u00060:1 K(see inset of Figure 6) that is in agreement\nwiththeestimatedvaluefromthelowfieldmagnetization\ndata.\nThe critical exponent \u000ein ideal case might satisfy the\nWidom scaling relation \u000e= 1 +\r=\f(Ref.30) which for \f\nand\r, provided by the Arrott-Noakes analysis, gives \u000e=\nFigure 6. Arrott-Noakes plots reflecting the equation of states\n(with\f= 0:31\u00060:03and\r= 0:81\u00060:04) for U4Ru7Ge6in\nthe magnetic field applied along the [001] direction.\nBk\u0016e\u000b(\u0016B=U)\u0012P(K)\u001f0\u000110\u00008\u0000\nm3\u0001mol\u00001\u0001\n[001] 1.43 7.5 5.8\n[111] 1.43 8.1 5.7\nTable II. Results of the modified Curie-Weiss fit in the tem-\nperature interval 30\u0000300 K.\n3:52\u00060:04that is in excellent agreement with the value\n\u000e= 3:55\u00060:04obtained from direct fitting of the critical\nisotherm that should follow M\u0018(\u00160H)1=\u000e. Note that\nthe critical exponents for the mean field approximation\nare\f= 0:5,\r= 1and\u000e= 3(Ref.31), however, weshould\nmention the work of Yamada32who has shown that the\nspin fluctuations in weak itinerant ferromagnets lead to\nthe Arrott plots linear in strong magnetic fields and bent\ndownwards at the region of small magnetizations.\nE. Paramagnetic susceptibility\nThe nearly identical temperature dependences of para-\nmagnetic susceptibility measured along the [111] and\n[100] direction, respectively document isotropic param-\nagnetic state of U4Ru7Ge6(see the 1=\u001fvs.Tplot in Fig-\nure 7). The susceptibility values at temperatures above\n30 Kcan be well fitted with a modified Curie-Weiss law\nin temperature range with parameters shown in Table II.\nF. Heat capacity\nHeat capacity data show a clear anomaly at 10:7 Kas\ndisplayed in Figure 8. This temperature coincides with\ntheTCvalue determined from magnetization data by6\nFigure 7. Temperature dependence of inverse susceptibility of\nU4Ru7Ge6in the magnetic field of 1 Tapplied along the [111]\nand [001] direction. The full curve represents the fit with a\nmodified Curie-Weiss law.\nFigure 8. Temperature dependence of the heat capacity ( C=T\nvs.Tplot) of U4Ru7Ge6. Inset: the C=Tvs.T2plot.\nArrott-Noakes plot analysis. The estimated magnetic en-\ntropyatTC(i.e. integratedfrom 0:3 KtoTC)of0:2\u0001R ln 2\nis much lower than R ln 2which is another evidence of\nitinerant electron magnetism in U4Ru7Ge6. The mag-\nnetic moment reorientation transition at Tris reflected\nin a tiny but clear peak at this temperature.\nThe gamma coefficient of the electronic specific heat\ndetermined from a standard C=Tvs.T2plot (see\nInset in Figure 8) constructed from data below 4 K\namounts to 362 mJ \u0001mol\u00001\u0001K\u00002. The value related to\none U ion = 90:5 mJ\u0001mol\u00001\u0001K\u00002reflects presence of\nthe U 5f-electron states at EFsimilar to numerous other\nU intermetallics with itinerant 5 f-electrons which usu-\nally exhibit elevated values somewhere between 30 and\n100 mJ \u0001mol\u00001\u0001K\u00002per U ion.\nFigure 9. Linear thermal expansion of U4Ru7Ge6along the\n[001] and [111] directions, respectively.\nG. Thermal expansion and magnetostriction\nThe linear thermal expansion \u0001L=Lmeasured along\nthe [001] ([111]) direction in zero magnetic field which\ncan be seen in Figure 9, shows two distinct anomalies\nwhichcanbeattributedtothemagneticphasetransitions\nrevealed by magnetization measurements. When cooling\nfrom higher temperatures a downturn (upturn) having\nonset in the vicinity of TCis followed by a steep increase\n(decrease) of the corresponding \u0001L=Lbelow \u00186 K(\u0018\nTr).\nMost of the thermal expansion studies of ferromagnets\nrevealing the spontaneous magnetostriction at tempera-\nturesT < T Cwere done by using the X-ray or neutron\ndiffraction33. We investigated the crystal structure of\nU4Ru7Ge6by X-ray powder diffraction at low tempera-\ntures down to 3 K. No change of diffraction within the\nexperimental error data has been observed below 11 K.\nFrom Figure 9 it is however evident that our ther-\nmal expansion data obtained by dilatometer on the\nU4Ru7Ge6single crystal clearly demonstrate the exis-\ntence of lattice distortions in ferromagnetic state. The\ndistortions here are, however, very small ( <10\u00005).\nThe dilatometer enables us to determine the crystal\ndistortions along the 3 perpendicular crystal axes [100],\n[010] and [001], respectively. To study the correspond-\ning linear spontaneous magnetostriction of a ferromag-\nnet with a dilatometer one should perform measurements\non a single crystal sample containing only one ferromag-\nnetic domain. Knowing magnetization data we studied\nthe phase with the easy magnetization along [001] sta-\nble at temperatures Tr< T < T Cin the following way.\nWe first cooled the crystal down to 6:2 K(the tempera-\nture just above the onset of the spin reorientation tran-\nsition) in the field of 1 Tparallel to the [001] direction,\nat this temperature decreased the magnetic field down to\n30 mTand then measured the thermal expansion in the7\nFigure 10. Linear thermal expansion of U4Ru7Ge6along the\n[001]and[100]directioninthemagneticfieldof 30 mTapplied\nalong [001] and the corresponding thermal volume expansion\nat temperatures from 6:2 KtoTC. The presented data are\nconsidered as the best estimation of corresponding sponta-\nneous magnetostriction (see text).\nlongitudinal (\u0001L=L)[001]and transversal (\u0001L=L)[100]ge-\nometry, respectively, with increasing temperature up to\nTC. In this experiment the thermal expansion along the\nc- anda-axis, respectively, of the ferromagnetic tetrag-\nonally distorted structure was measured. The field of\n30 mTapplied along [001] is the minimum field main-\ntaining the single-domain sample with the magnetic mo-\nment oriented along the c-axis. These (\u0001L=L)[001]and\n(\u0001L=L)[100]thermal expansion data can be taken as a\nreasonable approximation of spontaneous linear magne-\ntostriction (denoted as \u0015S[001]and\u0015S[100], respectively).\nOne can see that the values of the spontaneous tetrag-\nonal distortion is negative (the c-axis shrinks) and very\nsmall (\u0015S[001]\u0018 \u0000 9\u000110\u00006at6:2 K). Simultaneously\nthea-axis expands ( \u0015S[100] =\u0015S[010] \u00189\u000110\u00006at\n6:2 K). Figure 10 shows also the corresponding spon-\ntaneous thermal volume expansion usually denoted as\n!S\u0000\n= 2\u0001\u0015S[100] +\u0015S[001]\u0001\n. As expected for an itinerant\nelectron ferromagnet the volume of the U4Ru7Ge6lattice\nexpandsbelow TC. Reasonabledatacanbecollectedonly\ndown to 6:2 K. Below this temperature the magnetic-\nmoment-reorientation transition from [001] to [111] di-\nrection commences, the direction of magnetic moment\nbecomes uncertain despite the magnetic field of 30 mTis\napplied along [001].\nH. Electrical resistivity\nThe almost identical corresponding values of the elec-\ntrical resistivity measured for current along [111] and\n[100] direction, respectively, over the entire temperature\nrange 2\u0000300 Kindicates a quite isotropic electron trans-\nFigure 11. Temperature dependence of electrical resistivity\nof the U4Ru7Ge6measured for current along [111] and [100]\ndirection.\nport in U4Ru7Ge6(see Figure 11). The downward con-\ncave\u001a(T)curve in the paramagnetic state resembles\nrather the behavior of transition metal compounds. This\ntrend changes at TCto the low-temperature upward con-\ncave curve with the T2scaling. When inspecting the sec-\nond derivative we observe a deep minimum of @2\u001a=@2T\natTCand a local minimum at Tr.\nIV. DISCUSSION\nTheM(\u00160H)curve measured for the hard magnetiza-\ntion direction [100] merges with the easy-magnetization\ndirection [111] curve near 300 mTas seen in Figure 2.\nThis anisotropy field of U4Ru7Ge6is an unprecedentedly\nlow value ever observed for a uranium intermetallic com-\npound. Moreover, we observed an entirely isotropic para-\nmagnetic susceptibility and electrical resistivity which up\nto our best knowledge has not been reported for any ura-\nnium 5f-electron ferromagnet.\nOne may argue that the almost isotropic magnetism in\nU4Ru7Ge6is a consequence of the itinerant character of\nU 5f-electron magnetic moment. The spontaneous mag-\nnetic moment of this compound is \u00180:85\u0016B=f:u:, which\nprovides an average moment of \u00180:21\u0016B=Uion when\nsupposing negligible contributions from Ru and Ge ions.\nThis value is 4 times larger than the U-born magnetic\nmomentoftheitinerant5 felectronferromagnet UNi 27–10\nwhich, in contrary, exhibits huge magnetocrystalline\nanisotropy with the anisotropy field \u001d35 T10.\nBoth our magnetization data and ab initio calculations\nclearly show that the easy magnetization direction of\nU4Ru7Ge6in the ground state is [111]. As a result of\nmagnetoelastic interaction34, the ferromagnetic ordering\nat temperatures below TCis accompanied by the sponta-\nneous magnetostriction causing a distortion of a crystal8\nlatticerelatedtotheeasy-magnetizationdirection. These\ndistortions are in ferromagnetic materials possessing cu-\nbic crystal structure in paramagnetic state (at tempera-\nturesT >T C) the spontaneous magnetostriction leads to\na distortion lowering the paramagnetic cubic to tetrag-\nonal, orthorhombic and rhombohedral symmetry for the\n[001]-, [110]- and [111]-easy-magnetization direction, re-\nspectively. The expected rhombohedral distortion of the\ncubic U4Ru7Ge6latticeintheferromagneticgroundstate\nis so tiny that it falls within the experimental error of\na standard X-ray diffraction but is clearly indicated by\nthermal expansion results at low temperatures. As a con-\nsequence of the distortion the one equivalent crystallo-\ngraphic site common for all U ions in the cubic lattice\nsplits into two inequivalent ones which is confirmed by\nab initio calculations.\nOur magnetization results also reveal that the easy-\nmagnetization direction holds onto the [111] axis only at\nlow temperatures up to Tr(= 5:9 K) whereas at higher\ntemperatures up to TCthe easy-magnetization direction\nis unambiguously along [001] and the paramagnetic cubic\nlattice is tetragonally distorted along this direction. This\nfinding is in good agreement with the theoretical calcu-\nlations which reveal the excited state with the [001] easy\nmagnetization 0:9 meVabove the ground state.\nThe magnetic moment reorientation transition in\nU4Ru7Ge6atTris manifested in specific features\n(anomalies)whichweobservedinthetemperaturedepen-\ndencies of magnetization (Figure 4a), AC susceptibility\n(Figure 4b), specific heat (Figure 8), thermal expansion\n(Figure 9) and electrical resistivity (Figure 11). It is in\nfact an order-to-order magnetic phase transition accom-\npanied by structural distortion due to notable magne-\ntoelastic coupling. These phase transitions are known to\nbe of the first order type (e.g. HoAl 235). The thermal\nexpansion anomalies seen in Figure 9 at TCandTr, re-\nspectively, may be viewed as an illustrative examples of\nthe second and first order type phase transitions. How-\never, thefirst order phase transitionis to be accompanied\nby latent heat which we were unable to detect by detailed\nspecific-heat measurement and the Trrelated specific-\nheat anomaly is considerably broader than expected for\na first order phase transition. We attribute the lack of\nobservables pointing to the presence of latent heat to the\ncomplex domain structure processes during the spin re-\norientation in the multidomain sample in the vicinity of\nTrand our tentative determination has to be confirmed\nby a designed method allowing determination of order\ntype by other means (e.g. phase coexistence in \u0016SR).\nThe anisotropy field values in U ferromagnets are typi-\ncally hundreds T whereas in U4Ru7Ge6it is roughly 3 or-\ndersofmagnitudesmallervalue. Wheninspectingcrystal\nstructures we observe that in all cases of the U ferromag-\nnets characterized by high values of anisotropy field the\nU ions have some U nearest neighbors. Contrary, the in-\ndividualUionsin U4Ru7Ge6areburiedinsidetheRuand\nGepolyhedrapreventingdirectconnectiontoanynearest\nU ion which should have consequences for magnetism36.The direct 5 f-5foverlap of U electron orbitals is prob-\nably behind the huge magnetic anisotropy of U com-\npounds. The symmetry of the network of U nearest\nneighbors determines the type of magnetic anisotropy in\nthese materials11. The driving mechanism of magnetic\nanisotropy in U4Ru7Ge6is presumably the hybridization\nof U 5f-electron states with the 4 d-electron states of sur-\nrounding Ru ions which are in hexagonal arrangement,\nperpendicular to the easy magnetization direction [111].\nOnset of itinerant electron ferromagnetism is usually\naccompanied by a positive spontaneous magnetovolume\neffect37,38. Also our thermal expansion data show this\ntendency despite the negative value of \u0015S[001]. Unfortu-\nnately,themeasurementsusingdilatometercannotbeex-\ntended to temperatures lower than Trbecause the body\ndiagonals representing the [111] easy magnetization di-\nrection are not perpendicular and therefore an experi-\nment analogous to that for Tr<T <T Cis not accessible.\nV. CONCLUSIONS\nWe have demonstrated by experiment and by calcula-\ntions the almost isotropic ferromagnetism in U4Ru7Ge6\nwhich represents a case contrasting the huge magne-\ntocrystalline anisotropy in so far reported ferromagnetic\nuraniumcompoundsexhibitingbyruleatleast3ordersof\nmagnitude larger values of anisotropy field. The ground-\nstate easy-magnetization direction is along the [111] axis\nof the cubic lattice as found both by experiment and by\ncalculations. At Tr= 5:9 Kthe easy magnetization direc-\ntion changes for [100] which holds at temperatures up to\nTCas found by experiments. This is in agreement with\ncalculations revealing that the moment pointing to the\n[001]directionischaracteristicfortheexcitedstatewhich\nis only 0:9 meVabove the ground state. This spin reori-\nentation transition is significantly projected in low-field\nmagnetization, AC susceptibility and thermal-expansion\ndata and causes also a weak anomaly at Trvisible in the\ntemperature dependence of specific heat and electrical\nresistivity, respectively.\nThe magnetoelastic interaction induces a rhombo-\nhedral (tetragonal) distortion of the paramagnetic cu-\nbic crystal lattice in case of the [111]([001]) easy-\nmagnetization direction. The rhombohedral distortion\nleads to emergence of two crystallographically inequiv-\nalent U sites which is also in agreement with results of\ncalculations. We propose a scenario of the very weak\nmagnetic anisotropy of U4Ru7Ge6connected with the\nlack of direct 5 f-5foverlaps of orbitals of nearest U –\nU neighbors. These are prevented by the specific crys-\ntal structure in which the individual U ions are trapped\ninside of the Ru, Ge polyhedra.9\nACKNOWLEDGMENTS\nTheauthorsareindebtedtoJanProkleškaforchecking\nthe manuscript, providing critical insight and construc-\ntive suggestions. This work was supported by the CzechScience Foundation Grant No. P204/16/06422S and by\nCharles University, project GA UK No. 720214. Ex-\nperiments were performed in MLTL ( http://mltl.eu/ )\nwhichissupportedwithintheprogramofCzechResearch\nInfrastructures (Project No. LM2011025).\n\u0003michal.valiska@gmail.com\n1F. Bloch and G. Gentile, Zeitschrift für Physik 70, 395\n(1931).\n2J. H. van Vleck, Physical Review 52, 1178 (1937).\n3J. F. Herbst, Reviews of Modern Physics 63, 819 (1991).\n4M. D. Kuz’min and A. M. Tishin, “Chapter three theory\nof crystal-field effects in 3d-4f intermetallic compounds,” in\nHandbook of Magnetic Materials , Vol. Volume 17, edited\nby K. H. J. Buschow (Elsevier, 2007) pp. 149–233.\n5D. D. Koelling, B. D. Dunlap, and G. W. Crabtree, Phys-\nical Review B 31, 4966 (1985).\n6M. S. S. Brooks and P. J. Kelly, Physical Review Letters\n51, 1708 (1983).\n7V. Sechovský, Z. Smetana, G. Hilscher, E. Gratz, and\nH. Sassik, Physica B+C 102, 277 (1980).\n8J. M. Fournier, A. Boeuf, P. Frings, M. Bonnet, J. v.\nBoucherle, A. Delapalme, and A. Menovsky, Journal of\nthe Less Common Metals 121, 249 (1986).\n9L. Severin, L. Nordström, M. S. S. Brooks, and B. Johans-\nson, Physical Review B 44, 9392 (1991).\n10P. H. Frings, J. J. M. Franse, A. Menovsky, S. Zemirli, and\nB. Barbara, Journal of Magnetism and Magnetic Materials\n54, 541 (1986).\n11V. Sechovský and L. Havela, “Chapter 1 magnetism of\nternary intermetallic compounds of uranium,” in Handbook\nof Magnetic Materials , Vol. Volume 11 (Elsevier, 1998) pp.\n1–289.\n12G.BuschandO.Vogt,JournaloftheLessCommonMetals\n62, 335 (1978).\n13C. F. Buhrer, Journal of Physics and Chemistry of Solids\n30, 1273 (1969).\n14J. L. Smith and E. A. Kmetko, Journal of the Less Com-\nmon Metals 90, 83 (1983).\n15O. Eriksson, M. S. S. Brooks, B. Johansson, R. C. Albers,\nand A. M. Boring, Journal of Applied Physics 69, 5897\n(1991).\n16V. Sechovský, L. Havela, H. Nakotte, F. R. de Boer, and\nE. Brück, Journal of Alloys and Compounds 207-208 , 221\n(1994).\n17B. R. Cooper, R. Siemann, D. Yang, P. Thayamballi, and\nA. Banerjea, Hybridization-induced anisotropy in cerium\nand actinide systems (North-Holland, Netherlands, 1985).\n18B.R.Cooper, G.J.Hu, N.Kioussis, andJ.M.Wills,Jour-nal of Magnetism and Magnetic Materials 63, 121 (1987).\n19A. A. Menovsky, Journal of Magnetism and Magnetic Ma-\nterials 76-77, 631 (1988).\n20H. Rietveld, Journal of Applied Crystallography 2, 65\n(1969).\n21J. Rodriguez-Carvajal, Physica B: Condensed Matter 192,\n55 (1993).\n22T. Roisnel and J. Rodriguez-Carvajal, in EPDIC 7 - Sev-\nenth European Powder Diffraction Conference , edited by\nR. Delhez and E. Mittemeijer (Trans Tech Publications).\n23M.Rotter, H.Müller, E.Gratz, M.Doerr, andM.Loewen-\nhaupt, Review of Scientific Instruments 69, 2742 (1998).\n24B.Lloret, B.Buffat, B.Chevalier, andJ.Etourneau,Jour-\nnal of Magnetism and Magnetic Materials 67, 232 (1987).\n25S. A. M. Mentink, G. J. Nieuwenhuys, A. A. Menovsky,\nand J. A. Mydosh, Journal of Applied Physics 69, 5484\n(1991).\n26E. Du Trémolet de Lacheisserie, D. Gignoux, and\nM. Schlenker, Magnetism (Springer, 2005).\n27K. Koepernik and H. Eschrig, Physical Review B 59, 1743\n(1999).\n28A. Arrott, Physical Review 108, 1394 (1957).\n29A. Arrott and J. E. Noakes, Physical Review Letters 19,\n786 (1967).\n30B. Widom, J. Chem. Phys. 43, 3892 (1965).\n31P. Weiss, J. Phys. Theor. Appl. 6, 661 (1907).\n32H. Yamada, Physics Letters A 55, 235 (1975).\n33A. V. Andreev, “Chapter 2 thermal expansion anoma-\nlies and spontaneous magnetostriction in rare-earth inter-\nmetallics with cobalt and iron,” in Handbook of Magnetic\nMaterials , Vol. Volume 8 (Elsevier, 1995) pp. 59–187.\n34E. R. Callen and H. B. Callen, Physical Review 129, 578\n(1963).\n35P. Durga, K. P. Arjun, V. K. Pecharsky, and\nJ. K. A. Gschneidner, Journal of Physics: Condensed Mat-\nter25, 396002 (2013).\n36S. F. Matar, B. Chevalier, and R. Pöttgen, Solid State\nSciences 27, 5 (2014).\n37M. Shimizu, Physica B: Condensed Matter 159, 26 (1989).\n38P. Mohn, K. Schwarz, and D. Wagner, Physica B: Con-\ndensed Matter 161, 153 (1990)."    },    {        "title": "1611.08713v2.Specific_Heat_of_Spin_Excitations_Measured_by_FerromagneticResonance.pdf",        "content": "arXiv:1611.08713v2  [cond-mat.mtrl-sci]  15 Sep 2020Specific Heat of Spin Excitations Measured by Ferromagnetic Resonance\nBenjamin W. Zingsem\nFaculty of Physics and Center for Nanointegration (CENIDE) ,\nUniversity Duisburg-Essen, 47057 Duisburg, Germany and\nErnst Ruska Centre for Microscopy and Spectroscopy with Ele ctrons and Peter Grünberg Institute,\nForschungszentrum Jülich GmbH, 52425, Jülich, Germany\nMichael Winklhofer\nSchool of Mathematics and Science, Carl-von-Ossietzky Uni versity Oldenburg, 26129 Oldenburg\nSabrina Masur\nCavendish Laboratory, Department of Physics, JJ Thomson Av enue, Cambridge CB3 0HE, United Kingdom\nPaul Wendtland and Ralf Meckenstock\nFaculty of Physics and Center for Nanointegration (CENIDE) ,\nUniversity Duisburg-Essen, 47057 Duisburg, Germany\nRuslan Salikhov\nFaculty of Physics and Center for Nanointegration (CENIDE) ,\nUniversity Duisburg-Essen, 47057 Duisburg, Germany and\nHelmholtz-Zentrum Dresden-Rossendorf, Institute of Ion B eam Physics and Materials Research,\nBautzner Landstrasse 400, 01328 Dresden, Germany\nMichael Farle\nFaculty of Physics and Center for Nanointegration (CENIDE) ,\nUniversity Duisburg-Essen, 47057 Duisburg, Germany and\nKirensky Institute of Physics, Federal Research Center \"Kr asnoyarsk Science Centre,\nSiberian Branch of the Russian Academy of Science\nUsing ferromagnetic-resonance spectroscopy (FMR), we inv estigate the anisotropic properties of\nepitaxial 3nmPt/2nmAg/10nm Fe/10nm Ag/GaAs(001) films in fully saturated meta-stable states\nat temperatures ranging from 70Kto280K. By comparison to spin-wave theory calculations, we\nidentify the role of thermal fluctuation of magnons in overco ming the energy barrier associated with\nthese meta-stable states. We show that the energy associate d with the size of the barrier that bounds\nthe meta-stable regime is proportional to the heat stored in the magnonic bath. Our findings offer\nthe possibility to measure the magnonic contribution to the heat capacity by FMR, independent of\nother contributions at temperatures ranging from 0Kto ambient temperature and above. The only\nrequirement being that the selected sample exhibits magnet ic anisotropy, here, magnetocrystalline\nanisotropy.\nI. INTRODUCTION\nKnowledge of the magnetic contribution to the ther-\nmal properties of physical systems is essential for un-\nderstanding magnetocalorics[ 1], spintronics[ 2,3], and\nmagnonic[ 3,4] applications. Also, in biomedical appli-\ncations like hyperthermia [ 5,6], it is essential to assess\nthe coupling between the magnetic contributions and the\ncrystal system’s thermal properties. However, measur-\ning the contribution of magnons to the heat capacity of\nmagnetic materials at temperatures of technical interest\nis challenging. One approach is to suppress magnons by\nshifting the spin-wave dispersion to energies higher than\nthe thermal energy. The magnon contribution to the to-\ntal heat can then be separated by conventional calorime-\ntry experiments with and without an applied field. Doing\nso for a system like Yittrium Iron Garnet (YIG), how-\never, would already require fields as high as 30T[7] at atemperature of only 20K. Hence, it is unlikely that such\nexperiments can get close to ambient temperature. An-\nother option is to measure the thermal population of the\nspin-wave dispersion using, for example, Brillouin light\nscattering (BLS). Here we propose a new method for de-\ntermining the magnonic heat capacity using ferromag-\nnetic resonance (FMR). Our method is based on ferro-\nmagnetic resonance measurements at a fixed frequency,\nwhere a magnetic field is applied in different directions\nnear a hard axis of the magnetic anisotropy. First, the\nmagnet is fully saturated, then the direction of the field\nis swept, such that this saturated configuration becomes\nmetastable all the while FMR is measured at each angle\nstep. We then observe the transition from this metastable\nconfiguration to a stable configuration in the FMR signal.\nFrom these unconventional FMR measurements, we have\nevaluated the Zeeman energy in critical configurations of\nmetastable states. We find that in the observed temper-2\nature regime between 70Kand280K the temperature-\ndependent change of the critical Zeeman energy is pro-\nportional to the magnonic heat capacity.\nII. EXPERIMENTAL PROCEDURE\nFerromagnetic resonance measure-\nments were conducted on an epitaxial\n3nmPt/2nmAg/10nm Fe/10nm Ag/GaAs(001) thin\nfilm in which the (001)-direction of the Fe layer points\nout of the sample plane. For such systems the Helmholtz\nfree energy density is known to have the form\nF/parenleftBig\n/vectorB,/vectorM/parenrightBig\n=−/vectorM(θ,φ)·/vectorB(θB,φB)+1\n2µ0M2cos2(θ)\n+2K⊥\n2sin2(θ)+1\n4K4/parenleftbig\nsin2(2θ)+sin4(θ)sin2(2φ)/parenrightbig\n(1)\nincluding the Zeeman contribution /vectorM·/vectorBas discussed in\n[8]. The angles are given in spherical coordinates, where\nθis the polar out-of-plane, and φthe azimuthal in-plane\nangle of the magnetization, θBandφBare the polar and\nazimuthal angle of the external magnetic field /vectorBrespec-\ntively. The anisotropy parameter K4accounts for the\ncubic magnetocrystalline anisotropy, K⊥\n2for a uniaxial\nanisotropy. The thin-film shape-anisotropy is described\nby the demagnetization term1\n2µ0M2cos2(θ).\nFor the magnetization to be resonantly excited,\n/vectorM(θ,φ)must be a minimizer of this energy landscape.\nIn this case, the magnetization is in an equilibrium state\nand the resonance condition is described by the curva-\nture of the energy landscape Eq. 1[9]. According to [ 9],\nthis holds, even if it is a local minimum. In this case, the\nequilibrium is metastable, such that the energy landscape\npresents another - stable - equilibrium. Hence, FMR is\nexpected to occur in metastable states. To create such\nmetastable states in the experiment, we sweep the angle\nof the applied field at fixed field strengths. The effect\nof such an angle sweep on the energy landscape at suffi-\nciently low fields is illustrated in Fig. 1, which shows the\nin-plane component ( θ=θB=π\n2) of the Energy land-\nscape (Eq. 1) as a function of the in-plane angle φof\nthe magnetization. First, a magnetic field is applied in\nan easy direction at φB= 0(orange curve). Next, the\ndirection of this applied field is changed, i.e., φBis in-\ncreased. Once it is pointing parallel to the hard direction\natφB=π\n4(brown curve), two states of equal energy are\npresent. Pushing the angle of the applied field further\nbeyondφB=π\n4makes the state at φ <π\n4unfavorable\ncompared to the state at φ >π\n4. It is now metastable.\nHowever, it is still separated from the more favorable\nstate at φ >π\n4by an energy barrier. For even larger\nangles of φB, the minimum at φ <π\n4turns into a saddle\npoint (pink curve) and becomes unstable. The black dots\nmark the minimum, i.e., the magnetization’s equilibrium\nalignment for the state at φ <π\n4.\nThe experimental procedure for such an angle sweep is\nFigure 1. A plot of the energy landscape at different applied\nfield angles for a fixed field magnitude. The black points\nillustrate the position of the minimum that determines the\norientation of the magnetization. Black lines separate the un-\nstable (marked “Un”), metastable (marked “Meta”), and stab le\n(marked “Stable”) regions for this state.\nschematically shown in Fig. 2. First, a sufficiently large\nfield of300mT is applied along the magnetocrystalline\neasy axis of the sample to fully align the magnetization\nand overcome all anisotropies. Then the field is reduced\nto the desired field value for the angle sweep, and the\nangleφBis swept from the easy axis ( φB= 0) across\nthe hard axis ( φB=π\n4) in steps of 0.1◦toφB=π.\nOne angle step takes about 0.3sto measure. During this\nsweep, the FMR signal is recorded. The sharp disconti-\nnuity in the FMR signal, which is indicated as a black\nline in Fig. 2b), pinpoints the field-angle configurations\nat which the magnetization transitions from a metastable\nto a stable equilibrium. For the applied field magnitudes\nin this measurement, we find that the transition happens\nwithin a few degrees off the hard axis. The dashed curve\nin Fig. 2b) shows the points at which the minimum\ntransitions into a saddle-point, i.e., the point at which\nthe metastable equilibrium vanishes. The differentce be-\ntween the black curve and the dashed curve along the\nhorizontal axis corresponds to the thermal fluctuation\nfield [ 10]. These measurements were performed at var-\nious temperatures, and the curves that follow the mea-\nsured discontinuities at each temperature are depicted in\nFig.3. With increasing temperature, the discrepancy\nbetween the points at which the transition happens, and\nthose at which the metastable state vanishes, becomes\nlarger. Hence, with increasing temperature, the magne-\ntization overcomes larger barriers.\nIII. RESULTS AND DISCUSSION\nWe find that, as we decrease the temperature, the crit-\nical angle offset from the hard axis at which the magneti-\nzation transitions from its metastable into a stable equi-\nlibrium increases and approaches the angle predicted for\nzero temperature, as shown in Fig. 3. The field regime3\nFigure 2. a) Schematic representation of the measurement\nprocedure, using a measured spectrum (compare b) ) for illus -\ntration. b) Low field section of the measured FMR signal of a\n10nm Fe(100) film measured at 9.535GHz at room tempera-\nture. The straight line at 45◦indicates the magnetocrystalline\nhard direction. The curved line shows the critical angles at\nwhich the metastable regime ends, and the dashed curved line\nindicates the boundary of the metastable regime as predicte d\nusing the theoretical model in ref. [ 9]. The difference between\nthe black curve and the dashed curve along the horizontal axi s\ncorresponds to the thermal fluctuation field[ 10].\nfrom20mT to43mT was chosen because here, the an-\ngles could be well distinguished, and the sample is fully\nsaturated.\nFor all temperatures, the resonance field was extracted\nfrom the FMR spectra, and fitted, solving the commonly\nused eq. 2[8,11–13] to determine the anisotropy param-\neters.\n/parenleftbiggω\nγ/parenrightbigg2\n=1\nM2sin2(θ)/parenleftBigg\nd2\ndθ2Fd2\ndφ2F−/parenleftbiggd2\ndθdφF/parenrightbigg2/parenrightBigg\n(2)\nFor the fits, the experimental resonance field was used in\nthe calculation to determine the orientation of the mag-\nFigure 3. The boundary of the metastable regime for the sam-\nple measured in-plane from the hard-axis of the magnetocrys -\ntalline anisotropy. Points with error bars represent measu red\nvalues. Solid lines are linear interpolations used later to de-\ntermine the temperature derivative at different fields.\n5010015020025030048505254565860\nTemperature [K]K4[kJ/m3]\nFigure 4. Cubic crystal anisotropy K4as a function of tem-\nperature. Blue points with error bars are the values obtaine d\nfrom fitting. The red curve is a third-order polynomial inter -\npolation to serve as a guide to the eye.\nnetization vector that minimizes the free energy density.\nThis magnetization vector was then used in Eq. 2to cal-\nculate the resonance field position for the respective fre-\nquency at a given set of anisotropy parameters. The so\ndetermined cubic anisotropy shown in Fig. 4is in agree-\nment with literature data [ 8]. The out of plane uniaxial\nanisotropy K⊥\n2and the magnetization did not change\nsignificantly in the given temperature regime. We obtain\nM= (1.71±0.01)·106A/mandK⊥\n2= (16±0.6)·103J/m3,\nusing a g-factor of 2.09[14].\nWith these values, we then evaluated the contribution\nof the Zeeman-energy FZeeman=−/vectorM·/vectorBto the free en-\nergy density in Eq. 1for all temperatures using the criti-\ncal angles φBfrom Fig. 3as the in-plane applied field an-\ngle. In this calculation, we fixed θBandθto90◦as they\nare constrained by the sample’s shape anisotropy. The\nmagnetization angle φis determined numerically by min-\nimization eq. 1within the metastable regime. This crit-\nical Zeeman energy density is the energy density that is4\n280 230 180 1301109070024681012\nTemperature [K]\u0001FZeeman\u0002\u0003f,Bf\u0004\n\u0001T[J\nm3K]\nFigure 5. Magnon specific heat as a function of temperature.\nThe red points show the central numerical derivative of the\nZeeman contribution to the free energy including error bars .\nThe black dashed curve is the result of numerically evaluati ng\nthe magnon specific heat eq. 4. The parameters we used for Fe\narea= 286.65pm [19]ωZB= 18THz [15]γ= 2.92·10101/T·s\n[14] andαD= 0.87[17].\nprovided to the system in order to perform the transition.\nTherefore we analyze its change as a function of tem-\nperature. Considering that the magnetization has some\namount of heat available to it in the form of magnons, we\nwould expect that, as we decrease the temperature, more\nZeeman energy is required to make the transition. More-\nover, since the Zeeman contribution is defined negative,\nwe expect that its change is proportional to the change\nof the magnon heat in the system. Therefore we calcu-\nlated the change of the thermal energy of the magnons\nthat is the heat capacity. To calculate the magnonic heat\ncapacity, we proceed as described in [ 15]. As of [ 16] the\nmagnon dispersion can be approximated as\nω(k) =γB+ωZB(1−cos/parenleftbiggπ\n2k\nkm/parenrightbigg\n) (3)\nwhereγis the magnetogyric ratio, Bis the magnetic flux,\nkm=αD3√\n6π2\nais the radius of the Debye-Sphere with\nthe lattice constant aand the scaling factor αD[17] that\napproximates the Brillouin zone, and ωZBis the magnon\nfrequency at the zone boundary. According to [ 15], the\nmagnon specific heat can then be written as\ncmagnon\nV=1\n(2π)3kmˆ\n0d3k(/planckover2pi1ω(k))2\nkbT2exp/parenleftBig\n/planckover2pi1ω(k)\nkBT/parenrightBig\n/parenleftBig\nexp/parenleftBig\n/planckover2pi1ω(k)\nkBT/parenrightBig\n−1/parenrightBig2\n(4)by calculating the temperature derivative of the inner en-\nergy of the magnons. A comparison between the temper-\nature derivative of the Zeeman contribution and the nu-\nmerically calculated magnon heat capacity of Fe is shown\nin Fig. 5. We find that, for each applied field, the curves\nare proportional and the data is in line with the models in\n[18] and [ 15]. This leads us to conclude that this experi-\nment grants direct access to the heat capacity of magnons\nat elevated temperatures. This technique can be applied\nto any ferromagnet since any ferromagnet can be tailored\nto have any desired anisotropy simply by shape, even if\nthe intrinsic magnetocrystalline anisotropy is small.\nIV. SUMMARY\nIn a magnetization configuration non-collinear with the\nexternal field, we have shown that FMR modes exist in\nmetastable magnetic states as predicted in [ 9]. We also\nfind that the magnonic heat capacity of iron is propor-\ntional to the temperature derivative of the Zeeman en-\nergy at the critical points in the unconventional FMR\nangular dependence by comparison to spin-wave theory\ncalculations. We find a good agreement between the mea-\nsured data and the calculation. The magnitude of of the\nmagnon contribution to the specific heat shown in Fig.\n5is also in line with other works[ 7,16,17,20]. Our re-\nsults suggest that measuring the temperature-dependent\nsize of the energy barrier, which confines a saturated\nmeta-stable magnetic state, presents a new method for\ndetermining the magnon contribution to the specific heat.\nThe only requirement for these measurements is that the\nsample exhibits magnetic anisotropy. This can either\nbe magnetocrystalline anisotropy or shape anisotropy in\npatterned magnetic shapes.\nACKNOWLEDGMENTS\nIn part funded by the Deutsche Forschungsgemein-\nschaft (DFG, German Research Foundation) – Project-\nID 405553726 – TRR 270“. This study was supported in\npart by the Research Grant No. 075-15-2019-1886 from\nthe Government of the Russian Federation.\n[1] O. Gutfleisch, M. A. Willard, E. Brück, C. H. Chen,\nS. G. Sankar, and J. P. 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Salikhov et al., “Arrangement at\nthe nanoscale: Effect on magnetic particle hyperther-\nmia,”Scientific reports , vol. 6, p. 37934, 2016.\n[6] S. Liébana-Viñas, K. Simeonidis, U. Wiedwald, Z.-A.\nLi, Z. Ma, E. Myrovali, A. Makridis, D. Sakellari,\nG. Vourlias, M. Spasova et al., “Optimum nanoscale de-\nsign in ferrite based nanoparticles for magnetic particle\nhyperthermia,” Rsc Advances , vol. 6, no. 77, pp. 72 918–\n72925, 2016.\n[7] S. M. Rezende and J. C. López Ortiz, “Ther-\nmal properties of magnons in yttrium iron gar-\nnet at elevated magnetic fields,” Phys. Rev. B ,\nvol. 91, p. 104416, Mar 2015. [Online]. Available:\nhttp://link.aps.org/doi/10.1103/PhysRevB.91.104416\n[8] M. Farle, “Ferromagnetic resonance of ultrathin\nmetallic layers,” Reports on Progress in Physics ,\nvol. 61, no. 7, p. 755, 1998. [Online]. Available:\nhttp://stacks.iop.org/0034-4885/61/i=7/a=001\n[9] B. W. Zingsem, M. Winklhofer, R. Mecken-\nstock, and M. Farle, “Unified description of\ncollective magnetic excitations,” Phys. Rev. B ,\nvol. 96, p. 224407, Dec 2017. [Online]. Available:\nhttps://link.aps.org/doi/10.1103/PhysRevB.96.224407\n[10] S. Bance, H. Oezelt, T. Schrefl, M. Winklhofer,\nG. Hrkac, G. Zimanyi, O. Gutfleisch, R. F. L. Evans,\nR. W. Chantrell, T. Shoji, M. Yano, N. Sakuma,\nA. Kato, and A. Manabe, “High energy product\nin battenberg structured magnets,” Applied Physics\nLetters , vol. 105, no. 19, pp. –, 2014. [Online]. Available:\nhttp://scitation.aip.org/content/aip/journal/apl/10 5/19/10.1063/1.4897645\n[11] H. Suhl, “Ferromagnetic resonance in nickel ferrite\nbetween one and two kilomegacycles,” Phys. Rev. ,\nvol. 97, pp. 555–557, Jan 1955. [Online]. Available:\nhttp://link.aps.org/doi/10.1103/PhysRev.97.555.2[12] J. Smit and H. G. Beljers, “Ferromagnetic resonance ab-\nsorption in BaFe12O19, a highly anisotropic crystal,”\nPhilips Research Reports , vol. 10, pp. 113–130, 1955.\n[13] G. Giannopoulos, R. Salikhov, B. Zingsem, A. Markou,\nI. Panagiotopoulos, V. Psycharis, M. Farle, and D. Niar-\nchos, “Large magnetic anisotropy in strained Fe/Co\nmultilayers on AuCu and the effect of carbon doping,”\nAPL Mater. , vol. 3, no. 4, pp. –, 2015. [Online]. Available:\nhttp://scitation.aip.org/content/aip/journal/aplmat er/3/4/10.1063/1.4919058\n[14] Z. Frait and R. Gemperle, “The g-factor and surface mag-\nnetization of pure iron along [100] and [111] directions,”\nLe Journal de Physique Colloques , vol. 32, no. C1, pp.\nC1–541, 1971.\n[15] C. Kittel, Quantum theory of solids . New York: Wiley,\n1963.\n[16] S. M. Rezende, R. L. Rodríguez-Suárez, R. O.\nCunha, A. R. Rodrigues, F. L. A. Machado,\nG. A. Fonseca Guerra, J. C. Lopez Ortiz, and\nA. Azevedo, “Magnon spin-current theory for the\nlongitudinal spin-seebeck effect,” Phys. Rev. B ,\nvol. 89, p. 014416, Jan 2014. [Online]. Available:\nhttp://link.aps.org/doi/10.1103/PhysRevB.89.014416\n[17] S. M. Rezende, R. L. Rodríguez-Suárez, J. C.\nLopez Ortiz, and A. Azevedo, “Thermal properties of\nmagnons and the spin seebeck effect in yttrium iron\ngarnet/normal metal hybrid structures,” Phys. Rev.\nB, vol. 89, p. 134406, Apr 2014. [Online]. Available:\nhttp://link.aps.org/doi/10.1103/PhysRevB.89.134406\n[18] L. Wojtczak, A. Urbaniak-Kucharczyk, J. Mielnicki, an d\nB. Mrygon, “The magnetic contribution to specific heat\nin thin ferromagnetic films,” physica status solidi (a) ,\nvol. 76, no. 1, pp. 113–120, 1983. [Online]. Available:\nhttps://onlinelibrary.wiley.com/doi/abs/10.1002/pss a.2210760112\n[19] P. Kohlhaas, R. Donner and N. Schmitz-Pranghe, “Über\ndie temperaturabhangigkeit der gitterparameter von\neisen, kobalt und nickel im bereich hoher temperaturen,”\nZ. Angew. Phys. , vol. 23, pp. 245–249, 1967.\n[20] S. R. Boona and J. P. Heremans, “Magnon thermal mean\nfree path in yttrium iron garnet,” Physical Review B ,\nvol. 90, no. 6, p. 064421, 2014."    },    {        "title": "1611.08940v2.Multiscale_examination_of_strain_effects_in_Nd_Fe_B_permanent_magnets.pdf",        "content": "Multiscale examination of strain e\u000bects in Nd{Fe{B permanent magnets\nMin Yi,1,\u0003Hongbin Zhang,1,yOliver Gut\reisch,1,zand Bai-Xiang Xu1,x\n1Institute of Materials Science, Technische Universit at Darmstadt, 64287 Darmstadt, Germany\n(Dated: July 17, 2017)\nWe have performed a combined \frst-principles and micromagnetic study on the strain e\u000bects\nin Nd{Fe{B permanent magnets. First-principles calculations on Nd 2Fe14B reveal that the mag-\nnetocrystalline anisotropy ( K) is insensitive to the deformation along caxis and the abin-plane\nshrinkage is responsible for the Kreduction. The predicted Kis more sensitive to the lattice de-\nformation than what the previous phenomenological model suggests. The biaxial and triaxial stress\nstates have a greater impact on K. NegativeKoccurs in a much wider strain range in the abbiaxial\nstress state. Micromagnetic simulations of Nd{Fe{B magnets using \frst-principles results show that\na 3\u00004% local strain in a 2-nm-wide region near the interface around the grain boundaries and\ntriple junctions leads to a negative local Kand thus remarkably decreases the coercivity by \u001860%\nor 3\u00004 T. The local abbiaxial stress state is more likely to induce a large loss of coercivity. In\naddition to the local stress states and strain levels themselves, the shape of the interfaces and the\nintergranular phases also makes a di\u000berence. Smoothing the edge and reducing the sharp angle of\nthe triple regions in Nd{Fe{B magnets would be favorable for a coercivity enhancement.\nI. INTRODUCTION\nStrain can be utilized to tailor the magnetic properties\nof many materials, leading to either promising applica-\ntions or undesirable problems. For example, strain e\u000bects\nin soft magnetic materials can be used for the electric\ncontrol of magnetic properties by using the strain medi-\nated magnetoelectric coupling [1, 2]. In addition, strain\nmediated magnetization switching has been a potential\nway to revolutionize the spintronic devices that currently\nutilize power-dissipating currents [3{7]. In the perma-\nnent magnets which are featured by high coercivity and\nhigh maximal energy product, local strain around the\ngrain boundaries and triple junctions is thought to reduce\nthe local magnetocrystalline anisotropy and thus the\ncoercivity [8{14], degrading the magnetic performance.\nThese indicate strain as a double-edged sword in mag-\nnetic materials. Understanding its e\u000bects is prerequisite\nfor a wise application or avoidance of this double-edged\nsword.\nIn this work, we focus on the strain e\u000bects in a typi-\ncal permanent magnet Nd{Fe{B. In Nd{Fe{B magnets,\nstrain e\u000bects are inevitable. On one hand, sintering pro-\ncesses, post-thermal treatments, and hot pressing un-\navoidably induce a certain residual strain. Such strain\ncan be either at the bulk level or at the local level. On\nthe other hand, the coercivity of standard Nd{Fe{B mag-\nnets is only\u001820% of the theoretical upper limit from the\nStoner{Wohlfarth model. The huge deviation from the\ntheoretical prediction is believed to be mainly originated\nfrom the microstructural e\u000bects [8, 15{18]. The critical\nmicrostructural features that a\u000bect the coercivity are the\n\u0003yi@mfm.tu-darmstadt.de\nyhzhang@tmm.tu-darmstadt.de\nzgut\reisch@fm.tu-darmstadt.de\nxxu@mfm.tu-darmstadt.deintergrain phases and grain boundary phases. The struc-\ntural or crystal-orientation mismatch between Nd 2Fe14B\nmain phase and other phases will generate local strain\nnear the interfaces of di\u000berent phases or grains. It is pos-\nsible that such local strain results in regions of reduced\nanisotropy as nucleation sites for reversal domains.\nFor the theoretical study of strain e\u000bects in Nd{Fe{\nB magnets, by using the phenomenological theory re-\ngarding the magnetoelastic anisotropy [19], Hrkac et al.\n[9, 11{14] and Kubo et al. [10] used molecular dynamics\n(MD) to determine the strain induced anisotropy con-\nstant (Kme). However, depending on the interatomic po-\ntential used in MD, the value of calculated Kmecan di\u000ber\nin one order of magnitude. For example, based on a pair-\nwise interaction model for Nd 2Fe14B, Hrkac et al. con-\nsidered various crystal structures and crystal orientations\nof Nd and Nd oxides and evaluated maximum values of\nKme\u0018\u000010{\u00004000 MJ/m3in single atoms (average Kme\nfor all atoms:\u0018\u00001{\u000010 MJ/m3) in a\u00182-nm local re-\ngion [9, 11, 12]. In contrast, Kubo et al. [10] developed\na new angular-dependent potential model for Nd 2Fe14B\nand estimated Kmein the order of\u00000:1 MJ/m3within\na\u00182-nm region. However, no experimental results have\ndirectly veri\fed this 2-nm local region with extremely\nreduced magnetocrystalline anisotropy. In fact, early ex-\nperiments showed that the homogeneous thermal strain\npresent at the boundaries of Nd 2Fe14B grains has only\na small in\ruence on the coercivity [19]. More recently,\nMurakami et al. [20, 21] directly measured the strain\ndistribution around di\u000berent interfaces in sintered Nd{\nFe{B magnets. They demonstrated that the region with\na strain of \"c\u0018 \u0006 1% was extended over several tens\nnanometer (not the theoretical prediction of \u00182 nm con-\n\fned to a local region) away the interface. Similar to the\nearly experiments [19], they also speculated that the in-\nterfacial strains have limited in\ruence on the coercivity.\nOne plausible reason for the inconsistence between sim-\nulations and experiments is the experimental resolution\nlimitations, i.e. presently it is di\u000ecult to measure thearXiv:1611.08940v2  [cond-mat.mtrl-sci]  14 Jul 20172\nstrain within a\u00182-nm-wide local region in these experi-\nments [20]. Therefore, in terms of the inconsistence not\nonly between previous di\u000berent MD simulations them-\nselves but also between the simulations and experimen-\ntal measurements up to now, in the modelling aspect it\nis highly required that this issue be more precisely inves-\ntigated at a quantitative or multiscale level.\nIn the present work, we perform a combined \frst-\nprinciples and micromagnetic study on Nd{Fe{B mag-\nnets, in order to demonstrate a multiscale simulation\nframework for elucidating the strain e\u000bects on Nd{Fe{B\nmagnets and to clarify what kind of local strain can sig-\nni\fcantly reduce the coercivity. Previous \frst-principles\ncalculations have provided insights into the magnetic mo-\nments and the magnetocrystalline anisotropy based on\neither the crystal \feld of Nd ions [22{27] or the total\nenergy di\u000berence [28, 29]. Especially, Suzuki et al. [23]\nexplored the crystal \feld parameter of Nd ions in the\ncase of changing the length of a-axes andc-axis. Asali et\nal. [30] showed the dependence of magnetic anisotropy\nonc=aratio of X 2Fe14B (X=Y, Pr, Dy) and Torbatian et\nal. [31] examined triaxial-strain e\u000bects on the magnetic\nanisotropy in Y 2Fe14B. But they did not report results\nfor Nd 2Fe14B. So strain e\u000bects of Nd 2Fe14B in di\u000berent\nforms and magnitudes scrutinized from \frst principles\nare still of interests. By using the \frst-principles results\nas inputs, we carry out further micromagnetic simula-\ntions to elucidate the strain e\u000bects on the coercivity of\nsingle- and multi-grain Nd{Fe{B magnets.\nII. METHODOLOGY\nThe \frst-principles calculations were carried in the\nframework of the projector augmented-wave formalism\nas implemented in the Vienna ab initio simulation pack-\nage (VASP) [32]. The Perdew{Burke{Ernzerhof (PBE)\nexchange-correlation functional in the generalized gradi-\nent approximation (GGA) was employed [33]. According\nto the previous work [28], an energy cuto\u000b of 400 eV and\na Monkhorst{Pack k{mesh 5\u00025\u00024 were utilized to reach\na good convergence. The convergence criteria for the full\nstructure relaxation at di\u000berent stress states and strain\nlevels were set as 10\u00005eV and 10\u00003eV/\u0017A for the ener-\ngies and forces, respectively [28]. To obtain the magne-\ntocrystalline anisotropy ( K), 4f electrons are treated as\nvalance electrons [28]. Non-self-consistent calculations\nwith di\u000berent spin quantization axes were done by in-\ncluding spin-orbit coupling, starting from self-consistent\ncharge densities of spin-polarized calculations. In this\nway,Kwas evaluated as the change of such total en-\nergies when the magnetization was along di\u000berent axes,\ni.e.\nK=(\n[max (Ea;Eb)\u0000Ec]=V (Ea>EcandEb>Ec)\n[min (Ea;Eb)\u0000Ec]=V (Ea<EcorEb<Ec)\n(1)in whichVis the volume of the relaxed unit cell. Ea,Eb,\nandEcare the total energy when the magnetization was\nparallel toa,b, andcaxis, respectively. Positive Kindi-\ncates easy axis along caxis, while negative Kindicates\nan easyabplane.\nUsingKandMs(saturation magnetization) obtained\nfrom \frst-principles calculations as functions of stress\nstates and strain levels, micromagnetic simulations were\ncarried out by the 3D NIST OOMMF code [34] for solv-\ning the Landau{Lifshitz{Gilbert (LLG) equation [35{37].\nSingle- and multi-grain Nd{Fe{B magnets are discretized\nby cubic meshes with a size of 1 nm. For the single\ngrain, prisms with di\u000berent sectional geometry were con-\nsidered. For modelling the multigrain, we used the scan-\nning electron microscopy (SEM) image of a sintered Nd{\nFe{B magnet from the previous experiments [20]. The\nexchange constant is set to be a constant of 12.5 pJ/m\n[38]. Hysteresis curves were calculated by setting the\ninitial magnetization along the positive caxis and the\nexternal \feld along negative caxis.\nIII. RESULTS AND DISCUSSION\nA. First-principles calculations of strained\nNd2Fe14B\nFigs. 1 and 2 present the \frst-principles results of K\nandMsunder various stress states and strain levels up\nto\u00067%. Bulk-level homogeneous strain in the above-\nmentioned range in real Nd 2Fe14B magnet is not realistic.\nHowever, there are several plausible sources of the very\nlarge local strain, such as lattice or crystal orientation\nmismatch between Nd 2Fe14B grains and the intergranu-\nlar phases, thermal residual stress at triple regions, grain\nirregularity for stress concentration, symmetry breaking\nat the grain surface or near the interface, etc. In ad-\ndition to these possible experimental phenomena, other\nreasons for introducing large strain range in this theo-\nretical work lie in three aspects. Firstly, Hrkac et al.\n[9, 11, 12, 14] and Kubo et al. [10] adopted MD sim-\nulations and indeed predicted a signi\fcant change in K\nthat is caused by the very large local strain within a\n\u00182-nm narrow interface region. Secondly, up to now no\nexperimental data on the strain in this localized region\nare available. The question on the magnitude of the ex-\ntremely localized strain near the interface is still open\nfrom the experimental viewpoint. Thirdly, large strain\nis possible in the local region. It is well known in me-\nchanics, theoretical strength of a material (inversely pro-\nportion to the square root of the atom layer distance)\ncan be 3 orders higher than measurable fracture tough-\nness (inversely proportion to the square root of the mi-\ncrocrack length) [44]. Therefore the grain surface layer\nof 2 nm can sustain large deformation without fracture.\nAlso as shown in the previous MD simulations [9{12, 14],\nthe local atomic arrangements at the grain boundaries\nor interfaces experience dramatic change, but are still3\n-8-6-4-2 024681.51.511.521.531.541.55\n c (%)  Ms (MA/m) \n-8-6-4-2 02468345678910\n c (%)  K (MJ/m3) \n-4 -3 -2 -1 0 1 2 3 4345678910\n DFT results\n exp. Yamada, JMMM,1986\n exp. Otani, JAP, 1987\n exp. Sagama, JAP, 1985\n exp. Koon, JAP, 1985\n exp. Givord, Physica, 1985\n  E[100] - E[001] (x106J/m3)△c/c (%)\nc \na \n-8-6-4-2 024680246810\n a (%)  K (MJ/m3) \n-8-6-4-2024681.51.511.521.531.541.55\n a (%)  Ms (MA/m) \nc \na \n-20-15-10 -5 051015200123456\n ab (%)  K (MJ/m3) \n-20 -10 0 10 201.5151.521.5251.53\n ab (%)  Ms (MA/m) \na b 𝛾𝑎𝑏 \n0 2 4 61.521.531.541.551.56\nPressure (GPa) Ms (MA/m) \n0 2 4 64.44.64.855.2\nPressure (GPa) K (MJ/m3) (a) \n(c) \n(e) \n(g) (b) \n(d) \n(f) \n(h) \n-4 -3 -2 -1 0 1 2 3 4345678910\n DFT results\n exp. Yamada, JMMM,1986\n exp. Otani, JAP, 1987\n exp. Sagama, JAP, 1985\n exp. Koon, JAP, 1985\n exp. Givord, Physica, 1985\n  E[100] - E[001] (x106J/m3)△c/c (%)\n-4 -3 -2 -1 0 1 2 3 4345678910\n DFT results\n exp. Yamada, JMMM,1986\n exp. Otani, JAP, 1987\n exp. Sagama, JAP, 1985\n exp. Koon, JAP, 1985\n exp. Givord, Physica, 1985\n  E[100] - E[001] (x106J/m3)△c/c (%)\nFIG. 1. First-principles calculated KandMsin the stress state of (a) (b) cuniaxial stress, (c) (d) auniaxial stress, (e) (f)\npure shear in abplane, and (g) (h) hydrostatic pressure. The inset lines in (a) are corresponding to the experimental results of\nunstrained single crystal from the literatures [39{43].\nstable without fracture. The locally dramatic change in\natomic arrangements can be considered as large e\u000bective\nstrain which is con\fned to the vicinity of interfacial re-\ngion. Our theoretical calculations are supposed as a \frst\nstep to cover such a case in order to see what would occur\nthere. Based on the above considerations, we introduced\na large strain range (but still less than the strain valuespredicted by previous MD simulations) in this work. By\ninputting the strain-dependent \frst-principles results to\nthe micromagnetic model with a locally strained region\nof\u00182 nm, we attempt to reveal the local strain e\u000bects on\nthe coercivity. This theoretical work could be considered\nas a plausible \frst step towards a more accurate study\nby combining experimental local-strain measurement and4\n-7-6-5-4-3-2-101234567-7-6-5-4-3-2-101234567  \n a=b (%)   c (%) \n K (MJ/m3)\n-50510\n-7-6-5-4-3-2-101234567-7-6-5-4-3-2-101234567  \n a (%)   c (%) \n Ms (MA/m)\n0.811.21.4\na b c \n-7-6-5-4-3-2-101234567-7-6-5-4-3-2-101234567  \n a (%)   b (%) \n Ms (MA/m)\n1.441.461.481.51.521.54\n-7-6-5-4-3-2-101234567-7-6-5-4-3-2-101234567  \n a (%)   b (%)  K (MJ/m3)\n-20246810\na b \n-7-6-5-4-3-2-101234567-7-6-5-4-3-2-101234567  \n a=b (%)   c (%) \n Ms (MA/m)\n1.41.451.51.55\n-7-6-5-4-3-2-101234567-7-6-5-4-3-2-101234567  \n a (%)   c (%)  K (MJ/m3)\n-20246810\n c \na \n(a) \n(b) (c) \n(d) (e) \n(f) P P’ Q’ \nQ \nFIG. 2. First-principles calculated K and Ms in the stress state of (a) (b) abbiaxial stress, (c) (d) acbiaxial stress, and (e) (f)\nabctriaxial stress.\ntheoretical calculations of a interface-containing large su-\npercell with several hundreds of atoms. In the stress-free\nNd2Fe14B unit cell, our \frst-principles calculations show\naKvalue of\u00185.1 MJ/m3(\u001830.1 meV/unit cell) and a\nMsvalue of\u00181.525 MA/m. The calculated Kagrees well\nwith the experimental results [39{43], as indicated by the\n\fve horizontal lines in Fig. 1(a). The calculated Msis\n\u001838.3\u0016B/formula unit (f.u.), which also matches well\nwith the experimental value of \u001837.7\u0016B/f.u. [45] and\nother \frst-principle results [46{48]. The consistence be-\ntween our calculations and experimental results validates\nour \frst-principles study on Nd 2Fe14B.\nThe calculated Msin Figs. 1 and 2 shows that it is\nnot remarkably in\ruenced by the stress states and strain\nlevels, except the severer triaxial stress state in the lower\nleft corner of Fig. 2(f). However, the calculated Kis\nhighly dependent both on the stress states and strain\nlevels. In the cuniaxial stress state in which only the\ncrystal axis cof Nd 2Fe14B is stressed and other two crys-\ntal axesaandbare stress-free or free to relax, Kshows\na decreasing trend as the strain \"cis increased, as shown\nin Fig. 1(a). On the contrary, Fig. 1(c) indicates that\nKincreases with the strain \"awhen anauniaxial stress\nis applied. Fig. 1(e) shows that the pure shear in the ab\nplane has negligible e\u000bects on Kwhen the shear strain\n\rabis less than 10%. Only in the extremely sheared case\n(\rab>15%),Kis remarkably reduced. Since the hydro-static pressure up to \u00185.3 GPa induces a tiny shrinkage\nof the lattice, it only slightly reduces K, as shown in Fig.\n1(g) which is one special case of Fig. 2(e). For the hy-\ndrostatic pressure in Fig. 1(g), the stress in both three\ndirections is the same. While for the triaxial stress state\nin Fig. 2(e), the stress along a(b) axis and the stress\nalongcaxis can be either equal or not. The maximum\nhydrostatic pressure \u00185:3 GPa in Fig. 1(g) corresponds\nto a strain state of \"a=\"b\u00181:27% and\"c\u00181:47%.\nThe results in Fig. 1(g) are consistent with those in the\ntriaxial stress state shown in Fig. 2(e). The variation of\nKunder biaxial and triaxial stress states is presented in\nFig. 2. It is obvious that negative Koccurs in a much\nwider strain range in the abin-plane biaxial stress state,\nas shown in Fig. 2(a). The shrinkage in abplane can no-\ntably reduce K. For example, an abbiaxial stress state\nwith\"a=\"b=\u00003% and\u00004% reduce Kto\u00181.4 and\n\u0018\u00000.38 MJ/m3, respectively. In contrast, for the ac\nbiaxial stress state in Fig. 2(c), the strain range for neg-\nativeKis very small. Only in the case of negative \"aor\nlarge positive \"c,Kis reduced. For the abctriaxial stress\nstate in Fig. 2(e), negative Kappears for large negative\n\"a=\"b. Thecelongation and abplane shrinkage reduce\nK. For example, Kdecreases to\u00182.1 MJ/m3in the case\nof\"a=\"b=\u00003% and\"c= 3%. The results in Fig. 2(e)\nagree well with the previous work by calculating the crys-\ntal \feld parameters of Nd ions [23] and are qualitatively5\n<=0.02  >=0.03  e/Bohr3 (a) (b) (c) \nNd M \n5d electron cloud  4f electron cloud  \nFe/B  nucleus  \nK > 0 \nNd M \nFe/B  \nK < 0 (f) \n(g) \n0.0 0.5 1.0 1.5 2.0 2.5 3.00.0250.0500.0750.1000.600.650.70\n  Charge density (e/Bohr3)\nDistance ( Å) noStrain\n Strain-c\n Strain-inplane\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.020.040.060.080.81.2\n  Charge density (e/Bohr3)\nDistance ( Å) noStrain\n Strain-c\n Strain-inplaneNd(g)→B  Nd(f)→Fe(c)  (d) (e) Fe(c)  Nd(g) \nNd(f) Nd(f) \nB \nFe(c)  \nFe(c)  Fe(c)  \nNd(g) B Fe(c)  Nd(g) \nNd(f) \nNd(g) Nd(f) \nB B \nFe(c)  \nFe(c)  Fe(c)  \nFe(c)  Nd(g) \nNd(f) \nNd(g) Nd(f) \nB B \nFe(c)  \nFe(c)  Fe(c)  \n0.0 0.5 1.0 1.5 2.0 2.5 3.00.0250.0500.0750.1000.600.650.70\n  Charge density (e/Bohr3)\nDistance ( Å) noStrain\n Strain-c\n Strain-inplane\n(a) \n(b) \n(c) \n0.0 0.5 1.0 1.5 2.0 2.5 3.00.0250.0500.0750.1000.600.650.70\n  Charge density (e/Bohr3)\nDistance ( Å) noStrain\n Strain-c\n Strain-inplane(a) \n(b) \n(c) \nFIG. 3. Valence-electron-density distributions of Nd 2Fe14B in (001) plane: (a) \"a=\"b=\"c= 0; (b)\"a=\"b= 0 and\"c= 4%;\n(c)\"a=\"b=\u00004% and\"c= 4%. Dotted circles indicate regions where charge density distribution around Nd atoms apparently\nchanges. Charge density distribution along the lines (d) Nd(g) !B and (e) Nd(f)!Fe(c) indicated by arrows in (a)-(c). (f) and\n(g) Schematics for a possible explanation of the sign of K[25].\nconsistent with the results from MD simulations [9{12].\nFrom the results in Fig. 1(a) and (c), one might think\nthat the shrinkage along aandchas opposite e\u000bects on\nK. In fact, this is not the case. Due to the positive Pois-\nson e\u000bect, uniaxial tensile stress along a(c) axis induces\nshrinkage along c(a) axis. So under the uniaxial stress\nstate (Fig. 1(a) and (c)), the interference between the\nstrain along aandcaxis makes it di\u000ecult to judge the\nmain in\ruential factor for K. Then we consider the tri-\naxial stress state in which we can either set strain along\ncaxis by forcing zero strains along aandb, or set strain\ninabplane by forcing zero strains along caxis, as indi-\ncated by the two dotted lines in Fig. 2(e). For the line\nQQ0, i.e. the case of \"a=\"b= 0,Kdoes not change\nso much when \"cis larger than\u00006%. For the line PP0,\ni.e. the case of \"c= 0,Kgradually changes from \u00189.7\nMJ/m3to\u0018\u00004.5 MJ/m3when\"a=\"bdecreases from\n7% to\u00007%. These results from lines PP0and QQ0indi-\ncate thatKis insensitive to the deformation along c, but\nchanges apparently with the abin-plane deformation. In\nother words, the shrinkage in the abplane should be re-\nsponsible for the Kreduction. The decrease of Kwith\nincreasing\"cin Fig. 1(a) is ascribed to the celongation\ninducedabplane shrinkage through the positive Poisson\ne\u000bect. These also explain the results in Fig. 2 that neg-\nativeKalways appears in the region with negative \"aor\n\"band positive \"candabbiaxial stress state allows much\nlarger strain range for negative K.\nIn order to qualitatively understand the sign change of\nK, we analyzed the valence charge density. The densitymap in the (001) plane of Nd 2Fe14B is shown for three\ntypical cases in Fig. 3. In order to clearly display the\ncharge density di\u000berence, the legend is scaled down to\nthe range 0 :02\u00000:03 e/Bohr3. In this way, the charge\ndensity di\u000berence around Nd atomic sites can be easily\nidenti\fed by the color, as indicated by the dotted circles\nin Fig. 3(a)-(c). It can be found that the charge density\nat Fe(c) sites exhibits a distorted distribution towards\nB sites and forms an aspherical shape. The charge den-\nsity at Nd(f) sites and Nd(g) sites is slightly di\u000berent,\nbut both deviate from the spherical distribution. The\ncharge density at B sites is extremely anisotropic and\nis extended towards Nd(g) sites and Fe(c) sites [49, 50].\nDespite of these common feature of the charge density,\nstrain can induce some non-trivial changes. Comparison\nof Fig. 3(a) with no strain and Fig. 3(b) with \"a=\"b= 0\nand\"c= 4% reveals only a slight change of the charge\ndistribution around Fe(c) and B sites. Since no remark-\nable change of the charge distribution around Nd sites is\nobserved in Fig. 3(b), the sign of Kremains the same\nas that in Fig. 3(a). It indicates that deformation along\ncaxis without in-plane strain (i.e. \"a=\"b= 0) does\nnot remarkably change K, agreeing well with the above\nresults. In contrast, if an additional in-plane shrinkage\nstrain (\"a=\"b=\u00004%) is applied, the charge distri-\nbution around Nd sites is notably altered, as shown by\nthe dotted circles in Fig. 3(c). Moreover, charge density\ndistribution along the lines Nd(g) !B (Fig. 3(d)) and\nNd(f)!Fe(c) (Fig. 3(e)) also indicates apparent increase\nof charge density around Nd when in-plane compressive6\nstrain is applied. Due to the reduction of the distance\nbetween Nd sites and Fe/B sites, there is evidence of\nsome degree of hybridization between Fe/B atoms and\nNd atoms (Fig. 3(c)). Through the hybridization, the\n5d electron cloud of Nd atoms apparently extends to-\nwards Fe/B atoms. This relocates the 4f electron cloud\nperpendicular to the abplane in order to avoid the re-\npulsive force from the horizontally extended 5d electron\ncloud [25], thus leading to an easy abplane and nega-\ntiveK(Fig. 3(g)). Therefore, one possible explanation\nis that the in-plane shrinkage makes Fe/B atoms much\ncloser to Nd atoms and results in hybridization between\nthem, which further changes the 5d electron cloud sur-\nrounding the 4f electron cloud of Nd atoms and \fnally\nalters the sign of K[25].\nIt should be noted that several researchers [9{12] have\ndealt with the strain induced Kchange by using the\nphenomenological magneto-elastic coupling energy which\nwas derived by de Groot and de Kort [19]. They calcu-\nlated the strain induced anisotropy constant ( Kme) as a\nfunction of lattice strain and applied Kmeto estimate the\nchange ofKby using the elastic constants from isotropic\npolycrystals. For a qualitative and order-of-magnitude\nanalysis, we rewrite the Kmefrom de Groot and de Kort\nasKme\u0018B\"in whichBdenotes the magnetoelastic co-\ne\u000ecient and \"the strain level. By using the parameters\ngiven in the literature [19], our estimation of Bis shown\nto be in the order of 40 MJ/m3. It means that a large\nstrain in the order of 10% can only give a Kchange of\n\u00184 MJ/m3. For a negative K, a strain more than 12% is\nrequired. However, our \frst-principles calculations show\nthat a small strain around 4% can even reduce Kto neg-\native values (Fig. 2(a)). Hence our \frst-principles study\nindicates a much larger sensitivity of Kto the lattice\ndeformation. The underestimation of strain e\u000bects by\nthe phenomenological description could be attributed to\nthe assumption of one-ion magneto-elastic Hamiltonian\nwithout the two-ion one, because the two-ion magneto-\nelasticity is also related to the modi\fcation of the two-\nion magnetic interactions by the strains [51]. But in\nour \frst-principles calculations, both one-ion and two-ion\nmagnetic interactions, as well as the fully electron-lattice\ncoupling, are consistently included.\nB. Micromagnetic simulations of locally strained\nNd{Fe{B magnets\nPrevious experiments have demonstrated that homoge-\nneous small strain in Nd{Fe{B magnets has negligible ef-\nfect on the coercivity [19]. However, previous MD simula-\ntions veri\fed that a large strain is possible in a very local-\nized\u00182-nm-wide region near the interface [9{12]. They\nused the atomic displacement near the interface to calcu-\nlate the local strain, which is taken as the lattice strain as\ninputs for the phenomenological magneto-elastic theory\n[19] to estimate the Kchange. In the micromagnetic sim-\nulations here, we also follow the similar idea as shown inthese previous studies [9{12, 14, 52], i.e. the source of the\nlocal strain is not the focus and an e\u000bective lattice strain\nis assigned to the local region. The symmetry breaking\nand the change of chemical environments near the local\nregion are out of the scope in this work, although they\ncan also in\ruence the coercivity. However, unlike these\nprevious studies which used the phenomenological theory\n[19], here we directly take the lattice strains and stress\nstates associated with the \frst-principles calculations to\nde\fne the locally strained region in Nd{Fe{B magnets.\nThe local region is approximately set as 2 nm thick, as\ndemonstrated by the MD simulations [9{12, 14]. The pa-\nrametersKandMsof the locally strained region under\nvarious strain levels and stress states are taken from the\n\frst-principles results presented above.\n1234\na=-3%\na=-4%\nm0Hc  (T) −3% \n−4% ea=eb= 1234567\n-6 -4 -2 0 2 4 6\nt h locally strained region  \nh=200 nm, d=300 nm, t=2 nm  \n0.68               1  mc \n0.752               1  mc \n0.723               1  mc \n0.789               1  mc (a) (b) \n(c) (d) m0Hc  (T) \nlocally  ab biaxial stress state  \nea=eb  (%) \nlocally  ab biaxial stress state  \nFIG. 4. (a) Schematic of a single-grain Nd{Fe{B magnet with\na hexagonal section and with its surface covered by a locally\nstrained region. (b) Local strain dependent coercivity for the\ngrain in (a) under the local abbiaxial stress state. (c) mc\ndistribution at the remanent state ( \u00160Hex= 0) for grains with\ntriangular, square, hexagonal, and circular sections under a\nlocalabbiaxial strain of\u00004%. (d) Coercivity for the grains\nin (c) under local abbiaxial strains of \u00003% and\u00004%.\n1. Single-grain Nd{Fe{B magnets\nWe \frstly investigated the prism-shaped single grain\nwhich is covered by a locally strained surface with a thick-\nness oft. Fig. 4(a) displays the grain shape of a hexago-\nnal prism, with the geometry dimension of h= 200 nm,\nd= 300 nm, and t= 2 nm. If we assume that the grain\nsurface is under the local abbiaxial stress state, it can\nbe found that for the hexagonal prism, the coercivity de-\ncreases from 5.7 T to 1.96 T under an abbiaxial strain\nof\"a=\"b=\u00005% (4(b)). However, the coercivity is only\nslightly increased by 0.5 T in the case of \"a=\"b= 5%.7\n-7-6-5-4-3-2-101234567-7-6-5-4-3-2-101234567  \n a (%)   c (%) \nHc/Hc0 (%)\n-60-50-40-30-20-10010\n-7-6-5-4-3-2-101234567-7-6-5-4-3-2-101234567  \n a (%)   b (%) \nHc/Hc0 (%)\n-60-40-200\n-7-6-5-4-3-2-101234567-7-6-5-4-3-2-101234567  \n a=b (%)   c (%) \nHc/Hc0 (%)\n-80-60-40-200\n-8-6-4-202468-50-40-30-20-10010\n a or  c (%)  Hc/Hc0 (%) \n  \nuniaxial  a\nuniaxial  c\n-7-6-5-4-3-2-101234567-7-6-5-4-3-2-101234567  \n a (%)   c (%) \n-60-40-200(a) (b) (c) (d) \nFIG. 5. Micromagnetic simulation results on the coercivity change as functions of the local surface strain under the stress state\nof (a) uniaxial stress, (b) abbiaxial stress, (c) acbiaxial stress, and (d) abctriaxial stress. The micromagnetic mode is single\nNd{Fe{B grain with a hexagonal section in Fig. 4(a).\nThis indicates that the coercivity is more sensitive to the\nlocal region with abplane shrinkage and negative K.\nWe further studied the e\u000bects of grain shape of the\nlocally strained single grain. The motivation is to ex-\nplore the possible role of the place where strain/stress\nappears and the associated micromagnetic mechanism.\nThe grain shape e\u000bects have been recently investigated\nfor achieving high coercivity [53{56]. Here we considered\nfour types of prism grains with triangular, rectangular,\nhexagonal, and circular sections. The distribution of the\nccomponent of the unit magnetization vector ( mc) at the\nremanent state ( \u00160Hex= 0) is presented in Fig. 4(c). It\ncan be seen that the magnetization near the corners or\nedges has already rotated out of the easy direction even\nat the remanent state. More precisely, the minimum mc\nvalues in Fig. 4(c) are found to decrease in the order:\ncircular prism >hexagonal prism >square prism >tri-\nangular prism. This means that the local reversal occurs\nfastest in the triangular prism and slowest in the circular\nprism. Such a local reversal is due to the inhomogeneous\nstray \feld near the corners or edges in the nonellipsoidal\ngrains [53, 57]. By the local reversal, the inhomogeneous\nmagnetization can suppress magnetic surface charges and\ndecrease the stray-\feld energy with respect to the ho-\nmogeneous magnetic state. The di\u000berent local reversal\nbehavior could result in distinct coercivity. We \fnd in\nFig. 4(d) that at the same local stress states and strain\nlevels, the coercivity is shown to increase in the order:\ntriangular prism <square prism <hexagonal prism <\ncircular prism. For example, in the case of a local abbi-\naxial stress state with \"a=\"b=\u00004%, the coercivity is\nfound to signi\fcantly increase from 1.58 T in the trian-\ngular prism to 2.3 T in the circular prism. These results\nindicate that in addition to the local stress states and\nstrain levels themselves, where the locally strained region\nappears (e.g. grain shape, surface irregularity, edge cur-\nvature, etc.) also plays an important role in determining\nthe coercivity.\nBy using the hexagonal prism in Fig. 4(a), we carried\nout a detailed study on the sensitivity of the coercivity\nto the local stress states and strain levels. The coercivity\nchange distribution in Fig. 5 is similar to the Kdistri-bution in Figs. 1 and 2. We \fnd that in all the cases,\nthe coercivity enhancement is limited to \u001810%, while\nthe coercivity decrease can be as high as \u001880%. Again,\nthe coercivity decrease is found to be more sensitive to\nthe local strain than the coercivity enhancement. Fig.\n5(a) shows that the uniaxial stress state along caxis and\naaxis induce a maximum coercivity decrease of \u001820%\nand\u001850%, respectively. The local abbiaxial stress state\nallows a much larger strain range for the coercivity de-\ncrease by\u001860%, as shown in Fig. 5(b). In contrast, the\nlocalacbiaxial and abctriaxial stress states have much\nsmaller strain range for the coercivity decrease, as shown\nin Fig. 5(c) and (d).\n2. Multi-grain Nd{Fe{B magnets\nMicromagnetic simulations on the multigrain were fur-\nther performed. The multigrain model in Fig. 6(a) was\nbuilt by using the SEM image of a sintered Nd{Fe{B\nmagnet [20]. The size m= 280 nm and n= 300 nm is\nestimated from the SEM image. Around the triple junc-\ntion, the region with a strain of \"c=\u00061% is extended\nover several tens nanometer away the interface, as mea-\nsured in the previous experimental work [20]. An addi-\ntional locally strained region with 2-nm width is assumed\nin the interface, as did in the previous work [9, 11, 12].\nDue to the small size of the model, the simulated coerciv-\nity without local strain is as high as \u00185.84 T, as shown in\nFig. 7. The e\u000bect of the strain \"c=\u00061% extending over\nseveral tens nanometer on the coercivity is neglectable,\nfurther con\frming the statement in the previous experi-\nmental work [20]. It can be seen from Fig. 7 that in the\ncase of local uniaxial stress states, the strain has little\nin\ruence on the coercivity. However, the biaxial and tri-\naxial stress states remarkably reduce the coercivity. An\nabbiaxial stress state with \"a=\"b=\u00004% and\u00003%\ndecreases the coercivity from 5.84 T to 1.98 T and 2.8 T\nor by\u001866% and\u001852%, respectively. This means that\na moderate strain level in a suitable local stress state can\nreduce the coercivity in the multigrain Nd{Fe{B magnets\nby more than 3 T. The magnetic reversal process in Fig.8\n(b) \n1 \n \n \n \n-1 mc \n(a) \n𝜀𝑐≈−1% \n𝜀𝑐≈1% m-Nd \nCu-rich region  \nNd2Fe14B local strain  Nd2Fe14B \nFIG. 6. Micromagnetic simulation of a multigrain based on\nthe SEM image of the experimental work [20]. (a) Model\nillustration with the size of l=m= 280 nm and n= 300 nm.\nThe locally strained region is with a width of \u00182 nm. (b)\nMagnetic reversal process between external \felds of \u00001:96 T\nand\u00001:98 T when the local region is under the biaxial stress\nstate with\"a=\"b=\u00004%.\n123456\n0.5 11.5 22.5 33.5 44.5 55.5           1              2               3              4               5            uniaxial c uniaxial a \nbiaxial ac biaxial ab \ntriaxial abc Stress state  no local strain  \nea=eb=-3% \nea=eb=-4% ea=5%  \neb=-5% ea=eb=-5% \nea=eb=-4% \nec=4%  ea=eb=-3% \nec=3%  ea=-4%, ec=4%  \nea=-5%, ec=5%  ea=eb=-5% \nec=5%  \nStrain level (%)  m0Hc  (T) \nFIG. 7. Coercivity of the multigrain as functions of the local\nstress states and strain levels. The dotted line indicates the\ncoercivity in the case of no local strain. Each color denotes\none stress state of the local region. The strain values are\nmarked when the coercivity decrease is over 1 T.\n6(b) indicates several apparent nucleation sites in the lo-\ncally strained region. Then the reversal domain rapidly\nexpands and the whole grain is completely reversed in-\nstantly.\nBy using the \frst-principles results in Figs. 1 and 2, we\ncalculated the coercivity of the multigrain as functions\nof the local stress states and strain levels, as shown in\nm0Hex  (T) \nmc \n-1-0.500.51\n-3 -2 -1 0ab biaxial stress state  \n-3% -4% ea=eb= FIG. 8. Simulated reversal curves of the multigrain with the\nlocal strain region under the abbiaxial stress state ( \"a=\"b=\n\u00003% and\u00004%). Solid lines correspond to the model in the\ninset withl= 1:4\u0016m. Dotted lines correspond to the model\nin Fig. 6(a).\n-1-0.500.51\n-3 -2 -1 0\nm0Hex  (T) \nmc ab biaxial stress state  \n-3% -4% ea=eb= \nFIG. 9. Simulated reversal curves of the multigrain with the\nlocal strain region under the abbiaxial stress state ( \"a=\n\"b=\u00003% and\u00004%). Solid lines correspond to the model in\nthe inset with a circularly shaped triple region. Dotted lines\ncorrespond to the model in Fig. 6(a).\nFig. 7. It is found that the local uniaxial stress states\nalmost do not a\u000bect the coercivity. Only the strain values\nmarked in Fig. 7 and their associated stress states can\nreduce the coercivity by more than 1 T. Obviously, the\nlocalabbiaxial stress state possesses a higher possibility\nto result in more reduction in the coercivity. In the case\nof localabbiaxial and abctriaxial stress in Fig. 7, though\n5% strain induce a large negative value of Kthan 4%\nstrain, the coercivity is slightly increased. This is due to\nthe fact that a local negative Kfavors the formation of an\ninitial 90 degree domain wall between the locally strained\nregion and the strain-free region, but a more negative K\nincreases the \feld for the subsequent formation of a 180\ndegree domain wall. This also indicates that a larger9\nnegative value of local Kinducing more reduction in the\ncoercivity is not always correct.\nDue to the small mesh size (1 nm here) determined\nby the physical length in micromagnetic simulations, the\nsample size of the modeled multigrain is often much\nsmaller than that of the real magnets. In order to present\nan example for demonstrating the size e\u000bect, here we in-\ncrease the multigrain size by extending l(Fig. 6(a)) to\n1.4\u0016m, as show in the inset of Fig. 8. Such an exten-\nsion results in a mesh number of \u00180.12 billion, which is\nextremely computationally expensive. From the rever-\nsal curves in Fig. 8, it can be seen that in both local\nabbiaxial stress states with \"a=\"b=\u00004% and\u00003%,\nthe increase of lfrom 280 nm to 1.4 \u0016m makes the coer-\ncivity decrease by \u00180.3 T. The coercivity reduction can\nbe qualitatively understood from the demagnetization ef-\nfect. Thelextension favors a reduction and increase of\ndemagnetization factor along the landcdirection, re-\nspectively. This increases the in-plane shape anisotropy\nand exerts additional torque to make magnetization de-\nviate fromcaxis, thus resulting in premature nucleation\nand reduced coercivity.\nFinally it is worth mentioning that the single-grain re-\nsults in Fig. 5(d) inspire a strategy of increasing the\ncoercivity by tuning the shape or geometry of the lo-\ncal strain region in the multigrains. By this inspiration,\nwe then changed the model in Fig. 6(a) into an ideal\nmodel whose triple region is set as a circular prism, as\nshown in the inset of Fig. 9, in order to study the ef-\nfect of where the local strain appears. In contrast to\nFig. 6(a) in which the local strain is in the triangular\nedge of the triple region, the model in Fig. 9 puts the\nlocal strain in the circular edge. We can \fnd from Fig.\n9 that the coercivity is obviously enhanced if a circular\nprism is used to represent the triple region. This is con-\nsistent with the result from the single-grain study and\nindicates the shape of the triple region as an in\ruential\nfactor for the coercivity. Smoothing the edge and remov-\ning the sharp angle of the triple region are favorable for\nthe coercivity enhancement. It should be noted that the\ncircularly shaped triple region presented in the simula-\ntion is an ideal case, but it provide practical information\ntowards coercivity enhancement. In realistic condition,\nthough achieving a perfectly circular triple region is dif-\n\fcult, reducing the sharp angle of the triple regions or\nmaking them as smooth as possible in Nd{Fe{B mag-\nnets is possible by using gas atomised powders, controlled\ngrain boundary di\u000busion, or additive manufacturing.\nIV. CONCLUSIONS\nThe strain e\u000bects in Nd{Fe{B magnets are examined\nby a combined \frst-principles and micromagnetic study.In this way, we use the \frst-principles results on the stress\nstates and strain levels dependent KandMsas the input\nfor micromagnetic simulations of the coercivity in single-\nand multi-grain Nd{Fe{B magnets. The main conclu-\nsions are summarized in the following:\n(1) In Nd 2Fe14B phase, the stress states and strain\nlevels have negligible e\u000bects on Msbut signi\fcant e\u000bects\nonK.Kis sensitive to the abin-plane deformation rather\nthan thec-axis deformation. The abplane shrinkage is\nresponsible for the Kreduction. The biaxial and triaxial\nstress states have a greater impact on Kthan other stress\nstates. Negative Koccurs in a much wider strain range\nin theabbiaxial stress state.\n(2)Kis shown to be more sensitive to the lattice de-\nformation by the \frst-principles study than by the pre-\nvious phenomenological model [19] which only consid-\ners one-ion magneto-elastic Hamiltonian and underesti-\nmates the strain e\u000bects. An abbiaxial stress state with\n\"a=\"b=\u00003% and\u00004% reduces Kto\u00181.4 and\u0018\u00000:38\nMJ/m3, respectively.\n(3) In Nd{Fe{B magnets, the local abbiaxial stress\nstate in the locally strained region is more likely to induce\na large loss of coercivity. A coercivity decrease by 60%\nor by 3\u00004 T can be induced by a 3 \u00004% local strain in\na 2-nm-wide region near the interface around the grain\nboundaries and triple junctions.\n(4) In addition to the local stress states and strain lev-\nels themselves, the shape of the interfaces and the inter-\ngranular phases also makes a di\u000berence. Smoothing the\nedge and reducing the sharp angle of the triple regions\nin Nd{Fe{B magnets would be favorable for a coercivity\nenhancement.\nIt is anticipated that our multiscale results here based\non \frst-principles calculations and micromagnetic sim-\nulations provide quantitative information for that what\nkind of local stress state and how large local strain can\ninduce signi\fcant decrease in the coercivity. The results\nwill also be applied to the high-resolution experimental\nresearch of the local strain measurement in Nd{Fe{B per-\nmanent magnets.\nACKNOWLEDGMENT\nThe \fnancial supports from the German federal state\nof Hessen through its excellence programme LOEWE\nRESPONSE and the German Science Foundation (DFG\nXu121/7-1) are appreciated. The authors also greatly\nacknowledge the access to the Lichtenberg High Perfor-\nmance Computer of TU Darmstadt.\n[1] H. Zheng, J. Wang, S. E. Lo\rand, Z. Ma, L. Mohaddes-\nArdabili, T. Zhao, L. Salamanca-Riba, S. R. Shinde, S. 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Tsymbal1**    \n1 Department of Physics and Astronomy & Nebraska Center for Materials and Nanoscience,  University of \nNebraska, Lincoln, Nebraska 68588, USA  \n2 Department of Physics, University of Northern Iowa, Cedar Falls, Iowa 50614, USA  \n \nVoltage controlled magnetic anisotropy (VCMA) is an efficient way to manipulate the magnetization states in \nnanomagnets, promising for low -power spintronic  applications. The underlying physical mechanism for VCMA is \nknown to involve a change in the d-orbital occupation on the transition metal interface atoms with an applied electric \nfield. However, a simple qualitative picture of how this occupation controls  the magnetocrystalline  anisotropy (MCA) \nand even why in certain cases the MCA has opposite sign still remains elusive. In this paper , we exploit a simple  model \nof orbital  populations to elucidate a number of features typical for the interface MCA  and the effect of electric field on \nit, for 3d transition metal thin films used in magnetic tunnel junctions. We find that in all considered cases including the \nFe (001) surface, clean Fe 1-xCox(001)/MgO interface and oxidized Fe(001)/MgO interface, the ef fects of alloying and \nelectric field enhance  the MCA energy with electron depletion which is largely explained by the occupancy of the \nminority -spin dxz,yz orbitals.  On the other hand, the hole doped Fe(001) exhibits an inverse VCMA , where the MCA  \nenhancement is achieved  when electron s are accumulated at the Fe (001)/MgO interface with applied electric field. In \nthis regime we predict a significantly enhanced VCMA which exceeds 1pJ/Vm. Realizing this regime experimentally \nmay be favorable for a pra ctical purpose of voltage driven magnetization reversal.  \n \n \nMagnetocrystalline  anisotropy (MCA) is one of the critical \nparameters that determine the magnetization orientation, \nswitching dynamics, and thermal stability of magnetic media. \nFerromagnetic thin films with high magnetic anisotropy \npromise an easier operation scheme and hig her signa l stability. \nHowever, the large  coercivity of these materials require high \nelectric currents either to generate a magnetic field or produce \na spin -transfer torque to write the bit information, which \nimposes significant limitations on their applica tion in portable \nand high speed electronic devices. For example, in MgO -based \nmagnetic tunnel junctions (MTJs), which are nowadays used \nin magnetic random access memories (MRAMs), current \ndensities as large as 106 A/cm2 are required to reverse the \nmagnetiz ation of the magnetic bit by spin -transfer torque.   \nA possible approach to overcome this limitation is to use \nan electric field, rather than electric current, to control the \nmagnetization of a ferromagnet.1 Along these lines, voltage \ncontrolled magnetic anisotropy (VCMA) is considered as one \nof the most promising approaches to dramatically reduce the \nwriting energy of MRAMs.2 A large VCMA effect ( ~ 1 pJ/Vm) \nis, however,  required  for device application  to overcome a \nlarge coercively of the ferromagnetic film which is need ed for \nits thermal stability .  \nThe effect of an electric  field on the surface  (interface)  \nmagnetic anisotropy  has been extensively studied both \ntheoretically3-13 and experimentally .14-24 Very large changes in \nmagnetic anisotropy (VCMA ~ 5  pJ/Vm)  were observed due \nto ionic motion and chemical reactions  driven by applied \nvoltage.25-26 Ionic motion  however is a slow process and can \nhardly be used for device applications where an ultrafast \nswitching of magnetization is required. On the other hand,  if \nthe VCMA is caused purel y by electronic effects, the \ntimescale of the VCMA would lie on sub ns regime desirable \nfor applications.  It is known that VCMA in Fe (Co)/ MgO -\nbased MTJs is largely controlled by the effect of electric field \nscreening on the ferromagnetic metal surface  (interface)  \nresulting in electron  doping  induced by an external electric \nfield.3  It is underst ood that the main driving mechanism for \nVCMA  is a c hange in the 3d-orbital occupation on the \ntransition metal interface atoms with an applied electric field. \nTheoretical calculations  are capable to predict reasonabl y well \nelectric field induced changes in the surface MCA  energy \n(MAE) . However, a n intuitive  picture for VCMA is missing , \nalthough such a qualitative picture would be very helpful for \nthe experimentalists to design materials and interface \nstructures with enhanced VCMA.   \nIn this paper, we consider a simple model which allows us \nto q ualitatively  explain the surface (interface) magnetic \nanisotropy and the effect of electric field on it . For 3 d \ntransition metal thin films used in MTJs. We show that results \nof our first -principles density functional  theory (DFT)  \ncalculations for the  Fe (001) surface, clean Fe 1-xCox(001)/  \nMgO interface and oxidized Fe(001)/MgO interface, involving \neffects of alloying and electric field , can be understood in \nterms of changes in the population of the minority -spin dxz,yz \norbitals which enhance (reduce)  the surface  MAE with \nelectron depletion (accumulation) at the interface . On the other \nhand, the hole doped Fe(001) exhibits an inverse VCMA, \nwhere the MCA enhancement is achieved when electron s are \naccumulated at the Fe(001)/MgO interface. In this regime , we \npredict a significantly enhanced VCMA which exceeds \n1pJ/Vm. These results may be important for the search of the \nmaterials structures with enhanced VCMA which is critical for \ndevice applications.  \nThe microscopic origin of MCA is the relativistic spin -\norbit coupling (SOC) , \nLS , where L and S are the orbital \nand spin momentum operators  respectively , and ξ is the SOC \nconstant.  For not too thick films, the largest contribution to \nMCA comes from surfaces or interfaces due to the reduced \nsymmetry. The surface  (interface)  MAE is known to be very \nsensitive to the local environment and details of the electronic \nband structure.  \nFor 3 d transition metal thin film s, the MAE can be \nevaluated using  the second -order perturbation theory ,27-28 2 \n which  is applicable due to the rela tively small SOC constant ξ  \n(typica lly in the range of 10 -100 meV  per atom ) compared to \nthe band energies , the crystal field splitting , and the exchange \ncoupling . We define the MAE as the energy difference \nbetween the magnetization pointing along the x direction in \nthe plane of the film and the z direction perpendicular to the \nplane, so that the positive MAE implies the out -of-plane easy \naxis, known as perpendicular magnetic anisotropy (PMA). \nWithin the second -order perturbation theory the MAE is \ndeterm ined by the matrix elements of SOC between occupied \nand unoccupied states so that  \n22\n2\n,\n,o z u o x u\nMAE\nou uoLL\nE   \n\n   \n\n\n\n\n, (1)  \nwhere \no  and \nu  are unperturbed wave functions  for \noccupied and unoccupied states with energies \no  and  \nu  and \nspin states \n  and \n , respectively.  \n \nFIG. 1. (a) Schematic  of the minority 3d orbital splitting  by the \ncrystal field of tetragonal symmetry.  Broadening of the orbital levels \nmimics the energy bands. The vertical  black dotted lines  represent the \nFermi energies of Fe, Co and Ni. (b) Magnetocrystalline anisotropy \nenergy (MAE) as a function of orbital band filling n, for different \nvalues of the T2g level splitting Δ and broadening δ.  \nFor transition metal ferromagnets, such as bcc Fe, the \nMAE is determined by the energy bands formed from the 3 d \norbitals and depends on their occupations. The crystal field of \ncubic symmetry splits five  d orbitals into the Eg doublet and \nthe T2g triplet, as shown in Fig. 1a. At the (001) interface, the crystal field symmetry is reduced to tetragonal, which further \nsplits these levels: Eg into two singlets A1 (dz2) and B1 (dx2 –y2), \nand T2g into singlet B2 (dxy) and doublet E (dxz, dyz). Fig. 1a \nshows schematically the orbital order as a function of the \nenergy  typical for the Fe/MgO (001) interface. The exchange \nsplitting in Fe is sufficiently large, so that the majority -spin \nband is nearly fully occupied  (see Fig. 2S -a in the \nSupplementary Material ). We therefore ignore for simplicity \ncontribution to the  MAE from the majority -spin band. The \ndashed red line  (Δ = D,  = 0) in Fig. 1b shows the MAE as a \nfunction of orbital filling n in Fig. 1a. It is seen that first the \nMAE is growing due to population of the dxy orbitals  resulting \nin positive contribution to MAE in Eq. (1) through the \n22 xy z xyd L d\n matrix element. Then, it drops down and \nbecomes negative, due to the negative contribution from the\nyzd\norbitals29 through the matrix elements \n2 yz x zd L d  and \n22 yz x xyd L d\n. Finally, the MAE grows again due to filling  \nof the Eg states, reducing the latter contribution. Broadening of \nthe orbital states that mimics the energy bands, as indicated in \nFig. 1a, diminishes the sharp features of MAE versus n but \npreserves the qualitative behavior, as seen from the solid red \nline in Fig.  1b.       \nThere are a number  of important implications  which \nfollow from this simplistic  consideration. First, it is seen that \nthe largest positive MAE ( PMA ) occurs for n close to 6, \ncorresponding to Fe. It is worth mentioning that the large \nPMA does not  necessar ily require hybridization of Fe -dz2 and \nO-pz orbitals across  the Fe /MgO  interface , as is often \nthought,30-31 and can be explained purely based on the orbital \npopulation.  In particular, our first -principles calculations \npredict  a sufficiently large PMA of about 0.87 mJ/m2 for the \nclean Fe (001) surface with no MgO . Second, the MAE drops \nwhen moving from Fe to Co in terms of the band  filling  and i t \nbecomes negative for Co. This is consistent with the results of \nDFT calculation s performed for Fe 1-xCox/MgO interface as a \nfunction of x.32  Third,  since the VCMA is controlled b y a \nchange in the 3 d band population produced by applied electric \nfield, the slope of the curves in Fig. 1b determines the sign of \nVCMA. It is seen that for n changing from 6 to 7 (i.e., from Fe \nto Co) the slope is negative, indicating that the MAE decreases \nwith adding electrons , which is  due to the population of t he dyz \norbitals . This is consistent with our DFT calculation for the \nFe/MgO (001) interface, showing that the MAE increases with \nthe increasing electric field pointing from Fe to MgO  (electron \ndepletion ), as evident from Fig. 2a.33,34 The increase of PMA \nwith electron depletion is typical for most of the experime nts \non  Fe (Co)/MgO interfaces15-24 and is also known for the clean \nFe (001) surface.3 The dominant contribution comes from the  \ndyz band , which is again consistent with the qualitative  picture \npresented above . Finally, i t is notable from Fig. 1b that \nreducing n below 6 leads to the decrease of MAE, due to the \nreduction of the positive contribution from the dxy orbitals. \nThis changes  the sign of VCMA and has important \nimplications as discussed below.  \nEdz2 dxy dxzdyz dx2‒y2Eg T2g\nΔD\nFe Co Ni2δa\nb\n5 6 7 8 9 10-6-4-20246    = D  = 0 \n   = D   = D  \n  = D  = 0\n  =D  = D\n  \nnMAE (2/D)3 \n  \nFIG. 2.  Results of DFT calculations for Fe(001)/MgO  (a) and \nFe(001)/FeO/MgO ( b) interfaces. MAE as a function of applied \nelectric field  EMgO in MgO (red curves) and E-field induced changes \nin the orbital contribution to MAE (green and blue curves). In the plot \nthe electric field EMgO is scaled according to  the experimental \ndielectric constant of MgO  = 9.5.  \nFurthermore , this simple analysis explains a change in sign \nof the MAE , when Fe/MgO interface becomes oxidized.  Our \ncalculations predict th at the MAE alters  from about +1.67 \nmJ/m2 for the clean Fe/MgO  interface to –1.44 mJ/m2 for the \nFe(001) /FeO/MgO interface , where  the first monolayer of Fe \nis fully oxidized  (Fig. 2b) .33 This is due to  the hybridization of \nthe dxy orbitals  of Fe and px,y orbitals of O in the (001) plane, \nresulting in the formation of the bonding and antibonding \nstates . The minority -spin antibonding state is largely \ncomposed of the dxy orbitals and  pushed up in energy , as seen \nfrom Fig. 2 S-b. This behavior is captured by our simple model \nwith negative Δ, which puts the dxz,yz doublet at the lowest \nenergy. In this case, for n changing from 5 to 7, the positive \ncontribution to MAE from the dxy orbitals is eliminated, \nresulting in the increasing negative contribution coming from \npopulation of the dxz,yz states (Fig. 1b, blue curves). Moreover, \nthis qualitative analysis predicts t he VCMA sign  consistent \nwith our DFT calculation, showing that the MAE increases \nwith the increasing electric field pointing from Fe to MgO and \nis largely determined by the dyz contribution (Fig. 2 b).      \nNow we expand these considerations  to elucidat e the \neffects of electrostatic and chemical doping on VCMA of the \nFe(001) /MgO interface. First, we consider the effect of \nelectrostatic doping , assuming a n ultrathin Fe layer  which is \ntypical for PMA experiments.  A particular calculation is performed in a symmetric geometry of the  MgO(3MLs)/  \nFe(3MLs)/MgO(3MLs)/ vacuum (2nm) supercell. In the \ncalculation we vary the number of valence electron s by adding \na valence charge to the supercell  and neutralizing it by \nbackground of the constant charge  density of  opposite  sign.  \nThrough the self -consistent electronic structure calculation, \nthe valence ch arge in vacuum and in MgO  is reassembled in a \nway to minimize the electrostatic energy, i.e. it is deposited on \nthe Fe/MgO interface. The associated electric field  emerg es in \nvacuum and in MgO. This is equivalent to the calculati on in an \nexternal electric field , apart from the constant background \ncharge. The effect of the latter on the Fe metal is minimized \ndue to the sufficiently large size of the sup ercell (3. 91 nm) \ncompared to  Fe layer thickness (0.43 nm). Due to the imposed \nsymmetry, the two interfaces in the supercell are exposed to \nthe same field.33  \nThe results of the calculation are shown in Fig. 3a , where \nMAE is plotted against the excess valence charge  ne. For ne = \n0, w e find that MAE is about 1.57 mJ/m2. It increases with \nadding holes to the system correspond ing to an increasing \nelectric field pointing from Fe to MgO. This behavior is \nconsistent with the previous calculations6 and typical for \nexperiments15-18. For small values of ne, the MAE changes \nlinear ly with ne. The slope of the curve determines VCMA  \nwhich is shown in Fig. 3b.  Assuming that the electric field in \nMgO is given by EMgO = – ne /0a2 , where   is the dielectric \nconstant of MgO ,  0  is the vacuum permittivity  and a is the \nin-plane lattice parameter , we find the VCMA of about 0.25 \npJ/Vm. This value is comparable to that in Fig. 2a  (~0.15 \npJ/Vm). Interestingly, with increasing ne up to about 0.025  \n(EMgO  – 0.5 V/nm) the MAE curve becomes flatter  (also \nseen in Fig. 2a) and at ne  0.03 the VCMA is reduced . Such a \nnon-linear variation of MAE (inset in Fig. 3a) has been seen in \na number of experiments16,18. This behavior can be attributed  \nto the resonant minority -spin interface state which is well \nknown from spin -polarize d tunneling .35 In our calculation  the \ninterface state appears at about 0.05 eV above the Fermi \nenergy  and is largely composed of the dxz,yz orbitals  (see, e.g. , \nFig. 2 S-a). According to  Plucinski  et al. ,36 this state  lies in the \nenergy gap of the projected minority -spin bulk band s. Thus, \nelectric field induced occupation of this state does not lead to a \nsizable reduction in the MAE  (as expected for the  dyz orbital) \ndue to the absence of the unoccupied counterpart Eg states to \naffect the  MAE through \n2 yz x zd L d  and \n22 yz x xyd L d  \nmatrix elements.  \nThe most striking feature, however, which is evident from \nFigs. 3a,b is the maximum in MAE at ne  – 0.05 and a change \nof VCMA sign at ne < – 0.05. This behavior is consistent with  \nour simple analysis , according to which the inverse VCMA \noccurs when n < 6, due to  the reduc ed contribution from the \ndxy orbitals  (Fig. 1b, red lines) . It is notable  that in this inverse  \nregime, the VCMA  reaches a very large negative value of \nabout –1.1 pJ/ Vm at ne = – 0.075. The electric field in MgO \ncorresponding to the VCMA maximum is  EMgO  1.73 V/nm. \nRealizing this regime  experimentally may be favorable for a \npractical purpose of voltage driven magnetization reversal.  \na\nb\n-0.1 0.0 0.1 0.2-0.03-0.02-0.010.000.010.020.030.04\nMAE (mJ/m2)\n EMgO (V/nm ) total\n dyz\n other orbitals MAE (mJ/m2)\n-1.46-1.44-1.42-1.40\n \n-0.2 -0.1 0.0 0.1 0.2-0.03-0.02-0.010.000.010.020.030.04\nMAE (mJ/m2)\n EMgO (V/nm ) total\n dyz\n other orbitals MAE (mJ/m2)\n1.641.661.681.70\n Fe(001)/ MgO\nFe(001)/ FeO/MgO4 \n  \nFIG. 3 . Calculated MAE (a) and VCMA (b) as a function of the \nexcess number of valence electrons  ne (holes for negative ne) at the \nFe(001)/MgO interface. The inset shows the MAE as a function of \nelectric field in MgO. The electric field  is estimated from EMgO = ne \n/0a2. The positive field is assumed to be pointing from Fe to MgO \nand corresponds to electron depletion.   \nThe inverse  VMCA may be achieved  through the \nelectrostatic doping caused by a top metal layer cover ing the \nPMA ferromagnet . Due to the different work  function s of the \nPMA ferromagnet and the adjacent metal layer , charge \ntransfer occurs between the layers to equalize the chemical \npotentials , changing the number of valence electrons in the \nferromagnetic metal.  The effect may be sizable only for \nultrathin magnetic films due to the short screening length in \nmetals. This may explain the results of Shiota  et al. ,24 who \nobserved a change of VCMA from normal to inverse , when Ta  \nunderlay er was replaced with Ru in the M/CoFeB/MgO (M = \nTa, Ru) multilayer. The wor k function of Ta is about 0.2 5 eV \nlower than that of Fe, whereas the work function of Ru is 0.2 1 \neV higher37. Therefore, it is expected that Fe is electron doped \nat the interface with Ta, but hole doped at the interface with \nRu which may lead to the inverse VCMA. Also, the results of \nNozaki et al.18 demonstrate that VCMA is enhanced up to 0.29 \npJ/Vm when a sub -nm thick layer of Fe  is interfaced with Cr. \nAt such a small thickness the effects of intermixing may play a \nrole, resulting in the hole doping of Fe. According to Fig. 3b, a \nsmall hole doping r aises VCMA up to about 0.4 pJ/Vm (at ne \n= – 0.04), which may be the origin of the e nhanced VCMA \nobserved by Nozaki et al.               \nChemical doping is another way to manipulate VCMA.  \nHere we consider effects of Fe alloyed  with Co or Cr. Doping \nwith Co adds electrons to Fe, whereas doping with Cr adds \nholes. This is expected to have a n opposite effect on the MAE \nof the Fe1-xMx/MgO (M = Co, Cr) interface according  to the simple analysis of Fig. 1b. DFT c alculations are performed \nutilizing  the coherent  potential approximation (CPA)  within  \nthe full relativistic screened -Korringa -Kohn -Rostoker (KKR)  \nmethod .33  The results are shown in Fig. 4, where the MAE is \nplotted against the number of doped electrons (holes), \nassuming that ne = x for Co, and ne = – 2x for Cr.  In agreement \nwith the published results32 the MAE is decreasing when \nadding electrons to Fe  (FeCo/MgO) . On the other hand, when \nFe is doped with holes  (FeCr/MgO)  the MAE is decreasing . \nThis is consistent with our expectation, though a simple \npicture of doping fails at a very small doping level where \naccording to Fig. 3 we s ee an increase of MAE in the range of \nne down to – 0.05. This discrepancy  indicates that  doping with \nCr is not exactly the same  as the  rigid decrease of number of \nelectrons  in Fe . Nevertheless, we see again a qualitative \nagreement between our simple picture presented in Fig. 1b and \nthe accurate calculation shown in Fig. 4.      \n \nFIG. 4. MAE of the Fe 1-xMx/MgO interface (M = Co, Cr) versus \nnumber of doped electrons ne (holes for negative ne) calculated by \nKKR -CPA method .  \n \nIn conclusion, starting from a very simple picture of orbital \npopulation, we have qualitatively explained the available \nresults of the voltage controlled surface (interface) magnetic \nanisotropy of 3d transition metal thin films used in MTJs . The \nresults of the DFT calculations for the Fe (001) surface, clean \nFe1-xCox(001)/MgO interface and oxidized Fe(001)/MgO  \ninterface, involving effects of alloying and electric field, can \nbe understood in terms of changes in the population of the \nminority -spin dxz,yz orbitals which enhance the surface MAE \nwith electron depletion at the interface. On the other hand, t he \nhole d oped Fe(001) exhibits an inverse VCMA, where the \nMCA enhancement is achieved when electron s are  \naccumulated at the Fe(001)/MgO interface. In this regime , we \npredict a significantly enhanced VCMA which exceeds \n1pJ/Vm.  These results may be important to find the materials \nstructures with enhanced VCMA which is critical for device \napplications.  \nThis research was  supported by the Nanoelectronics \nResearch Corporation  (NERC), a wholly owned subsidiary of \nthe Semiconductor  Research Corporation (SRC), through the \nCenter for Nanoferroic Devices  (CNFD).   Computations were \na\nb\n-0.10 -0.05 0.00 0.05-1001020301.21.62.02.42.0 1.0 0.0 -1.0\n0.40.0-0.4-0.8-1.2-1.0 -0.5 0.0 0.51.41.61.8\n VCMA (mJ/m2/electron)\nne EMgO (V/nm)MAE (mJ/m2)\n VCMA (pJ/V/m) \nEMgO (V/nm)MAE (mJ/m2)\n-0.1 0.0 0.1 0.21.01.52.0 \n FeCo\n FeCrMAE (mJ/m2)\nne5 \n performed at the University of Nebraska Holland Computing \nCenter.   \n \n* E-mail: kevinjiazhang@gmail.com  \n** E-mail: tsymbal@unl.edu  \n \n-------------------  \n1 F. Matsukura, Y. Tokura, and H. Ohno, Nat. Nanotech. 10, 209 \n(2015).  \n2 E.Y. Tsymbal, Nat. Mater. 11, 12 (2012).  \n3 C.-G. Duan, J. P. Velev, R. F. Sabirianov, Z. Zhu, J. Chu, S. S. \nJaswal, and E. Y. Tsymbal, Phys. Rev. Lett. 101, 137201 (2008).   \n4 H. Zhang, M. Richter, K. Koepernik, I. Opahle, F. Tasná di, and H. \nEschrig, New J. Phys. 11, 043007 (2009).  \n5 K. Nakamura, R. Shimabukuro, Y. Fujiwara, T. Akiyama, T. Ito, \nand A. J. Freeman, Phys. Rev. Lett. 102, 187201 (2009).  \n6 K. Nakamura, T. Akiyama, T. Ito , M. 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Tulapurkar, T. Shinjo, M. Shiraishi, S. \nMizukami, Y. Ando, and Y. Suzuki, Nat. Nano tech. 4, 158 (2009).  \n16 A. Rajanikanth, T. Hauet, F. Montaigne, S. Mangin, and S. \nAndrieu, Appl. Phys. Lett. 103, 062402 (2013) .  \n17 S. Miwa, K. Matsuda, K. Tanaka, Y. Kotani, M. Goto, T. \nNakamura, and Y. Suzuki, Appl. Phys. Lett. 107, 162402 (2015).  18 T. Nozaki, A. Kozioł -Rachwał, W. Skowroński, V. Zayets, Y. \nShiota, S. Tamaru, H. Kubota, A. Fukushima, S. Yuasa, and Y. \nSuzuki, Phys. Rev. Appl. 5, 044006 (2016).  \n19 Y. Shiota, T. Maruyama, T. Nozaki, T. Shinjo, M. Shiraishi, and \nY. Suzuki, Appl. Phys. Exp. 2, 063001(2009).  \n20 T. Nozaki, Y. Shiota, M. Shiraishi, T. Shinjo, and Y. Suzuki, \nAppl. Phys. Lett. 96, 022506 (2010).   \n21 S. Kanai, M. Yamanouchi, S. Ikeda, Y. Nakatani, F. Matsukura, \nand H. Ohno, Appl. Phys. Lett. 101, 122403 (2012).   \n22 W.-G. Wang, M. Li, S. Hageman, and C. L. Chien, Nat. Mater. \n11, 64 (2012).   \n23 Y. Shiota, T. Nozaki, F. Bonell, S. Murakami, T. Shinjo, and Y. \nSuzuki, Nat. Mater. 11, 39 (2012).  \n24 Y. Shiota, F. Bonell, S. Miwa, N. Mizuochi, T. Shinjo, and Y. \nSuzuki, Appl. Phys.  Lett. 103, 082410 (2013).  \n25 C. Bi, Y. Liu, T. Newhouse -Illige, M. Xu, M. Rosales, J.W. \nFreeland, O. Mryasov, S. Zhang, S.G.E. te Velthuis, and W.G. \nWang, Phys. Rev. Lett. 113, 267202 (2014).  \n26 U. Bauer, L. Yao, A. J. Tan, P. Agrawal, S. Emori, H. L. Tuller, S. \nDijken, and G. S. D. Beach, Nat. Mater. 14, 174 (2015).   \n27 G. Van der Laan, J. Phys.: Condens. Matter 10 3239 (1998).  \n28 P. Bruno, Phys. Rev. B 39, 865 (1989).  \n29 We note that the specific role of the d yz orbital in the d xz,yz \nmanifold comes from our choice of the x axis to define the MAE \nin Eq. ( 1).  \n30 H. X. Yang, M. Chshiev, B. Dieny, J. H. Lee, A. Manchon, and K. \nH. Shin, Phys. Rev. B 84, 054401 (2011).   \n31 J. Okabayashi, J.W. Koo, H. Sukegawa, S. Mitani, Y. Takagi, and \nT. Yokoya ma, Appl. Phys. Lett. 105, 122408 (2014).  \n32 J. Zhang, C. Franz, M. Czerner, and C. Heiliger, Phys. Rev. B 90, \n184409 (2014).   \n33 Details of the calculations are outlined in the Supplementary \nMaterial.   \n34 We note that the VCMA sign is opposite to that in ref. 8 due to \nincorrect assignment of the electric field orientation given in that \npaper  when calculating the MAE . \n35 K. D. Belashchenko, J. Velev, and E. Y. Tsymbal, Phys. Rev. B \n72, 140404 (R) (2005).  \n36 L. Plucinski, Y. Zhao, C. M. Schneider, B. Sinkovic, and E. \nVescovo, Phys. Rev. B   80, 184430 (2009).  \n37 H. B. Michaelson, J. Appl. Phys. 48, 4729 (1977) . \n 1 \n SUPPLEMENTA RY MATERIAL  \nElucidating the  Voltage Controlled Magnetic Anisotropy  \n  \nJia Zhang,1 Pavel V. Lukashev,2 Sitaram S. Jaswal,1 and Evgeny Y. Tsymbal1    \n1 Department of Physics and Astronomy & Nebraska Center for Materials and Nanoscience,  \nUniversity of Nebraska, Lincoln, Nebraska 68588, USA  \n2 Department of Physics, University of Northern Iowa, Cedar Falls, Iowa 50614, USA  \n \n \n1. DFT  calculations  for clean and oxidized Fe(001)/MgO \ninterfaces  \nDensity-functional theory (DFT) calculations  are performed  \nusing  projector augmented wave method (PAW),1 \nimplemented in the Vienna ab initio  simulation package \n(VASP)2 within t he generaliz ed gradient approximation \n(GGA)  for the exchange -correlation potential .3 The \nintegration method4 with a 0.05 eV width of smearing is \nused, along with a plane -wave cut -off energy of 500 eV and \nconvergence criteria of 10-4 eV for ionic r elaxations and 10-3 \nmeV for the total energy calculations. The \nmagnetocrystalline anisotropy energy ( MAE ) is evaluated as \ndifference of energies calculated self -consistently \ncorresponding to magnetization pointing along the [100] and \n[001] directions, in t he presence of the spin -orbit coupling \n(SOC) . The latter is included in VASP as a perturbation \nusing the scalar -relativistic eigenfunctions of the valence \nstates. The site -projected contributions to the  MAE are \ncalculated by evaluating  the expectation values for the SOC \nenergies:        \n   \n   \n      , where c is the speed of light, \nr is the distance in each atomic sphere, V is the effective \npotential as a function of r, and L and S are orbital and spin \noperators, respectively.  For all considered systems , we use \nthe experimental in -plane lattice constant of bcc Fe, a = \n0.287 nm. The structures are relaxed in absence of electric \nfield until the largest force becomes less than 5.0 meV/Å. \nSummation over k points is performed using a  24×24×1 \nmesh in the first Brillion zone, which according to our tests \nis sufficient to provide the calculated MAE accuracy of \nabout 0.01meV .    The site and orbital contributions to the \nMAE are obtained in VASP by evaluating the SOC energy \ndifference between two magnetic axes (ΔESOC). 5 Within this \nindirect approach  it allows representing the MAE in terms \nof the sum over site and orbital contributions.  \nFor the Fe(001)/MgO interface , in our calculation we \nuse a MgO(4ML)/Fe(9ML)/MgO (4ML) /vacuum(2nm) \nsupercell  (Fig. 1S -a). The electric field is  introduced using  \nthe dipole layer placed in the vacuum  region of the \nsupercell.6  The positive electric field is defined as point ing \naway from the Fe layer to MgO. In this geometry we can \nevaluate effects of both positive and negative electric fields \nby performing calculations for only one direction of the \nfield and evaluating the layer resolved contribution to the \nMAE at the two Fe/MgO interfaces. Typically, one or two \ninterfacial monolayers of the  ferromagnetic metal provide \nthe dominant contribution to the MAE and VCMA due to the short -ranged electrostatic screening  (Figs. 1S-b,c). In \ncase of the oxidized Fe(001)/MgO interface  we assume  an \nadditional oxygen atom which is placed in the first \ninterfacial Fe monolayer atop the interfacial Mg atom , \nforming an Fe(001)/FeO/MgO interface .  \n \nFig. 1S: (a) MgO/Fe/MgO supercell structure  (vacuum layer is not \nshown).  (b,c) Contributions from different Fe sites (monolayers) to \nMAE (b) and to the change in MAE ( MAE) in applied electric \nfield Evac = 2V/nm (c).  \nFigs. 2S -a,b show the orbital resolved density of states \n(DOS) at the interfacial Fe atom for the clean Fe(001)/MgO  \nand oxidized Fe(001)/FeO/MgO interfaces , respectively. In \nthe latter case, the interfacial Fe atom lies in the FeO plane. \nFor the clean Fe(001)/MgO  interface, there is a pronounced \npeak in the DOS of the interfacial Fe atom just above the \nFermi energy. This peak is as sociated with the interface \nresonant state which is largely composed of the dxz,yz orbitals \nMgO MgO Fe\n0.00.51.0\n123456789\n  MAE (mJ/m2)\n-0.020.000.020.04\n \n  MAE (mJ/m2)\nFe monolayer a\nb\nc2 \n (Fig. 2S -a, bottom panel, blue curve). It is notable that the \nsizable portion of the minority -spin dxy states are occupied \nfor the clean interface (Fig. 2S-a, botto m panel,  red curve), \nwhereas for the oxidized interface these states are largely \nunoccupied and lie in the range of energies from 1 to 3 eV \nabove the Fermi level (Fig. 2S-b, bottom panel,  red curve).  \nThese are antibonding states resulting from the \nhybridization of the dxy orbitals of Fe and px,y orbitals of O in \nthe (001) plane .  \n \nFIG. 2 S. Orbital resolved density of states (DOS) at  the interfacial \nFe atom for clean  Fe(001)/MgO (a) and oxidized \nFe(001)/FeO/MgO ( b) interfaces. Top and bottom pan els \ncorrespond to majority - and minority -spin contributions, \nrespectively.  \nNote that in all our calculations we do not relax the \natomic structure in the in the presence of electric field. The \nelectric field in MgO layer is expected to be EMgO = Evac/, \nwhere  is the dielectric constant of MgO and Evac is the \nelectric field in vacuum. From our calculation we estimate \nthe dielectric constant to be    3.3 from the calculated ratio \nof potential slop between MgO and vacuum , which is less \nthan the experimental value of    9.5, due to the neglect of \nthe ionic response of MgO in the calculation. In order to \ntake into account this deficiency, we plot the MAE in Figs. \n2a and 2c against the expected electric field in MgO \ncorres ponding to experimental conditions, i.e. EMgO = \nEvac/9.5.  2. DFT calculations for the “electrostatically doped ” \nFe(001)/MgO interface  \nSimilar to the above, t he first -principles calculations  for the \nelectrostatically doped Fe(001)/MgO interface are \nperformed using VASP. A symmetric supercell \nMgO(3MLs)/Fe(3MLs)/  MgO(3MLs)/ vacuum (2nm) is used \nin the calculations. The structure is relaxed for the neutral \nsystem until the force on all atoms is less than 1 meV/ Å.  \nThe MAE  is evaluated using the force theo rem.7 Within this \napproach, f irst, the electronic structure is self -consistently \ncalculated in the absence of SOC  by using 16× 16× 1 k -points \nk-point grid in the Brillouin zone. Then,  the MAE is \nobtained by taking the band energy difference in the \npresence o f SOC between two magnetic ax es along [100] \nand [001] directions with a finer 32× 32× 1 k -points  mesh . \nElectrostatic doping is performed by changing the number \nof valence electrons in the whole system and neutralizing \nthis cha rge by background of the constant charge of \nopposite  sign.  By using this method, the excess charge  \ndensity in vacuum and MgO is redistribute d in a way to \ndeposit the most part of the charge to the metal surface \n(interface ), which is analogous to the electric fie ld effect. \nThe cons tant charge density background extending to the \nmetal region produces a “chemical” doping effect in \naddition to the “electrostatic” doping through the electric \nfield. This effect is minimized in our calculations due to the \nsufficiently large size of the supercell (3. 91 nm) compared \nto Fe layer thickness (0.43 nm).  Atomic relaxations are not \nperformed for the charged system. Fig. 3S shows results of \ncalculation for the excess  valence charge  ne = –0.025 (the \nnegative number corresponds to adding holes).  It is seen that \nthe electrostatic potential in vacuum  has a parabolic shape  \nwhich is due to a constant background charge  in vacuum . \nThe MAE and ne for the Fe(001)/MgO interface  (shown  in \nFig. 3 of the main text ) is considered to be a half of the \ncorresponding values for the supercell, which contains two \nidentical interfaces.    \n \nFig. 3S: Calculated electrostatic potential energy (black curve) and \nexcess charge distribution (blue curve) across the MgO(3MLs)/  \nFe(3MLs)/MgO (3MLs)/vacuum(2nm) supercell structure  for ne = – \n0.025.    \na\nb\n-6 -4 -2 0 2-101 dz2\n dx2y2\n dxy\n dxz,yz\nEF\n  DOS (states/eV/atom)\nEnergy (eV)\n-6 -4 -2 0 2-1012\n dz2\n dx2y2\n dxy\n dxz,yz\nEF\n  DOS (states/eV/atom)\nEnergy (eV)Fe(001)/ MgO\nFe(001)/ FeO/MgO\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-0.6-0.4-0.20.00.20.40.6Fe     MgO                         vacuum                          MgO    Fe U (eV)\nz (nm)add no. of electron -0.025\n-0.2-0.10.00.10.2\n(z)3 \n 3. DFT calculations for the chemically doped \nFe(001)/MgO interface  \nCalculations of the MAE for the Fe1-xMx/MgO (M = Co, \nCr) interfaces are performed using the full relativistic \nscreened -Korringa -Kohn -Rostoker (KKR)  method based on \ndensity functional theory , where t he spin-orbit coupling is \ntaken into account by solving the full  relativistic Dirac \nequation .8 The coherent  potential approximation (CPA) is \nutilized to describe the composition al dependence of the   \nFe1-xMx alloys . The potentials are described within the \natomic sphere approximation (ASA).  Particular calculations \nare performed using MgO(3MLs)/Fe M(3MLs)  supercell \ngeometry by imposing periodic boundary conditions. More \ndetails of the calculations  can be found in ref. 9. \n                                                           \n1  P. Blö chl, Phys. Rev. B 50, 17953 (1994).  \n2  G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).   \n3  J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, \n3865 (1996).  \n4  M. Methfessel and A. T. Paxton, Phys. Rev. B 40, 3616 (1989).  \n5  Note that the second order correction to the total MAE due to \nSOC is a half of the first order MAE , as discussed by V. \nAntropov, L. Ke, and D. Aberg, Solid State Communications \n194, 35 (2014).  \n6  J. Neugebauer and M. Scheffler, Phys. Rev. B 46, 16067 \n(1992).  \n7  A. Lehnert, S. Dennler, P. Błoński, S. Rusponi, M. Etzkorn, G. \nMoulas, P. Bencok, P. Gambardella, H. Brune, and J. Hafner, \nPhys. Rev. B 82, 094409 (2010).  \n8  J. Zabloudil, R. Hammerling , L. Szunyogh, and P. Weinberger, \nElectron Scattering in Solid Matter: A Theoretical and \nComputational Treatise, Springer Series in Solid -State \nSciences, Vol. 147 (Springer, Berlin, 2005).  \n9  J. Zhang, C. Franz, M. Czerner, and C. Heiliger, Phys. Rev. B \n90, 184409 (2014) . \n "    },    {        "title": "1612.04478v5.First_principles_study_of_intersite_magnetic_couplings_in_NdFe___12___and_NdFe___12__X__X___B__C__N__O__F_.pdf",        "content": "arXiv:1612.04478v5  [cond-mat.mtrl-sci]  18 Jul 2017Ver. 5.0.0\nFirst-principles study of intersite magnetic couplings\nin NdFe 12and NdFe 12X (X = B, C, N, O, F)\nTaro Fukazawa,1,2,a)Hisazumi Akai,3,2Yosuke Harashima,1,2and Takashi Miyake1,2\n1)CD-FMat, National Institute of Advanced Industrial Scienc e and Technology,\nTsukuba, Ibaraki 305-8568, Japan\n2)ESICMM, National Institute for Materials Science, Tsukuba , Ibaraki 305-0047,\nJapan\n3)The Institute for Solid State Physics, The University of Tok yo,\n5-1-5 Kashiwano-ha, Chiba, Japan\n(Dated: 11 October 2018)\nWe present a first-principles investigation of NdFe 12and NdFe 12X (X = B, C, N,\nO, F) crystals with the ThMn 12structure. Intersite magnetic couplings in these\ncompounds, so-called exchange couplings, are estimated by Liecht enstein’s method.\nIt is found that the Nd–Fe couplings are sensitive to the interstitial dopant X, with\nthe Nd–Fe(8j) coupling in particular reduced significantly for X = N. T his suggests\nthat the magnetocrystalline anisotropy decays quickly with rising te mperature in\nthe X = N system although nitrogenation has advantages over the o ther dopants in\nterms of enhancing low-temperature magnetic properties. The Cu rie temperature is\nalso calculated from the magnetic couplings by using the mean field app roximation.\nIntroduction of X enhances the Curie temperature, with both str uctural changes and\nchemical effects found to play important roles in this enhancement.\nPACS numbers: 75.50.Ww, 75.50.Bb, 71.20.Be, 71.20.Eh, 71.15.Mb\nKeywords: Hard magnet, Permanent magnet, First-principles calc ulation, NdFe 12N,\nThMn 12-type structure\na)E-mail: taro.fukazawa@aist.go.jp\n1I. INTRODUCTION\nThe magnetic compound NdFe 12N, which has the ThMn 12tetragonal structure, has at-\ntracted much attention since Hirayama et al.1,2successfully synthesized the compound—\nmotivated by the results of a first-principles calculation3—and found that it has higher\nspontaneous saturation magnetization and anisotropy field than N d2Fe14B. NdFe 12was first\nsynthesized on a substrate, and then nitrogen was introduced. T he nitrogenation greatly\nenhanced the magnetization and anisotropy field1; nitrogenation is known to have this effect\nin other similar magnetic compounds also. Although it has not been rep orted in the case\nof NdFe 12, typical elements can also enhance the Curie temperature of rare -earth magnets.\nFor example, simultaneous enhancement of saturation magnetizat ion and the Curie temper-\nature has been experimentally observed in R 2Fe17Cx(Ref. 4 and 5), R 2Fe17Nx(Ref. 5), and\nRTiFe 11Nx(Ref. 6 and 7), where R denotes a series of rare-earth elements.\nThe advantages of typical elements in magnets have also been stud ied theoretically.\nKanamori8,9discussed mixingofthe d-statesoftransitionmetalsiteswiththestatesofneigh-\nboring atoms in compounds, and he suggested that the strong fer romagnetism in Nd 2Fe14B\nis possibly attributable to mixing between the boron elements and the neighboring iron\nelements. The importance of this chemical effect has been confirme d by first-principles\ncalculation of light elements in iron lattices10. In a recent first-principles study11on the\ninterstitial X in NdTiFe 11X (X = B, C, N, O, F), enhancement of magnetization by the\nmagnetovolume effect and chemical effect was investigated.\nFinite-temperature magnetism in hard-magnet compounds has also been studied. It was\ndemonstrated in a study12of an ab initio spin model of NdFe 12N that intersite magnetic\ncoupling between rare-earth sites (R) and transition metal sites ( T) is a key factor, and\nmagnetic anisotropy above room temperature is expected to be en hanced significantly by\nstrengthening the R–T couplings.\nIn this paper, we investigate magnetic couplings in NdFe 12and NdFe 12X for X = B, C, N,\nO, F based on first-principles calculation. The theoretical framewo rk and the computational\nmethods are described in Section II. We analyze the effects of the in terstitial dopant X on\nthe magnetic couplings by using Liechtenstein’s formula. The obtaine d intersite magnetic\ncouplings are converted to a Curie temperature by using the mean fi eld approximation. The\ndependence of the R–T magnetic couplings on X is discussed in Section IIIA, and the values\n2of the Curie temperature are discussed in Section IIIB.\nII. THEORETICAL FRAMEWORK AND METHODS\nWe use AkaiKKR13—a program based on the Korringa–Kohn–Rostoker (KKR)14,15\nGreen’s function method, which is also known as MACHIKANEYAMA—fo r calculation\nof magnetic moments and intersite magnetic couplings. This calculatio n is performed based\non the local density approximation16,17.\nSpin–orbit coupling is considered at only the Nd site, with the f-electr ons treated as\nan open core of trivalent Nd with the configuration limited by Hund’s ru le (the open-core\napproximation18–20). We apply the self-interaction correction scheme proposed by Pe rdew\nand Zunger21to Nd-f orbitals under the self-consistent scheme. The contribut ion of the Nd-f\nelectrons to values of the magnetic moment is not included in our resu lts. The 1s–5s, 2p–4p,\n3d–4d orbitals at Nd; the 1s–2s, 2p–3p orbitals at Fe; and the 1s or bital at X and the 2s\norbitals at X = F are treated as core states. We consider up to d-sc attering ( lmax= 2) in\nthe systems, and sample 6 ×6×6 k-points in the full first Brillouin zone with reduction\nof computational tasks by exploiting the crystal symmetry. A com mon set of muffin-tin\nradii is used in all calculations for NdFe 12and NdFe 12X (X = B, C, N, O, F) such that the\ndomain volume of local potentials perturbed in Liechtenstein’s metho d does not depend on\nthe system. We place one X atom per formula unit at the 2b site.\nWe use lattice parameters obtained via QMAS22, which is a package for first-principles\ncalculation based onthe projector augmented-wave (PAW) metho d23,24, within a generalized\ngradient approximation. These values of the lattice parameters ar e summarized in Appendix\nA. We refer readers to Ref. 25 for details of the calculation setup.\nTo investigate the dependence of magnetic properties on lattice pa rameters, we also per-\nform calculations of hypothetical NdFe 12X systems with the lattice parameters fixed to the\nvalues of other systems. We write “A#B” to mean “the system havin g chemical formula\nA with the lattice parameters of system B.” For example, we examine N dFe12#NdFe 12X to\ninvestigate the effect of structural changes (lattice expansion, etc.) induced by the introduc-\ntion of X separately from the chemical effects. For further simplicit y, we use the ordinary\nchemical formulae NdFe 12and NdFe 12X to denote systems with the optimal structure unless\notherwise stated.\n3Our definition of intersite magnetic couplings is as follows. According t o Liechtenstein\net al.26, we map energy shifts caused by spin-rotational perturbations o ntoJi,jvalues in the\nfollowing classical Heisenberg Hamiltonian H:\nH=−/summationdisplay\ni/summationdisplay\njJi,j/vector ei·/vector ej, (1)\nwhere/vector eiis a unit vector taking the direction of the local spin moment at the ith site. It\nis also possible to determine a set of Ji,jvalues by comparing the total energies calculated\nfor several different magnetic structures (e.g., different anti-fe rromagnetic configurations).\nHowever, to obtain a required number of unique Ji,jvalues for a fixed magnetic structure,\nwe exploit Liechtenstein’s prescription. Our Hamiltonian can be trans formed formally into\nthe form of H=−/summationtext\ni,j/vectorSiJi,j/vectorSj, withJi,j=S0\niJi,jS0\njand/vectorSi=S0\ni/vector ei, whereS0\nidenotes\nthe local moment of the ith site in the ground state. Note, however, that Ji,jis an indirect\noutcome from Liechtenstein’s scheme, whereas Ji,jis more directly related to the energy\nshift under the perturbation at the ith andjth sites considered in the scheme. Therefore,\nwe discuss values of Ji,jinstead of Ji,jto maintain theoretical clarity.\nIII. RESULTS AND DISCUSSION\nA. Chemical effects of X on JR–T\nFigure 1 shows the magnetic coupling constants JR–T(Ji,jof R–T bonds) for NdFe 12\n(denoted by X = Vc) and NdFe 12X for X = B, C, N, O, F. There are three iron sublattices\n(8j, 8i, and 8f) in the ThMn 12structure, and values of JR–Tfor the shortest Nd–Fe(8j),\nNd–Fe(8i), and Nd–Fe(8f) bonds are shown. The three coupling co nstants have similar\nvalues in NdFe 12. The introduction of X changes the strengths of the couplings sign ificantly,\nwith the magnitudes of the changes differing considerably between t he different couplings.\nTheJNd–Fe(8j) coupling exhibits the strongest dependency on X, which is understa ndable\nbecauseFe(8j)istheclosestsitetoX.The JNd–Fe(8j)couplingisreducedinallcaseswhereXis\nintroduced, with the exception of X= B. The smallest valueis found at X= N. That value is\nlessthanhalfofNdFe 12. Incontrast, JNd–Fe(8i)andJNd–Fe(8f)areenhanced, althoughslightly,\nby N. Nitrogenation thus appears to effectively reduce the Nd–Fe c oupling strengths. This\nsuggests that nitrogenation exacerbates unfavorable thermal dumping of magnetocrystalline\nanisotropy, although it induces strong uniaxial anisotropy at low te mperatures12.\n4-2\n-1.8\n-1.6\n-1.4\n-1.2\n-1\n-0.8\n-0.6\n-0.4\n-0.2Vc B C N O FJij [meV]\nXJNd-Fe(8j)\nJNd-Fe(8i)\nJNd-Fe(8f)-2\n-1.8\n-1.6\n-1.4\n-1.2\n-1\n-0.8\n-0.6\n-0.4\n-0.2Vc B C N O FJij [meV]\nXJNd-Fe(8j)\nJNd-Fe(8i)\nJNd-Fe(8f)\nFIG. 1. (Color online)(Left) Values of JR–TforNdFe 12X(X =Vc, B,C,N,O,F), whereNdFe 12Vc\ndenotes NdFe 12. (Right) Values of JR–Tfor NdFe 12X#NdFe 12Z (Z = B, C, N, O, F). Values of\nJR–Tcalculated for the same element Z are connected by lines, wit h dashed lines indicating Z =\nF.\nTo examine the chemical effects separately from the structural e ffects, the results for\nNdFe12X#NdFe 12Z (Z=B, C, N, O, F) are plotted in the right panel of Fig. 1. It can be\nseen that the overall trends as a function of X are not highly depen dent on Z. This indicates\nthat chemical effects dominate this behavior. Looking at the result s in more detail, it can\nbe seen that each of the absolute values of the JR–Tconstants has its minimum at X = N\nnotwithstanding the choice of lattice parameters. Furthermore, the reductions caused by N\nare smaller than the fluctuations due to the structural changes in the cases of JNd–Fe(8i)and\nJNd–Fe(8f). However, the reduction in JNd–Fe(8j)due to N is too large to be overcome by the\nstructural effect.\nFigure 2 shows JNd–Fe(8j) for NdFe 12X#NdFe 12N with fractional changes to the atomic\nnumber of X, ZX. This perturbation on ZXserves as a theoretical probe to the system, and\nthe external potential with the fractional ZXcan also be interpreted as a model within the\nvirtual crystal approximation for random occupation of the X site by multiple elements with\nthe mean value of their atomic numbers coinciding with ZX. In Fig. 2, there is a significant\ndecrease of the magnitude at approximately ZX= 7 (X = N), with a minimum at ZX= 6.8.\nThe figure also shows the local spin moment at the Nd site as a functio n ofZX. This local\nmoment is strongly correlated with JNd–Fe(8j), so the reduction in JNd–Fe(8j)by nitrogenation\nis attributable to the reduction in the spin magnetic moment at the Nd site. Note that\nJNd–Fe(8j) does not necessarily change proportionally to the local moments be causeJi,jin\n5-2\n-1.8\n-1.6\n-1.4\n-1.2\n-1\n-0.8\n-0.6\n-0.4\n-0.2 5  5.5  6  6.5  7  7.5  8-0.3\n-0.25\n-0.2\n-0.15Jij [meV]\nLocal moment [ µB]\nZXJNd-Fe(8j) (left scale)\nLocal moment@Nd (right scale)\nFIG. 2. (Color online) JNd-Fe(8j) (left scale) for NdFe 12X#NdFe 12N as a function of ZX— the\natomic number of X — compared with the local moment of the Nd si te (right scale). Note that\nthe right scale for the local moments is inverted.\n 0\n 50\n 100\n 150\n 200\n 250\n 300-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1Zx=6.8Minority Spin\nEnergy [Ry] 0 50 100 150 200 250 300EFMajority Spin [states/Ry]Total DOS\nFe(8j)-d\nNd-d (x10)\nX-p (x10)\nFIG. 3. (Color online) Total and partial DOS of NdFe 12X#NdFe 12N atZX= 6.8. Note that the\npartial densities of the Nd-d states and the X-p states are ma gnified tenfold.\nJi,j=S0\niJi,jS0\njalso depends on the ground state of the system.\nFigure 3 plots the density of states (DOS) at ZX= 6.8. The X-p DOS has a peak at the\nFermi level in the majority-spin channel, but there is no feature in t he minority channel at\nthat point. This type of DOSdistribution has been discussed previou sly10,11,27to explain the\nchange in the total spin moment caused by X. The discussion is also us eful for understanding\nthereduction oftheNdmoment, andtheresults ofour calculationa lso show goodagreement\nwith the accompanying theory. The X-2p state hybridizes with stat es of the neighboring\nFe sites. An antibonding state between them appears above the Fe rmi level in the cases\nof X = B and X = C. As ZXincreases, the X-2p level becomes deeper. Consequently,\nthis hybridized state is pulled down (Fig. 4) and crosses the Fermi lev el atZX∼7 in the\n6majority-spin channel, which leads to enhancement of the magnetic moment of the whole\ncompound (although the magnetovolume effect also enhances the m agnetic moment, the\ndependence of the total moment on X comes mainly from the chemica l effect11).\nThis hybridized state inthe majority-spin channel hassome weight a t the Nd site. Hence,\nas the occupation number of this state increases, the majority-s pin density becomes larger.\nThis results in a decrease in the magnitude of the local spin moment at the Nd site because\nit is antiparallel to the total spin moment (electrons in the majority- spin channel of the\nentire system are a minority at the Nd site). This reduction leads to w eakening of the\nspin-rotational perturbation considered in Liechtenstein’s formu la and weakens JNd–Fe(8j).\nTo summarize, filling of the hybridized state affects both the magnet ization and the Nd–Fe\nmagnetic coupling, but in opposite ways.\nB. Chemical effect of X on the Curie temperature\nAll values of the Curie temperature are computed within the mean-fi eld approximation\nfromJi,j. We use two different sets of Ji,jfor comparison. The larger set, L, is composed of\nJi,jfor thebonds with lengths shorter than0.9 a, whereais theside length of thebase square\nfor the conventional unit cell. The smaller set, S, is the union of the following two subsets:\n(i)Ji,jfor the shortest bonds in the classes of the site combinations (e.g., Fe(8j)–Fe(8i),\nFe(8f)–Nd and X–X); and (ii) Ji,jfor the bonds shorter than 3.80 ˚A. The geometries of the\nT–T and R–T bonds in Sare shown in Fig. 5. As a result, all bonds between the Fe’s and\ntheir 3rd nearest Fe-neighbors are included in S. When deciding on this cutoff of 3.80 ˚A,\nwe referred to a previous Monte Carlo study of Nd 2Fe14B (Ref. 28), where results with a\ncutoff of 3.52 ˚A are in good agreement with results calculated with a cutoff of 15.8 ˚A.\nFigure 6 shows the Curie temperatures of NdFe 12and NdFe 12X obtained from the sets L\n(left)and S(right). ResultsforNdFe 12#NdFe 12Xarealsopresented. Theoverallbehavior of\nthe two curves is similar between the two panels, which would justify o ur use of the smaller\nsetSin the analysis of Ji,jdependence of the Curie temperatures appearing later. The\nCurie temperature is enhanced by the introduction of X in all cases s tudied. Comparing the\nresults for NdFe 12X with NdFe 12#NdFe 12X, it can be seen that the enhancement originates\nprimarily from the structure as for X = C. The same order of enhanc ement originating from\nthe structural change (mainly thevolume expansion) isalso seen in t hecases ofX = B, N, O,\n7 0\n 50\n 100\n 150\n 200\n 250\n 300-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1NdFe12Minority Spin\nEnergy [Ry] 0 50 100 150 200 250 300EFMajority Spin [states/Ry]Total DOS\nFe(8j)-d\nNd-d (x10)\n 0\n 50\n 100\n 150\n 200\n 250\n 300-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1NdFe12BMinority Spin\nEnergy [Ry] 0 50 100 150 200 250 300EFMajority Spin [states/Ry]Total DOS\nFe(8j)-d\nNd-d (x10)\nX-p (x10)\n 0\n 50\n 100\n 150\n 200\n 250\n 300-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1NdFe12CMinority Spin\nEnergy [Ry] 0 50 100 150 200 250 300EFMajority Spin [states/Ry]Total DOS\nFe(8j)-d\nNd-d (x10)\nX-p (x10)\n 0\n 50\n 100\n 150\n 200\n 250\n 300-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1NdFe12NMinority Spin\nEnergy [Ry] 0 50 100 150 200 250 300EFMajority Spin [states/Ry]Total DOS\nFe(8j)-d\nNd-d (x10)\nX-p (x10)\n 0\n 50\n 100\n 150\n 200\n 250\n 300-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1NdFe12OMinority Spin\nEnergy [Ry] 0 50 100 150 200 250 300EFMajority Spin [states/Ry]Total DOS\nFe(8j)-d\nNd-d (x10)\nX-p (x10)\n 0\n 50\n 100\n 150\n 200\n 250\n 300-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1NdFe12FMinority Spin\nEnergy [Ry] 0 50 100 150 200 250 300EFMajority Spin [states/Ry]Total DOS\nFe(8j)-d\nNd-d (x10)\nX-p (x10)\nFIG. 4. (Color online) Total and partial DOS of NdFe 12and NdFe 12X (X = B, C, N, O, F). Note\nthat the partial density values of the Nd-d states and the X-p states are multiplied by 10 to show\nthe detail of the curves.\nF. However, the structural change accounts only for approxima tely half of the enhancement.\nThe chemical effects are as important as the structural change.\nThe Curie temperature of NdFe 12N shown in Fig. 6 is much higher than the experimental\nvalue ofTC≈820 K obtained by Hirayama et al.1. This overestimation in our calculation\npresumably comes from the use of the mean field approximation, whic h almost certainly\n8FIG. 5. (Color online) Crystal structure of NdFe 12X. The left figure shows the conventional unit\ncell of the structure. The right figures show cross sections o f the cell with the ab- and ac-planes\nindicated by light blue and green, respectively, in the left figure. The T–T bonds (magenta) and\nthe R–T bonds (green) in the smaller set, S, are also shown. Solid lines indicate in-plane bonds,\nand dashed lines indicate out-of-plane bonds.\n 900 1000 1100 1200 1300 1400\nVc B C N O FTc [K]\nXNdFe12X\nNdFe12 with NdFe12X lattice\n 900 1000 1100 1200 1300 1400\nVc B C N O FTc [K]\nXNdFe12X\nNdFe12 with NdFe12X lattice\nFIG. 6. (Color online) The Curie temperature of NdFe 12X (X = Vc, B, C, N, O, F) with the\noptimized lattices (+ symbols) and NdFe 12#NdFe 12X (×symbols). NdFe 12Vc denotes NdFe 12.\nThe data in the left figure are calculated from the larger Ji,jset,L; the data in the right figure are\ncalculated from the smaller set, S.\n9overestimates TCof spin models. However, the calculated difference between TCbefore and\nafter the nitrogenation is comparable to the experimental value of TNdTiFe 11Nx\nC −TNdTiFe 11\nC≈\n200 K based on TNdTiFe 11Nx\nC from Ref. 6 and TNdTiFe 11\nCfrom Ref. 29.\nToseehowtheCurietemperaturedependsonthechangein Ji,jcausedbytheintroduction\nof X, we calculate the following quantity:\n∆T(k,l)\nC=TC[{JNdFe12X\ni,j}]−TC[{JLOU,(k,l)\ni,j}], (2)\nwhereTC[{Ji,j}] is the Curie temperature calculated from the set of magnetic coup lings\n{Ji,j}, andJLOU,(k,l)\ni,jis defined as follows:\nJLOU,(k,l)\ni,j=\n\nJNdFe12\ni,j[(i,j) is equivalent to ( k,l)]\nJNdFe12X\ni,j (otherwise), (3)\nwhere LOU stands for “leave-one-unchanged.” Note that this ∆ T(k,l)\nCis positive when the\nchange in Jk,lcaused by the introduction of X enhances the Curie temperature. We also use\na similar quantity ˜∆T(k,l)\nCby replacing Ji,jin the reference NdFe 12system in equation (3)\nwith the values from NdFe 12#NdFe 12X. In this analysis, we focus on the bonds in Sand\nuse only their Ji,jvalues to calculate ∆ T(k,l)\nC.\nThe left panel in Fig. 7 shows ∆ T(k,l)\nCas a function of X. There are two Fe(8i)–Fe(8i)\nbonds and two Fe(8j)–Fe(8i) bonds in the Ssubset. To distinguish one from the other, we\ndenote the shortest bonds by <i>and the second-shortest bonds by <ii>in Fig. 7 and\nthe following discussion. From Fig. 7, it can be seen that the values of ∆TT–T\nC(∆T(k,l)\nC\nfor T–T bonds) are much larger than the values of ∆ TR–T\nCand ∆TR–R\nC, which are denoted\nby “Others” in the figure. This indicates that changes in the T–T bon ds predominantly\ncause the enhancement of TC. The right panel in Fig. 7 shows ˜∆T(k,l)\nC. We do not find any\nsignificant differences between ∆ T(k,l)\nCand˜∆T(k,l)\nC, which indicates that the chemical effects\nsignificantly contribute to ∆ T(k,l)\nC, whereas the structure change plays a minor role.\nBoth∆TFe(8j)–Fe(8f)\nC and˜∆TFe(8j)–Fe(8f)\nC have the largest magnitude in the range of X = B–\nO. This implies that the change of JFe(8j)–Fe(8f) caused by the introduction of X has a strong\npositive effect on the enhancement of the Curie temperature. It is noteworthy that even at\nX = C, where the lattice expansion alone seems sufficient to explain the enhancement of\nthe Curie temperature, ˜∆TFe(8j)–Fe(8f)\nC is large and not very different from that at X = B, N,\nO. Therefore, whereas the change in JFe(8j)–Fe(8f) gives the largest contribution, it does not\n10-200-100 0 100 200 300 400\nB C N O F∆Tc [K]\nXOthersFe(8i)-Fe(8f)Fe(8j)-Fe(8f)Fe(8j)-Fe(8i) < ii >Fe(8j)-Fe(8i) < i >Fe(8j)-Fe(8j)\n-200-100 0 100 200 300 400\nB C N O F∆T~\nc [K]\nXOthersFe(8i)-Fe(8f)Fe(8j)-Fe(8f)Fe(8j)-Fe(8i) < ii >Fe(8j)-Fe(8i) < i >Fe(8j)-Fe(8j)\nFIG. 7. (Color online) (Left) Difference in Curie temperature ∆T(k,l)\nCas defined by equation (2),\nand (Right) ˜∆T(k,l)\nCin which the lattice parameters of the reference NdFe 12system [see the first\nline in Eq. (3)] are set to the parameters of NdFe 12X.\nexplain the dependence of the Curie temperature on X. In the case of X = B, the difference\nfrom X = C mainly comes from the increase in ˜∆TFe(8j)–Fe(8j)\nC , whereas in the case of X = N,\nO, F, it comes mainly from the increase in ˜∆TFe(8j)–Fe(8i) <ii>\nC and˜∆TFe(8i)–Fe(8f)\nC . This leads\nus to believe that the mechanism behind the enhancement of the Cur ie temperature differs\nbetween the case of X = B and those of X = N, O, F.\nIV. CONCLUSION\nWe studied and investigated the internal magnetic couplings of NdFe 12and NdFe 12X for\nX= B, C, N, O, Fby first-principles calculations andfoundthat theint roduction ofnitrogen\nto NdFe 12reduces the strength of R–T magnetic couplings owing to Nd–X hybr idization,\nwithJNd–Fe(8j) particularly reduced so significantly that lattice expansion due to th e ni-\ntrogen cannot compensate for the reduction. Although nitrogen is often used to enhance\nthe magnetic properties of magnetic compounds, our results sugg est that nitrogenation may\nhave countereffects on the anisotropy field of NdFe 12at finite temperatures.\nWe also evaluated the Curie temperatures of NdFe 12and NdFe 12X within the mean\nfield approximation and found that the volume expansion caused by t he introduction of X\ncannot explain all enhancement of TC. The introduction of X causes significant changes\nin the magnetic couplings of NdFe 12and has a significant effect on the Curie temperature.\nNitrogen was found to enhance the Curie temperature, as found e xperimentally in similar\n11compounds. Oxygen and fluorine were also found to enhance TCas much as nitrogen.\nAlthough boron also produced the same order of positive effect on TCwithin the framework\nabove (see also Appendix B2 for results with a model for the parama gnetic state), the\nmechanism appears to be different from that for the cases of X = N, O, F.\nACKNOWLEDGMENTS\nThe authors are grateful for support from the Elements Strate gy Initiative Project under\nthe auspices of MEXT. This work was also supported by MEXT as a soc ial and scientific\npriority issue (Creation of new functional Devices and high-perfor mance Materials to Sup-\nport next-generation Industries; CDMSI) to be tackled by using t he post-K computer. The\ncomputation was partly carried out using the facilities of the Superc omputer Center, the\nInstitute for Solid State Physics, the University of Tokyo, and the supercomputer of AC-\nCMS, Kyoto University. This research also used computational res ources of the K computer\nprovided by the RIKEN Advanced Institute for Computational Scie nce through the HPCI\nSystem Research project (Project ID:hp170100).\nAppendix A: Lattice parameters\nTable I shows the optimized lattice parameters for NdFe 12and NdFe 12X (X = B, C, N,\nO, F) that we used in our calculations. The parameters p8iandp8jcorrespond to the atomic\npositions described in table II.\nTABLE I. Optimized lattice parameters for NdFe 12X (X = Vc, B, C, N, O, F), where NdFe 12Vc\ndenotes NdFe 12. For the definitions of the inner parameters p8jandp8i, see table II.\nXa[˚A]c[˚A]p8ip8j\nVc 8.533 4.681 0.3594 0.2676\nB 8.490 4.933 0.3599 0.2683\nC 8.480 4.925 0.3606 0.2756\nN 8.521 4.883 0.3612 0.2742\nO 8.622 4.794 0.3608 0.2670\nF 8.782 4.720 0.3594 0.2487\n12TABLE II.Atomic positionsof theelements assumedinourcal culation forNdFe 12andNdFe 12X(X\n= B, C, N, O, F). The variables, x,y, andzdenote the point ( ax,ay,cz) in Cartesian coordinates.\nElement Site x y z\nNd 2a 0 0 0\nFe 8f 0.25 0.25 0.25\nFe 8i p8i0 0\nFe 8j p8j0.5 0\nX 2b 0 0 0.5\n 25 26 27 28 29 30\nVc B C N O FNdFe12XMagnetic moment [ µB / f.u.]\nXKKR-LDA+SIC\nPAW-GGA\nFIG. 8. (Color online) Total magnetic moment of NdFe 12X (X = Vc, B, C, N, O, F), where\nNdFe12Vc denotes NdFe 12.\nAppendix B: Total and local moments\n1. Comparison with full-potential calculation\nFigure 8 shows the total moment of NdFe 12and NdFe 12X (X = B, C, N, O, F); Fig.\n9 shows the local moments of NdFe 12and NdFe 12X (X = B, C, N, O, F). In both, the\nvalues from KKR-LDA+SIC are compared with those from the full-po tential calculation\nwith PAW-GGA. In both cases, the regions of integration to obtain t he local moments are\nset to the spheres with the muffin-tin radii used in the KKR calculation . The contribution\nfrom Nd-f orbitals are excluded in those plots as mentioned in section II of the main text.\n13 1.5 2 2.5 3\nVc B C N O FKKR-LDA+SICLocal moment [ µB]\nX@Fe(8j)\n@Fe(8i)\n@Fe(8f)\n 1.5 2 2.5 3\nVc B C N O FPAW-GGALocal moment [ µB]\nX@Fe(8j)\n@Fe(8i)\n@Fe(8f)\n-0.4-0.35-0.3-0.25-0.2-0.15-0.1\nVc B C N O FKKR-LDA+SICLocal moment [ µB]\nX@Nd\n-0.4-0.35-0.3-0.25-0.2-0.15-0.1\nVc B C N O FPAW-GGALocal moment [ µB]\nX@Nd\n-0.2-0.1 0 0.1 0.2\nVc B C N O FKKR-LDA+SICLocal moment [ µB]\nX@X\n-0.2-0.1 0 0.1 0.2\nVc B C N O FPAW-GGALocal moment [ µB]\nX@X\nFIG. 9. (Color online) Local magnetic moments at the Fe sites (top), the Nd site (middle), and\nthe X site (bottom) in NdFe 12X (X = Vc, B, C, N, O, F), where NdFe 12Vc denotes NdFe 12. The\ndata in the left figures are from the KKR-LDA+SIC calculation ; the data in the right figures are\nfrom the PAW-GGA calculation with the local moments defined a s integrated spin density within\nthe muffin-tin radii used in the KKR calculation.\n2. Local moment disorder\nWe also performed calculations for NdFe 12and NdFe 12X with the local moment disorder\n(LMD) model30to approximate the paramagnetic states. In the calculation, each of the\natomic sites, A(=Nd, Fe, X), is described by twofold atomic potentials, A↑andA↓, where\nA↑has opposite spin-polarization to A↓, and they are treated as potentials of distinct atoms\n14-3-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 3\nVc B C N O FLocal moment [ µB]\nX@Fe(8j)\n@Fe(8i)\n@Fe(8f)\nFIG. 10. (Color online) Values of the local magnetic moments at the Fe sites for NdFe 12(at X =\nVc) and NdFe 12X (X = B, C, N, O, F) in the state of local moment disorder30.\nthat can occupy the Asite with 50% probability. This randomness is treated with the\ncoherent potential approximation. In this hypothetical (Nd↑\n0.5Nd↓\n0.5) (Fe↑\n0.5Fe↓\n0.5)12X↑\n0.5X↓\n0.5\nsystem, the absolute value of the Fe(8j) moment is greatly reduce d for X = B and C as\nshown in Fig. 10. In contrast, for X = N, O, F, the reduction is much s maller.\nIntersite magnetic couplings in this system can be compared with Ji,jin the main text by\nusing Liechtenstein’s Ji,jbetween A↑\niandA↑\njembedded in this model system. To compare\nthem in terms of temperature, we show the Curie temperature for NdFe12and NdFe 12X\ncalculated from the thus defined Ji,jin Fig. 11. The cutoff bond length for this Ji,jis\n0.9a, which is identical to that for Lin Section IIIB. 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We \ndo not completely understand the origin of the ferroel ectric ordering at this point however t he \nsimultaneous presence of magnetic and ferroelectric ordering at room temperature in Fe 3Se4 \nalong with hard magnetic properties will open new research areas for devices.  \n \n* Email: p.poddar@ncl.res.in  \n \n \n \n \nKeywords: Room temperature multiferroics, Magnetoelectric, Iron selenide, Magneto -\nRaman , Dielectric spectroscopy.  Introduction:  \nSince the renaissance of multiferroics in previous decade1the search for single -phase  \nmultiferroic s with room tempera ture magneto -electric coupling has not yielded much result . \nThere are, in fact very few materials discovered so far that show the coexistence of \nferromagnetic  (FM)  ordering along with ferroelectric  (FE)one, though some compounds like \nBiFeO 3 became hugely p opular due to their antiferromagnetic  (AFM)property along with \nproperties2–4. Some of the  non-stoichiometric compounds have achieved some of the desired \nproperties through doping 5–7, although they are hard to reproduce and not suitable for large \nscale production. Some of the technical challenges a ssociated with most of the compounds are  \nvery low magnetic transition temperature, small polarization values and week coupling \nbetween them  etc. However, multiferroic compounds are full of surprises . Lately, there were \nreports for unconventional multiferroics where mechanisms for the origin of ferroelectricity \nwere discussed in centro -symmetric materials based on charge ordering and bond ordering8. \nFe3O4 has become the oldest known multiferroic to human kind after the discovery of \nspontaneous polarization  in Fe 3O4below 38 K arising from alternation of charge states and \nbond lengths9–11.  \nTransition metal based chalcog enides possesses huge potent ial as magneto -electric materials.  \nCdCr 2S4 and CdCr 2Se4with Curie temperature 85 K and 125 Krespectively  were reported  to \nshow multiferroic behavior,12 although the origin of this behavior was not clearly understood. \nRecently, high temperature multiferroicity was predicted in BaFe 2Se3 phase, which was quite \nunexpected13. Later, it was shown experimentally with the help of photoemi ssion spectroscopy \nthat two kinds of electrons (localized and itinerant) coexist in BaFe 2Se3 phase14.  \nBinary transition metal chalcogenide Fe3Se4 has gained attention  quite recently because of its \nhigh uniaxial magnetic anisotropy constant without the presence of any rare earth metal or noble metal atom leading to very high c oercivity at room temperature ( 4kOe)15–17. Fe 3Se4 is \nferrimagnetic at room temperature with Curie temperature 323 K  and have monoclinic structure \nwith a space group I2/m (12) 16. We have measured and discussed  the magnetic entropy and \nheat capacity ofFe3Se4 nanorods  in our previous work18.But,till now, dielectric study of this \nmaterial  has not been reported  either in bulk or nanoforms . \nIn this report we investigate a novel and unexpected perspective of Fe 3Se4 which establishes \nFe3Se4 as a room temperature multiferroic material.  Fe3Se4 shows magneto -electric coupling \nat room temperature along with very strong ferrimagnetic properties which is very unique for \nsingle phase  binary compound in undoped form  with a simple c rystal structure .These beh aviors \nare very surprising for Fe 3Se4, as, at room temperature the single crystals of this compound is \nknown to show metallic properties  due to the overlapping of cation -cation  forming  \nbands19.Metallic behavior and ferroelectricity are not mutually compatible as the conduction \nelectro ns screen the static internal electric field. This kind of dual behavior was predicted  \nverylong ago20but  found to exist recently in LiOsO 321.  \n \n \n \n \n \n \n \n Sample preparation:  \nIn this work, we have used the sample from our previous work mentioned in ref .18. The details \nof sample synthesis and basic characterizations are mentioned in the suppo rting information.  \nExperimental details and techniques:  \nMagnetization measurements were done using the VSM attachment of PPMS from Quantum \nDesign systems equipped with 9 T superconducting magnet  on powder samples  packed in \nspecial plastic holders designed  so that the dielectric contribution of the holder is negligible. . \nTemperature dependent dielectric spectroscopy was performed using Novocontrol Beta NB \nImpedance  Analyzer connected with home built sample holder to couple with a helium closed \ncycle refrige rator (Janis Inc.). The powdered sample was compressed in the form of circular \npellet of diameter 13 mm and a custom designed sample holder was used to form parallel plate \ncapacitor geometry. Ferroelectric hysteresis loop measurements were done on pellets m ade by \ncold pressing the sample powder in zero field. Raman spectra were recorded on an  HR-800 \nRaman spectrophotometer (JobinYvon -Horiba, France) using monochromatic radiation \nemitted by a He−Ne laser(633 nm), operating at 20 mW  and with accuracy in the ran ge between \n450 and 850 nm ± 1 cm−1 . An objective of 50× LD magni fication was used both to focus and \nto collect the signal from  the powder sample dispersed on the glass slide.  For magneto -Raman \nmeasu rements, a permanent bar magnet was used  to apply magnetic field near the sample. The \nintensity of magnetic field was measured  with Gaussmeter  by placing the hall probe as near as \npossible to the sample.  \n \nResults and discussion:  Nanorods of Fe 3Se4, grown by high temperature thermal decomposition  in organic solvents, \nwith average diameter 50 nm  (supporting information) . X-ray diffraction pattern showed that \nnanorods are monoclinic in structure with space group I2/m (12) (supporting information). The \nmagnetic property of these samples were studied in detail in our previous works16,18.  \nMagnetic property of Fe 3Se4 nanorods:  \nTemperature dependence of magnetization in zero field cooled (ZFC) and field cooled \nconditions (FC) shows bifurcation below 340 K and Curie transition temper ature around 323 \nK, below which it goes into a ferrimagnetic phase. Room temperature (300 K) hysteresis \nmeasurements yield a closed loop with coercivity 2.6 kOe and saturation magnetization 5 \nemu/g. As the temperature is lowered to 10 K, huge increase in c oercivity occurs reaching a \nvalue of 30 kOe. These values are well according to the values reported in earlier reports in \nFe3Se4 nanoparticles15–17. The hard magnetic property in Fe 3Se4 comes from large anisotropy \nconstant which is a result of ordered iron vacancies in alternative layers of iron in lattice22,23. \nTheorigin and behav ior of magnetic properties are discussed in detail in our early reports16,18. \nObservation of electrical polarization in Fe 3Se4 nanorods:  \nThe as synthesized sample pressed in the form of pellet shows the evidence of the presence of \nspontaneous and reversible polarization in Fe 3Se4 nanorods. Figure 1 shows the P-E loop taken \nwith frequency 500 Hz and at  a potential of 100 V  at room temperature ( 300 K). No electrical \npoling of the sample was done before the measurement. The loop displ ays clear ferroelectric \nhysteresis. The hysteresis loop is symmetrical and does not show clear saturation. The shape \nof the loop resembles the hysteresis loop observed for Fe 3O4 nanoparticles and its composites \nwith PVDF24. The value of polarization is very small but comparable to other nanoparticles \nsystems12,25,26. The maximum polarization achieved for 100 V is 0.012 μC/cm2at room \ntemperature .Within the accessible instrumental range of potential (100 V), no real saturation value of polarization is obtained. The polarization can be increased with electrically poling the \nsamples to orient the dipoles prior the hysteresis measurement.   \nThere are several methods to rule out the possibility of getting an artifact in the ferroelectric \nhysteresis measurements. One of the most straightforward way is to measure the polarization \nat different frequencies, as artifacts are usually highly frequency dependen t. Figure  4.3 shows \nthe polarization hysteresis loop taken with 100 V potential at various frequencies.27. A closed \nloop is observed in  all cases spanning a broad frequency region (100 -500 Hz). As it is clear \nfrom the figure 4.3 the area under the ferroelectric loop increases with decrease in the frequency \nbut the shape of the loop remains same at all frequencies. Therefore, the occurrenc e of artifacts \ncan be ruled out in this case.  \nImpedance  spectroscopy:  \nThe dielectric response from thesample was measured in a frequencyrange 1 Hz to 106 Hz \nspanning temperature range 150 K to 350 K  at 1 V rms  (see supporting information) .The \nfrequency de pendence of various  parameters is plotted as a function of frequency in \nFigure S3.As perceived  from the figure, both the real and out of phase part of permittivity ( ε’ \nand ε” respectively) values decreases with increasing frequency.  At low frequency, the \npermittivity values consists of contributions from all the dipolar, interfacial, atomic, ionic and \nelectronic polariza tion, which can be explained by Maxwell -Wagn er theory. The heavier \ndipoles are able to follow the external field at low frequency, so that values of ε are higher. As \nthe frequency starts to increase the dipoles lag behind the field and ε value decreases. In many \ncompounds large values of permittivity are seen owing to the presence of ionic impurities in \nthe sample. This is revealed by high values of tan δ in these cases. Here, tan δ values of the \norder 10-1 are achieved (see figure S3c). At frequency greater than 104 Hz, a frequency dependent feature is  seen in tan δ plot.  The origin of this frequency dependence in loss tangent \nis not clearly understood.  \nThe temperature dependence of ε’ is extracted from the above  figure ( Figure S3)and shown in \nfigure 2 at frequencies  107, 1587 and 11952 Hz .Asmall kink  was observed around temperature \n323 K in real part of  verses te mperature curve for all frequenc ies. For clear view the \ntemperature dependence of dielectric constant and loss tangent is plotted  in bottom panel of \nfigure 2. The kink around the transition temperature is encircled. Above 317  K, the value of  \ndecreases sharply indicative of a ferroelectric to paraelectric transition .Although the feature \nseen in very weak in nature, but the proximity of the kink observed with the magnetic Curie \ntransition  (Tc) in this compound  indicates towards the presence of magneto -electric coupling \nin the compound.  \nTo understand the nature of dielectric relaxation  in these nanorods,  complex Argand plane plot \n’’ and ’, also known as Cole -Cole plot, was examined  (see suppo rting information) . Figure S4 \nshows the plots between the real ( ’)and imaginary ( ’’) part of the impedance at different \ntemperatures. It can be realized from the Cole -Cole plot that the center  of the semicircle arcs \nare below the x -axisimplying that the electrical response from the sample departs from the ideal \nDebye's relaxation process. As can be inferred from the figure , at very low temperatures (plots \nat 185 K, 220 K) , there is only one semicircle  which comes from the contribution fr om the \ngrains. As the temperature increases and approaches near the Curie temperature another \nsemicircle ap pears at lower frequency region associated with the contribution from the grain \nboundary  (mostly the organic surfactants  used in the syn thesis) . At Tc (323 K), both the \ncontributions from grain and grain boundary is prominent but when the temperature is raised \nmuch above  Tc (plot s at 336 K and 350 K ), only the contributions due to grain boundary is \nprominent.  The resistance from the grain and grain bou ndaries are calculated from the Cole -Cole plots and plotted against temperature in figure 3. The parameter (spreading factor) \ncharacterizes  the distribution of relaxation time  signifying the departure from the ideal \nelectrical response . can be determined  from the expression  for the maximum value of the \nimaginary part of the permittivity given below.  \n∈𝑚𝑎𝑥′′= (∈𝑠−∈∞)tan [(1−𝛼)𝜋4⁄]\n2 \nHere, ∈𝑠 and ∈∞are the low and high frequency limit of ∈′respectively. The spreading factor \nwas calculated from the above equation for both th e semi circle arcs and the values fo r grain \nand grain boundary were found to be  0.54 and 0. 57, respectively  at 317 K. These non -zero \nvalues of shows the poly -dispersive nature of dielectric relaxation as observed in literatures \n28,29. \nIn order to study the coupling between spin and the phonons in the present system Raman \nspectrum was measured from temperature 295 to 333 K. Figure 4shows the Raman spectr a of \nas prepared Fe3Se4 nanoparticles taken at temperature sfrom 295 K  to 333 K . The spectrum \nconsists of sharp peaks at 224, 291, 409 cm-1. The peak at 224  and 291  cm-1 can be ascribed to \nthe Fe -Se vibration modes as it is close to the reported values of 220 and 285 cm-1 for the Fe -\nSe vibra tion in β-Fe7Se8 having similar monoclinic structure30,31.As the temperature is \nincreased, the peaks, 224, 291 cm-1, show significant change in the peak position and peak \nwidth (FWHM).  \nTo analyze the spectra, least square fit with Lorentzian  line shape was used to fit the peaks. \nWhen the peak position and FWHM is plotte d against temperature , a clear incongruity is seen \naround the magnetic /ferroeletric  transition temperature  ( 323 K) (see figure 6). From the \nfigure 4, it is appreciated that  both the Raman modes softens as temperature is increased from \n295 K.  As the temper ature further reaches the magnetic/ferroelectric ordering temperature the Raman mode starts hardening and peak shifts towards higher wavenumber and then \nimmediately after Tcdecreases sharply  towards lower wavenumber . This anomaly in Raman \nmodes observed near magnetic Tcprovided  a significant input  indicating the presence of spin-\nphonon coupling in the system. This kind of anomaly near the magnetic transition temperature \nhas been  observed previously in case of some rare earth chromites32, pure selenium \nelement33and the concurrent anomaly around T c is ascribed to the spin phonon coupling . \nFor Fe 3Se4 single crystals  it was obse rved that beyond magnetic ordering temperature the  \ninteratomic spacing rearranges such that the cation -cation overlap ping disappears partially or \ncompletely19. \nObservation of spin -phonon -charge coupling:  \nEvidence for the presence of spin -charge -phonon coupling in this system can been seen from \nfigure 5 where the magnetization (M), real part of dielectric permittivity ( ’), specific heat \ncapacity at constant pressure (C p) and the heat flow curves taken from thermo gravimetric  \nanalysis (TGA) measurements are plotted as a function of te mperature. The anom aly in these \nparameters are highlighted in the shaded region.     \nConclusion:  \nIn this work, we report the observation of ferroelectric order i n Fe 3Se4 nanoparticles  at room \ntemperature. These particles also show signatures of spin -charge  coupling as  an anom aly was \nobserved in dielectric permittivity around magnetic transition temperature 323 K. Ferroelectric \npolarization measurements revealed hysteresis loops in a broad frequency range.  The \nmicroscopic origin of the coupling between spin -charge -phonon is  not clearly understood. \nVigorous theoretical calculations are required to probe this mechanism in this compound. Our important observation about t he coexistence of both magnetic and charge ordering at room \ntemperature proposes Fe 3Se4 as a possible room tem perature multiferroic compound.  \nAcknowledgement:  \nP.P. acknowledges the Centre for Excellence in Surface Science at the CSIR -National \nChemical Laboratory , network project on Nano -Safety, Health & Environment (SHE) funded \nby the Council of Scientific and Indu strialResearch (CSIR), India, and the Department of \nScience & Technology (DST), India through and Indo -Israel grant to develop materials for \nsolar -voltaic energy devices DST/INT/ISR/P -8/2011 . M.S.B  acknowledges the support from \nthe DST, Indiafor providing Junior Research Fellowship ( SRF) through the INSPIRE program.   References:  \n(1)  Hill, N. a. Why Are There so Few Magnetic Ferroelectrics? J. Phys. Chem. B  2000 , \n104 (29), 6694 –6709.  \n(2)  Seidel, J.; Martin, L. W.; He, Q.; Zhan, Q.; Chu, Y. -H.; Rother, A.; Hawkridge, M. 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R.; Yalilov, N. Z.; Abdi nov, D. S. Electrical Properties of Fe7Se8, \nFe3Se4, and NiFe2Se4 Single Crystals. Phys. Status Solidi  1973 , 20 (1), K29 –K31.  \n(20)  Anderson, P. W.; Blount, E. I. Symmetry Considerations on Martensitic \nTransformations: “Ferroelectric” Metals? Phys. Rev. Lett. 1965 , 14 (13), 532 –532. \n(21)  Shi, Y.; Guo, Y.; Wang, X.; Princep, A. J.; Khalyavin, D.; Manuel, P.; Michiue, Y.; \nSato, A.; Tsuda, K.; Yu, S.; et al. A Ferroelectric -like Structural Transition in a Metal. \nNat. Mater.  2013 , 12 (11), 1024 –1027.  \n(22)  Andresen,  a. F.; van Laar, B.; Kvamme, E.; Ohlson, R.; Shimizu, A. The Magnetic \nStructure of Fe3Se4. Acta Chem. Scand.  1970 , 24, 2435 –2439.  \n(23)  Hirakawa, K. The Magnetic Properties of Iron Selenide Single Crystals. J. Phys. Soc. \nJapan  1957 , 12 (8), 929 –938. \n(24)  Prabhakaran, T.; Hemalatha, J. Ferroelectric and Magnetic Studies on Unpoled Poly \n(Vinylidine Fluoride)/Fe3O4 Magnetoelectric Nanocomposite Structures. 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In Physics of Ferroelectrics ; Springer \nBerlin Heidelberg: Berlin, Heidelberg, 2007; Vol. 105, pp 1 –30. \n(28)  Jaiswal, A.; Das, R.; Maity, T.; Vivekanand, K.; Adyanthaya, S.; Poddar, P. \nTemperature -Dependent Raman and Dielectric Spectroscopy of BiFeO 3 \nNanoparticles: Signatures of Spin -Phonon and Magnetoelectric Coupling. J. Phys. \nChem. C  2010 , 114 (29), 12432 –12439.  \n(29)  Jaiswal,  A.; Das, R.; Maity, T.; Poddar, P. Dielectric and Spin Relaxation Behaviour in \nDyFeO3 Nanocrystals. J. Appl. Phys.  2011 , 110 (12), 124301.  \n(30)  Campos, C. XRD, DSC, MS and RS Studies of Fe75Se25 Iron Selenide Prepared by \nMechano -Synthesis. J. Magn. Magn.  Mater.  2004 , 270 (1–2), 89 –98. \n(31)  Campos, C. E. M.; de Lima, J. C.; Grandi, T. a.; Machado, K. D.; Pizani, P. S. \nStructural Studies of Iron Selenides Prepared by Mechanical Alloying. Solid State \nCommun.  2002 , 123 (3–4), 179 –184. \n(32)  Gupta, P.; Poddar, P. Using Raman and Dielectric Spectroscopy to Elucidate the Spin \nPhonon and Magnetoelectric Coupling in DyCrO 3 Nanoplatelets. RSC Adv.  2015 , 5 \n(14), 10094 –10101.  (33)  Pal, A.; Shirodkar, S. N.; Gohil, S.; Ghosh, S.; Waghmare, U. V; Ayy ub, P. \nMultiferroic Behavior in Elemental Selenium below 40 K: Effect of Electronic \nTopology. Sci. Rep.  2013 , 3 (2051), 1 –7. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n \n \n-100 -50 0 50 100-40-2002040  100 Hz\n 200 Hz\n 500 HzP(C/cm2) 10-3\nV (V)\n  \n-100 -50 0 50 100-15-10-5051015\n  P(C/cm2) 10-3F= 500 Hz\nV= 100 V\nV (V/cm)Figure 6: Ferroelectric polarization loop of  Fe 3Se4nanoparticles at frequency 500 Hz and 100 \nV applied voltage (top panel). The frequency dependence of the loop taken at frequency 100, \n200 and 500 Hz (bottom panel).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n200 250 300 3505.86.06.26.46.6\ntan \n '\n  '\nT(K)1587 Hz0.1350.1400.1450.150\n tan \n \n5.76.06.36.6\n200 250 300 3507.27.68.05.05.25.4\n  \nT(K)1587 Hz\n  \n '\n107 Hz\n  \n11952 HzFigure 2(a) . Temperature dependent plot of real part of permittivity from 200 K to 350 K at \nfrequencies 107 Hz, 1587 Hz and 11952 Hz respectively, extracted from the figure 1 (a). The \nshaded region highlights the weak anomaly observed  \n(b) Temperature dependence of real part of permittivity and loss tangent at frequency1587 Hz. \nAn anomaly can be seen (encircled) in ’ around 317 K which is in close proximity of magnetic \ntransition temperature.  \n \n  \n \n \n200 240 280 320 36023456\nGrain \nboundary Grain\nRGRGBRGCGB CG\nZ”\nZ’ RGB\nRG  \nT(K)RG \n789 \n Figure 3: Temperature dependent of resistance contribution from grain and grain boundary \nextracted from Cole -Cole plot is plotted. Inset shows the circuit arrangements and parameters \nRG and R GB in a Cole -Cole plot.    \nFigure 4:Temperature dependence of Raman scattered signal from Fe 3Se4 nanoparticles \nrecorded from temperature 295 K to 333 K (top panel).The peak position and FWHM of two \nRaman modes were deduced from these plots and plotted against temperatures (Bottom panel). \n100 200 300 400 500300600333K \n Raman shift (cm-1)Intensity (a.u.)295 K\n(d)\n290 300 310 320 330 340280284288292\n  Peak position (cm-1)\nT(K)\n290 300 310 320 330 340216219222225\nT(K)Peak position (cm-1)\n290 300 310 320 330 3405101520\n  FWHM (cm-1)\nT(K)\n290 300 310 320 330 34051015202530FWHM (cm-1)\nT(K) (c)(b) (a)(a)and (b) shows the variation for 224 cm-1mode and (c) and (d) shows the variation for 291 \ncm-1mode  \n \n   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n200 250 300 35030313233\n  \nT(K)Cp(J/mol/K)\n200 250 300 3500.000.03100 Oe \nT(K)M(emu/g)\n200 250 300 350-0.45-0.40-0.35 Heat flow (W/g)\nT(K)\n200 250 300 3505.86.06.26.46.6\n ' '\nT(K)1587 Hz\n Figure 5:  Understanding the spin -phonon -charge coupling in Fe 3Se4 at room temperature. The \nshadowed region shows anomaly in magnetization, dielectric permittivity, heat capacity and \nheat flow measured from thermo -gravimetric measurements.  \n  Supporting information  \n \n \nSynthesis of Fe 3Se4 nanoparticles by one pot organic phase synthesis:  \n \nFe(acac) 3 (0.53 g, 1.5 mmol) and Se powder (0.158 g, 2 mmol) w ere added to  15 m l of \noleylamine in a 100 m l three -neck flask  under N 2 atmosphere . The mixture was heated to 120 \n°C and kept for 1 h. Then, temperature was increased up to 200 ° C and kept for 1 h. Finally, \nthe solution temperature was raised to 300 °C and ke pt for 1 h. After 1 h, the heat source was \nremoved and solution was allowed to cool down  naturally  to room temperature. The Fe 3Se4 \nnanoparticles were precipitated by the addition of 20 m l of 2-propanol. The precipitate was \nthen centrifuged and washed with solution containing hexane and 2 -propanol in 3:2  ratio.  \nStructural and morphology characterization of Fe 3Se4 nanoparticles:  \nExperimental details and techniques:  \nMagnetization measurements were done using the VSM attachment of PPMS from Quantum \nDesign syste ms equipped with 9 T superconducting magnet  on powder samples packed in \nspecial plastic holders designed so that the dielectric contribution of the holder is negligible. . \nTemperature dependent dielectric spectroscopy was performed using Novocontrol Beta NB  \nImpedance Analyzer connected with home built sample holder to couple with a helium closed \ncycle refrigerator (Janis Inc.). The powdered sample was compressed in the form of circular \npellet of diameter 13 mm and a custom designed sample holder was used to form parallel plate \ncapacitor geometry.  Ferroelectric hysteresis loop measurements were done on pellets made by \ncold pressing the sample powder in zero field. Raman spectra were recorded on an HR -800 \nRaman spectrophotometer (Jobin Yvon -Horiba, France) using monochromatic radiation emitted by a H e−Ne laser (633 nm), operating at 20 mW and with accuracy in the range \nbetween 450 and 850 nm ± 1 cm−1,. An objective of 50× LD magnification was used both to \nfocus and to collect the signal from the powder sample dispersed on the glass slide. For \nmagneto -Raman measurements, a permanent bar magnet was used  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure S1. Powder XRD pattern of as synthesized Fe 3Se4 nanoparticles.  \n \n \n \n \n20 40 60 80\n2\n040413\nB\n(003)(112)(400)(110)\n(310)\n(222)(601)(020)(311)(402)\n  \n2 (deg) Fe3Se4\n JCPDS # 657252(111)\nA\n02Intensity (a.u.) \nFigure S2.  M-H hysteresis loop measurement of Fe 3Se4 nanoparticles at 300 K.   \n-20 0 20-3-2-10123M(emu/g)\n  \nH(kOe)300 K \n \n \nFigure S3. Frequency dependence of (a) real part of permittivity (b) out of phase part of \npermittivity (c) loss factor (d) absolute permittivity measured with ac 1 V rms value at a \ntemperature range from 200 K to 350 K.  \n \n  \n1001011021031041051060.10.20.30.40.5\n145 K\n350 K\n  tan\nF (Hz)\n1001011021031041051060246\n350 K\n  ''\nF (Hz)145 K\n10010110210310410510603691215\n350 K\n  '\nF (Hz)145 K\n10010110210310410510603691215\n350 K\n  \nF (Hz)145 K(a)\n(d) (c)(b) \n \nFigure S4.  Cole -Cole plot of as synthesized Fe 3Se4 nanoparticles ( ’’ versus ’) measured by \nimpedance spectroscopy at temperatures 185, 220, 295, 300, 317, 310, 326, 336 and 350 K \nrespectively.  \n \n  \n0 2 4 6 810 1201234\n310 K\n5 10024220 K\n2 4 6 8 100.51.01.52.0\n317 K\n0 2 4 6 810 12024\n300 K\n2 4 6 8 10 12024336 K\n2 4 6 8 10 12024350 K\n0 2 4 6 810 12024\n  \n326 K\n0 2 4 6 810 12024295 K\n2 4 6 8 10 120123 \n185 K’’’Figure S5 Temperature dependent of resistance contribution from grain and grain boundary  \nextracted from Cole -Cole plot is plotted. Inset shows the circuit arrangements and parameters \nRG and RGB in a Cole -Cole plot.  \n \n \n \n \n \n \n \n \n \n \n \n \n200 240 280 320 36023456\nGrain \nboundary Grain\nRGRGBRGCGB CG\nZ”\nZ’ RGB\nRG  \nT(K)RG \n789 \n  \n \n \n \n \n \n \n \n \n \n "    },    {        "title": "1612.06802v1.Magnetocrystalline_anisotropy_of_Laves_phase_Fe__2_Ta___1_x__W__x__from_first_principles___the_effect_of_3d_5d_hybridisation.pdf",        "content": "Magnetocrystalline anisotropy of Laves phase Fe 2Ta1−xWxfrom first principles - the\neffect of 3d-5d hybridisation\nAlexander Edström\nDepartment of Physics and Astronomy, Uppsala University, Box 516, 75121 Uppsala, Sweden\nThe magnetic properties of Fe 2Ta and Fe 2W in the hexagonal Laves phase are computed using\ndensityfunctionaltheoryinthegeneralisedgradientapproximation, withthefullpotentiallinearised\naugmented plane wave method. The alloy Fe 2Ta1−xWxis studied using the virtual crystal approx-\nimation to treat disorder. Fe 2Ta is found to be ferromagnetic with a saturation magnetization of\nµ0Ms= 0.66 Twhile, in contrast to earlier computational work, Fe 2W is found to be ferrimag-\nnetic with µ0Ms= 0.35 T. The transition from the ferri- to the ferromagnetic state occurs for\nx≤0.1. The magnetocrystalline anisotropy energy (MAE) is calculated to 1.25 MJ/m3for Fe 2Ta\nand0.87 MJ/m3for Fe 2W. The MAE is found to be smaller for all values xin Fe 2Ta1−xWxthan for\nthe end compounds and it is negative (in-plane anisotropy) for 0.1≤x≤0.9. The MAE is carefully\nanalysed in terms of the electronic structure. Even though there are weak 5d contributions to the\ndensity of states at the Fermi energy in both end compounds, a reciprocal space analysis, using the\nmagnetic force theorem, reveals that the MAE originates mainly from regions of the Brillouin zone\nwith strong 3d-5d hybridisation near the Fermi energy. Perturbation theory and its applicability in\nrelation to the MAE is discussed.\nThe magnetocrystalline anisotropy energy (MAE) is\nthe intrinsic relativistic feature, originating from spin-\norbit coupling (SOC)1, of magnetic materials that the\nenergydependsonthedirectionofmagnetizationrelative\nto the crystal lattice. It is crucial in a wide range of ap-\nplications, from permanent magnets2–5to magnetic stor-\nage devices6. The SOC is strong in heavy elements such\nas rare-earths (REs) and actinides which consequently\nacquire large MAE, while in applications it is highly de-\nsirable to obtain a large MAE without such expensive or\ninaccessibleconstituentelements7. Onecompoundwhich\nhas gained much attention due to its huge MAE is tetrag-\nonal FePt8–12. This material acquires its magnetisation\nmainly from Fe, while the important factors resulting in\nthe large MAE include the strong SOC of the Pt atom,\nas well as the uniaxial crystal structure. The crystal\nstructure is crucial because highly symmetric, e.g. cu-\nbic, crystals tend to have at least one order of magnitude\nlower MAE. Nevertheless, FePt contains large amounts\nof the valuable element Pt, whereby alternative magnetic\n3d-5d composites in uniaxial crystal structures can be of\ngreat technological value. One such compound is hexag-\nonal Laves phase Fe 2W, which was initially reported by\nArnfelt and Westgren13and recently attracted some at-\ntention in the context of permanent magnet replacement\nmaterials14,15. Early electronic structure calculations16\nfailed to establish the existence of ferromagnetism in the\ncompound from the Stoner criterion. While it now seems\nclear that the compound is magnetically ordered14,15, a\nthorough understanding of the magnetism in this mate-\nrial appears to be absent in literature and some discrep-\nancies can be seen between recent computational14and\nexperimental work15. For example, calculations14over-\nestimated the saturation magnetization by nearly thirty\npercent and provided a vastly different MAE when com-\npared to experimental data from nanoparticles15. It is\ntherefore the purpose of this work to use state of the\nart electronic structure calculations to unambiguouslydetermine the magnetic ground state of the Fe 2W com-\npound and investigate the magnetic properties, including\nthe technologically important intrinsic properties of sat-\nuration magnetization ( Ms) and MAE. The closely re-\nlated compound Fe 2Ta is isostructural to Fe 2W17and\nalso studied. Some focus will be put on the MAE, which\nwill be carefully analysed in terms of the electronic struc-\nture. Furthermore, the possibility to tune the MAE by\nalloying W and Ta will be examined and a discussion of\nthe underlying physical principles provided.\nDensity functional theory (DFT) calculations in the\ngeneralized gradient approximation18(GGA) were per-\nformed with the full-potential linearized augmented\nplane waves (FP-LAPW) method as implemented in\nWIEN2k19. Initially, spin-polarized calculations were\nperformed in the scalar relativistic approximation, but\nto calculate the MAE, SOC must be included and this\nwas done in a second variational approach20. The size\nof the basis set used is typically described by the prod-\nuct of the smallest muffin-tin sphere and the largest re-\nciprocal lattice vector included, RKmax. For structure\noptimizations, this value was set to RKmax= 7, while\nfor MAE calculations a larger value of RKmax= 9was\nused. To obtain a well converged formation energy, a\nvalue as large as RKmax= 9.5was needed. Integration\nofk-points over the Brillouin zone was performed using\nthe improved tetrahedron method21and 700 k-points in\nthe full Brillouin zone (48 in the irreducible wedge of the\nBrillouin zone after considering the 24 symmetry opera-\ntions of the crystal) were used for structure optimization,\n1500 for the calculation of formation energy and as many\nas 30000 k-points were used in order to obtain well con-\nverged MAE values.\nOne unit cell of the relevant crystal structure contains\ntwo inequivalent Fe positions with multiplicity two and\nsix respectively, as well as two equivalent 5d sites. Calcu-\nlationswereperformedwiththeinitialspinstateeitherin\nferro or ferrimagnetic configurations, i.e. parallel or an-arXiv:1612.06802v1  [cond-mat.mtrl-sci]  20 Dec 20162\nTable I: Lattice parameters and parameters of the\ninternal atomic positions, magnetic moments, saturation\nmagnetization and formation energy of Fe2W and Fe2Ta\nas calculated in a scalar relativistic, spin polarized GGA\ncalculation, neglecting SOC, in WIEN2k.\nFe2Ta Fe 2W\na (Å) 4.811 4.674\nc (Å) 7.874 7.768\nxFe2 0.83192 0.82946\nz5d 0.06405 0.06924\nm(Fe1) (µB) 0.90 -1.14\nm(Fe2) (µB) 1.43 1.17\nm(5d) (µB) -0.24 -0.05\nmtot(µB/u.c.) 8.88 4.45\nµ0Ms(T) 0.66 0.35\nFormation energy (eV/u.c.) -2.82 -0.63\ntiparallel alignment of spins on the two different Fe posi-\ntions. In the case of Fe 2Ta, the total energy was found to\nbe approximately 1.8 eV per unit cell lower in the case of\nferromagnetic ordering compared to ferrimagnetic order-\ning. For Fe 2W, on the other hand, all calculations con-\nverged into the ferrimagnetic state, regardless of initial\nspin configuration and lattice parameters. Lattice pa-\nrameters were calculated by minimizing the total energy\nwith respect to volume and c/aand relaxing the inter-\nnal atomic positions in each step. The calculated lattice\nparameters are reported in Table I, which also contains\nspin magnetic moments and the corresponding satura-\ntion magnetizations as well as formation energies. For\nFe2Ta, the lattice parameters have been experimentally\nreported as a= 4.833Å andc= 7.868Å17and for Fe 2W,\na= 4.727Å andc= 7.704Å13, in close agreement with\nthe calculated values in Table I, although for Fe 2W,c/a\nis slightly larger in the calculated data. The Fe moments\nin Fe 2W are of similar size and opposite sign but as there\nare two and six of the respective Fe sites in one unit cell,\nthere is a net total of 4.45µB/u.c., corresponding to a\nsaturation magnetization of µ0Ms= 0.35T. Since in\nFe2Ta the Fe moments are parallel, the total magnetic\nmoment and corresponding saturation magnetization is\nsignificantly larger, reaching a value of µ0Ms= 0.66T.\nTa and W have a small induced moments of −0.24µBand\n−0.05µB, anti-parallel to the total magnetic moment, re-\nspectively, as is typical for these 5d atoms in a magnetic\n3d host22.\nSince a different magnetic ordering, with a magnetic\nmoment close to zero on the first Fe site and a larger\nmoment moment around 1.3µBon the second Fe atom,\nhas been reported in earlier computational work (pseu-\ndopotential DFT calculations in the GGA)14for Fe 2W,\nfurther investigation seems necessary to unambiguously\ndetermine the correct magnetic ground state within the\nGGA. Hence, fixed spin moment calculations, allowingthe total magnetic moment of the system to be con-\nstrained to a fixed given value, were performed. The\ntotal magnetic moment was varied around the value of\n6.8µB/u.c., previously reported14. Magnetic moments\nof−0.05µBand1.25µBwere then obtained on the two\nFe atoms, which is similar to the earlier computational\nresults14. Initially, the lattice parameters were set to the\nvalues mentioned in Ref14but then attempts were made\nat optimizing the crystal structure with fixed magnetic\nmoment to lower the energy further. However, all calcu-\nlations resulted in total energies which were higher than\nthose obtained for the structure given in Table I and no\nminimum could be located in the total energy as function\nof total magnetic moment. Based on these results, the\nmost probable conclusion appears to be that the authors\nof Ref14assumed a ferromagnetic order as initial state\nand reached a local energy minimum for the magnetic\nmoments were reported. The correct magnetic moments\ncorresponding to the global energy minimum, within the\nGGA, based on all results obtained here, are expected to\nbe those in Table I. The explanation given here is con-\nsistent with the observation that the previous computa-\ntional work presented a value of µ0Msas approximately\n0.56T which overestimated the experimental low tem-\nperature value of approximately µ0Ms= 0.44T. Nev-\nertheless, the value given in this work somewhat under-\nestimates the experimental result. A possible source of\ndiscrepancy is surface effects of the nanoparticles, where\nenhanced magnetic moments could appear near the sur-\nface.\nSomewhat surprisingly, a non-negligible difference is\nseen also in lattice parameters and total magnetic mo-\nment for Fe 2Ta, when comparing to previous computa-\ntional work14, wherea= 4.825 Åandc/a= 1.6329(cor-\nresponding to c= 7.879 Å) was reported. The difference\ninais merely 2% and might be expected for the two\ndifferent computational methods. The difference in total\nspin magnetic moment is, however, larger. For example,\nthe magnetic moment on the Fe 1is computed to 0.90µB,\nwhile the other authors reported a value well above 1µB.\nThe reason for this discrepancy is difficult to pinpoint\nexactly, as both sets of calculations are performed in the\nGGA18, but might partly be related to the difference in\nlattice parameters.\nBy comparing the total energy of Fe 2(Ta/W) in the\ncalculated ground state with that of bcc Fe and bcc Ta or\nW, the formation energy was calculated to −0.63eV/u.c\nfor Fe 2W, which is lower than the value close to zero\npreviously reported14. A negative formation energy is\nexpected for a stable phase and a possible scenario ap-\npears to be that the authors of Ref.14obtained a too high\nformation energy due to calculating a local energy min-\nimum and thus a too high total energy. For Fe 2Ta, the\nformation energy is lower and this compound may there-\nfore be expected to be more stable and form more easily\nin nature.\nIn order to compute the MAE, calculations were per-\nformed including SOC, which also results in a non-zero3\norbital magnetic moment, that is otherwise quenched.\nThe computed spin magnetic moments ( mS), orbital\nmagneticmoments( mL)andMAEsarelistedinTableII.\nWhen the magnetization is along the a-axis, the SOC re-\nsults in a lowering of symmetry so that the second Fe site\nwith initially six equivalent atoms are split into two types\nwith two and four Fe atoms of each, labelled Fe 2and Fe 3\nrespectively. Hence, the spin and orbital moments are\nsame for Fe 2and Fe 3when the magnetization is along\nthec-axis but not when it is along the a-axis. The MAE\nis calculated to EMAE = 1.24meV/u.c. = 1.25MJ/m3\nfor Fe 2Ta andEMAE = 0.79meV/u.c. = 0.87MJ/m3for\nFe2W, with easy magnetization axis along the c-direction\nof the crystal in both cases. The calculated uniaxial\nMAE for Fe 2W presented here is in better agreement\nwith the small uniaxial MAE recently presented in ex-\nperimental work15than the large in plane MAE previ-\nously computed for the Fe 2W compound14. Neverthe-\nless, the computed value found in this work is signif-\nicantly larger than the reported experimental value of\n286kerg/cm3= 28.6kJ/m3. However, measurements\nhave only been presented for nanoparticles, while unam-\nbiguous MAE measurements require single crystals. For\nFe2Ta, an experimental MAE has not been found in liter-\nature, but the value calculated here differs notably from\nthe value of EMAE =−1.4meV/u.c. previously calcu-\nlated14. This discrepancy is most likely related to the\ndifference in magnetic moments obtained, as mentioned\nabove, but could also be partially related to other com-\nputational details, such as the treatment of SOC or core\nelectrons.\nFig. 1 shows the spin polarized density of states (DOS)\nfor Fe 2Ta (a) and Fe 2W (b). The majority spin DOS is\nsimilar for the two compounds, with the Fermi energy\n(EF) at approximately the same location. However, as\nTa is exchanged for W more electrons are added into the\nsystem and the minority spin states become occupied,\nwhereby these are shifted more to the left in Fig. 1b) and,\nas a result, EFcoincides with the bottom of a valley in\nthe minority spin DOS of Fe 2W. Thus the DOS( EF) for\nFe2Wisdominatedbyminorityspinstates, incontrastto\nFe2Ta, where the opposite is true. This fact will be of im-\nportance later when analysing the relation between MAE\nand orbital moment anisotropy. It is also interesting to\nnote that the minority spin DOS of Fe 2W has a valley\natEF, resulting in a higher degree of spin polarization\nof the DOS( EF) compared to Fe 2Ta. In both cases, the\nDOS(EF) is dominated by Fe, with rather modest contri-\nbutions from the 5d atoms. This might be one important\nreason, together with other details in the band structure\naroundEF, why these compounds do not possess larger\nMAE. Even the L1 0phase of MnAl exhibits an MAE\nwell above 1MJ/m323without any constituent element\nheavier than a 3d atom. Heavier atoms, such as 5d’s,\nshould allow significantly larger MAE, e.g., 4MJ/m310\nor more24in FePt. However, this requires significant 3d-\n5d hybridisation around EF, as is seen in FePt25, but\nappears to be limited in the compounds studied here.Table II: Spin magnetic moments, mS, orbital magnetic\nmomentsmL, saturation magnetizations and MAE for\nFe2W as calculated in WIEN2k, including SOC with\nmagnetization either along 100 or 001 directions and\nusing the lattice parameters presented in Table I.\nFe2Ta m/bardbl100 m/bardbl001\nmS(Fe1) (µB) 0.943 0.932\nmS(Fe2) (µB) 1.433 1.432\nmS(Fe3) (µB) 1.427 1.432\nmS(Ta) (µB) -0.240 -0.238\nmL(Fe1) (µB) 0.070 0.109\nmL(Fe2) (µB) 0.091 0.099\nmL(Fe3) (µB) 0.101 0.099\nmL(Ta) (µB) 0.033 0.034\nµ0Ms(T) 0.69 0.69\nEnergy (meV/u.c.) 1.24 0\nEnergy (MJ/m3) 1.25 0\nFe2W m/bardbl100 m/bardbl001\nmS(Fe1) (µB) -1.148 -1.150\nmS(Fe2) (µB) 1.163 1.172\nmS(Fe3) (µB) 1.172 1.172\nmS(W) (µB) -0.044 -0.044\nmL(Fe1) (µB) -0.066 -0.151\nmL(Fe2) (µB) 0.045 0.039\nmL(Fe3) (µB) 0.074 0.039\nmL(W) (µB) 0.002 0.002\nµ0Ms(T) 0.36 0.35\nEnergy (meV/u.c.) 0.79 0\nEnergy (MJ/m3) 0.87 0\nNevertheless, the contribution from Ta (3.2 states/eV for\nboth spin channels summed) is greater than that of W\n(1.8 states/eV). This is consistent with the observation\nthat the MAE is greater in the compound containing Ta,\nalthough other differences in the electronic structure are\nalso expected to play a role. One more interesting ob-\nservation in the DOS is that the minority spin DOS of\nFe2W has a valley at EF, resulting in a higher degree of\nspin polarization of the DOS( EF) compared to Fe 2Ta.\nIn a system with weak SOC, such as 3d-based itinerant\nmagnets, where ξis significantly smaller than the band-\nwidth (less than 100meV compared to several eV), it is\nreasonable to describe the effect of SOC in terms of per-\nturbation theory and important insights can be gained\nby doing so26–28. For a uniaxial crystal the leading term\nis of second order while for cubic crystals it is fourth or-\nder. Andersson et al.28discussed the case of having sev-\neral atomic types and hybridisation between these in a\ntight-binding description. One can consider unperturbed\nsingle particle states at the point kin the Brillouin zone4\nE - EF(eV)-5 0 5States / eV\n201001020\nMinority spinMajority spinTot\nFe1\nFe2\nTaa)\nE - EF(eV)-5 0 5States / eV\n201001020\nMinority spinMajority spinTot\nFe1\nFe2\nWb)\nFigure 1: Spin polarized DOS for Fe 2Ta in a) and Fe 2W\nin b).\nas\n|k,i/angbracketright=/summationdisplay\nq,µck,i,q,µ|k,q,µ,σi/angbracketright, (1)\nwith summation over atomic sites qand orbital states µ,\nbut not over the spin σnsince the unperturbed states\neach have well defined spin. With on site SOC, The shift\nin the energy eigenvalue Ek,iassociated with|k,i/angbracketrightis\n∆Ek,i(ˆn) =−/summationdisplay\nj/negationslash=i/summationdisplay\nqq/prime/summationdisplay\nµµ/primeµ/prime/primeµ/prime/prime/primenk,i,qµ,q/primeµ/prime/prime/primenk,j,q/primeµ/prime/prime,qµ/prime\n·/angbracketleftqµσi|ξqˆl·ˆ s|qµ/primeσj/angbracketright/angbracketleftq/primeµ/prime/primeσj|ξq/primeˆl·ˆ s|q/primeµ/prime/prime/primeσi/angbracketright\nEk,j−Ek,i,(2)\nwith occupation numbers nk,i,qµ,q/primeµ/prime/prime/prime=c∗\nk,i,q,µck,i,q/prime,µ/prime/prime/prime\nand spin and orbital angular momentum operators ˆ sand\nˆl. For a given qand k, it is clear that the effect of\nthe SOC is determined by matrix elements of the form\n/angbracketleftµi,σi|ˆl·ˆ s|µj,σj/angbracketrightand for convenience these are listed\nwith respect to spin and d-orbitals in the appendix. ˆnis\nthe spin quantization axis (magnetization direction) and\nthe dependence of ∆Ek,i(ˆn)on this quantity comes from\nthe SOC matrix elements. For the total shift in Ek,i, the\ncoupling between all states j/negationslash=ishould be considered.\nHowever, if both iandjdenote occupied states there will\nbeacancellationwhenthesearesummedovertocompute\nthe total energy. Therefore, only coupling between occu-\npied and unoccupied states are relevant, except possiblyin the small regions of the Brilloiun zone where deforma-\ntions of the Fermi surface occur, as was pointed out by\nKondorskii and Straube26. This leads to the important\nand well established conclusion that the MAE is deter-\nmined by the electronic band structure near the Fermi\nenergy, in particular by the coupling between occupied\nand unoccupied states. One more important observation\nfrom Eq. 2 is that regions in the band structure with\nsignificant Fe-Ta hybridisation will allow MAE contri-\nbutions of orderξTaξFe\nEk,j−Ek,i, which is significantly larger\nthanξ2\nFe\nEk,j−Ek,i, sinceξTais several times larger than ξFe,\nor similarly for W instead of Ta.\nFrom the discussion above it is motivated to perform\na careful analysis of the electronic band structure near\nthe Fermi energy to obtain a better understanding of the\nMAE. Fig. 2a)-f) shows the spin polarized band struc-\nture through various high symmetry points in the Bril-\nlouin zone, without SOC, for Fe 2Ta, with spin up states\non the left side and spin down states on the right side.\nColor coding is used to show the orbital character of the\nbands with red, green and blue indicating m= 0(dz2),\nm= 1(dxzor dyz) andm= 2(dxyor dx2−y2) character,\nrespectively, for different atomic types in the different\nrows. A black region on a band indicates that the given\natomic type is not significantly contributing to the band\nin that region. The large number of bands present, even\nwithin one electronvolt from the Fermi surface, and com-\nplicated band structure with further complication due\nto hybridisation, makes analysis of the MAE in terms\nof the band structure difficult. Some observation can,\nnevertheless, directly be made. The Γpoint is often of\nparticular importance since it has the highest symmetry.\nHere there are occupied and unoccupied spin up states\nvery near the Fermi energy at this point, potentially al-\nlowing very strong effect from the SOC, especially since\nthese states both show strong Ta contributions and Ta\nhas the largest SOC constant. However, the unoccupied\nband is largely of m= 0character, while the occupied\none is ofm= 1character. Such states do not couple\nvia SOC (see Table IV), whereby the potentially strong\nMAE contribution at Γis absent.\nTo obtain information about which regions in recip-\nrocal space are particularly important to the MAE, the\nband structures after applying SOC with magnetization\nalong either 100 or 001 directions are plotted in Fig. 2g).\nFrom these bands the MAE contribution per k-point can\nbe evaluated using the magnetic force theorem29, by tak-\ning the difference of the sum over occupied energy eigen-\nvaluesfordifferentmagnetizationdirections,whichisalso\nplotted(redline, right y-axis)inFig.2g). SincetheMAE\nis positive in Fe 2Ta, regions with positive MAE contri-\nbutions are expected to outweigh the negative regions.\nIn agreement with the observation mentioned about Γ\nabove, there is a rather weak MAE contribution from\nthe region around that point. Instead, it is clear that\nthe most important region is that around the A-point\nwhere a large and positive MAE contribution is seen,5\nwhile other regions show smaller values of varying sign,\nwhich one might expect to nearly cancel out in a Bril-\nlouin zone integration. From a first look at the bands\nin Fig. 2a)-f), the most important bands for the MAE\natAshould be the highest occupied and lowest unoccu-\npied ones, which are in both cases spin down with four-\nfold degeneracy. However, in Fig. 2g) one can identify\nthe strongest positive MAE contribution where occupied\n001-bands (blue dashed line) are shifted well below the\ncorresponding 100-bands (black dash-dotted line). This\noccurs mainly for the highest occupied (also four-fold de-\ngenerate) spin up bands at A, whereby these should also\nbe considered. The three sets of band which thus far ap-\npear most important at Aall have significant contribu-\ntionsfromseveralatomictypesandorbitals, inparticular\nTa and Fe 1,m= 1andm= 2states, but for the lowest\nunoccupied spin down bands, Fe 2m= 1andm= 2are\nalso important. This means that detailed analysis of the\nMAE contribution from the A-point is complicated since\na large number of terms from Eq. 2 must be considered.\nIt is clear, however, that there is a significant Fe-Ta hy-\nbridisation in the relevant region and as was pointed out\nabove, this allows for significant additions to the MAE.\nFig. 3 contains the same type of information as Fig. 2,\nbut for Fe 2W. Since Fe 2W also has a uniaxial (posi-\ntive) MAE, positive regions are expected to dominate\nthe MAE contributions in Fig. 3g). In similarity with\nthe Fe 2Ta case, there are large regions of small contri-\nbutions with varying sign, which one would expect to\nnearly vanish in an integration. In particular, the im-\nportant Γ-point provides a weak contribution, which can\nbe understood from the relatively large separation in en-\nergy between the highest occupied and lowest unoccu-\npied states, compared to other regions. The most im-\nportant positive contributions to the MAE stem from\ntheL-neighbourhood, as well as a region along the path\nA−H, while there is a significant negative region around\nM, which might partially explain why the MAE of Fe 2W\nis weak. In the important region along the A−Hpath,\nthere are two spin up bands nearly parallel to each other.\nThese are on opposite sides of the Fermi energy where\nthek-point resolved MAE is strongest, and can therefore\ncontribute to the MAE. Both bands are mainly of W and\nFe1m= 1character. From the SOC matrix elements in\nthe appendix, one finds that states of same spin and m\nvalueyieldapositive(uniaxial)contributiontotheMAE.\nFurthermore, the Fe-W hybridisation allows the large W\nSOC to make the coupling strength large and this ex-\nplains the large positive MAE coming from that part of\ntheA−Hpath.\nAt theL-point, a significant positive source of MAE is\nfound in the highest occupied spin up states which are\nmainly Ta and Fe 1m= 1, since the 001 bands are shifted\nbelow the 100 bands. This situation is reversed as one\nmoves along the L−Mpath and the change of sign in\nthek-resolved MAE appears to coincide with the spin\ndown bands which are unoccupied at Lbecoming occu-\npied nearM. The presence of many bands with signif-icant hybridisation effects makes it difficult to pinpoint\nstates coupling via SOC which are particularly impor-\ntant to the MAE along the L−Mpath. Nevertheless,\nit should be pointed out that once again there is signifi-\ncant Fe-W hybridisation, so that the strong W SOC can\nincrease the MAE. Since there is a limited 5d contribu-\ntion to the DOS at the Fermi energy, there can only be\nsignificant 3d-5d hybridisation near the Fermi energy in\na limited region of the Brillouin zone. Nevertheless, the\nreciprocal space analysis of the electronic structure and\nMAE contributions reveals that the MAE is mainly de-\ntermined by those regions in the Brillouin zone where\nthere is notable 3d-5d hybridisation, in both Fe 2Ta and\nFe2W.\nAs both quantities are due to the SOC, Bruno27\npointed out the close relation between magnetocrys-\ntalline anisotropy and orbital moments and showed, us-\ning perturbation theory on a tight binding model, that if\ndeformations of the Fermi surface can be neglected and\nthe MAE is dominated by spin-diagonal coupling, the\nMAE and orbital magnetic moment anisotropy are pro-\nportional. If coupling between minority spin states dom-\ninates the SOC, a maximum orbital magnetic moment\nis expected in the easy direction of magnetization, as is\nseen in the case of Fe 2Ta in Table II. If, on the other\nhand, the SOC is dominated by the coupling between\nmajority spin states, a maximum orbital magnetic mo-\nment is expected along the hard magnetisation axis, as\nis seen in the case of Fe 2W. This is consistent with the\nobservation made in Fig. 1, that the Fe 2Ta DOS(EF) is\ndominated by minority spin states, while the opposite is\ntrue for Fe 2W. For a further analysis of the relation be-\ntween MAE and mLin the studied systems, energy and\norbital moments have been computed as functions of the\nangleθ(withφ= 0) when the magnetization is along\nˆn= (sinθcosφ,sinθsinφ,cosθ). The result for the en-\nergy as function of θis shown in Fig. 4. The second\norder perturbation theory for a uniaxial system leads to\nthe conclusion that the energy as function of θfollows\nthe relation\nE(θ) =K0+K1sin2θ, (3)\nwith isotropic energy K0. This is merely the first part of\nthe longer expansion\nE(θ,φ) =K0+K1sin2θ+K2sin4θ+\n+K3sin6θ(1 +k3,3cos 3φ+k3,6cos 6φ) +...\n(4)\nvalidforauniaxialcrystalwiththree-foldrotationalsym-\nmetry about the z-axis, such as the one studied here. For\na system where the MAE is well described by second or-\nder perturbation theory, one expects that the energy is\nwell fitted by Eq. 3 and that Kiis vanishingly small for\ni>1. AsseeninFig.4a), fittingtheenergyasfunctionof\nangle between magnetization direction and 001-direction\ntoK1sin2θprovides an unsatisfactory curve for E(θ)\nfor both Fe 2Ta and Fe 2W, while including also the term6\nL M Γ A H KE-EF (eV)\n-101\n(a) Fe 1, spin up.\nL M Γ A H KE-EF (eV)\n-101 (b) Fe 1, spin down.\nL M Γ A H KE-EF (eV)\n-101\n(c) Fe 2, spin up.\nL M Γ A H KE-EF (eV)\n-101 (d) Fe 2, spin down.\nL M Γ A H KE-EF (eV)\n-101\n(e) Ta, spin up.\nL M Γ A H KE-EF (eV)\n-101 (f) Ta, spin down.\nL M Γ A HE-EF (eV)\n-0.400.4\nMAE (10-2eV/k-point)\n-8-4048\n(g) Band structure including SOC with magnetisation along 100 (black dash-dotted line) or 001-direction (blue\ndashed line) as well as the MAE contribution per k-point (red solid line).\nFigure 2: Atomic type and spin resolved band structure of Fe 2Ta with the colors red, green and blue indicating the\ncontribution of m= 0(dz2),m= 1(dxzor dyz) andm= 2(dxyor dx2−y2) states respectively, in (a)-(f). Black\nbands mean that the d-orbitals of given atomic type do not contribute significantly to the band in that region. (g)\nshows bands with SOC as well as k-point resolved MAE contributions obtained via the magnetic force theorem.\nK2sin4θyields an excellent fit (for the fit to K1sin2θ,\nK1was simply set to E(π/2)−E(0), while the fit to\nK1sin2θ+K2sin4θwas done with the method of least\nsquares). This indicates that second order perturbationtheoryprovidesaquantitativelyinaccuratedescriptionof\nthe MAE in the studied compounds, while fourth order\nterms should provide an accurate description with higher\n(than fourth) order corrections being small. Clearly, the7\nL M Γ A H KE-EF (eV)\n-101\n(a) Fe 1, spin up.\nL M Γ A H KE-EF (eV)\n-101 (b) Fe 1, spin down.\nL M Γ A H KE-EF (eV)\n-101\n(c) Fe 2, spin up.\nL M Γ A H KE-EF (eV)\n-101 (d) Fe 2, spin down.\nL M Γ A H KE-EF (eV)\n-101\n(e) W, spin up.\nL M Γ A H KE-EF (eV)\n-101 (f) W, spin down.\nL M Γ A H KE-EF (eV)\n-0.500.5\nMAE (10-2eV/k-point)\n-6-3036\n(g) Band structure including SOC with magnetisation along 100 (black dash-dotted line) or 001-direction (blue\ndashed line) as well as the MAE contribution per k-point (red solid line).\nFigure 3: Atomic type and spin resolved band structure of Fe 2W with the colors red, green and blue indicating the\ncontribution of m= 0(dz2),m= 1(dxzor dyz) andm= 2(dxyor dx2−y2) states respectively, in (a)-(f). Black\nbands means that the d-orbitals of given atomic type do not contribute significantly to the band in that region. (g)\nshows bands with SOC as well as k-point resolved MAE contributions obtained via the magnetic force theorem.\nfit toK1sin2θis significantly better in the case of Fe 2W\nthan for Fe 2Ta. This indicates that restriction to sec-\nond order perturbation theory, rather than fourth, is a\nbetter approximation for the W compound, which mightbe related to the smaller contribution of the 5d atom to\nthe DOS(EF), making the assumption of a small ξmore\nrealistic.\nThe anisotropy constants obtained from the fitting to8\nθ0 π/4 π/2Energy(meV/f.u.)\n00.511.5Ta:K1sin2(θ)\nTa:K1sin2(θ)+K2sin4(θ)\nTa:Calculateddata\nW:K1sin2(θ)\nW:K1sin2(θ)+K2sin4(θ)\nW:Calculateddataa)\nϕ0 π/3 2π/3Energy (µeV/f.u.)\n-505Ta calculated data\nTa fit\nW calculated data\nW fitb)\nFigure 4: Energy as a function of the polar angle θ\nbetween the c-axis and the magnetization direction in\na) and Energy as a function of the azimuthal angle φ\nwithθ=π\n2in b). The fit in b) is to a function\nE(θ=π\n2,φ) =C1+C2cos 3φ+C3cos 6φand\nC2cos 3φ+C3cos 6φis plotted.\nK1sin2θ+K2sin4θarelistedinTableIII.Aswasalready\nanticipated from Fig. 4a), K2is of more importance in\nFe2Ta and in fact it is of opposite sign and significantly\nbigger than K1. In the case where K1andK2have the\nsamesign, the θ-derivativeof E(θ) =K1sin2θ+K2sin4θ\nhasonlytwozerosforreal Ki, namelyθ= 0andθ=π/2,\nwhereby the easy and hard magnetization directions will\noccur at these angles. For opposite signs of K1andK2,\nan additional zero occurs at\nθ= sin−1(/radicalbigg\n−K1\n2K2) (5)\nand for Fe 2Ta there is a minimum in the energy at ap-\nproximately θ= 0.15 = 8.8◦. The easy magnetization\ndirection is thus expected at this angle rather than at\nθ= 0, so the material strictly speaking does not have\na uniaxial magnetization. For Fe 2W, both constants are\npositive so θ= 0is the easy axis. Although in this caseTable III: Anisotropy constants K1,K2and\n˜K3=K3(1 +k3,3+k3,6)from least squares fitting of\nE(θ,φ= 0)toK1sin2θ+K2sin4θor\nK1sin2θ+K2sin4θ+˜K3sin6θ(see Fig. 4).\nK1(meV/f.u. )K2(meV/f.u. ) ˜K3(meV/f.u. )\nFe2Ta -0.27 1.50\nFe2Ta -0.19 1.23 0.19\nFe2W 0.50 0.30\nFe2W 0.45 0.46 -0.11\nthe magnitude of K1is greater than K2, the latter is not\nnegligible.\nTable III also contains parameters from a fit to\nK1sin2θ+K2sin4θ+˜K3sin6θ. This indicates non-\nnegligible values of ˜K3for both compounds and, in the\ncase of Fe 2Ta, it is of the same magnitude as K1. How-\never, it is not clear how many fitting parameters are\nreasonable to include with the given numerical accuracy.\nComparison to a fit from a calculation with only 2×104\nk-points yields a value smaller by a factor of one third for\nFe2Ta, indicating that the numerical accuracy might be\ninsufficient. However, more accurate calculations become\nprohibitively computationally demanding.\nTypically, in uniaxial systems which do not possess\nstrong SOC, the variation in energy for rotations of the\nmagnetisation direction in the plane is small. This makes\nit challenging and computationally heavy to compute the\nin plane magnetic anisotropy (this might differ in, for\nexample, actinide systems, where even cubic materials\ncan have enormous MAE30). Nevertheless, the energy as\nafunctionof φwithθ=π\n2wascomputedandtheresultis\nshown in Fig. 4b). The calculated points have been fitted\ntoE(θ=π\n2,φ) =C1+C2cos 3φ+C3cos 6φ(C2andC3\nshould correspond to K3k3,3andK3k3,6, respectively)\nandC1has been subtracted from the calculated points\nand fitted curves. As expected, the variations in Fig. 4b)\nare much smaller, by nearly three orders of magnitude,\nthan the variations seen in Fig. 4a). It is difficult to\nsay whether the deviations between the computed points\nand the fitted lines are mainly due to limitations in the\nnumerical accuracy or because of neglecting higher order\nterms.\nFig. 5 shows how the orbital magnetic moments vary\nwith magnetization direction for Fe 2Ta (a) and Fe 2W\n(b). In both materials the greatest contribution to the\norbital moment anisotropy is due to the Fe 1atom. The\nFe2and Fe 3atoms have identical orbital magnetic mo-\nments atθ= 0, as expected from symmetry, while they\ndeviate from one another at other directions. The com-\npounds differ in the sign of the variation of the orbital\nmagnetic moment with θ, although they both have same\nsign ofK1+K2. In Fig. 5c), which shows a plot of\nthe energy as function of θvs the anisotropy in total\norbital magnetic moment as function of θ, this appears\nas a difference in the sign of the slope of the curves.9\nAs was previously mentioned, this can be understood in\nterms of the DOS( EF) which is mainly due to the ma-\njority spin channel in Fe 2W and mainly due to the mi-\nnority spin channel for Fe 2Ta. According to the work of\nBruno27, this should lead to approximate proportional-\nity between ∆mL(θ)andE(θ), but with opposite signs in\nthe proportionality constants. However, that was based\non second order perturbation theory, and as was seen\nabove, fourth order perturbation theory is expected to\nbe necessary for a quantitative description of the mag-\nnetic anisotropy in these materials, especially in Fe 2Ta.\nFig. 5c) also shows a linear fit to the curves for E(θ)\nvs∆mL(θ). For Fe 2W, the linear fit provides a reason-\nabledescriptionofthecurve, whileinFe 2Tathedeviation\nfrom linearity is more pronounced. This might largely be\nbecause of strong spin polarisation of the DOS at EFfor\nFe2W, which makes the approximation that only spin di-\nagonal SOC contributes to the magnetic anisotropy more\nreasonable. Although the DOS( EF) in Fe 2Ta is domi-\nnated by minority spin states, the contribution from the\nmajority spin channel is significant, whereby neglecting\nspin-off diagonal contributions is questionable. Further-\nmore, the stronger contribution from the 5d states could\nalso affect the relation between MAE and orbital mo-\nment anisotropy in that direction, consistent with previ-\nousobservations28ofnon-proportionalitybetweenorbital\nmagnetic moment and anisotropy in energy systems with\nsignificant 3d-5d hybridisation.\nAs the MAE depends sensitively on the band structure\naround the Fermi energy, it can be controlled by tuning\nthe band structure around the Fermi energy. In practice\nthis can be done, for example, by alloying, which will be\nexplorednextbyconsideringthealloyFe 2Ta1−xWx. Due\nto the complicated electronic structure, which was illus-\ntrated in Fig. 2g) and Fig. 3g), it is difficult to predict\ntheeffectofalloyingonpropertiessuchastheMAEwith-\nout explicitly doing calculations to evaluate the proper-\nties. For the system studied here it is also of interest\nto investigate where the transition from ferro- to ferri-\nmagnetism occurs. The virtual crystal approximation31\n(VCA), in which the alloyed atoms are exchanged for vir-\ntual atoms with non-integer effective atomic numbers, Z,\nwhich on average have the right ionic charge and number\nofelectronsforagivenalloyconcentration,willbeusedto\ntreat the disorder. The VCA, although simple compared\nto more sophisticated single site approximations, such\nas the coherent potential approximation (CPA), often\nprovides a good average description for properties such\nas magnetic moments3,32–34, especially for neighbours in\nthe periodic table and small alloy concentrations31. For\ndelicate properties, like the MAE, on the other hand,\nthe VCA has often been seen to result in quantitative\noverestimates compared to CPA calculations34,35, super\ncellcalculations36,37orexperiments34,38,39. Nevertheless,\none should still be able to observe correct qualitative\ntrends in the MAE from the VCA and it will be applied\nalso for this property.\nCalculations were performed for values of xin incre-\nθ0 π/4 π/2∆mL (µB)\n00.050.10.150.20.250.30.35\nTotal\nTa\nFe1\nFe2\nFe3(a) Orbital magnetic moment as function of θfor Fe 2Ta.\nθ0 π/4 π/2∆mL (µB)\n-0.15-0.1-0.0500.05\nTotal\nW\nFe1\nFe2\nFe3\n(b) Orbital magnetic moment as function of θfor Fe 2W.\n∆mL(θ) (µB)0 0.02 0.04 0.06 0.08 0.1 0.12E(θ) (meV/f.u.)\n-1.5-1-0.500.51\nIncreasing θ →\n← Increasing \nθTa: linear fit\nTa: Calculated data\nW: linear fit\nW: Calculated data\n(c) Change in energy versus change in orbital moment as θis\nvaried from 0toπ/2.\nFigure 5: Energy and orbital magnetic moments as\nfunction of the angle θbetween then magnetization\ndirection and the 001 axis.\nments of 0.1. A calculation for x= 0.1revealed that this\nis enough for the magnetic ordering to transition into the\nferrimagnetic ordering observed also for Fe 2W. A com-\nplete structural relaxation, using spin polarized calcula-\ntions neglecting SOC, was thus performed for x= 0.1.10\nThe resulting lattice parameters are a= 4.771Å and\nc= 7.847Å. Lattice parameters for 0.2≤x≤0.9were\ncalculated by linear interpolation between the values ob-\ntained forx= 0.1andx= 1.0. Calculations including\nSOC were then performed for the whole range of alloys\nand the resulting spin magnetic moments (for magnetiza-\ntion along the c-axis) and MAEs are presented in Fig. 6.\nA large decrease in total spin magnetic moment is seen\nwhen going from x= 0tox= 0.1, due to the change in\nsign of the Fe 1spin moment, but also because of an ac-\ncompanying reduction in size of the Fe 2moment. For x\ngreater than 0.1, the total spin magnetic moment mono-\ntonically increases until x= 1.0. This appears to be from\na combination of decrease in size of the Fe 1moment and\nincrease in size of the Fe 2moment. The MAE decreases\nwithxuntil it reaches a minimum at x= 0.5and then\nincreases until x= 1.0. Hence, the largest positive values\nof the MAE are obtained for the end compounds and it\ncannot be increased by the alloying considered here. A\nnegative in-plane anisotropy of very large magnitude is\nseen forx= 0.5. However, it is important to remember\nthat the VCA is expected to overestimate the magnitude\nof the MAE, whereby the real value might be of smaller\nmagnitude.\nx0 0.2 0.4 0.6 0.8 1mS(µB)\n-20246810\nTotal\nW/Ta\nFe1\nFe2a)\nx0 0.2 0.4 0.6 0.8 1MAE (meV/f.u.)\n-6-4-202b)\nFigure 6: a) Spin magnetic moments and b) MAE\n(computed as total energy differences for\nmagnetizations along 100 and 001 directions) as\nfunctions of xin Fe 2Ta1−xWx.\nA comprehensive computational study has been per-\nformed for the hexagonal Laves phase compounds Fe 2Ta\nand Fe 2W, with focus on the important intrinsic mag-\nnetic properties saturation magnetization and MAE. For\nFe2W, a new ferrimagnetic ground state has been sug-gested, different from that found in earlier computational\nwork14. In the case of Fe 2Ta, a similar magnetic ordering\nis found as in preceding calculations14, but an opposite\nsign is found in the MAE. The discrepancies in compari-\nsonwithearliercalculationscallsforfurtherexperimental\nefforts to unambiguously determine the magnetic prop-\nerties of these compounds.\nThe MAE has been carefully analysed in terms of the\nelectronic structure and by using the magnetic force the-\norem to compute k-point resolved contributions to the\nMAE. Because the density of states at the Fermi energy\nis dominated by 3d states, 5d-states can only contribute\nnotably to the MAE in small regions of the Brillouin\nzone. Nevertheless, it is found that the MAE originates\nmainly from regions in the Brillouin zone where there is\na strong 3d-5d hybridisation, allowing the strong SOC of\nthe 5d atoms to increase the MAE.\nThe main motivation to study uniaxial 3d-5d com-\npounds is the possibility to have a very large MAE, such\nas the value of 6.6MJ/m34observed in FePt. When a\nsignificant amount of magnetic 3d elements is included,\nthiscanbecombinedwithlargesaturationmagnetisation\nand a high Curie temperature. Among the compounds\nstudiedhere, theMAEscalculatedarequitemodestcom-\npared to that seen in FePt. In addition, for Fe 2W, a ferri-\nmagnetic ordering is found, resulting in a low saturation\nmagnetisation. Nevertheless, whether a material is use-\nful for a given application depends on a combination of\nthe mentioned intrinsic parameters. For example, in the\ncontext of permanent magnets, the hardness parameter\nκ=/radicalBigg\nK\nµ0M2, (6)\nwith MAE Kand saturation magnetisation M, can be\nused to determine whether a material has potential to\nexhibit a reasonable coercive field and be used as a per-\nmanent magnet4,5.κis required to be greater than unity\nbut the microstructural engineering to obtain the de-\nsired properties of a permanent magnet should be eas-\nier with larger κand Hirosawa5suggestedκ>1.4to be\ndemanded from potential permanent magnet materials.\nForthematerialsstudiedhereonefinds κ= 1.8forFe 2Ta\nandκ= 2.9for Fe 2W, from the data in Table II, well\nabove the requirement put forward by Hirosawa. These\nlarge values of κappear largely because of the modest\nsaturation magnetisations and the energy product of a\npermanent magnet will be limited by this. In both ma-\nterials the saturation magnetisation is below the value of\nµ0Ms= 1.6T4found in the powerful Nd 2Fe14B magnet.\nHowever, at least in Fe 2Ta the saturation magnetisation\nis greater than 0.48T seen in BaFe 12O19ferrite magnets,\npotentially making the compound technologically inter-\nesting as an intermediate alternative between rare-earth\nand ferrite magnets.\nExperimental work has reported a Curie temperature\nof 550 K in Fe 2W15, which should be sufficient for many\ntechnological applications. As a useful extension of the11\ncurrent work, it would be interesting to compute the\nCurie temperatures of Fe 2W and Fe 2Ta, e.g., by calculat-\ning the Heisenberg exchange parameters from first prin-\nciples and using these as input to the mean field approx-\nimation or Monte Carlo simulations. This would reveal\nwhether Fe 2Ta also has a high enough Curie temperature\nto be technologically interesting and might also shed fur-\nther light on the issue regarding the magnetic ordering\nof Fe 2W.\nTo investigate the possibility of enhancing the relevant\nproperties, alloying of W and Ta has been considered in\ncalculations for Fe 2Ta1−xWx, with the disorder treated\nin the virtual crystal approximation. These calculations\nindicate that the transition from ferro- to ferrimagnetic\nordering occurs for xsmaller than 0.1 and that the MAE\nis significantly reduced and mainly strongly negative in\nthealloy. Fortechnologicalpurposesthisdoesnotappear\npromising. However, there are various isostructural 3d-\n5d compounds, such as Mn 2Ta, Co 2Ta or Fe 2Hf17,40and\none might also consider alloys among these. Allowing for\n3d or 4d atoms to substitute the 5d atom gives further\npossibilities40. As a next step, it should be worthwhile\nto investigate ternary or quaternary phase diagrams for\nmagnetic 3d elements combined with 5d and other ele-\nments in uniaxial crystals. Numerous such phases which\nhave not been properly characterized in terms of mag-\nnetic properties should exist and the type of computa-\ntional methods used in this work should be of great value\nin identifying interesting materials.\nI would like to thank Yaroslav Kvashnin for criti-\ncally reading and providing useful comments on the\nmanuscript. I’malsogratefultoJánRuszandOlleEriks-\nson for discussions and for encouragement to pursue this\nwork. Computational work has been performed with\nresources from the Swedish National Infrastructure for\nComputing (SNIC) at the National Supercomputer Cen-\ntre (NSC) in Linköping.\nAppendix A: Matrix elements of the spin-orbit\noperator\nIf|i/angbracketrightis a single particle eigenstate to an unperturbed\nHamiltonian with no SOC, the total shift in the energyEidue toHSOC =ξˆl·ˆ sin second order perturbation\ntheory is\n∆Ei=−ξ2/summationdisplay\nj/negationslash=i/vextendsingle/vextendsingle/vextendsingle/angbracketleftn|ˆl·ˆ s|k/angbracketright/vextendsingle/vextendsingle/vextendsingle2\nEj−Ei, (A1)\nIf the unperturbed Hamiltonian commutes with the spin\noperator,|i/angbracketrighthas a well defined spin σibut can be con-\nsidered a superposition of different orbitals µso in the\nsimplest case (ignoring other quantum numbers, e.g., k)\n|i/angbracketright=/summationdisplay\nµci,µ|µ,σi/angbracketright. (A2)\nFor d-electron magnetism, which is of focus here, it is\nsuitable to consider µas dz2, dxz, dyz, dxyor dx2−y2.\nThe numerator in Eq. A1 then contains matrix elements\n/angbracketleftdi,σi|l·s|dj,σj/angbracketright, which determine the effect of the\nSOC. For convenience these matrix elements are explic-\nitly listed in Tabel IV, with θandφdenoting the polar\nand azimuthal angles of the spin quantization axis rela-\ntive to the crystal lattice.\nAs mentioned in the main text, only coupling between\nstates|i/angbracketrightand|j/angbracketrightwith energies EiandEjsuch that\nEi< EF< Ejwill contribute to the MAE and clearly\nthen ∆Ei≤0according to Eq. A1. In terms of the ma-\ntrix elements in Table IV this means that any coupling\ncontaining cosθwill lower the energy for θ= 0, i.e. fa-\nvoring a uniaxial magnetization (positive MAE), while\nsinθlowers the energy for θ=π/2which favors in-plane\nmagnetization(negativeMAE).Thesituationtakinginto\naccount multiple atomic types and hybridisation in Eq. 2\nis somewhat more complicated and contains a product\nof matrix elements for possibly different atomic types.\nNevertheless, the MAE is still determined by the matrix\nelements in Table IV.\n1J. H. V. Vleck, Physical Review 52, 1178 (1937)\n2O. Gutfleisch, M. a. Willard, E. Brück, C. H. Chen, S. G.\nSankar, and J. P. Liu, Advanced materials 23, 821 (2011)\n3D. Niarchos, G. Giannopoulos, M. Gjoka, C. Sarafidis,\nV. Psycharis, J. Rusz, A. Edström, O. Eriksson, P. To-\nson, J. Fidler, E. Anagnostopoulou, U. Sanyal, F. Ott,\nL.-M. Lacroix, G. Viau, C. Bran, M. Vazquez, L. Reichel,\nL. Schultz, and S. Fähler, JOM 67, 1318 (2015)\n4J. M. D. 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Asdente, Physical Review 140, A1303\n(1965)"    },    {        "title": "1612.08679v1.Giant_atomic_magnetocrystalline_anisotropy_from_degenerate_orbitals_around_Fermi_level.pdf",        "content": "arXiv:1612.08679v1  [cond-mat.mtrl-sci]  27 Dec 2016Giantatomicmagnetocrystallineanisotropyfrom\ndegenerateorbitalsaroundFermilevel\nRuiPang,†Bei Deng,‡and XingqiangShi∗,‡\nSchoolof PhysicsandEngineering,ZhengzhouUniversity,H enan 450001,China, and\nDepartmentof Physics,SouthUniversityofScienceand Tech nologyofChina, Shenzhen518055,\nChina\nE-mail: shixq@sustc.edu.cn\nKEYWORDS: Giant magnetic anisotropy energy, atomic dimer, spin-orbital coupling, de-\nfected graphene\nAbstract\nNano-structures with giant magnetocrystalline anisotrop y energies (MAE) are desired in\ndesigning miniaturized magnetic storage and quantum compu ting devices. Through ab ini-\ntio and model calculations, we propose that special p-element dimers and single-adatom on\nsymmetry-matched substrates possess giant atomic MAE of 72 -200 meV with room temper-\nature structural stability. The huge MAE originates from de generate orbitals around Fermi\nlevel. More importantly, we developed a simplified quantum m echanical model to understand\ntheprincipleonhowtoobtaingiantMAEforsupportedmagnet icstructures. Thesediscoveries\nand mechanisms provide aparadigm to design giant atomic MAE in nanostructures.\nMagnetocrystalline anisotropic energy (MAE), a key proper ty of magnetic materials,1is of\ncrucial importance to create the energy barrier that locks t he magnetic moment of the recording\n∗To whomcorrespondenceshouldbeaddressed\n†SchoolofPhysicsandEngineering,ZhengzhouUniversity,H enan450001,China\n‡DepartmentofPhysics,SouthUniversityofScienceandTech nologyofChina,Shenzhen518055,China\n1unit and prevents its transition to other directions. Its va lue is accessible for several experiments\nsuchas X-ray absorptionspectra, X-raymagneticcirculard ichroismand inelasticelectron tunnel-\ning spectroscopy.2–4A large MAE couldguarantee along-time-stabilityand a ther mal stabilityof\nthe magnetic moment.5For this reason, systems with large MAE attract interests of tremendous\ntechnological fields such as high-density-recording, quan tum computation and molecular spin-\ntronics.1,6–8To maintain thedirection ofthemagneticmomentoveroneyea r, the totalMAE for a\nrecording unit should be at least 40 kBT.3However, the typical MAE of a single 3 datom is 0.01\nmeV in bulk crystal and 0.1 meV in surface.9Therefore, to achieve the desired large total MAE,\na large magnetic cluster is required, which limits the minia turization of the magnetic recording\ndevices.10\nIn systemswithlowersymmetrysuch assingleadatoms,dimer s,nanowiresand clusters,giant\natomic MAE is more realizable.11–26Among these systems, an Os adatom on MgO(100) was\nreportedtohavearecord-breakingMAEof208meV.13However,thespin-flipscatteringbetween\nelectrons in substrates and adatoms limits the spin relaxat ion time, despite of a giant MAE.11,12\nRecently, an increasing number of investigations focused o n supported transition-metal-dimers,\ngivingthatspecificdimersonsubstratescouldformvertica lstructures. Theoutermostatomofthe\nsupported dimer could generate giant MAE as large as 100 meV a nd might be increased to 150\nmeV with the tuning of external electrical fields, which lead s to a new research field of building\nmagneticrecording devices at the molecularscale.27–36As thespin-orbitalcoupling constant λis\nproportionaltoZ4,whereZistheatomicnumber,3dimerscontaining5 d-elementswerepredicated\ntogivethelargestMAEinthe delectronsystems.27Themaingrouplate p-elements,whichlieon\nthe right hand side of d-elements in the periodic table, typically have larger λthan the transition\nmetals in the sameperiod.37Normally, these pelectron systems are not prototypesystemsrelated\nto magnetism so that only few organic radicals˛ a ´r MAE have been paid attention.37However, the\nisolatedpatoms such as Bi and Tl are magnetic. By designing a suitable s urface ligand filed, the\np-element˛ a ´rs magnetizationcan be preserved on surfaces. More interes tingly,there are only six p\norbitalswhilethe numberof dorbitalsis ten, hence the couplingbetween orbitals associ ated with\n2SOIissimplertohandleinthe pelectronsystems,whichmakesthedesignof p-elementadsorption\nstructuresmucheasierforthepurposeofoptimizingMAE.To ourknowledge,noattempthasbeen\nmadetoconstructsupportedmagneticstructureswiththese pelements.\nTo address the above issue and confirm our expectation, in thi s letter, we construct a series of\nstablemolecularmagneticstructureswith p-elementsbyplacingaBi adatomorBi-X dimmers(X\n=Ga,Ge,In,Sn,TlandPb)onsymmetrymatchedsurfaces,i.e. ,p-elementdimersonnitrogenized\ndivacancies (NDV) of graphene (FIG. 1, the NDV has been propo sed and synthesized38,39), and\np-element adatoms on MgO(100) (Figure 1c). We discover that m ost of these structures have\nextremelylargeMAE(upto200meV)withvertical-easy-magn etization-axis,suggestingthatthey\ncan be applied in magnetic storage and quantum spin processi ng. Based on these discoveries, we\ndevelop a quantum mechanical model and propose that in order to get giant MAE with heavy p\nelements,onecantrytocreate px/ydegenerateorbitalsaroundFermilevelandmovethe pzorbital\nfrom Fermi levelas far as possible. Additionally,weargue t hat thesimilarideais applicableto d-\nandf-electron systems. Theseprovideusaparadigm foratomicma gnetic-storage-devicedesign.\nX=Ga,Ge,In,Sn,Tl,Pb Bi \n(a) \n(d) (c) (b) y\nx\nxyN\nC\nBi \nOMg Maxium MAE=203 meV \nFigure 1: Top- and side-views of Bi-X on nitrogenized divaca ncy of graphene (a)-(b), and of Bi\non MgO(100)(c)-(d).\nToelaboratetheprinciplestogenerategiant pelectronMAE,firstwefocusonsystemsofBi-X\non NDV (Bi-X@NDV)ofgraphene, and then wedemonstratethat t hisprinciplecan beappliedto\nothermagneticsystems.\nFor Bi-X@NDV, by examiningthetotal energies of a set ofposs iblestructures,40we find that\nfor X = Ga, Ge, In and Sn, the vertical structure (shown in Figu res 1a and 1b) is the most stable.\n3For X = Tl, the vertical structure is metastable but protecte d by an energy barrier of 3.6 eV which\nis sufficiently high to keep the structure at room temperatur e.40For X = Pb, the vertical structure\nis unstable (See Supporting Information). Nevertheless, t he properties of all the six vertical Bi-X\nstructures will be discussed on the same footing in the follo wing, in order to gain more insights\nintotheprincipleof p-electron magnetismforgiantMAE.\nTable1: Spinmoments(M S,inµB)ofBiandX,andmagnetocrystallineanisotropyenergies(M AE,\nin meV) of Bi-X on nitrogenized divacancy of graphene, posit ive MAE indicates a vertical easy\naxis.\nX Ga Ge In Sn Tl Pb\nMS(Bi) 0.85 0.64 0.85 0.55 0.77 0.55\nMS(X) 0.02 -0.01 -0.01 0.00 -0.01 0.01\nMAE -179.0 72.2 109.4 81.7 202.7 170.6\nWe present the spin moments (M S) of Bi and X alongside with the MAE of Bi-X dimers in\nTable1. TheMAEisdefinedasMAE =Ex−EzwhereEz(x)isthetotalenergyofthesystemwhen\nthemagnetizationaxisisalongthesurfacenormal z(alongxshowninFigure1). Themagnetization\nis mainly distributed in Bi atom while the atom X has nearly no MSin all these systems. The M S\nof Bi with 3rd group elements are 0.85, 0.85 and 0.77 µBfor Ga, In and Tl, while those with the\n4th group elements are 0.64, 0.55 and 0.55 µBfor Ge, Sn and Pb, respectively. More importantly,\nwe note that all these systems have giant values of MAE togeth er with vertical easy axis except\nfor Bi-Ga. The vertical easy-axis is crucial for applicatio ns because it is unique and the energy\ndifference between magnetization along in-plane directio ns is typically small (no more than 10%\nofthecorrespondingMAE inourcases). TheMAEsofBi-Ge, Bi- In and Bi-Sn are comparableto\nthe largest MAE reported for d-element dimers.35To the knowledge of us, the MAE of the Bi-Tl\ndimerof203meV is thelargestvalueeverreported forthesup porteddimers,and thisMAE value\nisvery closeto therecord-breaking valueofaOsadatom onMg O.13\nNotealsofromTable1thattheMAEvaluesincreasealongwith theincreasingatomicnumber\nof element X for X in the samegroup. However, as X is not spin-p olarized, theMAEs are mainly\nfrom Bi. In order to understand the origins of these giant MAE s as well as their dependence on\nthe atomic number of X, in Figure 2 we plot the projected densi ty of states (PDOS) of Bi in Bi-\n4(a) \n(e) (d) (c) (b) \n(f) Spin-Poarized PDOS of Bi(States/eV) \nE-E F(eV) E-E F(eV) py \npz \npx Ga \nPb Tl Sn In Ge \nUp \nDown \nUp \nUp Up Up \nUp \nDown Down Down Down Down \nFigure 2: (a) to (f): Spin-polarized PDOS of porbitalsof Bi in Bi-X@NDV (X = Ga, Ge, In, Sn,\nTland Pb) withoutSOI. Thedashed verticallinesindicateFe rmi level.\nX@NDV calculated without SOI and those with SOI in Figure 3. A s thesorbital (l= 0) has no\ncontributiontotheSOI, only porbitalsareconsidered inDOS plotting.\nFigure 2 shows the spin-polarized PDOS of Bi. A clear charact er found in these PDOS is that\nthepxandpyorbitalsarequasi-degenerate. Thedegeneracyisassociat edwiththeapproximateC 4v\nsymmetry of the NDV substrate in which the nearest neighbor N -N distances are 2.71 and 2.81\n. The degeneracies of Bi bonded with the 3rd group elements ar e higher than the ones bonded\nwith the 4th group elements. In the latter cases Bi gains more electrons from the X atom, which\nmay lift the degeneracy of Bi orbitals. These degenerate orb itals are fully spin-polarized. The pz\norbitalsplitsintotwolevels,oneisoccupiedandtheother isunoccupiedwithsmallspin-splitting.\nExceptforBi-Ga,thequasi-degenerateminority-spin px/yorbitalsliearoundtheFermilevelinall\ntheotherfiveconsideredsystems.\nThe total PDOS of Bi with SOI are shown in Figure 3. The total PD OS with magnetization\nin vertical ( zaxis) and parallel ( xaxis) directions are presented. When the magnetization is i n\nthe parallel direction, thePDOS are almostthe same as theon es withoutSOI. However, when the\n5(a) \n(d) \n(e) (c) (b) \n(f) Total PDOS of Bi(States/eV) \nE-E F(eV) E-E F(eV) py \npz \npx Ga \nPb Tl Sn In Ge \nFigure3: TotalPDOSof porbitalsofBiinBi-X@NDVwithSOIformagnetizationalongp arallel-\nand vertical-directions. Thedashedvertical linesindica teFermi level.\nmagnetization is in the vertical direction, the pxandpyorbitals split in energy. For these orbitals\njustliearoundtheFermilevel,theenergy-splittingcause dPDOSchangehasadirectrelationwith\nMAE(see below). Incontrary, thechangeof pzorbitalsisnegligiblein allcases dueto lm=0.\nSuch above behaviors of the porbitals with SOI in Figure 3 and the origin of the giant MAE\ncan be understood by a simplified quantum mechanical model, i .e., a combination of a two-level\ndegenerate perturbation theory and a second-order nondege nerate perturbation theory.41The SOI\nHamiltoniancanbeapproximatelywrittenas ˆV=λˆL·ˆSwhereλisanorbitalandelementdepen-\ndent parameter containing the radial part of the orbital wav e function. The angular and spin parts\nofpwavefunctionscan beexpressedas42\npx(±) =Y1\n1−Y−1\n1√\n2χ±1/2(1)\npy(±) =Y1\n1+Y−1\n1√\n2iχ±1/2(2)\npz(±) =Y0\n1χ±1/2(3)\n6where±labels themajority and minority spin, χis the wave function in the spin space, Ym\nlis the\nspherical harmonics. So theHamiltonian ˆH0expandedin apairofdegenerated px/yorbitalsofthe\nminority spin can be expressed as a diagonalized matrix with diagonal elements being 0 and Δ,\nwhereΔis the energy difference between the minority spin pxandpyorbitals around Fermi level\ncalculated byDFT withoutSOI(as showninFigure2).\nWhen the magnetic moment is in the vertical direction, the SO I Hamiltonian is λˆLzˆSz, calcu-\nlatingthematrix elementsoneget ˆH=ˆH0+ˆV\nˆH=\n0iλ/2\n−iλ/2Δ\n (4)\nBy diagonalizingthisHamiltonian,theSOI-induced newene rgy levelsareobtained:\nE1,2(degenerate )=Δ±√\nΔ2+λ2\n2(5)\nIfΔ=0,E1,2=±λ\n2. So including SOI will lift the degeneracy of the px/yorbital pair in vertical\nmagnetizedcases. Theas-splitorbitalsmoveup-anddown-w ardsbyanamplitudeofλ\n2,separately.\nForBi-X (X=In, Tland Pb), thedegenerateorbitalswithoutS OIlocateattheFermilevel(as can\nbe seen from Figure 2c, 2e and 2f), such lift of the degeneracy will directly lower their energy\nbyλ\n2. For the Bi-X (X = Ge and Sn), Δ/negationslash=0, butΔis quite small so the above argument is still\nhold qualitatively. When the magnetization is along the xaxis, the SOI Hamiltonian is λˆLxˆSx=\nλˆL+ˆS++ˆL+ˆS−+ˆL−ˆS++ˆL−ˆS−\n4, whereˆL±andˆS±are angular moment ladder operator and spin ladder\noperator. Bycalculatingthematrixelementsbetween px(−)andpy(−),wefindthatallthesevalues\nare zero. So including SOI will not have much influence in ener gy for parallel magnetization.\nSuch mechanism requires the degenerate orbitals locating n ear the Fermi level. For Bi-Ga (see\nFigure 2a), the degenerate orbitals are far away from the Fer mi level. The lowest unoccupied\nmolecular orbital (LUMO) and highest occupied molecular or bital (HOMO) are at 0.65 and -0.8\neV, respectively. However, since the λof Bi is approximately 0.8 eV,43the splitting by including\n7SOI isλ/2 = 0.4 eV which is insufficient to change the occupation numbe r. This is verified in\nFigure 3a. Thus the degenerate orbital coupling has negligi ble influence on the MAE of Bi-Ga.\nInstead, the coupling between the nondegenerate orbitals s hould givethe dominatingcontribution\nto the MAE in this case. In the framework of a second-order non degenerate perturbation,44,45the\nSOIis\nESOI(nondegenerate )=−λ2∑\nu,o|/angbracketlefto|L·S|u/angbracketright|2\nEu−Eo(6)\nSo uptothesecond-orderperturbation, ESOI≤0. BecausetheSOIisinverselyproportionaltothe\nenergygap,fromFigure2athemostdominatingcouplingmaya risefromtheoccupied pzaround-\n2eVandtheoccupied px/yorbitalsat-0.8eV,couplingtotheunoccupied px/yaround0.6eV.When\nthemagnetizationisinthe zdirection,alltheabovementionedmatrixelementsarezero ;butwhen\nthe magnetization is along the xaxis,/angbracketleftpz(+)|LxSx|py(−)/angbracketright=iλ\n2. Therefore, the system is more\nstable when being magnetized in-plane. We should mention th at such effects also exist in other\nBi-X cases, which makesit reasonableto interpret thevaria tionofMAE overtheelement number\nof atom X as shown in Table 1. As mentioned above, the nondegen erate part is contributed from\nthecouplingbetween pzandpx. Insuchcases,fromEq. ( ??)onecanseethat ESOIisproportional\ntotheoccupancyof pz. By integratingthePDOSof pzshowninFigure2,wefindthatforthoseX\nin the same group, the occupation of Bi pzorbital become smaller with the increasing of element\nnumber.40Thereductionof pzmaybeattributedtothereducingofcovalencyinBi-Xbondwh ich\nis resulted from the increasing of metallicity of X. Finally , we tired DFT+U with U = 2 eV on Bi\nin Bi-Tl but found negligible influence on the PDOS of Bi, whic h means MAE is unlikely to be\ndependenton theparameterU..\nThe above two mechanisms are illustrated in Figure 4. In the p resence of degenerate px/yor-\nbitals, the SOI between the pxandpyhas no effect when Msis in-plane; but splits the degenerate\noribtals when Msis out-of-plane (M ⊥) and hence lowers the energy of the system. For the non-\ndegenerate orbitals, the SOI between pzandpx/yhas negligible effect when Msis out-of-plane;\nbut has significant effect on the energy lowering of the syste m whenMsis in-plane (M /bardbl). The\nfinal results depend on the competition between these two eff ects from both degenerate and non-\n8degenerate orbitals. As λ<Eu−Eotypically, the coupling between nondegenerate orbitals is a\nhigher-order smaller contribution compared to the couplin g between the degenerate orbitals. So\nusually the degenerate case dominates. We noticed that in so me of the available investigations,\nthe coupling between the degenerate orbitals were overlook ed in the interpretation of MAE, in\nspite of the presence of the degenerate orbitals.27,35,36Our work points out the necessity to con-\nsider the effects of degenerate orbitals and makes the pictu re on the physical origin of giant MAE\ncompleted.\nFigure4: Illustrationofthecompetitionoftwomechanisms ofMAEfromdegenerate(upperpart)\nand thenondegenerate(lowerpart)orbitals. Thered linesi ndicateinitiallydegenerate px/yenergy\nlevels without SOI, the orange arrow indicates spin, and the blue letters indicate the effective\ncontributionstoMAE(M ⊥forthedegeneratecaseand M /bardblforthenondegeneratecase).\nThe abovetwo mechanisms suggest a general picture to realiz e giant MAE. In order to gener-\nate vertical magnetization, degenerate orbitals around Fe rmi level are recommended. This can be\nobtained through manipulating an adatom or dimer on a symmet ry matched substrate by electric\nfields orbydopingthesubstrate. In orderto reducethein-pl aneMAE(associated withnondegen-\nerate orbital coupling), the orbital splittingof lm=0 need to be enlarged. As atom X lies abovethe\ngraphene surface, the Bi atom which is responsible for the gi ant MAE has a larger distance from\nthe substrate than an adaom has in adatom/MgO.11This implies that the influence of the surface\nspin-flip scattering will be smaller in our cases so the syste m we proposed is hopeful to have a\nlonger spin relaxation time. Note that in a certain crystal fi eld, the splittingof degenerate orbitals\nunder SOI could also take place for dandforbitals. The orbital of lm=0 always contributes to\nin-planeMAE.Thustheseconclusionsshouldalsoholdforab roadrangeofmagneticsystems. To\n9further demonstratethemechanismsshownin Figures 4 are ge nerally applicable, we investigatea\nsingleBi adatomonMgO(100). Themoststablegeometricstru ctureissubjecttotheatopadsorp-\ntiononoxygen(Figure1cand 1d),andaccordinglythequasi- degenerateorbitalsappearedaround\nFermi-level with an MAE being 75 meV, which again suggests th e proposed mechanisms hold in\notheranalogicalsystems.40\nInsummary,wediscoveredthatmostoftheverticalstructur esofBi-X@NDVofgraphenecan\nbestableatroomtemperature,andBiisspin-polarizedinth esesystems. Mostofthesedimersshow\ngiant MAE with easy axis in the out-of-plain direction. In pa rticular, the MAE of Bi-Tl@NDV is\ncomparable to the largest MAE ever reported. The origin of gi ant MAE can be attributed to the\nlift of the degeneracy with SOI of the orbitals near the Fermi level without SOI. Such degeneracy\nis relevant to the local C 4vsymmetry of the system. These finding provides a novel and gen eral\nmethodtodesigngiantMAEwith p-,d-andeven felectronmagnetism: onecouldfindasubstrate\nwith a proper symmetry (C 4vforpin this paper) to create quasi-degenerate orbitals and then\ntune these orbitals close to Fermi level; and simultaneousl y move the orbitals with lm=0 far from\nFermi level. These discoveries and insights pave a pathway f or further development of robust\nnanomagneticunits.\nAcknowledgement\nThisworkissupportedbytheNationalNaturalScienceFound ationofChina(Grants11474145and\n11334003), the Young Teachers Special Startup Funds of Zhen gzhou University, and the Special\nProgram for Applied Research on Super Computation of the NSF C-Guangdong Joint Fund (the\nsecond phase).\nSupporting InformationAvailable\nThisfilecontainsthediscussiononthecomputationaldetai l,stabilityofstructures,porbitaloccu-\npacies and geometricalinformationofthestructuresinves tigated.\nThismaterial isavailablefree ofchargeviatheInternetat http://pubs.acs.org/ .\n10References\n(1) Plumer,M.L.;VanEk,J.;Weller,D. ThePhysicsofUltra-High-DensityMagneticRecording ;\nSpringer, Berlin, Heidelberg,2012;Vol.41.\n(2) Kuch, W.; SchÃd’fer, R.; Fischer, P.; Hillebrecht, F. U. Magnetic Microscopy of Layered\nStructures ;Springer, 2015.\n(3) StÃ˝ uhr, J.; Siegmann, H. C. 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Res. 2015,48,\n3080–3087.\n13GraphicalTOCEntry\nBismuth \nAtom X \n14"    },    {        "title": "1702.00220v1.Superzone_gap_formation_and_low_lying_crystal_electric_field_levels_in_PrPd__2_Ge__2__single_crystal.pdf",        "content": "arXiv:1702.00220v1  [cond-mat.str-el]  1 Feb 2017Superzone gap formation and low lying crystal electric field levels\nin PrPd 2Ge2single crystal\nArvind Maurya,∗S. K. Dhar, and A. Thamizhavel\nDepartment of Condensed Matter Physics and Materials Scien ce,\nTata Institute of Fundamental Research,\nHomi Bhabha Road, Colaba, Mumbai 400 005, India.\n(Dated: September 18, 2018)\nAbstract\nThe magnetocrystalline anisotropy exhibited in PrPd 2Ge2single crystal has been investigated\nby measuring the magnetization, magnetic susceptibility, electrical resistivity and heat capacity.\nPrPd2Ge2crystallizes in the well known ThCr 2Si2-type tetragonal structure. The antiferromag-\nnetic ordering is confirmed as 5.1 K with the [001]-axis as the easy axis of magnetization. A\nsuperzone gap formation is observed from the electrical res istivity measurement when the current\nis passed along the [001] direction. The crystal electric fie ld (CEF) analysis on the magnetic sus-\nceptibility, magnetization and the heat capacity measurem ents confirms a doublet ground state\nwith a relatively low over all CEF level splitting. The CEF le vel spacings and the Zeeman splitting\nat high fields become comparable and lead to metamagnetic tra nsition at 34 T due to the CEF\nlevel crossing.\nPACS numbers: 81.10.Fq, 75.30.Kz, 75.50.Ee, 75.10.Dg\n1I. INTRODUCTION\nPraseodymium(Pr3+),beinganon-Kramer’sionexhibitsavarietyofinteresting magnetic\nbehaviour in its compounds. Pr3+often orders magnetically with a doublet ground state,\nwhile it behaves as a Van Vleck paramagnet down to the lowest temper ature when the\ncrystal electric field split ground state is a singlet (e. g. in PrRhAl 4Si21). There is a\nsurge in the study of Pr compounds in recent times after the obser vation of heavy fermion\nsuperconductivity in PrOs 4Sb12, PrV 2Al20, and PrTi 2Al20at ambient or under pressure,\noriginating from the quadrupolar Kondo order of Pr f-orbitals2,3.\nPrPd2Ge2is a member of the well known large family of compounds crystallizing in\nthe tetragonal ThCr 2Si2-type structure (space group I4/mmm, # 139). A previous report\non polycrystalline sample provided evidence for an antiferromagnet ic transition at 5 K5.\nFurther, neutron diffraction indicated a magnetic cell three times la rger than the chemical\ncell and the Pr moments aligned along the c-axis with a spontaneous magnetization of\n∼2.0µBat 2 K5. Three possible configurations of the moments in the antiferromag netic\nstate, compatiblewiththetriplingofthemagneticcell, werepresent ed. However, theauthors\nof that work felt that neutron diffraction on a single crystal was ne cessary to determine\nunambiguously the magnetic configuration. In the present work, w e explore the magnetic\nproperties of a single crystal of PrPd 2Ge2, using the techniques of magnetization, electrical\nresistivity and heat capacity. Our data on single crystalline sample su ggest that the easy\ndirection of magnetization may not lie exactly along the c-axis. The crystal electric field\n(CEF) analysis reveals an interesting possibility of metamagnetism at high fields ( ≃35 T)\nalong the [001] direction due to the crossover among the split energ y levels with magnetic\nfield.\nII. CRYSTAL GROWTH\nWe used the Czochralski method to grow a single crystal of PrPd 2Ge2, in a tetra-arc\nfurnace, as the compound melts congruently. First, a homogeneo us polycrystalline ingot,\nweighing ∼10 gm was prepared by repeated arc melting of the high pure metals in the\nstoichiometric ratio 1 : 2 : 2, which was subsequently used as startin g material for the\ncrystal growth. To start with, a tungsten rod was used as seed t o pull the single crystal\n2out of molten charge. The pulling speed was maintained at 10 mm/hour , after the initial\nnecking and stabilization. In order to estimate the magnetic part of the heat capacity and\nto estimate the magnetic entropy, a polycrystalline sample of the no n-magnetic LaPd 2Ge2\nwas also prepared in a home built mono arc furnace.\nTo confirm the phase purity of grown crystal, a small portion of the specimen was sub-\njected to powder x-ray diffraction (XRD). The XRD (not shown her e for brevity) revealed\na clear pattern without any impurity peaks suggesting the single pha se nature of the grown\ncrystal. ALe-bail fit wasperformedonthex-raydiffractionpatte rnandthelatticeconstants\nwere estimated to be a= 4.336(8) ˚A andc= 10.050(8) ˚A, respectively, which are in good\nagreement with the previously reported values4,5. The composition of the crystal was further\nconfirmed by energy dispersive analysis by x-ray (EDAX). In order to study the anisotropic\nphysical properties, the grown single crystal was cut along the pr incipal crystallographic\ndirections viz., [100] and [001] using a spark erosion cutting machine and back-refl ection\nLaue diffraction. Well defined Laue diffraction spots together with t he four fold symmetry\nascertain the good quality of the grown single crystal.\nIII. MAGNETIC SUSCEPTIBILITY AND MAGNETIZATION\nMagnetic susceptibility of PrPd 2Ge2shows a clear anomaly at TN= 5.1 K, with field\nparallelto[100]and[001]directions, respectively(see, Fig.1),inco nformitywiththereported\nTNof polycrystalline sample. Similar to isotypic CePd 2Ge26, PrPd 2Ge2also exhibits a\nsignificant anisotropy in the magnetic susceptibility at low temperatu res, measured in a\nfield of 0.1 T. Magnitude of χalong [100] is smaller than that of [001], and remains nearly\ntemperatureindependentintheantiferromagneticstatebelow TN,markingitasthehardaxis\nof magnetization, while χalong [001] increases sharply at lower temperature after exhibiting\na cusp at TN, unlike the conventional behaviour for a simple two sublattice antife rromagnet.\nThe increase in the magnetic susceptibility along the [001] direction at temperatures below\nTNreveals that themagneticstructure iscomplex asdetermined from theneutrondiffraction\ndata by Welter and Halich5.\nThe Curie-Weiss fit to the inverse susceptibility data gives µeff= 3.70 (3.76) µB/Pr,\nandθp= -14.9 (4.3) K in [100] and [001] directions, respectively. µeffvalues are close to\n3.58µB, predicted by the Hund’s rules for free Pr3+. A negative value of θpcorresponds\n31.0 \n0.5 \n0.0  χ(emu/mol) \n300 200 100 0\nTemperature (K) 0.8 \n0.4 \n0.0  χ(emu / mol) \n10 5 0\nT (K) χ [001] χ [100] x 5 \n H || [100] \n H || [001] PrPd2Ge 2\n H = 0.1 T (a)M ( µB/Pr) \nMagnetic Field (Tesla) PrPd2Ge 2(b)3\n2\n1\n0\n8 6 4 2 0       1.8 K  H || [100] \n H || [001] \nFIG. 1: (a) Magnetic susceptibility of PrPd 2Ge2between 1.9 and 300 K. Inset shows low tem-\nperature data on expanded scale, in which χalong [100] is multiplied by a factor of 5 to show its\ncomparison with that of [001], (b) The isothermal magnetiza tion data at 1.8 K along [100] and\n[001].\nto antiferromagnetic correlations among Pr ions; however θphas a positive, albeit small,\nvalue along [001], which is tentatively ascribed to ferromagnetic inter action among the next\nnearest neighbours in the (001) plane.\nThe isothermal magnetization at 1.8 K along [100] varies linearly with fie ld, which is in\nconformity with the basal plane being the hard plane of magnetizatio n. On the other hand\nthe magnetization along the tetragonal [001] direction, undergoe s a distinct spin flip transi-\ntion at 1.9 T. A change in the antiferromagnetic configuration presu mably happens in the\nlow field region ( ∼1 T) as well which is marked by a distinct hysteresis. The magnetizatio n\nattained at 7 T is 2.22 and 1.50 µB/Pr for field parallel to [001] and [100] directions, respec-\ntively. These values are significantly lower than the saturation value of 3.20µBfor Pr3+ion\ncorresponding to the total angular momentum quantum number J= 4 and Land´ e g-factor\n4(gJ= 4/5). A higher magnetic field is required to populate all CEF split energy le vels in\norder to attain the full moment value (see section VI). It may be no ted that for a bipartite\ncollinear antiferromagnet, the susceptibility along the easy axis gra dually decreases to zero\nas the temperature is gradually decreased to zero. The isotherma l magnetization should be\nnearly zero up to spin flop transition. We do not see such behavior in o ur single crystal. We\nbelieve our data show that the Pr moments do not lie parallel to [001] a xis and the neutron\ndiffraction results reported in Ref. 5 are oversimplified.\nIV. ELECTRICAL RESISTIVITY\nFrom the electrical resistivity data, recorded between 2 and 300 K , shown in Fig. 2,\nPrPd2Ge2is metallic down to 2 K. The resistivity is anisotropic and exhibits higher v alues\nfor the current density Jparallel to [100] direction. Note that below TN, forJ/bardbl[100],\na faster drop resulting from the loss of magnetic disorder scatter ing is observed. But, for\nJ/bardbl[001],ρ(T) shows an upturn at TN. Such a feature observed in some antiferromagnets\nis generally attributed to a reduction in electron density caused by t he opening of a gap\n(superzone gap) in the Fermi surface at TNwhen the magnetic periodicity is incongruent\nwith the periodicity of the chemical unit cell. Fig. 2(b) shows the data forJ/bardbl[100] and\nH/bardbl[001]. It is noticed that the decrease of resistivity below TNbecomes less prominent as\nthe field increases such that at intermediate transverse field (1.40 -2.05 T), a characteristic\nsignature of superzone gap is observed (Fig. 2(b)). Whether the superzone gap persists to\nhigher fields can be ascertained only by data taken at temperature s lower than 2 K. This\ninteresting observation shows that in PrPd 2Ge2, the superzone gap doesn’t depend only on\ndirection, but it is also a function of magnetic field. On the other hand , forJ/bardbl[001] and\nH/bardbl[100], the zero field superzone gap persists at least up to 10 T (Fig. 2 (c)). However, the\noverall magnitude of resistivity, the upturn at TNandTNdecrease. It is to be mentioned\nhere that the jump in the electrical resistivity at the superzone ga p in zero field is very small,\nroughly estimated to be 0.12 µΩ·cm indicates Fermi surface gap is very small in this case. In\norder to estimate the magnitude of Fermi surface gapping, we follo wed the technique used\nby Mun et al.7for YbPtBi by calculating the relative change in the conductivities us ing the\nrelation: ( σn−σg)/σn, whereσn(= 1/ρn) andσg(= 1/ρg) are the conductivities of normal\nand the gapped states. The conductivity values obtained below and above the superzone\n560\n40\n20\n0 ρ(µΩ. cm)\n300 200 100 0\nTemperature (K)15\n14\n13ρ (µΩ cm)\n15 10 5 0PrPd2Ge2\n J || [100]\n J || [001]\n9.0\n8.8\n8.6\n8.4\n8.2 ρ(µΩ. cm)\n15 10 5 0\nTemperature (K)8.9\n8.7\n8.5ρ (µΩ  cm)\n10 5 0\nT (K)PrPd2Ge2 J || [001]\nH || [100]\n0 T\n4 T\n6 T\n8 T\n10 T2 T17\n16\n15\n14 ρ(µΩ. cm)\n8 6 4 20 T1.9\n0.30.51.6\n0.811.21.41.81.952.2 T\n2.05PrPd2Ge2\n J || [100]\nH || [001]\n2.1\n0.15 2ρ x 1.52 \nTemperature (K)(a)  (b)\n(c)\nρn + ρ0ρg + ρ0\nFIG. 2: (a) Variation of zero field electrical resistivity ρwith temperature, with current density J\nparallel to principal crystallographic directions in PrPd 2Ge2. Inset zooms up the low temperature\npart, for J/bardbl[001] direction the data is multiplied by 1.52 to show up in th e samey-axis scale. (b)\nand (c) show the ρ(T) data at selected transverse field in both configurations, re spectively. Various\niso-field traces of resistivity in (b) are shifted verticall y for clarity. The inset in (c) shows the low\ntemperature electrical resistivity for J/bardbl[001], the solid lines are linear fit to the resistivity data\nto estimate the ρnandρg, the resistivities of the normal and the gapped state.\ngap are shown in Table I. It is obvious from the table that the Fermi s urface gapping is\nonly 3% and it is almost constant for fields up to 4 T. In order to get mo re insight into the\nbehaviour of the Fermi surface gapping, experiments down to low t emperature are necessary.\nV. HEAT CAPACITY\nThe heat capacity of PrPd 2Ge2and non magnetic reference compound LaPd 2Ge2mea-\nsured between 1.9 K and 20 K, is shown in Fig. 3. The antiferromagnet ic transition at 5.1 K\nis marked by a sharp peak. The low temperature anomaly observed o n a polycrystalline\n6TABLE I: The electrical resistivity values obtained from th e linear fit (refer to inset of Fig. 2(b))\nto the electrical resistivity data at temperatures below an d above the superzone gap in different\nmagnetic fields. The last column gives the degree of Fermi sur face gapping estimation.\nMagnets Field\n(T)Resistivity\nρg\n(µΩ·cm)Conductivity\nσg= 1/ρg\n(µΩ·cm)−1Resistivity\nρn\n(µΩ·cm)Conductivity\nσn= 1/ρn\n(µΩ·cm)−1∆σ=σn−σg\nσn\n0 8.6229 0.115970 8.3555 0.11968 0.03099\n1.6 8.6294 0.11588 8.3600 0.11962 0.03126\n2.0 8.6260 0.11593 8.3670 0.11952 0.03004\n2.6 8.6165 0.11606 8.3558 0.11968 0.03025\n4.0 8.6189 0.11602 8.3258 0.12011 0.03405\nsample of PrPd 2Ge2by Welter and Halich5at 3 K is not observed in our heat capacity data,\nindicating the good quality of our sample. The heat capacity of non ma gnetic iso-structural\nLaPd2Ge2was subtracted from that of PrPd 2Ge2, to estimate the contribution by Pr-4 f\nelectrons only ( C4f), assuming identical phonon contribution to the heat capacity in tw o\ncompounds. Entropy S4fwas calculated by the same process as described for CePd 2Ge2.\nA smooth extrapolation of C(T) data below 1.9 K in PrPd 2Ge2was done as shown by the\nsolid line in Fig. 3, to get an approximate value of the heat capacity at lo w temperatures\nfor which the experimental data are not available. The error in the e stimation of entropy\ndue to extrapolation is at the most a few % of its true value. At TN,S4fis comparable to\nRln2 corresponding to a doublet ground state with effective spin S= 1/2. Here R is the\nuniversal gas constant. From the C4fvalues at higher temperatures, the CEF split levels\nhave been derived, which is the subject of next section.\nVI. CRYSTAL ELECTRIC FIELD ANALYSIS\nA crystal electric field (CEF) analysis was performed on the magnet ization and heat\ncapacity data of PrPd 2Ge2to derive possible informationabout the crystal electric field level\nsplittings. For the case of Pr ( J= 4,L= 5,S= 1), CEF Hamiltonian can be expressed as,\nHCEF= (B0\n2O0\n2+B0\n4O0\n4+B4\n4O4\n4+B0\n6O0\n6+B4\n6O4\n6)+gµBJ.H, (1)\n7C, S4f  (J/ K mol) PrPd 2Ge 2\nLaPd 2Ge 2S4f \nR ln2\nTemperature (K) 15 \n5\n0\n20 10 0C4f \nFIG. 3: Heat capacity (C) as a function of temperature in PrPd 2Ge2and non magnetic analog\nLaPd2Ge2. The 4fcontribution C4fand calculated entropy S4fare are also shown on the same\nscale. Dashed horizontal line marks the Rln2 value of entrop y, where R is the gas constant.\nwhereBm\nl’s are CEF parameters and Om\nl’s are Steven’s operators8,9, respectively. Second\nterm represents Zeeman energy in presence of magnetic field.\nPr is a non Kramer’s ion (integral spin S). In the tetragonal D4hpoint symmetry, the\nCEF splits the 9 fold degenerate state of Pr3+ion into two doublets and 5 singlets10. The\ncompounds which exhibit singlet ground state are non magnetic (for instance the recently\ndiscovered PrRhAl 4Si2)1. Since PrPd 2Ge2orders at 5.1 K, the ground state is definitely\ndoublet, as indicated strongly by the entropy (see above). There fore, the excited states\ncomprise of a ground state doublet and 5 singlets and another doub let at some higher\nenergy.\nAfter diagonalizing the (2 J+ 1)×(2J+ 1) i.e. 9 ×9 dimensional matrices, the CEF\nparameters Bm\nl, which fit the anisotropic susceptibility, isothermal magnetization ( Fig. 4c)\nand Schottky heat capacity of PrPd 2Ge2, are listed in Table II and the computed curves are\nshown in Fig. 4 along with the experimental data. It may be noted tha t the heat capacity of\nboth PrPd 2Ge2and LaPd 2Ge2was measured up to 80 K for this purpose. The ground state\nat zero magnetic field is an admixture of | −1/angbracketrightand|+3/angbracketrightpure states. There is a good degree\nof agreement between theory and observation, which gives crede nce to the correctness of the\nCEF parameters and energy levels derived from our analysis. Howev er, advanced techniques\nlike inelastic neutron scattering are required to confirm CEF splitting s.\nThe main panel of Fig. 4(d) shows the calculated magnetization for a pplied field up to\n8200\n100\n0 (χ−χ 0)-1 (mol/emu) \n300 200 100\nTemperature (K)  H || [100] \n H || [001] \n CEF values PrPd2Ge 2\n H = 0.1 T 15 \n10 \n5\n0C4f  (J/ K mol) \n80 60 40 20 0 Experimental \n CEF fit \nMagnetic Field (Tesla) M ( µB/Pr) \nM ( µB/Pr) (a) (b) \n016 24 27 79.280.280.4\n3\n2\n1\n0\n8 6 4 2 0PrPd2Ge 2\n H || [001] \n H || [100] \n 1.8 K Temperature (K) \nMagnetic Field (Tesla) (c) 4\n2\n0\n100 80 60 40 20 03\n2\n1\n0M ( µB/Pr) \n100 50 0\nH (T) 10 mKPrPd 2Ge2 H || [100] \n H || [001] \n1.8 K (d) \nFIG. 4: A comparison of calculated (a) inverse susceptibili ty at 0.1 T, (b) heat capacity and (c)\nisothermal magnetization derived from CEF eigenstates and eigen values with the corresponding\nexperimental data. (d) Calculated magnetization of PrPd 2Ge2up to 100 Tesla at 1.8 K (main\npanel) and at 10 mK (inset)\n200\n100\n0\n-100\n-200Energy (K) \n100 80 60 40 20 0\nMagnetic Field (Tesla) PrPd2Ge 2\nH || [001]\nFIG. 5: Evolution of energy eigen values of CEF and Zeeman spl it Pr-energy levels with magnetic\nfield along [001]-direction in PrPd 2Ge2. Arrow marks the ground state level crossing, which leads\nto metamagnetism (cf. Fig. 4(d)).\n9TABLE II: CEF parameters, energy levels and the correspondi ng wave functions for PrPd 2Ge2at\nzero field.\nCEF parameters\nB0\n2(K) B0\n4(K) B4\n4(K) B0\n6(K) B4\n6(K)\n−1.1889 0 .0147 0 .0147 3 .9490×10−41.9220×10−5\nλ[100]= 0.1569 mol/emu, λ[001]=−1.28424 mol/emu\nenergy levels and wave functions\nE(K)| −4/angbracketright | −3/angbracketright | −2/angbracketright | −1/angbracketright |0/angbracketright |+1/angbracketright |+2/angbracketright |+3/angbracketright |+4/angbracketright\n80.40 -0.2449 0 0 0 0.9381 0 0 0 -0.2450\n80.23 0 0 0.7071 0 0 0 -0.7071 0 0\n79.26 0 0 0 0.9507 0 0 0 -0.3101 0\n79.26 0 -0.3101 0 0 0 0.9507 0 0 0\n27.17 0 0 0.7071 0 0 0 0.7071 0 0\n24.06 -0.7071 0 0 0 0 0 0 0 0.7071\n16.37 0.6633 0 0 0 0.3464 0 0 0 0.6633\n0 0 0.9507 0 0 0 0.3101 0 0 0\n0 0 0 0 0.3100 0 0 0 0.9507 0\n100 T. The magnetic field removes the degeneracy, and their energ y evolution with magnetic\nfield is governed by Eq. 1. The magnetization in [100] direction increas es gradually with\nfield and attains a marginally lower than full moment value of Pr (3.2 µB) at 100 T where\nit has not yet achieved saturation. Interestingly, along the [001] d irection, there is a meta\nmagnetic jump at around 34 Tesla and the magnetization attains the theoretical saturation\nvalue at higher fields. The metamagnetic jump at 34 T has a different o rigin than the spin-\nfloptransition thatoccurs at 1.9T due tothe change intheantiferr omagnetic configuration\nwith field. The metamagnetic jump at 34 T originates from the level cr ossing of the ground\nand the first excited CEF levels at that field (see Fig. 5) due to the do minance of Zeeman\nsplitting over CEF splitting. When the magnetization is calculated for a temperature of 10\nmK (inset of Fig. 4(d)), the metamagnetic transition due to level cr ossings is nearly vertical\nas one would expect it to be in the ground state, free of effects due to thermal fluctuations\n10at higher temperature. It would be interesting to measure the mag netization at higher fields\nto confirm the results of our CEF based calculations.\nVII. CONCLUSION\nThe anisotropic magnetic properties of PrPd 2Ge2have been investigated by growing a\nsingle crystal, in a tetra-arc furnace. The phase purity and the cr ystal composition were\nconfirmed by x-ray diffraction and EDAX. The transport and magne tic properties reveal\nlarge anisotropy along the two principal crystallographic directions viz., [100] and [001].\nThe antiferromagnetic order is confirmed at TN= 5.1 K. The increase in the magnetic\nsusceptibility below TNforH/bardbl[001], the easy axis direction, clearly reveals the complex\nmagnetic structure of this compound. The electrical resistivity da ta of PrPd 2Ge2show\nthe existence of anisotropic superzone gap which, interestingly, is dependent on both the\ncrystallographic direction and applied magnetic field. The N´ eel temp erature was found to\ndecrease with increasing field, as expected for a typical antiferro magnet system. Our crystal\nfield calculation explains the anisotropy in the magnetic susceptibility a nd magnetization of\nPrPd2Ge2. The energy levels determined from CEF analysis clearly explain the Sc hottky\nanomaly in the 4 f-derived part of the heat capacity. The CEF levels in PrPd 2Ge2are\ncomparatively low lying, which makes Zeeman interaction comparable t o CEF splitting at\na magnetic field of 34 Tesla leading to metamagnetic behaviour for H/bardbl[001] due to level\ncrossing. Its confirmation by high field magnetization on a single crys talline sample is\ndesirable.\n∗Electronic address: arvindmaurya.physics@gmail.com\n1A. Maurya, R. Kulkarni, A. Thamizhavel, and S. K. Dhar, Solid State Commun. 240, 24 (2016)\n2E. Bauer, N. Frederick, P.-C. Ho, V. Zapf, and M. Maple, Phys. Rev. B65, 100506 (2002).\n3A. Sakai, and S. Nakatsuji, J. Phys. Soc. Jpn. 80, 063701 (2011).\n4D. Rossi, R. Marazza, and R. Ferro, J. Less-Common Met. 66, 17 (1979).\n5R. Welter, and K. Halich, J. Phys. Chem. Solids 67, 862 (2006).\n6A. Maurya, R. Kulkarni, S. K. Dhar, and A. Thamizhavel, J. Phy s.: Condens. Matter 25,\n435603 (2013).\n117E. D. Mun, S. L. Bud’ko, C. Martin, H. kim, M. A. Tanatar, J. -H. Park, T. Murphy, G. M.\nSchmiedeshoff, N. Dilley, R. Prozorov, and P. C. Canfield, Phys . Rev. B 87, 075120 (2013).\n8M. T. Hutchings 1965 Solid State Physics: Advances in Research and Applications Vol.16\nedited by F. Seitz and B. Turnbull (New York: Academic) p.227 .\n9K. W. H. Stevens Proc. Phys. Soc. (London) Sect.A65 , 209 (1952).\n10W. A. Runciman, Phil. Mag. Ser. 81, 1075 (1956).\n12"    },    {        "title": "1702.05414v1.Magnetic_anisotropy_in_Permalloy__hidden_quantum_mechanical_features.pdf",        "content": "Magnetic anisotropy in Permalloy: hidden quantum mechanical features\nDebora C M Rodrigues,1, 2Angela B Klautau,2Alexander Edstr om,1Jan Rusz,1Lars Nordstr om,1Manuel\nPereiro,1Bj orgvin Hj orvarsson,1and Olle Eriksson1\n1)Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala,\nSweden\n2)Faculdade de F\u0013 \u0010sica, Universidade Federal do Par\u0013 a, CEP 66075-110, Bel\u0013 em-PA,\nBrazil\n(Dated: 11 October 2018)\nBy means of relativistic, \frst principles calculations, we investigate the microscopic origin of the vanishingly\nlow magnetic anisotropy of Permalloy, here proposed to be intrinsically related to the local symmetries of\nthe alloy. It is shown that the local magnetic anisotropy of individual atoms in Permalloy can be several\norders of magnitude larger than that of the bulk sample, and 5-10 times larger than that of elemental Fe\nor Ni. We, furthermore, show that locally there are several easy axis directions that are favored, depending\non local composition. The results are discussed in the context of perturbation theory, applying the relation\nbetween magnetic anisotropy and orbital moment. Permalloy keeps its strong ferromagnetic nature due to the\nexchange energy to be larger than the magnetocrystalline anisotropy. Our results shine light on the magnetic\nanisotropy of permalloy and of magnetic materials in general, and in addition enhance the understanding of\npump-probe measurements and ultrafast magnetization dynamics.\nPACS numbers: 75.30.Gw,71.15.Mb,71.70.Ej,71.20.Be\nKeywords: Permalloy; magnetocrystalline anisotropy; orbital moment anisotropy\nIntroduction { Random alloys can be viewed as a dis-\ntribution of clusters of di\u000berent composition, that have\nan underlying crystal structure in common. The con\fg-\nurational space is enormous for these systems and any\nmacroscopic property is the result of averaging of a im-\nmense amount of local clusters with di\u000berent con\fgura-\ntion and composition1. Random alloys often have prop-\nerties that stand out from the pure elements they are\nbuild up from, i.e. the mixing of elements may produce\nproperties that are completely unexpected.\nOne of the most prominent examples is Permalloy\n(Py), the common name for Fe xNi1\u0000xalloys with x\u00180:2\nand fcc crystal structure. These alloys are characterized\nby strong ferromagnetism, high permeability, vanishingly\nlow magnetic anisotropy energy (MAE), and low damp-\ning parameter2. These attributes elevate Py to a stan-\ndard material in magnetism and advantageous soft mag-\nnet for technological applications.\nOne might ask what the mechanism of the vanishing\nMAE of Py really is. One attempt to explain it is the\nresultant MAE picture3, which reinforces that an appro-\npriate mixture of two elements with distinct easy axis\n(as bcc Fe and fcc Ni with h001iandh111idirection,\nrespectively) would result in a material without strong\npreferential easy magnetization axis, i.e. a low MAE.\nNevertheless, recent experiments suggest that the Fe-Ni\nhybridization in the alloy environment is the major cause\nof low MAE in Py, rather than the easy axis of its sepa-\nrated constituents.4\nOther experiments show the existence of orbital mo-\nments at the individual chemical species in Py5. Addi-\ntionally, it is known that the magnetic anisotropy is pro-\nportional to the anisotropy of orbital moment for transi-\ntion metals6,7. Thus, the anisotropy at atomic level may\nexist, although diminished at the bulk. Considering this,as a random alloy, Py may be viewed as a huge ensem-\nble of interconnected clusters of Fe-Ni atoms distributed\non an fcc lattice, in which the macroscopic properties re-\n\rect the con\fgurational average of di\u000berent such clusters.\nThen di\u000berent parts of the alloy would have competing\nlocal anisotropies, that e\u000bectively average out, leading to\na fairly isotropic state. This microscopic scenario has, to\nthe best of our knowledge, not been considered so far as\na possible mechanism for low MAE in Py.\nFor that, in this Letter, we present \frst principles cal-\nculations to investigate this local MAE. However it is not\neasy to directly study the MAE, particularly not from\n\frst principles theory as it is hard to uniquely and ac-\ncurately decompose the energy into local contributions\nand the numerical challenges are countless. Instead we\nwill study another related quantity induced by the spin-\norbit coupling (SOC), namely the local anisotropy of the\norbital moments, since it is known to reveal also infor-\nmation about the MAE6. With this information we in-\nvestigate the role of these local competing anisotropies\nand how they reveal information about the soft magnetic\nbehavior of Py.\nMethod { The study was designed as the following:\n\frst, we performed ab-initio calculations of a fcc ma-\ntrix (\u001812500 atoms) of a Virtual Crystal Approximation\n(VCA) medium of Py (Py-VCA) and lattice parameter of\n3.54\u0017A5. The fcc unit cell was considered to have the same\nnumber of valence electrons as Py (9.6 e\u0000). After the\nself-consistent procedure, clusters composed by Fe and\nNi, with di\u000berent con\fgurations, were embedded in the\nPy-VCA matrix. The cluster region was self-consistently\nupdated while the potential parameters of the Py-VCA\nmatrix were kept \fxed. Finally, the magnetic spin and\norbital moments were computed for every site in the clus-\nter.arXiv:1702.05414v1  [cond-mat.mtrl-sci]  17 Feb 20172\nAs the con\fguration space for the clusters is vast, our\ninvestigation does not cover all possible con\fgurations.\nWe have, however, investigated a large number of geome-\ntries (84 excluding con\fgurations that are symmetrical\nto these), and we illustrate as an example a few typi-\ncal geometries in Fig. 1c. These clusters present distinct\ncon\fgurations due to the Fe distribution. Note that, lo-\ncally in a cluster, the number of Fe and Ni atoms can\nvary, although a con\fgurational average over all clusters\nof the material would naturally result in a concentration\nof Fe and Ni that re\rected the alloy concentration, i.e.\n20 % Fe and 80 % Ni (see dashed lines in Fig. 1b).\n  \nxyz\n0.20.40.60.81.0\n0.0\n0.20.40.60.81.0\n0.0\n 1   2   3  4   5   6  7    8    9  10  11  1213 14  1516 17  18 19\nc)a)\nb)\nsite8 configurations\n84 configurationsFeNi\nFe\nNiprobability probability\n1\nFIG. 1. (Color online) Distributions of Fe (red) and Ni (grey)\natoms at cluster sites (1: central site, 2-13: \frst neigbours and\n14-19: second neigbours), considering (a) 8 or (b) 84 con\fgu-\nrations. Three examples of con\fgurations that may be found\nin Permalloy are illustrated in (c). The dashed (dot-dashed)\nline represent the average Fe (Ni) concentration.\nIn our investigations we choose to study clusters with\n19 atoms, of which 4 or 5 are Fe atoms and the rest\nbeing Ni atoms. The atoms are sorted from a central\nsite (labeled 1), followed by its \frst (labeled 2-13 ) and\nsecond neighbors (labeled 14-19 ).\nThe electronic structure and magnetism of VCA-\nPy and the clusters were evaluated using the \frst-\nprinciples real-space linear mu\u000en-tin orbital method\nwithin the atomic sphere approximation (RS-LMTO-\nASA)8{11. This method follows the steps of the LMTO-\nASA formalism12but uses the recursion method13to\nsolve the eigenvalue problem directly in real space. The\ncalculations presented here are fully self-consistent, the\nexchange and correlation terms were treated within thelocal spin density approximation (LSDA)14, and the SOC\nterm was included at each variational step15,16. The RS-\nLMTO-ASA method is particularly designed to treat low\nsymmetry systems as the embedded clusters presented\nhere, without the need of periodic boundary conditions.\nRegarding the calculations for the matrix of VCA-Py,\nthe resulting spin moment ( ms) is 1:12\u0016Bper atom,\nwhich is in acceptable agreement with the experimental\nvalue of approximately 1 :0\u0016Bper atom4,5and previous\ncalculations4,17,18. Therefore, we conclude that the ef-\nfective medium that is considered to host the di\u000berent\nclusters reproduces the main features of Py.\nFor the di\u000berent clusters in this investigation we have\nestimated the local anisotropy from a well de\fned quan-\ntity { the orbital moment anisotropy, which is the di\u000ber-\nence of the orbital moment projection Lfor two di\u000berent\nglobal quantization axes \u0001 L=L^ n1\u0000L^ n2. Since \u0001Lis\nde\fned as a local quantity, it is numerically easy to evalu-\nate from \frst principles theory in contrast to the tiny en-\nergy di\u000berence needed for the MAE. It is established that\nthe energy di\u000berence between two states with the magne-\ntization direction along two di\u000berent global directions is\nEMAE =\u0000\u0018\n4\u0016B\u0001L6, where\u0018is the SOC constant. One\nof the key assumptions in deriving this relation is that\nspin diagonal matrix elements of the spin orbit coupling\nshould dominate the contribution to the MAE19. Since\nPy is a strong ferromagnet (the majority spin band is\nessentially \flled) only minority spin states contribute to\nthe density of states at the Fermi energy, and this cri-\nterion is expected to be ful\flled. When minority spin\nstates dominate the MAE, the easy axis is parallel to the\ndirection of maximum orbital magnetic moment. To ex-\nemplify the numerical advantage of the approach adopted\nhere, we note that values of 1 \u0016Ry for the MAE are re-\nlated to orbital anisotropies of 10\u00004\u0016B, which are values\nwell de\fned by the method's precision. Thus, it serves\nwell as the relevant quantity to evaluate and to quantify\nthe local anisotropies in alloys.\nResults { Before we discuss the results of the MAE, we\nnote that for all con\fgurations investigated here, the cal-\nculated individual moments were close to mFe\ns= 2:30\u0016B\nandmNi\ns= 0:61\u0016B, for spin, and LFe= 0:045\u0016Band\nLNi= 0:031\u0016Bfor orbital ones. These values are in\nagreement with previous theoretical17,18and experimen-\ntal5studies.\nFrom each cluster of our investigation, we estimated\nthe \u0001L, between two magnetization directions, for all\natoms. Therefore, the global direction ^ n1= [001]\nwas considered as reference and the orbital moment\nanisotropy computed as \u0001 L=L[001]\u0000L^ n2, with ^ n2=\n[110] and [111] directions. Comparing the \u0001 Lvalues one\ncan obtain the direction of maximum L, i.e. the local easy\naxis. This information is summarized in Fig. 2, which\nshows the distributions of easy axis directions per site.\nIn Fig. 2a we show results formed from an average over 8\ndi\u000berent clusters and in Fig.2b the average is made over\n84 di\u000berent con\fgurations. We note that the number of\ncon\fgurations favoring the [100] easy axis is larger than3\nthe [110] and [111] easy axis directions. We will return\nto this fact below.\n  \n0.20.40.60.81.0\n0.0\n0.20.40.60.81.0\n0.0a)\nb)\n 1   2   3  4   5   6  7    8    9  10  11  1213 14  1516 17  18 19\nsite8 configurations\n[001] [110] [111]84 configurationsprobability probability\nFIG. 2. (Color online) Likelyhood of di\u000berent easy axis di-\nrections for each of the 19 atomic sites considered for each\ncluster. Averages are formed from 8 con\fgurations (a) and\n84 con\fgurations (b). The easy axis direction are represented\nby the color bars.\nIn addition to these values of \u0001 Lwe also estimated the\nsite resolved E MAE, for each type of cluster. For that, we\nused the calculated SOC constants for \u0018Fe= 4:0 mRy\nand\u0018Ni= 6:7 mRy. The values of E MAE and the direc-\ntion of the easy axis for each atom are shown in Fig. 3\n(for sake of simplicity only 8 con\fgurations are shown).\nNote from the \fgure that we show local easy axis di-\nrections that in general is di\u000berent for each atom in a\ncluster, and sometimes even have di\u000berent local easy axis\ndirections. Furthermore, the 8 clusters considered in this\n\fgure all show rather di\u000berent behaviors when it comes\nto the MAE. For some of them, e.g., site 3 in one cluster\ncan have the [100] easy axis direction, but other con\fg-\nurations could for this site favor the [110] or the [111]\neasy axis direction. As is clear from the \fgure we \fnd\nvalues that are typically 5-10 times larger compared to\nthe values of bcc Fe ( \u00180:1\u0016Ry) or fcc Ni (\u00180:2\u0016Ry).20\nFurther, these local MAE values are, remarkably, orders\nof magnitude larger compared to the almost vanishing\nvalue of the MAE of bulk Py.\nA key point in the Fig. 3 is that the symmetry of each\ncluster is not cubic. Hence, spin-orbit e\u000bects enter as a\nlocal uniaxial anisotropy and it depends of second-order\nanisotropy terms instead of fourth-order as cubic envi-\nronments have. This is the primary reason why the local\nMAE values of Py are bigger than those of bcc Fe and\nfcc Ni.\nIt is interesting to compare the results of Fig. 2 and\n-6.0-5.0-4.0-3.0-2.0-1.00.01.02.03.04.05.06.0\n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19MAE (µRy)\nsite[001] [110] [111]FIG. 3. (Color online) MAE per site with the easy axis di-\nrection represented by squares (for [100]), circles (for [110])\nand triangles (for [111] direction). Negative values of MAE\nsymbolize an [001] easy axis. The 8 di\u000berent con\fgurations\nconsidered here are represented by di\u000berent colors. Note that\nsome data points are superposed, since the same MAE value\nis found for an atom placed in a given site of di\u000berent con\fg-\nurations.\nFig. 3 to recent supercell calculations for the MAE of\nFeCo based alloys21,22. There it is also found that the\nlocal anisotropy of various atomic con\fgurations varies\nstrongly, and even changes sign, while the alloy MAE\nis described by the average over many con\fgurations.\nThose systems are, however, very di\u000berent in that they\nhave a large MAE, meaning that one direction of magne-\ntization should be over-represented among di\u000berent clus-\nter con\fgurations. Py di\u000bers in that the MAE is vanish-\ningly small, meaning that there must be a balance from\ndi\u000berent local anisotropy contributions.\nDiscussion and Conclusion { We consider here the\nmagnetic anisotropy of a macroscopic sample as a con-\n\fgurational average of local anisotropies, for a diverse\ndistribution of clusters like the ones shown in Fig. 1c.\nEach cluster may have several atoms with large local\nanisotropies directed in any of the common crystallo-\ngraphic axes (h001i,h110iandh111i), but since the\ninter-atomic exchange interaction of Py (not shown here)\nis much stronger and ferromagnetic, the resulting mag-\nnetic con\fguration is a collinear ferromagnet, where, af-\nter a proper con\fgurational average is made, the result-\ning MAE is expected to be vanishingly small.\nIn the presented study, were investigated only clus-\nters with approximately the same concentration of Py\n(Fe0:2Ni0:8). In a real sample such constrain does not4\nexist, and con\fgurations involving, e.g., 1 Ni and 18 Fe\natoms and vice-verse must also be considered. Once a\nproper con\fgurational average of a huge set of clusters\nis considered, the proper macroscopic MAE can be ob-\ntained, and we suggest this leads to a vanishingly small\nMAE for Py.\nThe scenario proposed here is principally di\u000berent than\nsimply making a linear interpolation of anisotropy con-\nstants of bcc Fe and fcc Ni and adopting an interpolated\nvalue for all atoms of the alloy. We have shown that the\nlocal anisotropy is orders of magnitude larger than the\nobserved anisotropy in Py. We therefore argue that the\nvanishing anisotropy in bulk Py arises from the cancella-\ntion of these local anisotropies. It is likely that the sce-\nnario put forward here also applies to other magnetic pa-\nrameters, like the damping parameter or potentially the\nasymmetric exchange (like a local Dzyaloshinskii-Moriya\ninteraction). We also note that experiments showed\nthat amorphous materials present orbital induced mag-\nnetic anisotropy23,24explained by the random anisotropy\nmodel. Note that in amorphous materials the lack of\nsymmetry (chemical and crystalline) allows the emer-\ngence of orbital anisotropy.\nAs a \fnal comment, we note that the local anisotropy\ne\u000bects discussed here might a\u000bect the magnetization dy-\nnamics in thin \flms of Py25. For that, adopting a sce-\nnario of locally unique information, as proposed here,\nwould be relevant for the interpretation of pump-probe\nmeasurements and crucial to simulations involving an ef-\nfective spin-Hamiltonian.\nAcknowledgments { D.C.M.R thanks to D. Thonig\nfor insightful discussions. The authors acknowledge\nsupport from the Swedish Research Council (VR), the\nKAW foundation (grants 2012.0031 and 2013.0020),\nSTANDUPP and eSSENCE. A.B.K acknowledges sup-\nport from CNPq (Brazil). 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Kay, Applied\nPhysics A 49, 439 (1989)."    },    {        "title": "1702.06882v1.Missing_magnetism_in_Sr___4__Ru___3__O___10____Indication_for_Antisymmetric_Exchange_Interaction.pdf",        "content": "Missing magnetism in Sr 4Ru 3O10: Indication for Antisymmetric Exchange Interaction\nFranziska Weickert\u0003\nMPA-CMMS, Los Alamos National Laboratory, Los Alamos, NM 87544, USA and\nNHMFL, Florida State University, Tallahassee, FL 32310, USA\nLeonardo Civale, Boris Maiorov, and Marcelo Jaime\nMPA-CMMS, Los Alamos National Laboratory, Los Alamos, NM 87544, USA\nMyron B. Salamon\nUniversity of Texas at Dallas, Richardson, TX 75080, Dallas, USA\nEmanuela Carleschi and Andre M. Strydom\nDepartment of Physics, University of Johannesburg, Auckland Park 2006, South Africa\nR. Fittipaldi, V. Granata, and A. Vecchione\nCNR-SPIN Institute Sede Secondaria di Salerno and University of Salerno, Via Giovanni Paolo II, I-84084 Fisciano, Italy\n(Dated: November 28, 2021)\nWe report a detailed study of the magnetization modulus as a function of temperature and\napplied magnetic \feld under varying angle in Sr 4Ru3O10close to the metamagnetic transition at\nHcw2:5 T forH?c. We con\frm that the double-feature at Hcis robust without further splitting\nfor temperatures below 1.8 K down to 0.48 K. The metamagnetism in Sr 4Ru3O10is accompanied\nby a reduction of the magnetic moment in the plane of rotation and large \feld-hysteretic behavior.\nThe double anomaly shifts to higher \felds by rotating the \feld from H?ctoHkc. We compare\nour experimental \fndings with numerical simulations based on spin reorientation models caused by\nintrinsic magnetocrystalline anisotropy and Zeeman e\u000bect. Crystal anisotropy is able to explain\na metamagnetic transition in the ferromagnetic ordered system Sr 4Ru3O10, but a Dzyaloshinskii-\nMoriya term is crucial to account for a reduction of the magnetic moment as discovered in the\nexperiments.\nI. INTRODUCTION\nSr4Ru3O10belongs to the Ruddlesden-Popper family\nof ruthenium oxide perovskites Sr n+1RunO3n+1. This\nclass of metallic compounds caught much attention in\nrecent years due to its rich variety of ground states.\nSr2RuO 4then= 1 member, is discussed as an exam-\nple of rare p-wave superconductivity.1A quantum criti-\ncal endpoint covered by a high entropy phase was found\nin then= 2 layer system Sr 3Ru2O7.2The compound\nSr4Ru3O10(n= 3) discussed here shows ferromagnetism\nbelowTC= 105K.3Neutron di\u000braction experiments in\nzero magnetic \feld reveal ordering of the Ru moments\nalong thec-axis. No ferromagnetic (FM) or antiferromag-\nnetic (AFM) correlations are observed in the ab-plane.4,5\nSr4Ru3O10contains four inequivalent Ru sites with two\ndi\u000berent magnetic moments of 0.9 \u0016Band 1.5\u0016Bsitting\non outer and inner RuO layers, respectively. The mag-\nnetic unit cell contains 8 of the smaller and 4 of the big-\nger magnetic moments averaging to 1.1 \u0016Bper Ru. The\nhigher order ruthenate SrRuO 3withn=1orders also\nFM at a Curie temperature of 165 K.6,7. Srn+1RunO3n+1\nare strongly 2-dimensional electron systems with the\ntrend to become more isotropic for higher n, because of\ntheir layered structure. Two-dimensionality is re\rected\nin anisotropic transport properties as seen for Sr 4Ru3O10\nin the ratio of the electrical resistivity \u001ac=\u001aab'4008and\ncon\frmed by optical conductivity experiments.9Metamagnetism is a phenomenon observed in magnetic\nmaterials, where hidden magnetism is suddenly uncov-\nered by the application of an external \feld. The origin\nof metamagnetism ranges from spin \rip transitions in\nantiferromagnets,10,11to the reconstruction of the Fermi\nsurface in metallic materials,12and in the vicinity of a\nquantum critical point.13In a general description, meta-\nmagnetism is a phase transition or crossover from a mag-\nnetically disordered or ordered state with small net mag-\nnetization to a \feld polarized (FP) or partially FP state.\nIn the case of Sr 4Ru3O10the term metamagnetism refers\nto the sudden increase in the magnetization when the\n\feld applied in the ( ab) in-plane of this layered com-\npound exceeds 2 T. Magnetism is hidden only because\nthe spontaneous moment is mainly aligned with the easy\nc-axis at smaller \felds; we continue to refer to this as\nmetamagnetic (MM) transition.\nWhile metamagnetism of H?cin Sr 4Ru3O10was\ndiscovered from early on in \rux grown single crystals,3it\ntook more than a decade to improve the crystal quality\nto a level to see a double step in the magnetization at\nthe MM transition.14This strong dependence of physical\nproperties on the crystal purity is a characteristic sig-\nnature of strontium ruthenates Sr n+1RunO3n+1and was\nalso observed in the sister compound Sr 3Ru2O7.2,15{17\nThe MM transition in Sr 4Ru3O10develops below 68 K\nas a double-transition close to zero \feld and shifts grad-\nually toHc\u00182:5 T with temperatures down to 1.7 K.14arXiv:1702.06882v1  [cond-mat.str-el]  22 Feb 20172\nCarleschi et al.14speculate that the double transition\noriginates either in the ordering of Ru magnetic mo-\nments on two inequivalent crystallographic sites or in\nthe presence of two van Hove singularities in the den-\nsity of states close to the Fermi level. A transport study\nbased on electrical resistivity18,19reveals steps in the\nmagnetoresistance at various critical \felds around Hc\naccompanied by pronounced hysteresis. Fobes et al.19\ninterpret the transport data as domain movement of\nregions with high and low electronic spin polarization.\nAnomalous behavior at the MM transition was also ob-\nserved in speci\fc heat experiments up to 9 T20and in\nthermopower investigations.21Neutron di\u000braction exper-\niments up to 6 T reveal a change of lattice parameters at\nthe critical \feld Hc.4Field and pressure dependent Ra-\nman measurements22as well as a recent study of thermal\nexpansion and magnetostriction23con\frm strong magne-\ntoelastic coupling in Sr 4Ru3O10.\nIn this work we focus on magnetization measurements\nup to 7 T and down to lowest temperatures of 0.46 K\nand under rotational \felds between the c-axis and the\nab-plane as well as ( ab) in-plane rotation. Our inves-\ntigations include a detailed analysis of the behavior of\nthe magnetization modulus Mat the MM transition and\nits individual components MabandMcsimultaneously.\nIn the following, we analyze and interpret our data in\na localized picture, meaning the magnetic moments are\nmainly con\fned on the Ru4+sites in the crystal struc-\nture of Sr 4Ru3O10. This scenario is supported by neutron\ndi\u000braction experiments which have determined the spin\nand orbital momentum distribution in great detail.4,5\nOur main discovery is the observation of a reduced mea-\nsured moment at the MM transition caused by a spin\ncomponent pointing out of the rotational plane which we\nassert can best be explained by signi\fcant anisotropic\nexchange interactions in Sr 4Ru3O10.\nII. RESULTS\nAt \frst, we focus on the magnetization measured for\nH?cat temperatures below 2 K. Figure 1 shows\n\u001fAC=@M=@H between 0.5 T and 3.5 T. We observe a\nclear double MM phase transition with a main anomaly\natHc1= 2:3 T and a second anomaly at Hc2= 2:8 T\nfor increasing \feld as observed by Carleschi et al. .14Our\nnew experimental data down to 0.46 K clarify that nei-\nther transition sharpens to lower temperatures nor is\nthere splitting into more distinct anomalies. The inset in\nFig. 1 shows the H\u0000Tphase diagram with near-vertical\nphase boundaries for T!0 at the MM transition. Both\nanomalies are shifted by \u00000:3 T for measurements in de-\ncreasing magnetic \feld. The size of the hysteretic region,\nmarked as striped pattern, remains similar for all tem-\nperatures below 1.8 K.\nThe operation mode of the SQUID magnetometer al-\nlows the simultaneous collection of longitudinal Mlong\nand transversal component Mtrans of the sample mag-\nFIG. 1. Field derivative of the magnetization @M=@H at\n1.8 K, 0.65 K and 0.46 K. Solid lines show measurements dur-\ning increasing \feld sweeps and dashed lines during decreasing\n\feld sweeps as labeled by arrows. The inset shows H\u0000T\nphase diagram close to the MM transition with regions of\nhysteresis marked as striped patterns.\nFIG. 2. The inset shows a geometrical sketch of the sample\nSr4Ru3O10mounted inside the SQUID magnetometer.  is\nthe rotation angle of the applied \feld ~Hand\u0012the angle of\nthe magnetization ~M, both in respect to the magnetic easy\nc-axis.Mlong andMtrans are measured components of ~M\nparallel and perpendicular to the applied \feld in the plane\nof rotation. The component Mperpparallel to the axis of ro-\ntation is not captured during the measurement. The main\npanels compares the di\u000berent components of the magnetiza-\ntionMlong,Mtrans,Mab,Mc, and the modulus Mrotversus\nmagnetic \feld Hmeasured at 1.8 K for  = 81:6\u000e.\nnetization in respect to the applied magnetic \feld ~H\nas sketched in the inset of Fig. 2. The magnetization\ncomponent Mperp occurring perpendicular to the rota-\ntional plane is not recorded during the measurements.\nThis geometry allows us to calculate the magnetiza-\ntionM2\nrot=M2\nlong+M2\ntrans in the rotational plane.\nThe knowledge of the rotation angle  and relation\ntan( \u0000\u0012) =Mtrans=Mlongenables the determination3\nFIG. 3.ab-plane magnetization Mabas a function of Hab=\nHsin is shown for angles  between 89\u000eand 12.6\u000emeasured\nat 1.8 K.\nof the angle of the magnetization \u0012with respect to the\nmagnetic easy axis cin the rotational plane. We can\nnow estimate Mab=Mrotsin\u0012andMc=Mrotcos \nmagnetization, again occurring in the plane of rotation.\nNote, we follow closely the notation of angles used for\nmagnetic anisotropic materials. Figure 2 illustrates the\ndi\u000berent components of the magnetization for one par-\nticular measurement with  = 81:6\u000etaken at 1.8 K.\nProminent feature is the hysteresis loop around \u00061 T,\ncaused by FM domain dynamics. The longitudinal mag-\nnetizationMlongincreases moderately in small \felds and\nshows a sudden rise at Hcw3 T at the MM transition.\nMtrans on the other hand consists mainly of the Mccom-\nponent with a sudden decrease of the magnetization at\nthe same critical \feld Hc.Mtrans6= 0 above Hcindi-\ncates incomplete \feld polarization meaning that ~Mis\nnot perfectly aligned with ~H. This observation points\nto the presence of magnetic anisotropy. The calculated\nmagnetization modulus Mrotin the plane of rotation is\ndepicted as black line in Fig. 2. We \fnd a maximum\nmoment of 1 :5\u0016Bslightly higher than obtained in neu-\ntron experiments,4,5but in good agreement with previous\nmagnetization studies.24,25Most peculiar is that Mrot\ndrops suddenly below 1.2 \u0016Bat the MM transition and\nonly recovers partially to 1.2 \u0016Bup to maximum applied\n\feld of 7 T. This missing component of the magnetic mo-\nment in Sr 4Ru3O10was never recognized before. Fur-\nthermore, we observe strong hysteresis at the MM tran-\nsition between up and down measurements as reported\nin previous investigations.8,14,19,24,25\nGeometrical e\u000bects can distort magnetic properties\nduring magnetization experiments. Therefore, we plot\nin Fig. 3Mabas a function of the \feld component in the\nab-planeHab=Hsin to examine how Hc1;2change\nwith . In contrast to previous results by Jo et al.26\nobtained by torque magnetometry, we observe a clear si-\nmultaneous increase of both critical \felds Hc1;2to higher\nvalues while rotating from H?ctoHkc. In fact,\nFIG. 4. Angular  dependent shift of the double anomaly at\nthe MM transition in Sr 4Ru3O10. Solid points mark positions\nfor increasing and empty points decreasing \feld sweeps. The\ndotted line is a quadratic \ft to the data. The inset shows the\nreduction of the magnetization step \u0001 Mabat the MM tran-\nsition as a function of  including a quadratic extrapolation\nmarked as solid line.\nFIG. 5. Magnetization modulus Mrotin the rotational plane\nversus magnetic \feld Hfor angles between 85.3\u000eand 70.5\u000e\nin decreasing \felds. Arrows mark anomalies at Hc1andHc2\nfor the measurement at  = 85:3\u000e.\nHc1;2move out of the observable \feld range of 7 T max-\nimum for .72\u000e. Fig. 4 summarizes the critical \felds\nHc1;2in theH\u0000 phase diagram for \feld up and down\nsweep measurements. The di\u000berence Hc1\u0000Hc2increases\nslightly with smaller  . As mentioned above, the double\nanomaly is accompanied by signi\fcant hysteresis. The\ninset of Fig. 4 shows the evolution of combined step size\n\u0001Mabof both MM transitions for decreasing  . It fol-\nlows a quadratic \ft function marked as solid line and\nextrapolates to zero at about 65\u000e.\nThe magnetization modulus Mrotrecorded in the plane\nof rotation for  between 85.3\u000eand 70.5\u000eis plotted in\nFig. 5. We only show \feld-down sweep measurements for4\nclarity. Striking is the occurrence of a drop from about\n1.5\u0016Bto below 1.2 \u0016Bat the critical \feld of the main\nanomalyHc1followed by a minimum and a small step\nat the second anomaly at Hc2. The described features\nare marked in Fig. 5 by arrows for the measurement at\n = 85:3\u000e. The MM anomaly broadens and moves to\nhigher \felds for decreasing angles  .\nIII. DISCUSSION\nThe \"loss\" of magnetic moment in the rotational plane\ncan be explained either by partial AFM alignment or by\na momentMperpoccurring perpendicular to the rotation\n(parallel to rotation axis of  ). First scenario can be\nexcluded based on neutron experiments where no short\nor long range AFM coupling neither in zero nor in mag-\nnetic \felds H >H cwas observed in the ab-plane.4,5The\nsecond scenario is rather unexpected since magnetic mo-\nments tend to align with \feld and stay within the rota-\ntional plane, if no further coupling is present. We want\nto focus in our discussion on two mechanisms that po-\ntentially lead to a Mperpcomponent in the magnetiza-\ntion. First one is based on general magnetocrystalline\nanisotropy in tetragonal symmetry, with an easy c-axis\nand 4-fold in-plane anisotropy. The second mechanism\nis antisymmetric exchange between spins, also called\nDzyaloshinskii-Moriya (DM) interaction, causing a cant-\ning of the spins ~Si\u0002~Sj.\nA. General Anisotropy\nWe have to have a closer look at the crystal struc-\nture of Sr 4Ru3O10in order to understand and model\nits magnetic anisotropy caused by spin-orbit coupling.\nSr4Ru3O10crystals consist of three layers of corner shar-\ning RuO 6octahedra separated by a double layer of\nSr-O. Primary Bragg re\rections in synchrotron experi-\nments can be indexed assuming a tetragonal unit cell\nwith space-group I4=mmm , but a more detailed analy-\nsis of secondary re\rections reveals orthorhombic Pbam\nsymmetry.3The lower symmetry originates in c-axis ro-\ntation of the RuO 6octahedra that are correlated between\ndi\u000berent layers, meaning +11 :2\u000eclockwise rotation for in-\nner and\u00005:6\u000ecounterclockwise rotation for outer layers.\nThe free energy Faccounting for magnetocrystalline\nanisotropy in a tetragonal lattice, can be modeled27by\nF=F0+K1cos2\u0012+ (K2+K3sin(2')) sin4\u0012\u0000FZ:(1)\nF0is a constant background contribution independent of\n~Hor~M.~Mis expressed in polar coordinates ( \u0012;'), with\n\u0012= 0 along the crystal c-axis and'= 0 de\fning the in-\nplane hard axis for K3<0. Here,K1<0 de\fnes the\neasy direction and the Zeeman term FZcan be written\nFIG. 6. The experimental magnetization moduli Mrotfor 3\ndi\u000berent angles  (dots) measured at 1.8 K in decreasing mag-\nnetic \feld are compared with numerical simulations (lines) of\nthe general anisotropy model as described by Eq. (1).\nas\nFZ=MH(sin\u0012sin (cos!cos'+sin!sin')+cos\u0012cos ):\n(2)\nThe applied \feld has the polar coordinates (  ;!), with\n!= 0 corresponding to \feld rotation from the c- axis to\nthe in-plane hard direction and !=\u0019=4, \feld rotation\nto the easy direction.\nWe used numerical minimization of equation (1) to de-\ntermine\u0012and'as functions of the applied \feld. In the\nuniaxial case ( K3= 0), the MM behavior in Maband\nMcat the critical \feld Hcis reproduced by choosing cor-\nrect parameters K1andK2(data not shown). However,\nuniaxial anisotropy is unable to reproduce any reduction\nof the magnetization in Mrot, since the moment always\nstays in the rotational plane.\nNote, we do not know precisely the in-plane orientation\nof our sample. However, the rectangular shape suggests\nthat rotation axis is parallel to one of the principal axes\nsuch as [100] or [110]. For 4-fold tetragonal symmetry ei-\nther one of them would be the intermediate or hard axis,\nrespectively. We consider in the following a projection\nof~Honto the magnetic hard axis in the ab-plane with\n!= 0, because tilting of ~Mtowards the hard axis forces\nthe magnetic moments to align spontaneously toward ei-\nther one of the intermediate axes, which are 45\u000eapart\nfrom the hard axis. This spontaneous alignment \u000645\u000e\nis energetically degenerated and could lead to domain\nformation with an overall smaller net magnetization as\nobserved in our measurements.\nFig. 6 compares numerical results of  = 78\u000e;81\u000e;84\u000e\nbased on equation (1) with experimental data of the mag-\nnetization modulus Mfor = 77:9\u000e;81:6\u000e;83:4\u000e. We\nare able to reproduce i) a critical \feld value Hcw2:5 T\nthat increases with smaller  , ii) a drop \u0001 MatHc\nthat is comparable in size with the experimental data,\nand iii) a gradual slope M(H) forH > H c. The dou-5\nFIG. 7. Decreasing \feld Hmeasurements of the magnetiza-\ntion modulus Mtaken at 1.8 T at three angles  w87\u000e;82\u000e,\nand 71\u000eare shown for two di\u000berent in-plane angles != 0\nand 45\u000e. The almost identical results for 0 and 45\u000edisable\nmagnetocrystalline anisotropy being the cause for the loss of\nmagnetic moment in Sr 4Ru3O10.\nble feature at the MM transition is missing due to the\nsimplicity of the model. We obtain anisotropy param-\netersK1= 3:1 K,K2= 0:1 K andK3=\u00002:2 K in\nKelvin energy scale which convert to the following val-\nues 300 kJ/m3, 10 kJ/m3, and -210 kJ/m3, respectively,\nin units widely used in magnetic anisotropy tables. The\n4th order parameter K2being more than 10 times smaller\nthanK1implies that it is irrelevant for the description of\nthe anisotropy in Sr 4Ru3O10. For comparison, the 3 dFM\nmetal cobalt has anisotropy constants of K1= 450 kJ/m3\nandK2= 150 kJ/m3, which are of similar size as K1in\nSr4Ru3O10.28\nDespite the reasonable agreement between experiment\nand model, it is necessary to check in a subsequent exper-\niment our initial assumption of tilting the spins towards\nthe magnetic hard axis in the ab-plane. Therefore, we\nrotate the sample by != 45\u000ein the plane and measure\nagainMat three di\u000berent angles  as shown in Fig. 7.\nWe anticipate the 45\u000echange would bring the intermedi-\nate anisotropy axis into the rotation plane. Speci\fcally,\nthe magnetization would rotate toward the intermediate\naxis with the total magnetization remaining in the rota-\ntion plane and therefore no \"loss\" of magnetic moment\ne\u000bect. Surprisingly, the magnetization Mrotshows ex-\nactly the same behavior as for the != 0 experiments\nwithin experimental uncertainty. Even if in both experi-\nments!= 0 and 45\u000e, the plane of  -rotation would not\ninclude exactly the principal axes, we would at least ex-\npect the observation of a reduced anomaly in MrotatHc.\nBased on our last \fnding, we exclude general magnetic\nanisotropy as sole cause for the reduction of moment at\nthe MM transition in Sr 4Ru3O10.\nFIG. 8. Experimental magnetization modulus Mrotin the\nplane of rotation as a function of magnetic \feld \u00160Hup to\n7 T (\u0004) and numerical results including tetragonal magne-\ntocrystalline anisotropy and a DM alike energy term (broken\nline) for w82\u000e.\nB. Antisymmetric Exchange Coupling\nAnisotropic exchange interactions caused by spin-orbit\ncoupling under certain symmetry constraints were \frst\nconsidered by Dzyaloshinsky and Moriya29,30to explain\nweak FM ordering inside an AFM phase in transition\nmetal oxides. Recently, Bellaiche et al.31pointed out\nthat tilting of oxygen octahedra in perovskites can be\ndescribed by a pseudo-vector \u0017iat spinSipositionithat\nleads to an energy reduction\n\u0001E=KX\ni;j(\u0017i\u0000\u0017j)\u0001(~Si\u0002~Sj) (3)\nin analogy to DM antisymmetric exchange coupling. The\nsummation is done over nearest-neighboring spins ~Si;j\nandKis a constant. Consequently, we approximate a\nDM interaction between the in-plane component of the\nRu center magnetization and that of the two nearest-\nneighbor Ru atoms, whose spins are assumed to remain\nparallel. This gives rise to an e\u000bective energy term\nFDM=\u0000Dsin2(\u0012) sin(2') (4)\nin replacement of the K3term of the magnetocrystalline\nanisotropy. Angle 'is interpreted as the angle between\nin-plane magnetic moments sitting on one inner and two\nouter layers with '= 0 being the direction of in-plane\nmagnetic \feld. The change in the dependence on angle '\nbetween the magnetization vector and the c-axis prevents\nan accurate minimization of the total energy. Nonethe-\nless, the approximate solution shown in Fig. 8 for  = 82\u000e\ndoes produce a \"missing\" portion of the total magneti-\nzation, with the minimum moving to higher \feld with\ndecreasing  . The parameter Dis approximately 5.3 K\nand comparable to the energy scale of the magnetocrys-\ntalline anisotropy K1.6\nWe want to point out that the opening of 2 'be-\ntween the inner and outer magnetic moments can be\nseen as AFM order if it occurs periodically along the\nc-axis. Neutron scattering experiments on holmium-\nyttrium superlattices32e.g., were able to distinguish be-\ntween di\u000berent types of AFM periodicity along [00 l]\nwith increasing in-plane magnetic \feld, such as helical,\nhelifan-shaped, fan-shaped and FM order. However, the\nparticular crystal structure of Sr 4Ru3O10with three lay-\ners of RuO 6octahedra connected through a double layer\nof Sr-O doesn't give rise to additional periodicity, even\nif the magnetic moments follow a repetitive pattern such\nas\u0000';+';\u0000'alongcinside the triple layer. Consis-\ntently, in-plane AFM ordering was excluded by neutron\nscattering studies.4\nIV. SUMMARY\nIn summary, the MM transition in Sr 4Ru3O10as a\nfunction of magnetic \feld and rotation angle between\nH?candHkchas been studied in great detail by mag-\nnetization measurements down to lowest temperatures in\na SQUID magnetometer. Our experimental results re-\nveal a reduced magnetic moment in the plane of rotation\nwhich was never recognized before. It is robust to ( ab) in-\nplane rotation. We \fnd furthermore that the double step\nat the MM transition is stable down to lowest tempera-\ntures of 0.46 K. Our experimental results are interpreted\nin a strict localized picture with magnetic moments of dif-\nferent sizes sitting on inner and outer RuO 6layers in the\ncrystal structure. We completed our study with numeri-\ncal calculations based on energy minimization including\nZeeman e\u000bect, magnetocrystalline anisotropy and anti-\nsymmetric exchange and compared them with the exper-\nimental data. We conclude that all three contributions\nare essential ingredients to understand the magnetization\nbehavior and establish that a Dzyaloshinsky and Moriya\nlike component is essential to understand the occurrenceof a reduced magnetic moment in Sr 4Ru3O10.\nV. METHODS\nSr4Ru3O10single crystals were grown in an image fur-\nnace by a \roating zone technique33and characterized\nby energy dispersive spectroscopy, scanning electron mi-\ncroscopy, electron backscattering di\u000braction and x-ray\ndi\u000braction techniques.\nMagnetization measurements in \felds up to 7 T were\ncarried out in a Quantum Design MPMS SQUID mag-\nnetometer equipped with a standard4He setup for mea-\nsurements between 1.7 K and 100 K and in an iQuantum\n3He insert that \fts inside the MPMS sample space for\nthe temperature range 0.46 K to 2 K. Excellent agree-\nment between the3He and4He data was observed in\nthe temperature range of overlap. Angular dependent\nmeasurements at 1.8 K were obtained with a mechanical\nrotator mounted inside the MPMS magnetometer. Note\nthat the MPMS operates with a pair of coils for signal de-\ntection mounted parallel (longitudinal coil) and perpen-\ndicular (transversal coil) to the applied magnetic \feld.\nThe rotator is aligned with the rotation axis normal to\nthe plane de\fned by both SQUID coil axes.\nThe measured single crystal has a rectangular shape\nof (1.87 x 2.99)mm2inaband 0.54 mm along the crys-\ntallographic c-direction. We considered demagnetiza-\ntion e\u000bects by approximating the sample shape with an\nellipsoid34and estimated small correction \felds of - 30\nmT forH?cand - 140 mT for Hkc.\nACKNOWLEDGMENTS\nWork at Los Alamos was supported by the National\nScience Foundation, Division of Material Research under\nGrant NSF-DMR-1157490 and the U.S. Department of\nEnergy, O\u000ece of Science, Basic Energy Sciences, Mate-\nrials Sciences and Engineering Division.\n\u0003weickert@lanl.gov\n1K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z. Q.\nMao, Y. Mori, and Y. 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Osborn, Physical Review 67, 351 (1945)."    },    {        "title": "1702.07222v2.Human_brain_ferritin_studied_by_muon_Spin_Rotation__a_pilot_study.pdf",        "content": "arXiv:1702.07222v2  [physics.bio-ph]  22 May 2017Human-brain ferritin studied by muon Spin\nRotation: a pilot study\nLucia Bossoni1, Laure Grand Moursel2,3, Marjolein\nBulk2,3,4, Brecht G. Simon1, Andrew Webb3, Louise van\nder Weerd2,3, Martina Huber1, Pietro Carretta5,\nAlessandro Lascialfari6, Tjerk H. Oosterkamp1\n1Huygens-Kamerlingh Onnes Laboratory, Leiden University, 2333 CA Leiden,\nThe Netherlands\n2Department of Human Genetics, Leiden University Medical Ce nter, Leiden,\nThe Netherlands\n3Department of Radiology, Leiden University Medical Center , Leiden, The\nNetherlands\n4Percuros BV, Leiden, The Netherlands\n5Department of Physics, Pavia University, Pavia, Italy\n6Dipartimento di Fisica, Universit` a Degli Studi di Milano, Milano, Italy\nE-mail:bossoni@physics.leidenuniv.nl\nAbstract. Muon Spin Rotation is employed to investigate the spin dynam ics of\nferritinproteins isolated fromthe brainofan Alzheimer’s disease (AD)patient and\nof a healthy control, using a sample of horse-spleen ferriti nas a reference. A model\nbased on the N´ eel theory of superparamagnetism is develope d in order to interpret\nthe spinrelaxation rate ofthe muons stopped by the core ofth e protein. Using this\nmodel, our preliminary observations show that ferritins fr om the healthy control\nare filled with a mineral compatible with ferrihydrite, whil e ferritins from the AD\npatient contain a crystalline phase with a larger magnetocr ystalline anisotropy,\npossibly compatible with magnetite or maghemite.\nPACS numbers: 75.20.-g, 75.30.Gw, 75.75.Jn,76.75.+i,87. 80.Lg\nKeywords : muon Spin Rotation, nanomagnetism, ferritin, Alzheimer’s disease\n1. Introduction\nFerritin is a protein that attracts much interest, not only because of its crucial role in\niron storageand ferroxidaseactivity [ 1,2], but alsobecause ofits magnetic properties.\nFerritin isananoscopichollowproteinmade ofashell (apoferritin)of molecularweight\n450kDa, containingacoreoftrivalentiron(Fe(III)) inthemineral formofferrihydrite,\nanano-crystalquiteelusivetoX-raydiffraction. Ferritinacquires Fe(II), catalyzesiron\noxidation, and induces mineralization within its cavity [ 3]. The outer diameter of the\nshell is 12 nm, regardless of the iron loading, whereas the iron core d iameter can vary\nbetween 2-3 nm and 7 nm [ 4], depending on the number of stored ions.\nIt is generally agreed that the ferritin core is antiferromagnetic (A FM) below a\ntemperature in the range of 340-500 K [ 4,5,6]. However, some AFM sublattices doHuman-brain ferritin studied by muon Spin Rotation: a pilot study 2\nnot cancel out completely due to the small particle size. This results in an excess of\nspin orientation, giving rise to a magnetic moment of 225-400 µB[7,4,8]. Because of\nits hexagonal crystal structure (space group P63mc), the particle possesses a unique\neasyaxisofmagnetization[ 9]. The”giant”magneticmomentoftheparticlecanrotate\nabout the crystal easy axis, if it overcomes an energy barrier Ea, which depends on\nthe volume of the particle Vand on the magnetocrystalline anisotropy constant, K\n[10]. If, upon decreasing the temperature, the dynamic time constan t of the moment\ncrossingthe barrier( τc) is greaterthan the measuringtime ofthe specific experimental\ntechnique ( τm), the magnetic moment is said to be blocked [ 11]. This formalism was\nintroduced by N´ eel to describe the magnetism of nanoscopic single -domain particles\nwith an internal magnetic order. The magnetic behaviour of an asse mbly of these\nultra-fine particles was termed superparamagnetism (SPM). This b locking occurs at\nabout 12 K, for ferritin, when τm∼100 s [12], in a DC magnetometry measurement.\nIn addition to the AFM and SPM phases, it was also proposed that a Cu rie-Weiss-like\nbehavior could be found at the core-shell interface, as a result of the reduced Weiss\nfield [7,10]. More recently, other models have been proposed, yet the exact spin\nstructure of the nanoparticle and its magnetic properties are still a matter of debate\n[7,13,4,8].\nFerritin has also been extensively studied by neuroscientists, due t o its central\nrole in cellular iron homeostasis. Ferritin is at the center of many deba tes concerning\niron toxicity in relation to neurodegeneration, and in particular to Alz heimer’s disease\n(AD). InthebrainofADpatients, irondis-regulationhasbeenrepo rted[14]. Thismay\nindicateamalfunctionofthestorageprotein[ 15], leadingtooxidativestressviaFenton\nand Haber-Weiss reactions [ 1,16,17,18]. Microscopy studies revealed the existence\nof ferritins with a poly-phasic structure: ferritins found in the bra in of AD patients\nseem to contain a higher amount of cubic crystalline phases consiste nt with magnetite\nand w¨ ustite [ 19,20], whereas the ’healthy type’-ferritins may be more abundant in the\nhexagonal ferrihydride phase. In this scenario, pathological fer ritin would be better\ndescribed by magnetoferritin, an artificial complex made of apofer ritin containing a\nmagnetite or maghemite crystal [ 21]. However, these observations were not confirmed\nby nuclear magnetic resonance (NMR) [ 22]. Since magnetoferritin carries a larger\nmagnetic moment than ferrihydrite, i.e. typically between 2000 µBand 9000 µB[23],\none would expect a 200-fold (or larger) increase in the longitudinal a nd transverse\nrelaxation rates of the water protons surrounding the protein. H owever, no significant\nenhancement of the relaxation rate was observed.\nAdditionally, Pan et al.showed that in human-liver ferritin, an increasing percentage\nof octahedrally coordinated Fe(III) migrated to tetrahedral sit es and was partially\nreducedtoFe(II),uponincreasingtheelectrondoseinelectronm icroscopyexperiments\n[24]. Although it is rather unlikely that such alterations would happen only on\nthe ’pathological’ ferritin [ 25], the poly-phasic composition of physiological versus\npathological ferritins remains a debated issue. In order to unrave l these controversies,\nhere we propose a muon Spin Rotation ( µSR) experiment, as an alternative\ninvestigation technique.\nµSR has been successfully employed in the past to study the magnetis m of fine-\nparticles systems. In particular, horse-spleen ferritin [ 26,27] and similar artificial\ncompounds [ 28] have shown a two-component relaxation of the muon Asymmetry:\nwhile the fast-decaying Asymmetry (exponential- or Kubo-Toyabe -like) reflects the\nstatic order/collective excitation and the superparamagnetism of the iron core\noccurring respectively below and above the blocking temperature ( TB), the slowHuman-brain ferritin studied by muon Spin Rotation: a pilot study 3\nexponential tail has been ascribed to the interaction ofthe muon s pin with the protons\nof the organic shell. However, a study of ferritin purified from huma n tissue has not\nbeen undertaken yet.\nIn this manuscript, we present a µSR study of a ferritin sample isolated from\nthe brain of an AD patient, and an age- and gender-matched health y control (HC).\nAs a third sample and reference, a lyophilized commercial horse-sple en ferritin sample\n(HoSF) was used. Firstly, we evaluate the feasibility of µSR on a human sample. Then\nwe propose a model to interpret the spin dynamics of the ferritin iro n core, associated\nwith the SPM effect. Secondly, we draw some preliminary conclusions o n the mineral\ncomposition of the protein core based on the magnetocrystalline an isotropy constant\nderived from our model, showing that all steps of the analysis are fe asible. Finally,\nwe discuss the limitations and the improvements that should be under taken in future\nstudies.\n2. Sample preparation and characterization\nOne freshly frozen human-brain hemisphere from an AD case (Braa k stage 6C, age\n90 yrs, female) and one from a healthy age- and gender-matched c ontrol (age 88\nyrs, female) were obtained from the Netherlands Brain Bank (NBB) of Amsterdam.\nPatient anonymity was strictly maintained and informed consent was obtained from\nall prospective donors by NBB in accordance with EU regulations. All tissue samples\nwere handled in a coded fashion, according to Dutch national ethica l guidelines (Code\nfor Proper Secondary Use of Human Tissue, Dutch Federation of M edical Scientific\nSocieties).\nBefore the tissue was processed for protein isolation, a section fr om the temporal\nregion was selected from both the AD and the control individual for an MRI study\n(see Supplementary Information). Ferritin used in the µSR experiment was isolated\nfrom 2/3 of the hemispheres of the two individuals by following a modifie d version of\nthe method reported by Cham et al. [29]. The original protocol requires only a few\nsteps and it is practicalwhen dealingwith a largeamountofbrain mate rial. Moreover,\nthis method retains most of the iron inside the protein, that is our ma in concern in\nthis experiment. One part of the brain was homogenized on ice with th ree parts of\nPhosphate-buffered saline solution (PBS), with a tissue homogenize r (Omni B style\nTH Motor 220 V, LA Biosystems). Subsequently, the nuclei and unb roken cells were\nremoved by low-speed centrifugation: 2000g for 15 min, at 4◦C. The supernatant was\nretained and put on ice again. This solution was further diluted with pu re methanol\n99.8%(J. T.Baker,productnum.: 8045.2500)inordertoreachafin alconcentrationof\n40 % v/v. This new solution was heated for 10minutes at 75◦C, and afterward put on\nice again, and finally centrifuged at 2000 g for 15 min, at 4◦C. The ferritin-containing\nsupernatant (final volume: 1.5-2 liters) was retained in order to be further purified,\ndesalted and concentrated with an Amicon Ultra-15 Centrifugal Filt er Unit with an\nUltracel-100 membrane (UFC910008) with a molecular cutoff of 100 k Da. Aliquots\nof the supernatants were assayed for ferritin by SDS gel electro phoresis, Western\nBlot, and Transmission Electron Microscopy. The remaining solution w as freeze-dried\nfor the magnetic characterization. In order to prevent contamin ation with magnetic\nmaterial, no metal tools or containers were used in the dissection an d handling of the\ntissue. Ceramic and plastic materials were used instead.\nSDS-gel electrophoresis was done in parallel to Western Blot analys is: the results\nshow the presence of both ferritin sub-bands in the human samples (Fig.1).Human-brain ferritin studied by muon Spin Rotation: a pilot study 4\nFigure 1: SDS Page gel and Western Blot of the human ferritin samples\ntogether with the horse-spleen ferritin. (left) SDS Page gel. The heavy ( ∼22\nkDa) and light ( ∼19 kDa) chains of human-brain ferritin are shown by the orange\narrows. AD refers to the Alzheimer’s patient, HC to the healthy con trol, and HoSF\nto the horse-spleen ferritin sample. (right) Western Blot of the sa me human sample\nshowing a bright green band corresponding to the heavy chain of fe rritin.\nOne part of the gel was stained to assess purity and protein conce ntration, while the\nother part was used for Western Blot to determine the presence a nd molecular weight\nof ferritin. Protein solutions were loaded on the lanes of a 4-20% Crit erion TGXTM\npre-cast gel purchased from Bio-Rad. As a reference, Precision Plus ProteinTMDual\nXtra Prestained Protein Standards (product num.: 1610377) and ferritin from equine\nspleen purchased from Sigma-Aldrich (productr num.: F4503) were employed. Horse-\nspleen ferritin was loaded as a reference sample in three different co ncentrations. The\ngel was run using a standard Laemmli sample buffer and Tris/glycine/ SDS running\nbuffer. The gel was stained with PageBlueTMprotein staining solution (ThermoFisher\nScientific, product num. 24620) and imaged with the Li-Cor OdysseyTMimaging\nsystem. The human ferritin concentration was estimated from the heavy and light\nchain bands: 227.3 µg/g for the AD sample and 210.5 µg/g for the HD. Taking into\naccount the solution volume after the concentration step, and th e powder obtained\nafter freeze-drying, we estimated that the ferritin concentrat ion loaded onto the µSR\nsample holder was 11 % and 14.3 % of the total sample mass for the AD a nd HC\nferritin, respectively. However, a spread of values in the concent ration of different\nbatches cannot be excluded.\nThe presence of ferritin was verified by Western Blot. Anti-human- ferritin goat\nantibody NBP1-06985 purchased from Novus Biologicals was used as a primary\nantibody. Fluorescent donkey antigoat antibody was used as a sec ondary antibody.\nAntibody solutions and blocking solutions were prepared with tris-bu ffered saline\nTween(TBST)andnon-fat5%powdermilk. Thenitrocellulosemembra newasimaged\nwith the same Li-Cor OdysseyTMimaging system.Human-brain ferritin studied by muon Spin Rotation: a pilot study 5\nIn order to confirm the presence of the ferritin iron core, we perf ormed\nTransmission Electron Microscopy (TEM) on the purified protein solu tion. TEM\nimages were acquired using a JEOL1010TransmissionElectron Micros cope, at 70 kV.\nA 10µl-solution containing human-brain ferritin was drop-cast on a carbo n coated\ncopper grid (200 mesh, Van Loenen Instruments) and let dry for a few minutes. The\nsolution that did not absorb into the grid was removed with dust-fre e paper. The\nferritin solution obtained after the isolation protocol was imaged be fore freeze-drying\n(Fig.2) and later the freeze-dried powder used for the µSRexperiment was re-\nsuspended inmilliQwaterandimagedwiththe sameprotocol. Noapprec iablechanges\nin the iron core size distribution, as a result of freeze-drying and µSR experiments,\nwere observed (data not shown). Fig. 2shows dark dots of electron-dense material\nindicating the iron-filled ferritin cores. The size distribution of the fe rritin particles,\nobtained by the analysis of several TEM micrographs, follows a log-n ormal curve.\nMoreover, our results suggest that the AD ferritins have a smaller core size, with a\nmean of 4.39 nm and a standard deviation of 2.17 nm, than the HC ferr itins which\nshow a mean-core size of 6.64 nm and a standard deviation of 3.11 nm. These values\nare reasonably in agreement with the literature [ 30]. No larger, i.e. several tens of\nnm, particles were found in the analyzed solution by TEM inspection.\nFreeze-dried powders of human-brain ferritin were characterize d with a 7 T-\nFigure 2: TEM study of the ferritin nanoparticles isolated from human -\nbrain. (A) TEM micrograph of the ferritin isolated from the Alzheimer’s pat ient. (B)\nHistogramrepresentingferritinsizedistributionfromtheAlzheimer ’spatient’sferritin.\n(C) TEM micrograph of the healthy control’s ferritin. (D) Histogram representing\nferritin size distribution from the healthy control’s ferritin. The solid line is a fit to\nthe log-normal distribution. Scale bars are 100 nm in panel (A) and 2 00 nm in panel\n(C).\nMPMS SQUID magnetometer. In these experiments, the static spin susceptibility\nis measured at 50 G after cooling the sample in two different ways. In t he Zero-\nField-Cooled (ZFC) case, the sample is cooled from room temperatur e, down to 2 K\nwithout applying a magnetic field. Then the field is increased to 50 G and the staticHuman-brain ferritin studied by muon Spin Rotation: a pilot study 6\nsusceptibility is measured. In contrast, in the Field-Cooled (FC) cas e, the sample is\ncooled with the field of 50 G applied continuously. The static spin susce ptibility ( χ)\nof the human ferritin samples is shown in Fig. 3. The ZFC χshows the typical peak\nof a superparamagnet after subtracting a Curie-Weiss trend fro m the data [ 31,12]\n(inset Fig. 3). The AD sample χshows a broad peak, centered around 8.8 ±0.9\nK, indicating a blocking temperature ( TB) smaller than that observed in the HoSF\nsample (12 ±1 K,χdata are shown elsewhere [ 32]).TBof the HC sample is the same\nas the HoSF sample. These values of TBsuggest that the average core size of the AD\nsample is smaller than the one of the healthy control and horse-sple en samples. This\nobservation is in agreement with the TEM characterization. Please n ote that the FC\ncurve does not show this peak, as 50 G is enough to block the superp aramagnet at\nlow temperatures. Furthermore, the presence of a peak in the ZF C curve in the range\nof∼9-12 K rules out the possibility that a substantial part of our sample contains\nmagnetite particles of tens of nm [ 33,34,35,32].\nFinally, the presence of iron in all the powder samples was confirmed b y Laser-\n1 10 1000510\n1 10 1000.00.30.60.9\n1 10 10002468\n1 10 1000.00.20.4\nZFC\nFC\nCWfitT(K)HC\nT(K)χ−χCW(µemu/gOe)\nχ−χCW(µemu/gOe)\nT(K)\nZFC\nFC\nCWfitχ(µemu/gOe)\nT(K)AD\nFigure 3: Static Spin Susceptibility of human ferritin, as a function of the\ntemperature. Zero-Field-Cooled (ZFC) and Field-Cooled (FC) experiments were\nperformed at 50 G. The left panel indicates the data of the Alzheime r’s (AD) patient’s\nferritin, while the right panel refers to the healthy control’s (HC) f erritin. The inset\nshows the ZFC curve, after correction for the Curie-Weiss fit (da shed line), where the\npeak due to the blocking of the superparamagnetic particles becom es visible. The\narrow shows the position of the peak.\nablation Inductively coupled Mass spectrometry.\n3. Results\nFerritin powders of different origin were loaded onto a kapton holder , with a thin\n(25µm) kapton window to allow the muons to reach the sample. Experiment s were\ncarried out in longitudinal geometry, both in Zero-Field and longitudin al field mode.\nThe experiments were performed at the GPS beamline of the Paul Sc herrer Institute,\nwhere a continuous wave (CW) muon beam is available. CW muons allow a h igh time\nresolution, which in turn enables strongly inhomogeneous internal fi elds to be probed.\nA forward and a backward detector were used at all times. The time -dependentHuman-brain ferritin studied by muon Spin Rotation: a pilot study 7\nAsymmetry A(t) function is calculated as:\nA(t) =F(t)−αB(t)\nF(t)+αB(t)(1)\nwhere F(t) and B(t) are the positron counts in the forward and ba ckward detector\nrespectively,while αisaninstrumentalparameterwhichcompensatesforthedifferenc e\nin efficiency between the two detectors. Our estimated αranged between 0.787 and\n0.859 and it was calibrated in a transverse field of 50 G, at high temper ature (170 K).\nData were bunched with the ’Constant error’ binning option of the W imda software,\nin which the bin length exponentially increases with time from the initial v alue so that\nthe counts per bin and the resulting error remained fixed. Data wer e fitted between\n0 and 6µs. Different fitting functions were tested, such as the Uemura func tion [36],\nvarious combinations of Lorentzian and Gaussian Kubo-Toyabe [ 37]. The dynamical\nKubo-Toyabeand Keren functions [ 38] were also tested, but they could not capture all\nthe features of the data, as the empirical model discussed in this m anuscript. Muon\nSpin Rotation raw data were processed with Wimda, and fitted with Wim da, Matlab\n2016a and OriginLab(R) Pro 2016.\n3.1. Horse-spleen ferritin Zero-Field Asymmetry\nBefore carrying out the µSR study on the human-brain ferritin, a sample of freeze-\ndried horse-spleen ferritin was investigated. Horse-spleen ferrit in was chosen because\nit is similar to human ferritin and therefore a good reference. Zero- FieldµSR on HoSF\nshows a change in the muon Asymmetry decay A(t) as a function of the temperature\n(Fig.4). We observed an initial Asymmetry ( A0) of 15 %, which equals 2/3 of the\nfull Asymmetry. This signal loss will be discussed later in this section.\nIn the Zero-Field measurement data, two regimes were identified. A t temperatures\nhigher than 20 K, the Asymmetry is well described by the sum of two c ontributions,\nnamely an exponential term, decaying in the first 300-500 ns, and a slower stretched-\nexponential component, evolving from a Gaussian at high temperat ure, to a single\nexponential at low temperature. The decay of A(t) was modeled by assuming that a\nfraction of muons probes the local magnetic field of the protein she ll (fshell), and the\nrest probes the core ( fcore):\nA(t) =A0(fcoree−σt+fshelle−(λt)β)+B (2)\nWe observed a fraction ratio offcore\nfshell∼1\n4. A similar value was found by Cristofolini\net al., who suggested that such a fraction may originate from the core/ shell mass\nratio [26]. If we consider the mean values for the diameter of the core (5-6 n m) and\nof the shell (12 nm), we would expect to observefcore\nfshell∼0.08−0.14, instead. The\ndiscrepancy was ascribed to conformational changes of the prot ein shell, as suggested\nfor example for freeze-dried proteins [ 39]. Note that the experimental ratio could be\ndifferent also because the muon implantation is not random.\nIn Eq.2,σrepresents the spin-lattice relaxation rate of the ’core muons’, an alogous\nto the NMR 1 /T1[40,41]. As it will be shown, the fast decaying component is weakly\ndependent on the application of a longitudinal magnetic field. Finally, λis the spin\nrelaxation rate of the muons probing the protein shell, and the stre tched exponent in\nEq.2can be associated with either a distribution of muon sites or with an an isotropic\nhyperfine coupling, leading to a distribution of relaxation rates.\nBelow 20 K, the Asymmetry shows an increase at times longer than ∼4µs (Fig.Human-brain ferritin studied by muon Spin Rotation: a pilot study 8\n0 1 2 3 4 5 6 7051015\n0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7\n0 1 2 3 4 5 6 7051015\n0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 70 5 10 15 20 2502468150K\nfitAsymmetry(%)\ntime(µs)50K\nfit\ntime(µs)30K\nfit\ntime(µs)\n14K\nfitAsymmetry(%)\ntime(µs)6K\nfit\ntime(µs)4K\nfit\ntime(µs)Bµ(G)\nT(K)\nFigure 4: Asymmetry decay of the horse-spleen ferritin at different\ntemperatures. The dots represent binned raw data acquired in Zero-Field and the\nsolid line is the fit accordingto Eqs. 2and3. The recoveryofthe Asymmetryat longer\ntimes is visible in the bottom row of plots. The A= 0 line is shown for reference.\nThe temperature is shown in each panel. The inset is the fitted intern al fieldBµ\nresponsible for the increase in the Asymmetry at long times. The line is a guide to\nthe eye. For further details on the fit, please refer to the Supplem entary Information.\n4), indicating the onset of a static magnetic phase. We note that in co ntrast to the\nwork of Cristofolini et al.on HoSF [ 26], here the Gaussian decay is not reached at\nany temperature below 20 K. The best fit to the data was obtained b y adding a slow\noscillating term, with a small amplitude ( fsm∼6%):\nA(t) =A0(fcoree−σt+fshelle−(λt)β+fsmcos(2πγBµ+φ))+B,(3)\nwhere the baseline was deduced from the high-temperature regime ,γ= 8.516×104\nrads−1G−1is the muon gyromagneticratio, Bµis the local field probed by the muons,\nφis a phase constant and fsmrepresents the fraction of ’shell muons’ probing static\nmagnetism. The internal field Bµof∼8 G, at low temperature (Fig. 4(inset)) can\nbe explained in terms of the local dipolar field produced by blocked iron magnetic\nmoments of 5 µB, at an average distance of 1.8 nm from the muon implantation site\n(See the Supplementary information for a more detailed description of the fitting).\nThis internal field is of the same order of magnitude as that derived b y Cristofolini et\nal.[26], on horse-spleen ferritin, at low temperature.\n3.2. Human-brain ferritin Zero-Field Asymmetry\nThe Asymmetry decay of the human-brain ferritin is remarkably simila r to that of\nHoSF, as far as the high-temperature limit is concerned (Fig. 5), therefore we fitted\nthe data with the high-temperature model of Eq. 2. However, by a more carefulHuman-brain ferritin studied by muon Spin Rotation: a pilot study 9\n0 1 2 3 4 5 6 705101520\n0 1 2 3 4 5 6 705101520\n0 1 2 3 4 5 6 705101520\n(a)Asymmetry(%)50K\nfit\ntime(µs)HoSF (b) (c)\n50K\nfit\ntime(µs)HC\n50K\nfit\ntime(µs)AD\nFigure 5: Asymmetry decay for the horse-spleen ferritin and human-br ain\nferritin at 50 K. Equation 2fits equally well the three data sets. Human data have\nbeen baseline corrected. Baseline = 2.1 for the healthy control (HC ) and 3.54 for the\nAlzheimer’s (AD) sample, respectively. The Asymmetry of the horse -spleen ferritin\n(a)is marked with a dashed line and reported in the two other panels (b)and(c), for\ncomparison. The dots represent binned raw data and the solid line is t he fit according\nto Eq.2. TheA= 0 line is shown for reference.\ninspection of the data, some differences can be observed between the human and the\nhorse data sets. The most remarkable are that the human ferritin Asymmetry does\nnot increase at long times and at low temperature, as described by E q.3. This\nmay be due to the lower ferritin concentration in the human sample. S econdly, the\nstretched exponent does not decrease below 1.4 in both human sam ples, showing that\nthe mono-exponential limit, in the slow terms, is never reached (Fig. 6). Finally, the\ninitial Asymmetry of the human data is smaller than in the HoSF data (F ig.5), as it\nwill be discussed in the next section.\nFig.6summarizes the best-fit parameters for the relaxation rates ( λ,σ) and stretched\nexponent ( β), asafunction ofthe temperature, forthe threesamples. The A symmetry\ndecay of human ferritins was described by a model with three fitting parameters, while\nthe ratio of the core-shell muon fraction was fixed. The baseline wa s also left free to\nvary, but it did not show significant variations with the temperature .\n3.3. Longitudinal µSR field measurements on human-brain and horse-spleen ferri tin\nLongitudinal-Field (LF) µSR on the horse and human ferritin show two effects in\nthe muon Asymmetry. Firstly, the application of a weak field enhance s the initial\nAsymmetry A0, asalsoobservedby otherauthorsin aHoSF sample[ 26,27]. Secondly,\nthe stretched exponential relaxation is partially quenched by the lo ngitudinal field\n(Fig.7). The loss of A0was earlier ascribed to muonium formation (a radioactive\nisotope of hydrogen) after the µ+captures an electron, as often occurs in organic\nmatter andin polymers[ 42,43]. In this case, the polarizationcanbe fully recoveredby\nthe applicationofa longitudinal magnetic field, largeenough to decou ple the hyperfine\ninteraction between the electron and the muon. Given the large amo unt of organic\nmaterial in our sample and the partial recovery of A0with the application of an LF,\nit is reasonable to assume that the signal loss at t= 0 is due to muonum formation.\nMoreover, since the human sample is about seven times more dilute th an the HoSF,\none may ascribe the larger loss of initial Asymmetry to a larger fract ion of otherHuman-brain ferritin studied by muon Spin Rotation: a pilot study 10\n1 10 1000.00.20.40.60.8\n1 10 1001.01.21.41.61.82.01 10 1000.00.20.40.60.8\n1 10 1001.01.21.41.61.82.01 10 1000.00.20.40.60.8\n1 10 1001.01.21.41.61.82.02.2\n1 10 100051015\n1 10 1000510152025\n1 10 1000510152025λ(µs-1)\nHoSF(a)β(d) (f) (e)(c) (b)\nAD\nT(K) T(K)HC\nT(K)σ(µs-1)(g) (h) (i)\nFigure 6: Best fitting parameters for the horse and human ferritin data sets.\nMuon spin relaxation rate λ, stretched exponential βand spin-lattice relaxation rate\nσobtained from the fit of different data sets: for HoSF, see panels (a),(d)and(g);\nfor the AD patient’s ferritin, see panels (b),(e)and(h); for the healthy control’s\nferritin (HC), see panels (c),(f)and(i). The error bars represent the standard error\nof the estimated coefficients.\nproteins in the human sample (Fig. 5).\nThe quenching of the stretched exponential relaxation by small fie lds is in agreement\nwith the hypothesis that this decay originates from the muons implan ting in the\nprotein shell, and probing weak dipolar fields of the proton nuclei. Mor eover, the fast\nexponential term is not affected by the LF, thus suggesting that in this case, the local\nfield is either static and strong (i.e. a few Tesla) or dynamic, as expec ted for the iron\ncore.\n4. Discussion\nAmongthedifferentmagneticresonancetechniques, muonSpin Rot ationbenefits from\nan almost 100 % spin-polarized muon beam, leading to a signal which is no t limited\nby the Boltzmann polarization. Since the spin probe (the muon itself) is implanted\ninto the sample, the technique is neither limited by the sensitivity nor b y the natural\nabundance of the internal probe, as it is in the case of NMR. Addition ally, the signal\nintensity is not dependent on the strength of the external magne tic field. In µSR,\nZero-Field experiments are possible and the muon current is so small (∼pA) that no\nor minor sample damage is expected.\nMoreover, muon Spin Rotation offers several advantages for the study of nanoscopic\nmagnetic systems: it directly probes the stray field produced by th e metallicHuman-brain ferritin studied by muon Spin Rotation: a pilot study 11\n0 1 2 3 4 5 6 705101520\n0 1 2 3 4 5 6 705101520\n0 1 2 3 4 5 6 705101520\n(c) (b)\n150K,100G\n150K,50G\n150K,0GAsymmetry(%)\ntime(µs)HoSF(a) AD\n1.8K,50G\n1.8K,0G\ntime(µs)HC\n50K,100G\n50K,0G\ntime(µs)\nFigure 7: Asymmetry decay for different applied longitudinal fields an d\ndifferent temperatures of horse and human ferritin. (a) Asymmetry measured\nat Zero-Field and increasing field on the horse-spleen ferritin (HoSF ) sample, at\nhigh temperature, showing the quenching of the stretched expon ential term. (b)\nAsymmetry measured at Zero-Field and 50 G on the Alzheimer’s (AD) h uman ferritin\nsample, at 1.8 K, showing that both the initial Asymmetry and the slow -decaying\nterm are affected by the increase of the longitudinal field. (c)Asymmetry measured\nat Zero-Field and 100 G on the human healthy control (HC) ferritin s ample, showing\na similar behavior, at 50 K. Data have been baseline corrected.\nnanoparticle, and it captures its spin dynamics in a broad dynamic ran ge (between\n10−12s and 10−5s). On the other hand, obtaining structural information from\nrelaxation rates still remains a challenge. In the next section, we pr opose a model\nto interpret the spin-lattice relaxation rate of the muon fraction s topping in the iron\ncore of the protein ( σ).\nThere are at least two channels for the spin relaxation of muons impla nting in\nferritin: (i) spin excitations of the AFM ordered lattice, which would a ppear at low\ntemperatures, in the formofapowerlaw[ 44,45] and(ii) N´ eelrelaxationofthe particle\ngiant moment, activated by thermal energy. The correlation time f or the latter spin\nfluctuations is typically described by the N´ eel-Arrhenius relation, f or non-interacting\nparticles [ 11]:\nτc(T,V) =τ0eEa/kBT(4)\nwhere 1/τ0is the attempt frequency, Ea=VK[46,47] is the energy barrier, and kB\nis the Boltzmann constant. Such jumps of the particle magnetic mom ent are random\nbetween opposite values, therefore the spin-spin correlation fun ction becomes [ 48]:\n< Sz(0)Sz(t)>=S2\n0e−t/τc(T,V)(5)\nHence, one can borrow the formalism from the NMR theory of the sp in-lattice\nrelaxation rate, for a mono-dispersed particle distribution [ 40]:\nσ(T,V)∝γ2< h2\n0>/integraldisplay\n< Sz(0)Sz(t)> e−iωLtdt\n=γ2< h2\n0>τc\nτ2cω2\nL+1(6)\nwhere< h2\n0>is the static rmsvalue of the hyperfine field, ωL=γBµand\nBµis the local field probed by the muons implanting in the core [ 49]. Since theHuman-brain ferritin studied by muon Spin Rotation: a pilot study 12\nferritins’ diameters are log-normally distributed, as observed by T EM, Eq. 6should\nbe integrated over the volume distribution:\nσ(T)∝/integraldisplayV1\nV0σ(T,V)exp(−(ln(V)−<ln(V)>)2\n2∆2)\n∆V√\n2πdV (7)\nwhere ∆ and <ln(V)>are respectively the standard deviation and the mean of the\nlog-normal distribution. In order to interpret σ, we choose τ0andKfrom literature\n/s40/s98/s41/s32/s40 /s115/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s40/s97/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121\n/s86/s111/s108/s117/s109/s101/s32/s40/s110/s109/s51\n/s41/s72/s67\n/s65/s68/s70/s114/s101/s113/s117/s101/s110/s99/s121\n/s86/s111/s108/s117/s109/s101/s32/s40/s110/s109/s51\n/s41/s32/s50/s46/s55\n/s75/s32/s61/s32/s49/s32\n/s84/s32/s40/s75/s41/s32/s48/s46/s55\n/s32/s49\n/s32/s49/s46/s53\n/s32/s50\n/s32/s50/s46/s53/s70/s101/s114/s114/s105/s104/s121/s100/s114/s105/s116/s101/s77/s97/s103/s110/s101/s116/s111/s102/s101/s114/s114/s105/s116/s105/s110\nFigure 8: Simulation of the spin-lattice relaxation rate σ. (a)Simulated spin-\nlattice relaxation rate, using the volume particle distributions taken from electron\nmicroscopy (TEM) data (inset). The simulated σfor the AD sample is shown in\nred and the HC in blue. Parameters used for the simulation are K= 2.3 K/nm3\n(ferrihydrite), τ0= 0.1 ps [ 21,50],Bµ= 8 G. The insets report the experimental\ndistributions of the core-particle size from TEM and fit according to the log-normal\ndistribution (solid curve). (b)σsimulation for different Kvalues compatible with\nferrihydrite (solid lines in shades of blue) and magnetoferritin (in sha des of red). Here\nonlyoneparticlevolumedistribution, compatiblewith the HC,isconside red.Kvalues\nused for the simulations are reported in units of K/nm3. For the magnetoferritin\nsimulations, the range of Kvalues is 2.7-7.7 K/nm3, in steps of 1 K/nm3. The\namplitude of the peaks is normalized.\nvalues for ferrihydrite (see Fig. 8and its caption).\nThe simulation in Fig. 8(a)shows that particles with a larger volume (as observed\nfor HC) display a peak in σat higher temperature, and that particles with a broader\nparticle-size distribution display a broader peak in σ. Fig.8(b)shows a positive\ncorrelation between the value of Kand the temperature of the peak in σ. Here,\ninstead of choosing one single value for K, we present a range of values, as reported in\nliterature. Indeed, Kdepends on several factors such as chemical composition, kind\nof magnetic ion, particle shape, amount of frustrated spins at the particle surface,\ndistribution of nucleation sites inside the core, and level of crystallin ity. Plausible\nvalues for K, in the case of the ferrihydrite mineral, range from 0.7 to 2.5 K/nm3\n[51,52,53], while for magnetoferritin, and magnetite nanoparticles they can b e found\nbetween 2.7 and 9.7 K/nm3or higher [ 54,21]. However, intermediate values of 2.97\nK/nm3have also been measured for HoSF [ 50]. In Fig. 8(b)the chosen Kvalues\nare limited to 7.7 K/nm3, for illustration purposes. Moreover, in all the following\nsimulations, Bµhas been fixed to 8 G. This assumption, based on the experimentalHuman-brain ferritin studied by muon Spin Rotation: a pilot study 13\nobservation on the HoSF data, may be an over-simplification in the ca se of the human\ndata.\nA comparison between the experimental and simulated σis shown in Fig. 9, for\nthe three investigated samples. All the samples show a broad peak in σ, in the\n/s49 /s49/s48 /s49/s48/s48/s48/s53/s49/s48/s49/s53\n/s49 /s49/s48 /s49/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s49 /s49/s48 /s49/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s84/s32/s40/s75/s41 /s84/s32/s40/s75/s41/s65/s68/s32/s32/s32/s40 /s115/s45/s49\n/s41/s72/s111/s83/s70 /s72/s67\n/s84/s32/s40/s75/s41/s40/s97/s41 /s40/s98/s41\n/s32/s32\n/s40/s99/s41/s32\nFigure 9: Comparison between simulation and experimental data of the\nspin-lattice relaxation rate σ. (a)Experimental σfor the horse-spleen ferritin\nand simulated σaccording to the model discussed in the text and K= 2.5 K/nm3\n(light blue curve) and 2.7 K /nm3(dark brown curve). (b)Experimental σfor AD\nferritin and simulated curve for K= 4.7 and 5.7 K/nm3(red solid curves). Here the\nAD particle size distribution is taken into account. (c)Experimental σfor HC ferritin\nand simulated curve for K= 2 K/nm3(blue curve).\nsame temperature range observed in other works on HoSF and its s ynthetic analogues\n[26,28]. In the case of HoSF, the acceptable values for Kfall between 2.5 and 2.7\nK/nm3(Fig.9(a)), which is in line with the literature findings for HoSF, as discussed\nabove. We note that Kmay be overestimated, if the actual mean particle size was\nlarger than considered here. The HC sample shows a broader peak c entered in a\ntemperature region compatible with K∼2 K/nm3, which suggests the presence\nof a single ferrihydrite phase (Fig. 9(c)), in agreement with the expectations for\nferrihydrite and possibly characterized by a lower degree of cryst allinity with respect\nto the HoSF. Finally, the AD sample shows a narrower peak in σ, which results from\na narrower particle-size distribution (Fig. 9(b)), and a significantly larger anisotropy\nconstant of K∼4.7−5.7 K/nm3, corresponding to a ∼63 % increase of the magnetic\nanisotropy, which is more than what can be expected from a differen ce in particles size\nbetween AD and HC [ 54]. This constant falls in the experimentally observed range of\nmagnetoanisotropy for magnetoferritin nanoparticles.\nIt is worth stressing that this is the first attempt to use muon Spin R otation to\ndetermine the mineralization form of the ferritin-iron core in human- brain ferritin,\nas a complementary tool to electron microscopy. Therefore, in or der to improve the\ncurrent data, and confirm our preliminary results, the human ferr itin concentration\nand purity should be improved. A possibility would be to add a further p urification\nstep, such as affinity chromatography, at the end of our protoco l. As many of these\nsteps can be further improved, future experiments are certainly feasible and will be\nsensitive enough to determine differences with greater accuracy.Human-brain ferritin studied by muon Spin Rotation: a pilot study 14\n5. Conclusions\nWe isolated ferritin from the brain of an AD patient and a healthy age- and\ngender-matched individual. The protein solution was characterized by biochemical\nand physical techniques assessing the protein concentration and verifying the\nsuperparamagnetic properties, and the presence of iron in the sa mple. The\ncharacterizationshowsthattheisolationprotocolneedsfurthe rimprovements,inorder\nto significantly increase the concentration and purity of the ferrit in solution.\nµSR wasthen used to probethe spin dynamics ofthe ironcoreofhuma n-brainferritin,\nas a function of the temperature. Our pilot experiment showed Asy mmetry spectra\nthat significantly resembled those of commercial and highly pure hor se-spleen ferritin,\nwhich wasused as a reference in this study. A broad peak in the spin- lattice relaxation\nrate of the muons stopping in the core allowed us to identify the block ing temperature\nof ferritin. We proposed a model to interpret the spin-lattice relax ation rate, based on\nthe N´ eel theory of superparamagnetism, and on the experiment al size distribution of\nthe obtained ferritin particles. The comparison between simulation a nd experimental\ndata gaveus an indication ofthe magnetocrystallineanisotropycon stant ofthe ferritin\niron core. Our analysis suggests that ferritin isolated from the con trol subject is in\nagreement with a mineral phase compatible with ’physiological’ ferrihy drite, while the\nferritin isolated from the Alzheimer’s patient contains an iron mineraliz ation form\nwith a larger Kconstant, in the range observed for magnetoferritin, when the s ize\ndistribution obtained from electron microscopy is taken into accoun t. However, to\ndraw further conclusions about the composition of ferritin in relatio n with AD, ferritin\nisolatedfromdifferent ADpatientsshouldbe investigated. Thispilot s tudyshowsthat\nµSR is a promising approach to the study of iron-loaded human-brain p roteins.\nAcknowledgements\nWe are grateful to M. de Wit, A. Amato and C. Baines for proving help during the\nmuon Spin Rotation experiment and for useful discussion. We thank L. Cristofolini\nand J. Wagenaar for fruitful discussions. G. Lamers, J. Willemse, L . van der Graaf,\nJ. Aarts, N. Lebedev and C. Koeleman for technical assistance du ring ferritin freeze-\ndrying and characterization. We thank W. Breimer for helping during the MRI data\nacquisition.\nThis work was supported by the Dutch Foundation for Fundamenta l Research on\nMatter (FOM), by the Netherlands Organization for Scientific Rese arch (NWO)\nthrough a VICI fellowship to T. H. O. and through the Nanofront Pr ogram. One\nof us (M. B.) was supported by the FP7 European Union Marie Curie IA PP Program,\nBRAINPATH, undergrantnumber612360. 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Gross1\n1 Max-Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle, Germany\n2 Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan\n3 Department of Physical Chemistry, Faculty of Chemistry and Pharmacy, So\fa University, 1126 So\fa, Bulgaria.\n4 Lehrstuhl f ur Theoretische Festk orperphysik, Staudtstr. 7-B2, 91058 Erlangen, Germany.\n5 Max Planck Institute for Chemical Physics of Solids,\nN othnitzer Strasse 40, 01187 Dresden, Germany and\n6 Department of Physics, Indian Institute of Technology, Roorkee, 247667 Uttarkhand, India\n(Dated: November 26, 2021)\nWith a view to the design of hard magnets without rare earths we explore the possibility of\nlarge magnetocrystalline anisotropy energies in Heusler compounds that are unstable with respect\nto a tetragonal distortion. We consider the Heusler compounds Fe 2YZ with Y = (Ni, Co, Pt), and\nCo2YZ with Y = (Ni, Fe, Pt) where, in both cases, Z = (Al, Ga, Ge, In, Sn). We \fnd that for the\nCo2NiZ, Co 2PtZ, and Fe 2PtZ families the cubic phase is always, at T= 0, unstable with respect\nto a tetragonal distortion, while, in contrast, for the Fe 2NiZ and Fe 2CoZ families this is the case\nfor only 2 compounds { Fe 2CoGe and Fe 2CoSn. For all compounds in which a tetragonal distortion\noccurs we calculate the MAE \fnding remarkably large values for the Pt containing Heuslers, but\nalso large values for a number of the other compounds (e.g. Co 2NiGa has an MAE of -2.11 MJ/m3).\nThe tendency to a tetragonal distortion we \fnd to be strongly correlated with a high density of\nstates at the Fermi level in the cubic phase. As a corollary to this fact we observe that upon\ndoping compounds for which the cubic structure is stable such that the Fermi level enters a region\nof high DOS, a tetragonal distortion is induced and a correspondingly large value of the MAE is\nthen observed.\nPACS numbers:\nI. INTRODUCTION\nUnderpinning a diverse range of modern technologies,\nfrom computer hard drives to wind turbines, are hard\nmagnets. These are magnets in which the local moments\nall preferentially align along a certain crystallographic\ndirection, and may be characterized by the energy dif-\nference with an unfavorable spatial direction, known as\nthe magnetocrystalline anisotropy energy (MAE). Evi-\ndently the local moments of such magnets are very stable\nand so make excellent permanent magnets, hence their\ncentral role in various technologies1. Current produc-\ntion of hard magnets relies on alloys of rare earth el-\nements, in particular neodymium and dysprosium, e.g.\nthe \\neodymium magnet\" Nd 2Fe14B. Such alloys, due to\nthe localized nature of the open shell f-electrons of the\nrare earths, possess a very large spin orbit coupling and,\nas a consequence, very high MAE values. However, as\nthe rare earths are both costly and highly polluting to\nextract from ore there is a current focus on the design\nof hard magnets without rare earths2{5. Magnetic mate-\nrials having a low crystal symmetry evidently possess a\nnatural spatial anisotropy, and this in turn can lead to\nvery large values of the MAE. Such low symmetry mag-\nnets therefore o\u000ber a promising design route towards the\nnext generation of hard magnets. Accurate calculation of\nthe MAE requires sophisticated and computationally ex-\npensive \frst principles calculations, making di\u000ecult the\nkind of high throughput search that might be expected to\nyield interesting high MAE materials. In this paper weshow that for a promising materials class - the Heusler al-\nloys - the density of the states at the Fermi level provides\na very good indicator of the propensity to distortion, and\ntherefore of the likelihood of \fnding a high MAE mate-\nrial within this class. The use of such material mark-\ners for high MAE can, we believe, signi\fcantly alleviate\nthe computation bottleneck preventing high throughput\nsearch.\nThe Heusler materials have attracted sustained atten-\ntion due both to their exceptional magnetic properties as\nwell as a huge variety of possible compounds that may\nbe experimentally realized6,7; reviews may be found in\nRefs. 8{11. These materials, which consist of 4 inter-\npenetrating face centred cubic lattices, often exhibit a\nsymmetry lowering structural transition to a tetragonal\nor hexagonal phase12{16, raising the possibility of a crys-\ntal symmetry lowering induced large MAE. For Mn rich\nHeusler alloys this has previously been explored13,17; here\nwe consider this possibility in the Heusler families Fe 2YZ\nwith Y = (Ni, Co, Pt), and Co 2YZ with Y = (Ni, Fe,\nPt) where, in both cases, Z = (Al, Ga, Ge, In, Sn).\nOur principle \fndings are that (i) the Co 2NiZ and\nCo2PtZ classes naturally distort to a tetragonal struc-\nture with c=avalues in the range 1.3-1.5; (ii) the Fe rich\nHeuslers generally do not distort, with the exceptions of\nFe2CoZ where Z = Ge or Sn and the Fe 2PtZ family; (iii)\nthis distortion can induce a very high MAE - of up to\n5 MJ/m3for the Pt containing Heuslers, comparable to\nthe best known transition metal magnet L1 0-FePt, and\nof up to 1 MJ/m3for the Co rich but Pt free Heuslers. InarXiv:1702.08150v1  [cond-mat.mtrl-sci]  27 Feb 20172\nCo2NiAl\nCo2NiGa\nCo2NiGe\nCo2NiIn\nCo2NiSn\nCo2FeAl\nCo2FeGa\nCo2FeGe\nCo2FeIn\nCo2FeSn\nFe2NiAl\nFe2NiGa\nFe2NiGe\nFe2NiIn\nFe2NiSn\nFe2CoAl\nFe2CoGa\nFe2CoGe\nFe2CoIn\nFe2CoSn123456789DOS at Fermi level (states/eV)Inverse cubic\nInverse tetragonal\nRegular cubic\nRegular tetragonal\nCo2PtAl\nCo2PtGa\nCo2PtGe\nCo2PtIn\nCo2PtSn\nFe2PtAl\nFe2PtGa\nFe2PtGe\nFe2PtIn\nFe2PtSn123456789\nFIG. 1: Calculated total density of states at the Fermi energy for the Heusler compounds Co 2NiZ, Co 2FeZ, Co 2PtZ, Fe 2NiZ,\nFe2CoZ and Fe 2PtZ where Z = Al, Ge, Ga, Sn, or In. In each case the structure is indicated by the caption; and the tetragonal\nphase is shown when it is the lowest energy. Evidently, the instability of the cubic phase strongly correlates to the density of\nstates at the Fermi energy. For almost all cases when DOS at Fermi level >4.5 states/eV (indicated by the dashed horizontal\nline) the cubic phase is unstable; two exceptions to this are indicated by arrows.\neach case where a distortion occurs the volume change is\nfound to be very small (a few percent at most), with the\nexception of Fe 2PtGe in which a 6% increase of volume\noccurs upon distortion.\nWe furthermore \fnd that this tendency to tetragonal\ndistortion strongly correlates to a rather simple material\ndescriptor, namely the density of states (DOS) at the\nFermi level. A high DOS favours tetragonal distortion\nand, on this basis, we consider the possibility of inducing\na tetragonal distortion by moderate doping (via a virtual\ncrystal approximation) that shifts the Fermi energy from\na low to a high DOS position. Consistent with the va-\nlidity of this material descriptor we \fnd that the Heusler\nalloys Co 2FeAl and Co 2FeSn - in which the Fermi energy\nlies far from and close to a high DOS region respectively\n- all spontaneously su\u000ber tetragonal distortion upon dop-\ning.\nII. CALCULATION DETAILS\nFor structural relaxation we use the Vienna ab ini-\ntio simulation package (VASP)18with projector aug-\nmented wave (PAW) pseudopotentials19, a plane-wave-\nbasis set energy cuto\u000b of 400 eV, and the Perdew-Burke-Ernzerhof (PBE) functional20. Reciprocal space integra-\ntion has been performed with a \u0000-centered Monkhorst-\nPack 10x10x10 mesh. The structural optimization has\nbeen converged to a tolerance of 10\u00005eV, whereas the\nMAE values were obtained with a tolerance of 10\u00007eV.\nAll calculations are performed in the presence of spin-\norbit coupling term. For the calculation of the MAE\nwe have also deployed the all-electron full-potential lin-\nearized augmented-plane wave (FP-LAPW) code Elk21.\nThe de\fnition of MAE adopted in this study is the fol-\nlowing:\nMAE = Etot\n[100]\u0000Etot\n[001]; (1)\nwhere Etot\n[100](Etot\n[001]) represents the total energy with spin\norientation in the [100] ([001]) direction. A positive value\nof the MAE therefore indicates that out-of-plane spin\ncon\fguration is energetically favourable, whereas a neg-\native one that the in-plane direction is favourable.\nThe Heusler structure is described by the X 2YZ gen-\neral formula, in which the species X and Y are transition\nmetal elements whereas the Z atom is p-orbital element\nwith metal character (from III or IV main groups). The\ncrystal structure consists of four inter-penetrating face3\n-2 0 2-8-4048DOS (states/eV)\n-2 0 2 -2 0 2-8-4048\n-2 0 2\nEnergy (eV)-8-4048DOS (states/eV)\n-2 0 2\nEnergy (eV)-2 0 2\nEnergy (eV)-8-4048Co2NiAl\nFe2NiGeCo2FeAl\nFe2CoGeCUBICTETRAGONAL\nCUBIC TETRAGONALCUBIC\nCUBIC\nFIG. 2: Calculated total density of states for Co 2NiAl,\nCo2FeAl, Fe 2NiGe and Fe 2CoGe in cubic and (where it is\nthe lowest energy structure) the tetragonal phase. The Fermi\nenergy is set to 0 and positive (negative) value of the DOS\nrepresents the minority (majority) spin projection.\ncentred cubic lattices and belongs to the 225 (Fm-3m)\nsymmetry group for the regular Heusler structure, and\n216 (F-43m) for the inverse Heusler; Wycko\u000b positions\nof the atoms are presented in Table. I.\n4a 4c 4b 4d\n(0,0,0) (1/4,1/4,1/4) (1/2,1/2,1/2) (3/4,3/4,3/4)\nRegular Z X Y X\nInverse Z Y X X\nTABLE I: Structural order of regular and inverse Heusler\nstructures\nIII. STRUCTURAL DISTORTION\nWe \frst consider the stability with respect to tetrag-\nonal distortion of the Heusler alloys X2Y Zin which the\nXsub-lattices are occupied by either Fe or Co, the Y\nsub-lattice by Fe, Co, Ni, or Pt, and Zsub-lattice by\nAl, Ge, Ga, Sn, or In. This represents 30 materials in\ntotal, of which 10 have been previously experimentally\nsynthesized; for details we refer the reader to Table I and\nII of Appendix A. In Fig. 1 we present the DOS at the\nFermi energy of each of these Heusler materials for both\nthe high symmetry cubic phase and, where it exists, the\ntetragonal structure. For the Co rich Heuslers Co 2NiZ\nand Co 2PtZ the high symmetry phase is always unstable\nwith respect to tetragonal distortion while, in contrast, in\nthe case of the Fe containing Heuslers the cubic phase is\ngenerally stable. There are two exceptions to this latter\nrule: Fe 2NiGe and Fe 2NiSn, and the Fe 2PtZ family. For\nall cases where the tetragonal phase is stable we \fnd the\nc=aratios in the range 1.3-1.5 with the high end c=ara-\n-505Cubic\nTetragonal\nCubic\nTetragonal\nCubic\nTetragonal\nCubic\nTetragonal-303DOS (states/eV/spin)eg\nt2g\n-303\n-2 -1,5 -1 -0,5 0 0,5 1 1,5 2\nEnergy (eV)-303Total DOS\nCo-1\nCo-2\nNiFIG. 3: Majority spin projected and as well as Co- t2gand\negprojected DOS for Co 2NiGe; the cubic majority spin DOS\nis assigned a positive value, and the tetragonal majority spin\nDOS a negative value. The reduction in spectral weight near\nthe Fermi energy that occurs due to the tetragonal distortion\nmay clearly be seen. As may be seen from the lower panels,\nthe redistribution of spectral weight upon tetragonal distor-\ntion consists primarily of (i) a reduction in the egpeak of\nCo-1 character and (ii) a reduction in the Co-2 egpeak at\nthe Fermi level. A similar picture is found for other Co 2NiZ\ncompounds as well as the Co 2PtZ compounds.\ntios found for the Fe 2YZ Heuslers in which Z is either Ge\nor Sn (curiously, as we will see, these are also the Heusler\ncompounds that have the desired positive MAE). Of the\nHeuslers in Fig. 1 that have been experimentally synthe-\nsized only one, Co 2NiGa, is observed in the tetragonal\nstructure, in agreement with our calculations; all others\nare found to be cubic, also in agreement with our cal-\nculations (with the exception of Fe 2NiGe for which we\npredict a tetragonal structure, this case will be discussed\nin detail below).\nFor each structure we have also determined whether\nthe material takes on a regular or inverse occupation of\nthe sub-lattices. As may be seen in Fig. 1 most of the\nstructures are inverse Heusler except for the Co 2FeZ fam-\nily where the regular cubic structure has a lower ground\nstate energy. This \fnding is in a good agreement with\nan empirical rule \frst stated in Ref. 10: when the elec-\ntronegativity of the Y element is larger than that of the\nX element the system prefers the inverse Heusler struc-\nture, with otherwise the regular Heusler structure real-\nized. There are, however, two deviations from this rule\nin our results. We \fnd that Fe 2NiGe and Fe 2NiSn adopt\na tetragonally distorted regular structure, in agreement\nwith previous theoretical work22, but in contrast to the\ninverse structure expected on the basis of the empirical\nrule (the electronegativity of Ni is higher than that of\nFe).\nIn Ref. 23 experiment reports, in agreement with\nthe semi-empirical rule, a cubic inverse structure for\nFe2NiGe. Accompanying theoretical calculations23, how-\never, \fnd that the energy change due to antisite disorder4\nCo2NiAl\nCo2NiGa\nCo2NiGe\nCo2NiIn\nCo2NiSn\nCo2FeAl\nCo2FeGa\nCo2FeGe\nCo2FeIn\nCo2FeSn\nFe2NiAl\nFe2NiGa\nFe2NiGe\nFe2NiIn\nFe2NiSn\nFe2CoAl\nFe2CoGa\nFe2CoGe\nFe2CoIn\nFe2CoSn02004006008001000Saturation moment (kA/m)Saturation moment\nCo2PtAl\nCo2PtGa\nCo2PtGe\nCo2PtIn\nCo2PtSn\nFe2PtAl\nFe2PtGa\nFe2PtGe\nFe2PtIn\nFe2PtSnMagnetocrystaline anisotropy\n-14-12-10-8-6-4-20246\nMAE (MJ/m3)1.07 1.09\n-11.33\n-`12.00-1.010.425.19\n2.07\n0.86\n-6.94-2.22\n-2.11\n-2.38-3.57\n-5.41\nFIG. 4: Magnetic moments and MAE for the Heusler compounds Co 2NiZ, Co 2FeZ, Fe 2NiZ, and Fe 2CoZ where Z = Al, Ge, Ga,\nSn, or In. Values of the MAE are presented only for tetragonally distorted Heusler compounds; the MAE for the cubic phase is,\nin comparison, negligibly small. A positive value of the MAE indicates an out-of-plane easy axis (i.e., the moments are aligned\nwith the distortion axis), while a negative value an in-plane easy axis (i.e., the moments lie within the plane perpendicular to\nthe distortion direction). For comparison, the value of MAE for Nd 2Fe14B is 4.4 [MJ/m3], and saturation magnetization is\n1280 [kA/m].\nis always much smaller than the thermal energy avail-\nable due to annealing (which takes place at 650 K in\nthe experiment). The authors of Ref. 23 therefore con-\nclude that annealing will control the state of order for the\nFe2NiZ family. The mismatch between experiment and\nour results, calculated for fully ordered structures, there-\nfore likely has its origin in thermal induced substitutional\ndisorder. It is worth pointing out that the energy di\u000ber-\nence between the tetragonally distorted regular structure\n(our lowest energy structure) and the inverse cubic struc-\nture is 120 meV, i.e. about double the thermal energy\ndue to annealing. This indicates that the presence of an-\ntisite disorder has a signi\fcant impact on the propensity\nof this material towards tetragonal disorder and that, at\nleast for the Fe 2NiZ family, the coupling between disorder\nand tetragonal distortion is a subject worthy of further\ninvestigation.\nWe now consider the electronic origin of the instability\nof the cubic phase with respect to a tetragonal distortion.\nSuch instability of the high symmetry phase has been\nobserved in many Heusler compounds, in particular the\nMn rich Heuslers13,17,24, and has been attributed to a\nnumber of di\u000berent mechanisms: a Jahn-Teller e\u000bect24,\na \\band\" JT e\u000bect25, a nesting induced Fermi surface\ninstability26, and anomalous phonon modes27,28.\nIn Fig. 2 we present the total density of states for four\nrepresentative examples of the set of Heusler compoundswe investigate. For all four cases (and for all Heuslers we\nstudy in this work) the minority spin channel is not signif-\nicantly involved in the mechanism of distortion, having\na very low DOS near the Fermi energy. For the cases\n(Co2NiAl, Fe 2NiGe) in which the cubic phase is unsta-\nble we see a clear redistribution of spectral weight near\nthe Fermi energy, such that a high DOS near EFis low-\nered by the opening up of a \\valley\" near EFin the\ntetragonal phase. On the other hand, for the materials\nin which the cubic phase is stable the DOS at EFis al-\nready very low (see the right hand panels of Fig. 1 for\nthe representative cases of Co 2FeAl and Fe 2CoGe). As\nmay be seen in Fig. 3 for the case of Co 2NiAl this redis-\ntribution of weight occurs in all species and momentum\nchannels, but with states of Co character being the more\nimportant. Interestingly, it is seen that the redistribution\noccurs particularly in states of egcharacter.\nIV. MAGNETIC MOMENTS AND\nMAGNETOCRYSTALLINE ANISOTROPY\nIn Fig. 4 we present the total magnetic moment, sat-\nuration magnetization M sand MAE for the Co 2YZ and\nFe2YZ Heusler families. In all systems the magnetic or-\nder is found to be ferromagnetic. To a good approxima-\ntion the values of the saturation magnetization M sfall5\n-2 0 2-8-4048DOS (states/eV)\n-2 0 2 -2 0 2-8-4048\n-2 0 2\nEnergy (eV)-8-4048DOS (states/eV)\n-2 0 2\nEnergy (eV)-2 0 2\nEnergy (eV)-8-4048Co2FeAl\nCo2FeSnCo2FeAl\nCo2FeSnCUBICCUBIC DOPED\nCUBIC CUBIC DOPEDTETRAGONAL\nTETRAGONAL\nFIG. 5: Calculated density of states of Co 2FeAl and Co 2FeSn\nfor the cubic phase, the cubic phase with doping of 1.5e and\n0.3e respectively, and the tetragonally distorted phase. In\neach case the electron doping shifts the Fermi energy to a\nregion of high density of states and drives a tetragonal dis-\ntortion of the cubic phase, which is otherwise stable. Positive\n(negative) value of DOS represents the minority (majority)\nspin projection.\ninto four distinct bands: (i) Msclose to 500 kA/m for\nCo2NiZ; (ii) Msclose to 900 kA/m for Co 2FeZ, Fe 2NiZ,\nand Fe 2CoZ; (iii) Msclose to 500 kA/m for Co 2PtZ; and\n(iv)Msclose to 800 kA/m for Fe 2PtZ. From the view-\npoint of hard magnetic applications a high value of the\nsaturation magnetization is desired, and from this point\nof view the Co 2FeZ, Fe 2NiZ, Fe 2CoZ, and Fe 2PtZ com-\npounds are most interesting. For comparison we recall\nthat the two \\standard\" hard magnets have saturation\nmagnetizations of 970 kA/m for SmCo 5and 1280 kA/m\nfor Nd 2Fe14B.\nWe now turn to a discussion of the MAE values real-\nized in the cases for which a tetragonal distortion occurs\n(see also Fig. 4). We \frst note that a positive value of the\nMAE indicates that the magnetic moments all align with\nthe symmetry axis of the tetragonally distorted material:\nthis is essential for hard magnetic applications. When the\nMAE takes on a negative value this indicates that the\nmoments are in the plane perpendicular to this symme-\ntry axis. We have checked the energy required to rotate\nspins in-plane \fnding, as expected, a very soft energy\ndependence. This freedom to rotate the spin structure\nobviously renders such cases entirely unsuitable for hard\nmagnetic application. We will therefore focus on those\ncases for which the MAE is positive.\nOf the 15 compounds that su\u000ber a tetragonal dis-\ntortion only 6 have EMAE >0. Curiously, these are\nthe compounds for which the Z element is either Ge or\nSn: Co 2NiGe, Fe 2NiGe, Fe 2NiSn, Co 2PtSn, Fe 2PtGe,\nand Fe 2PtSn. The values of the MAE for the Pt free\ncompounds are all, as expected, modest as compared to\nthe Pt containing compounds. The maximum positiveMAE for Pt free compounds are found in Fe 2NiGe and\nFe2NiSn, with an MAE of \u00191 MJ/m3, while for the Pt\ncontaining compounds we \fnd a much higher MAE of\n5.19 MJ/m3for Fe 2PtGe. This value is close to the cur-\nrently highest value observed for an MAE in a rare earth\nfree material (a value of 7 MJ/m3for L1 0-FePt29). The\nrather high M svalue of 516 kA/m suggests this material\nmight be interesting to further explore in the context of\nspecialist application as a hard magnet.\nV. DISTORTION CONTROL\nThe previous two sections lead us to conclude that (i)\nthe propensity to tetragonal distortion strongly correlates\nwith a high DOS at the Fermi energy in the cubic phase\nand (ii) that if a tetragonal distortion occurs, high values\nof the MAE are possible. This raises the possibility of,\nwith a view to engineering a high MAE, inducing such a\ndistortion by doping.\nTo this end we consider the two materials presented\nin Fig. 2 in which the Fermi energy lies in the valley\nbetween the two high DOS regions, and dope the cubic\nphase within the virtual crystal approximation (VCA).\nIn the case of Co 2FeAl a doping of 1.5 electrons is re-\nquired to shift the Fermi energy into the high DOS re-\ngion, with a more modest 0.3 electrons required in the\ncase of Fe 2CoGe. In both cases we \fnd that upon such\ndoping, the cubic phase becomes unstable with respect to\na tetragonal distortion; structural details may be found\nin Table II. A subsequent calculation of the MAE \fnds\nvalues comparable to those obtained for the naturally\ntetragonally distorting Heusler compounds. It is also in-\nteresting to note that the mechanism of the distortion ap-\npears to be somewhat di\u000berent from the \\natural\" cases:\nwhile in Fig. 2 it is clearly seen that the distortion results\nin a signi\fcant redistribution of spectral weight away\nfrom the Fermi energy via the opening of a \\repulsion\nvalley\" in Fig. 5 this e\u000bect is seen to be much weaker.\nThis of course, may be an artifact of the VCA.\nVI. CONCLUSION\nWe have addressed the question of whether we may\nobtain large magnetocrystalline anisotropy energies in\nHeusler compounds that adopt a low symmetry tetrag-\nonal structure. To this end we have investigated the\nHeusler compounds Fe 2YZ with Y = (Ni, Co, Pt), and\nCo2YZ with Y = (Ni, Fe, Pt) where, in both cases, Z =\n(Al, Ga, Ge, In, Sn). We \fnd that the cubic phase of 15\nof these 30 Heusler compounds is unstable with respect\nto tetragonal distortion, in particular for the Co 2NiZ,\nCo2PtZ, and Fe 2PtZ families the cubic phase is always,\natT= 0, unstable. In contrast, for the Fe 2NiZ and\nFe2CoZ families this is the case for only 2 compounds\n{ Fe 2CoGe and Fe 2CoSn. The mechanism behind this\ndistortion involves a signi\fcant redistribution of spectral6\nstructure \u0016tot\nB acalc, ccalc[\u0017A]MAE [MJ/m3]\nCo2FeAl regular cubic 5.08 a=c=5.69 \u0000\n1.5e\u0000doped-Co 2FeAl regular tetragonal 5.43 a=6.16 c=6.88 \u00000.94\nCo2FeSn regular cubic 5.66 a=c=5.64 \u0000\n0.3e\u0000doped-Co 2FeAl regular tetragonal 5.41 a=5.97 c=6.45 \u00001.30\nTABLE II: Calculated material properties of Co 2FeAl, Co 2FeSn, and their electron doped systems. The most stable structure\nis shown in the 2nd column. The calculated magnetic moments per formula unit, lattice parameters, aandc, and the\nmagnetocrystalline anisotropy energies for the tetragonal cases are also listed.\nweight near the Fermi energy, such that a \\valley\" in the\nDOS at the Fermi energy is opened up in the tetragonal\nphase leading to a reduction in the number of states near\nthe Fermi energy. Curiously, we \fnd that for the com-\npounds we investigate a good rule of thumb exists that if\nthe DOS at the Fermi level is greater than 4.5 states/eV,\nthe cubic phase is unstable.\nOf the 15 compounds that su\u000ber tetragonal dis-\ntortion the magnetocrystalline anisotropy energies are\nfound to range in values from -12 MJ/m3(Co2PtAl) to\n+5.19 MJ/m3(Fe2PtGe). As expected, the values of the\nMAE for the Pt free Heuslers are more modest in mag-\nnitude, and range in value from -2.38 MJ/m3(Co2NiGa)\nto 1.09 MJ/m3(Fe2NiSn). For hard magnet application\nonly positive values of the magnetocrystalline anisotropy\nenergies, which correspond to moments aligned with the\ntetragonal symmetry axis, are interesting. Interestingly,\nwe \fnd that the MAE takes on a positive value for all\ncases in which the Z element is either Ge or Sn.\nFinally, we have considered the possibility of doping\nthe Heusler compounds in which the cubic phase is sta-\nble in order to induce a tetragonal distortion. Using the\nvirtual crystal approximation we \fnd that this is indeedpossible, and the doping induced distortion results in\nmagnetocrystalline anisotropy energies values compara-\nble to those obtained in the naturally distorting Heusler\ncompounds.\nVII. ACKNOWLEDGMENTS\nYM, GM and S. Sharma would like to thank the\nHeusler project funded by the MPG.\nAppendix A: Details of the structural and magnetic\nproperties of the Heuslers investigated in this work\nIn this Appendix we present structural details of the\nHeusler compounds calculated in the manuscript along\nwith experimental structure data where this exists. In\nTable III we present the Heusler compounds Co 2NiZ,\nCo2FeZ, and in Table IV the compounds Fe 2NiZ, and\nFe2CoZ where in each case Z = Al, Ge, Ga, Sn, or In.\n1J. Winterlik, S. Chadov, A. Gupta, V. Alijani, T. Gasi,\nK. Filsinger, B. Balke, G. Fecher, C. Jenkins, F. Casper,\nJ. Kubler, G.-D. Liu, L. Gao, S. Parkin, and C. Felser,\nAdv. Mater. 24, 6283 (2012).\n2R. W. McCallum, L. H. Lewis, R. Skomski, M. J. Kramer,\nand I. E. Anderson, Annu. Rev. Mater. Res. 44, 451 (2014).\n3J. M. D. Coey, IEEE transactions on magnetics 47, 4671\n(2011).\n4J. M. D. Coey, Scripta Materialia 67, 524 (2012).\n5M. J. Kramer, M. R. W., A. I.A., and S. Constantinides,\nJOM 64, 752 (2012).\n6C. S. Lue and Y.-K. Kuo, Phys. Rev. B 66, 085121 (2002).\n7V. Alijani, S. Ouardi, G. H. Fecher, J. Winterlik, S. S.\nNaghavi, X. Kozina, G. Stryganyuk, C. Felser, E. Ikenaga,\nY. Yamashita, S. Ueda, and K. Kobayashi, Phys. Rev. B\n84, 224416 (2011).\n8C. Felser, G. H. Fecher, and B. Balke, Angew. Chem. Int.\nEd.46, 668 (2007).\n9T. Graf, C. Felser, and S. S. P. Parkin, Progress in Solid\nState Chemistry 39, 1 (2011).\n10T. Graf, L. Winterlik, L. Muchler, G. H. Fecher, C. Felser,\nand P. S. P. P., Handbook of Magnetic Materials 21, 1\n(2013).11G. Kreiner, A. Kalache, S. Hausdorf, V. Alijanin, J.-F.\nQian, U. Burkhardt, S. Ouardi, and C. Felser, Z. Anorg.\nAllg. Chem 640, 738 (2014).\n12T. Roy and A. Chakrabarti, arXiv:1603.08350 (2016).\n13L. Wollmann, S. Chadov, J. K ubler, and C. Felser, Phys.\nRev. B 92, 064417 (2015).\n14A. Talapatra, R. Arr\u0013 oyave, P. Entel, I. Valencia-Jaime,\nand A. H. Romero, Phys. Rev. B 92, 054107 (2015).\n15L. Hongzhi, J. Pengzhong, L. Guodong, M. Fanbin,\nL. Heyan, L. Enke, W. Wenhong, and W. Guangheng,\nSolid State Communications 170, 4447 (2013).\n16J. Winterlik, S. Chadov, A. Gupta, V. Alijani, T. Gasi,\nK. Filsinger, B. Balke, G. H. Fecher, C. A. Jenkins,\nF. Casper, J. Kbler, G.-D. Liu, L. Gao, S. S. P. Parkin,\nand C. Felser, Advanced Materials 24, 6283 (2012).\n17L. Wollmann, S. Chadov, J. K ubler, and C. Felser, Phys.\nRev. B 90, 214420 (2014).\n18G. Kresse and J. Furthm uller, Comput. Mat. Sci. 6, 15\n(1996).\n19P. Bl ochl, Phys. Rev. B 50, 17953 (1994).\n20J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett. 77, 3865 (1996).\n21K. Dewhurst, S. Sharma, and et al.,7\nstructure Moment (\u0016B) acalc, ccalc[\u0017A]Ms[kA/m] MAE [MJ/m3] expt.\nCo2NiAl inv. tet. 2.78 (Co=1.4, 1.2 Ni=0.2) a=5.20 c=6.76 564 \u00002.11\nCo2NiGa inv. tet. 2.79 (Co=1.6, 1.4 Ni=0.2) a=5.19 c=6.80 562 \u00002.38\u0016tot\nB=2.807, a=3.669 c=7.33130\nCo2NiGe inv. tet. 2.46 (Co=1.3, 1.0 Ni=0.2) a=5.12 c=6.97 498 0.86\nCo2NiIn inv. tet. 2.99 (Co=1.6, 1.3 Ni=0.2) a=5.43 c=7.13 527 \u00002.22\nCo2NiSn inv. tet. 2.58 (Co=1.4, 1.1 Ni=0.2) a=5.37 c=7.22 458 0\nCo2FeAl reg. cubic 5.08 (Co=1.2, 1.2 Fe=2.8) a=c=5.69 1016 \u0000\u0016tot\nB=4.82-5.22, a=c=5.7431,32\nCo2FeGa reg. cubic 5.07 (Co=1.2, 1.2 Fe=2.8) a=c=5.70 1005 \u0000 \u0016tot\nB=5.035, a=c=5.72733\nCo2FeGe reg. cubic 5.60 (Co=1.4, 1,4 Fe=2.9) a=c=5.74 1099 \u0000\u0016tot\nB=5.54-5.70, a=c=5.70234\nCo2FeIn reg. cubic 5.23 (Co=1.3, 1.3 Fe=2.8) a=c=5.64 908 \u0000\nCo2FeSn reg. cubic 5.66 (Co=1.4, 1.4 Fe=2.9) a=c=5.64 974 \u0000\nCo2PtAl inv. tet. 2.84 (Co=1.5, 1.2 Pt=0.1) a=5.36 c=7.13 516 \u000012\nCo2PtGa inv. tet. 2.88 (Co=1.6, 1.3 Pt=0.1) a=5.36 c=7.21 516 \u000011.33\nCo2PtGe inv. tet. 2.52 (Co=1.5, 1.0 Pt=0.1) a=5.32 c=7.31 453 \u00001.01\nCo2PtIn inv. tet. 3.08 (Co=1.6, 1.4 Pt=0.1) a=5.50 c=7.67 493 \u00006.94\nCo2PtSn inv. tet. 2.59 (Co=1.5, 1.1 Pt=0.1) a=5.50 c=7.63 416 0.42\nTABLE III: Calculated material properties of Co-family Heusler materials in cubic and tetragonal structures. The most stable\nstructure in regular Heusler (denoted by \"reg.\") and inverse Heusler (denoted by \"inv.\") with cubic or tetragonal (denoted\nby \"tet.\") symmetry are shown in the 2nd column. The calculated total energy and magnetic moments per formula unit,\ndistortion parameter c/a, saturated magnetic moment M s, atom resolved moments, and MAE values are also listed. Note that\ntwo X atoms are not equivalent in inverse Heusler with cubic or tetragonal symmetry. Therefore, two values of atom resolved\nmoments for X elements are listed in inverse Heusler materials.\nstructure Moment (\u0016B) acalc, ccalc[\u0017A]Ms[kA/m] MAE [MJ/m3] expt.\nFe2NiAl inv. cubic 4.86 (Fe=2.6, 1.8 Ni=0.5) a=c=5.74 954 \u0000\u0016tot\nB=4.00-4.46, a=c=5.7535{37\nFe2NiGa inv. cubic 4.91 (Fe=2.6, 1.9 Ni=0.4) a=c=5.77 955 \u0000\u0016tot\nB=4.29-4.89, a=c=5.8023,35\nFe2NiGe reg. tet. 4.86 (Fe=2.3, 2.3 Ni=0.3) a=5.02 c=7.59 944 1.07\u0016tot\nB=4.20-4.38, a=c=5.7623,35\nFe2NiIn inv. cubic 5.22 (Fe=2.7, 2.2 Ni=0.4) a=c=6.07 885 \u0000\nFe2NiSn reg. tet. 5.02 (Fe=2.4, 2.4 Ni=0.3) a=5.22 c=8.02 853 1.09\nFe2CoAl inv. cubic 5.14 (Fe=2.5, 1.6 Co=1.0) a=c=5.71 1026 \u0000 \u0016tot\nB=4.91, a=c=5.76637\nFe2CoGa inv. cubic 5.28 (Fe=2.5, 1.8 Co=1.1) a=c=5.76 1041 \u0000 \u0016tot\nB=5.09, a=c=5.76736\nFe2CoGe inv. cubic 5.13 (Fe=2.7, 1.5 Co=1.0) a=c=5.72 1019 \u0000\u0016tot\nB=5.06, a=c=5.775-5.76438\nFe2CoIn inv. cubic 6.18 (Fe=2.7, 2.3 Co=1.3) a=c=6.03 1053 \u0000\nFe2CoSn inv. cubic 5.54 (Fe=2.7, 1.9 Co=1.1) a=c=5.99 959 \u0000\nFe2PtAl inv. cubic 5.04 (Fe=2.8, 2.1 Pt=0.2) a=c=5.97 885 \u0000\nFe2PtGa inv. tet. 5.06 (Fe=2.7, 2.3 Pt=0.2) a=5.50 c=7.10 873 \u00005.41\nFe2PtGe reg. tet. 5.27 (Fe=2.6, 2.6 Pt=0.1) a=5.40 c=7.38 893 5.19\nFe2PtIn inv. tet. 5.27 (Fe=2.8, 2.5 Pt=0.1) a=5.64 c=7.56 812 \u00003.57\nFe2PtSn reg. tet. 5.26 (Fe=2.6, 2.6 Pt=0.1) a=5.64 c=7.58 802 2.97\nTABLE IV: Calculated material properties of Fe-family Heusler materials in cubic and tetragonal structures. The most stable\nstructure in regular Heusler (denoted by \"reg.\") and inverse Heusler (denoted by \"inv.\") with cubic or tetragonal (denoted\nby \"tet.\") symmetry are shown in the 2nd column. The calculated total energy and magnetic moments per formula unit,\ndistortion parameter c/a, saturated magnetic moment M s, atom resolved moments, and MAE values are also listed. Note that\ntwo X atoms are not equivalent in inverse Heusler with cubic or tetragonal symmetry. Therefore, two values of atom resolved\nmoments for X elements are listed in inverse Heusler materials.\nhttp://elk.sourceforge.net/ (2016).\n22M. Gillessen and R. Dronskowski, Journal of Computa-\ntional Chemistry 31, 612 (2010).\n23T. Gasi, V. Ksenofontov, J. Kiss, S. Chadov, A. K. Nayak,\nM. Nicklas, J. Winterlik, M. Schwall, P. Klaer, P. Adler,\nand C. Felser, Phys. Rev. B 87, 064411 (2013).\n24C. Felser and A. E. Hirohata, Springer Series in Materials\nScience 222 (2016).\n25J. Suits, Solid State Commun. 18, 423 (1976).\n26S. R. Barman, S. Banik, A. K. Shukla, C. Kamal, and\nA. Chakrabarti, EPL (Europhysics Letters) 80, 57002\n(2007).27A. T. Zayak, P. Entel, K. M. Rabe, W. A. Adeagbo, and\nM. Acet, Phys. Rev. B 72, 054113 (2005).\n28S. Paul, B. Sanyal, and S. Ghosh, Journal of Physics:\nCondensed Matter 27, 035401 (2015).\n29O. A. Ivanov, L. V. Solina, V. A. Demshira, and L. M.\nMagat, Phys. Met. Metallov. 35, 92 (1973).\n30T. Fichtner, C. Wang, A. Levin, G. Kreiner, C. Mejia,\nS. Fabbrici, F. Albertini, and C. Felser, Metals 5, 484\n(2015).\n31M. S. Gabor, T. Petrisor, C. Tiusan, M. Hehn, and\nT. Petrisor, Phys. Rev. B 84, 134413 (2011).\n32S. Husain, A. Akansel, S.and Kumar, P. Svedlindh, and8\nS. Chaudhary, Scienti\fc Reports 432, 28692 (2016).\n33M. Zhang, E. Brck, F. R. de Boer, Z. Li, and G. Wu,\nJournal of Physics D: Applied Physics 37, 2049 (2004).\n34N. Uvarov1, Y. Kudryavtsev1, A. Kravets, Y. Vovk,\nR. Borges, M. Godinho, and V. Korenivski, J. Appl. Phys.\n112, 063909 (2012).\n35Y. J. Zhang, W. H. Wang, H. G. Zhang, E. K. Liu, R. S.\nMa, and W. G. H., Physica B: Condensed Matter 87, 86(2013).\n36K. Buschow, P. van Engen, and R. Jongebreur, J. Magn.\nMagn. Mater. 38, 1 (1983).\n37M. Csanad, T. Csorgo, and B. Lorstad, Nukleonika 49,\nS49 (2004).\n38Z. Ren, S. T. Li, and H. Z. Luo, Physica B: Condensed\nMatter 405, 2840 (2010)."    },    {        "title": "1703.01406v2.Impact_of_Anisotropy_on_Antiferromagnet_Rotation_in_Heusler_type_Ferromagnet_Antiferromagnet_Epitaxial_Bilayers.pdf",        "content": "arXiv:1703.01406v2  [cond-mat.mtrl-sci]  22 Mar 2017Impact of Anisotropy on Antiferromagnet Rotation in Heusle r-type\nFerromagnet/Antiferromagnet Epitaxial Bilayers\nT. Hajiri,1,∗M. Matsushita,1Y. Z. Ni,1and H. Asano1\n1Department of Crystalline Materials Science, Nagoya Unive rsity, Nagoya 464-8603, Japan\n(Dated: September 24, 2018)\nWe report the magnetotransport properties of ferromagnet ( FM)/antiferromagnet (AFM)\nFe2CrSi/Ru 2MnGe epitaxial bilayers using current-in-plane configurat ions. Above the critical thick-\nnessof theRu 2MnGe layertoinduceexchangebias, symmetric andasymmetri c curves were observed\nin response to the direction of FM magnetocrystalline aniso tropy. Because each magnetoresistance\ncurve showed full and partial AFM rotation, the magnetoresi stance curves imply the impact of the\nFe2CrSi magnetocrystalline anisotropy to govern the AFM rotat ion. The maximum magnitude of\nthe angular-dependent resistance-change ratio of the bila yers is more than an order of magnitude\nlarger than that of single-layer Fe 2CrSi films, resulting from the reorientation of AFM spins via the\nFM rotation. These results highlight the essential role of c ontrolling the AFM rotation and reveal\na facile approach to detect the AFM moment even in current-in -plane configurations in FM/AFM\nbilayers.\nINTRODUCTION\nAntiferromagnets (AFMs) show great potential to re-\nplace ferromagnets (FMs) in spintronic applications [1–\n3]. Compared with FMs, AFMs have the advantages of\nmuch faster spin dynamics [4, 5], more stability against\ncharge and external field perturbations [6], and no stray\nfield [7, 8]. However, since the AFM spins align in al-\nternating directions of magnetic moments on individ-\nual atoms, the resulting zero net magnetization makes\nhard to control AFM magnetic moments. Recently,\nthere have been several reports regarding the control of\nAFM momentsby applyingan electroniccurrent in AFM\nfilms [9] and FM/AFM bilayers [10, 11], by field cool-\ning (FC) [12, 13] and by applying an external field via\nthe exchange-springeffect [14–17]. These studies demon-\nstrated that the AFM moments can be controlled and\ndetected using electronic transport measurements with-\nout the need for large-scale facilities such as synchrotron\nand neutron facilities [18].\nAFM rotation is of interest because a more than\n100% spin-valve-like signal has been achieved in tun-\nneling anisotropic magnetoresistance (TAMR) stacks by\ncontrollingthe AFM spin configurationviathe exchange-\nspring effect of FM on AFM [15]. The exchange cou-\npling has been widely used in spintronic devices such\nas spin-valve-type magnetic memory devices to pin the\nFM magnetization [19, 20]. In contrast, TAMR utilizes\nthe rotating AFM exchange-coupled to FM [14, 21, 22].\nThe rotating AFM can be linked to the shift of hystere-\nsis loops (exchange bias) and broadening of the coerciv-\nity in magnetization measurements [18]. Although sev-\neral studies have been reported regarding the rotating\nAFM [15, 18, 23, 24], almost all of the studies have been\nperformed on polycrystallinestacksusing AFM for IrMn.\nSince the AFM moments rotate with exchange-coupled\nFM, the AFM rotation behavior is expected to be af-\nfectedbytheFMmagnetizationswitchingprocess. Thus,the effect of FM magnetocrystalline anisotropy resulting\nfrom the full epitaxial growth is more interesting. In ad-\ndition, all the studies have been performed using typical\n3dmetal FMs such as NiFe and Co. Similar to successful\nstudies on giant magnetoresistance (GMR) and tunnel-\ning magnetoresistance (TMR) [25, 26], there is a clear\nneed to study high-quality heterostructures with more\nadvanced compounds such as those with high-spin polar-\nization needed for spintronic devices [27].\nFor the advanced materials, we focused on the Heusler\ncompound Fe 2CrSi (FCS), which is theoretically ex-\npected to be half-metallic FM [28]. The four-fold mag-\nnetocrystalline anisotropy constant of FCS is known to\nbe 266 J/m3[29]. In addition, since only fully epitaxial\nstacks effectively provide the advanced properties, the\nHeusler compound Ru 2MnGe (RMG) was selected for\nAFM; wesucceeded in growingfully epitaxialFCS/RMG\nbilayers [30]. RMG exhibits the highest N´ eel temper-\nature,TN= 353 K, among Heusler compounds [31].\nIn addition, RMG has a nearly half-metallic electronic\nstructure [32]. Since the TAMR depends on spin con-\nfiguration of AFM, such an AFM electronic structure is\ninteresting.\nIn this letter, we systematically studied the magnetic\nand magnetotransport properties of FCS/RMG bilay-\ners using current-in-plane (CIP) configurations. The\nangular-dependent resistance change (∆ R) ratio and\nexchange bias ( Hex) exhibit a similar RMG thickness\n(tRMG) dependence, indicating that RMG spin reorienta-\ntion via FCS rotation is dominant in ∆ Rabove a critical\nthickness ( tc) to induce Hexeven in CIP configurations.\nAbovetc, the magnetoresistance (MR) curves along the\nhard axes of FCS magnetocrystalline anisotropy exhibit\na full rotation of AFM moments. However, a partial ro-\ntation of AFM moments is observed along the easy axes,\ndemonstrating the effect of the FM magnetization rever-\nsal process on AFM rotation. Although previous studies\nhave used current-perpendicular-to-plane configurations,2\nthese results indicate that CIP magnetotransport mea-\nsurements in FM/AFM bilayer provide a facile approach\nto detect AFM moments and promote their application\nin AFM spintronics.\nEXPERIMENTAL DETAILS\nRMG/FCS bilayers were deposited on MgO (001) sub-\nstrates by DC magnetron sputtering at a base pressure\nof approximately 5 ×10−8Torr. RMG thin films were\ndeposited at a substrate temperature of Ts= 500◦C and\nthen cooled to room temperature. Next, FCS was de-\nposited on RMG at room temperature. After the FCS\nwas deposited, the FCS/RMG bilayers were annealed at\n500◦Cfor30minutestoachieve L21orderingoftheFCS.\nIn addition, we also deposited FCS at Ts= 500◦C. The\nresults were the same in both cases. The crystal struc-\nture was analyzed using both in-plane and out-of-plane\nX-ray diffraction (XRD) measurements with Cu Kαra-\ndiation. The magnetic properties were characterized us-\ning vibrating sample magnetometry and superconduct-\ningquantuminterferencedevice(SQUID)magnetometry.\nThemagnetotransportmeasurementswereperformedus-\ning the standard DC four-terminal method in the CIP\nconfiguration. To induce exchange coupling, the bilayers\nwere annealed at 350 K for the SQUID measurements\nand at 375 K for the magnetotransport measurements\nfor 30 minutes with applying field of +10 kOe, and then\ncooled to T= 4 K with applying field of +10 kOe [33].\nLog intensity (arb. unit)\n2θ/θ (deg.)(a)\n(b)*\n***\n**Ru 2MnGe (002) MgO sub.Fe 2CrSi (002) \nRu 2MnGe (004) \nRu 2MnGe (006) Fe 2CrSi (004) \n-180 -90 0 90 180Intensity (arb. unit)\n(deg.)Ru 2MnGe (111)\nMgO (111)Fe 2CrSi (111)\nφ120 100 80 60 40 20\nFIG. 1. (Color online) Out-of-plane 2 θ/θscans (a) and in-\nplaneφ-scans of FCS/RMG bilayers.III. RESULTS AND DISCUSSIONS\nXRD patterns of the FCS/RMG bilayersare presented\nin Fig. 1(a) and 1(b). As observed in Fig. 1(a), only the\n(00l) FCS and (00 l) RMG peak series show Bragg peaks\nin the out-of-plane XRD pattern. Epitaxial growth is\nconfirmed by the in-plane φ-scans presented in Fig. 1(b).\nBoth RMG (111) and FCS (111) peaks are observed\nwith shifts of 45◦relative to the MgO (111) peaks.\nThese results indicate that their epitaxial relationship\nis FCS(001)[100]//RMG(001)[100]//MgO(001)[110]. In\naddition, since (111) reflection peaks of the Heusler al-\nloy originate from superlattice reflections in the L21or-\ndered structure [34], these XRD results indicate that\nhigh-quality FCS/RMG bilayers were obtained.\nFigures 2(a)–2(c) shows the magnetic hysteresis loops\nof FCS (5 nm)/RMG ( tRMGnm) bilayers measured at\nT= 4 K after FC. The measurements were performed\nalong the easy axis of FCS /angbracketleft100/angbracketright. ThetRMG= 5 and\n15 nm bilayers exhibit narrow hysteresis loops with a co-\nercive field Hcof approximately 100 Oe, which is sim-\nilar to single-layer FCS films. The tRMG= 20 nm\nbilayers exhibited a much wider hysteresis loop with\nHcof approximately 860 Oe. On the other hand, the\ntRMG= 5 nm bilayer exhibited no hysteresis loop shift,\nwhereas the tRMG= 15 and 20 nm bilayers exhibited\nHex= 32 and 155Oe, respectively. The tRMG-dependent\nHcandHexresults are summarized in Fig. 2(d) at\nT= 4 K. Hexis confirmed at above tRMG= 10 nm,\nindicating that tcis between tRMG= 5 and 10 nm. A\nmaximum Hexappears at tRMG= 20 nm; then, Hex\ndecreases with increasing tRMG. The same tRMGthick-\n4 4\n-1.0-0.50.00.51.0(c) (d) T = 4 K T = 4 K\n-4 -2 0\nH (kOe) Mr/M s\n2 4t RMG = 20 nm-1.0-0.50.00.51.0(a) T = 4 K\n-4 -2 0\nH (kOe) Mr/M s\n2t RMG = 5 nm\n-1.0-0.50.00.51.0(b) T = 4 K\n-4 -2 0\nH (kOe) Mr/M s\n2t RMG = 15  nm\nHex Hc1000\n800\n600\n400\n200\n0Hc (Oe)\n40 30 20 10 0\nRu 2MnGe thickness (nm) Critical thickness 200\n150\n100\n50\n0\nHex  (Oe) \nFIG. 2. (Color online) (a)–(c) Magnetic hysteresis loops of\ntRMG= 5, 15 and 20 nm at T= 4 K after FC, respectively.\n(d)HcandHexas a function of tRMGatT= 4 K.3\nH (kOe)0.0\n-4 -2 4 200.20.40.6\nMR (%)20  nm \n30  nm \n0\n-4 -2 4 20246\nH (kOe)0\n-1 \n-2 \n-3 \n-4 \n-5 \n-6 ΔR (%)\n40 30 20 10 0\nRu 2MnGe thickness (nm)ZFC\nFCt RMG = H // I\nFC \nH (kOe)0.0\n-4 -2 4 200.20.4\n10  nm 0.6t RMG  = 15  nm t RMG  = 20  nm t RMG  = 0 nm \n-0.06-0.04-0.020.00\nΔR (%)\n360 270 180900\nAngle  θ (deg.)360 270 180900\nAngle  θ (deg.)360 270 180900\nAngle  θ (deg.)-6-4-20\n-0.2\n-0.4\n-0.60.0Ru 2MnGe ( t RMG  nm)  / Fe 2CrSi (5 nm) \n0 nm \n15  nm \n+0.20 %tc(a)\n(c)(b)\n0.150.100.050.00\n151050\nFIG. 3. (Color online) (a) ∆ Ras a function of the angle between the current and field for sin gle-layer FCS films and FCS\n(5 nm)/RMG ( tRMGnm) bilayers. The measurements were performed at T= 4 K with an applied field of +4 kOe. (b) ∆ Ras\na function of tRMGafter both ZFC and FC at T= 4 K. (c) MR curves as a function of applied field on single-lay er FCS films\nand FCS (5 nm)/RMG ( tRMGnm) bilayers at T= 4 K.\nness dependence was observed at T= 77 K for a wider\ntRMGrange [35]. Hcshows no significant thickness de-\npendence below tRMG= 15 nm; then a jump is observed\nattRMG= 20 nm. However, tRMG= 20 nm shows much\nlargerHcandHexthantRMG= 25 nm at T= 4 K, of\nwhich results are unusual behavior. The possible reason\nwill be discussed later.\nNext, we focused on the tRMG-dependent magne-\ntotransport properties using CIP configuration . Fig-\nure 3(a) plots the ∆ Rratio as a function of the rel-\native angle θbetween the current and FM magnetiza-\ntion direction of FCS(5 nm)/RMG( tRMGnm) bilayers\natT= 4 K under an applied field of +4 kOe after FC.\nAttRMG= 0 nm, a typical anisotropic magnetoresis-\ntance (AMR) ratio is confirmed with a negative value\nof approximately -0.04 %. The negative AMR sign may\noriginate from the half-metallic electronic structure of\nFCS, as discussed in recent theoretical and experimental\nstudies [36, 37]. At tRMG= 15 nm, the amplitude of ∆ R\nincreases, and its angular dependence shifts by approx-\nimately 45◦. AttRMG= 20 nm, the amplitude of ∆ R\nis more than an order of magnitude larger than that of\nsingle-layer FCS films.\nThe ∆Rratios with respect to tRMGatT= 4 K after\nboth zero-field cooling (ZFC) and FC are summarized\nin Fig. 3(b). After FC, the ∆ Rratios are independent\noftRMGbelowtc. Above tc, the ∆Rratios increase at\n10 and 15 nm. At tRMG= 20 nm, the ∆ Rratio drasti-\ncallyincreasesandthendecreasesuponfurtherincreasing\ntRMG. These tRMG-dependent ∆ Rratios are similar to\nthe exchange bias, as observed in Fig. 2(d). Moreover,\nas observed in Fig. 3(b), the ∆ Rratios are enlarged by\nFC, demonstrating the effect of exchange coupling on the\n∆Rratio. Note that since the exchange coupling mightexist even without FC [38, 39], the ∆ Rratio after ZFC\nis larger than that for single-layer FCS films.\nIn addition, a relationship is observed between the ∆ R\nratios, the shape of the MR curves and exchange bias.\nThe MR curves measured as a function of the applied\nmagnetic field are presented in Fig. 3(c). The MR curves\nofthe single-layerFCS films ( tRMG= 0nm [40]) aresym-\nmetric, which originatesfrom the FCS AMR. In contrast,\nabovetc, the curves of the bilayers differ from typical\nAMR curves. The tRMG= 10 and 15 nm bilayers exhibit\nsmallasymmetriccurves,andtheasymmetryincreasesat\ntRMG= 20 and 30 nm. According to previous FM/AFM\nstudies [15, 18, 24], the asymmetric MR curves originate\nfrom the partial rotation of the AFM moments due to\nthe applied external field via the FM rotation, whereas\nthe symmetric MR curvesoriginate from the full rotation\nof the AFM moments.\nFinally, we would like to discuss the origin of the\nanomalous magnetic properties of the bilayers. Fig-\nures 4(a) and 4(b) compare different measurement con-\nditions; the sensing current Iwas applied in directions\nparallel to FCS/RMG [100] and [1-10], respectively. As\nobserved in Fig. 4(a) for I/bardbl[100], asymmetric MR\ncurves are obtained along H/bardbl[100] (θ= 0◦) and [010]\n(θ= 90◦). On the other hand, symmetric MR curves are\nobtained along H/bardbl[110] (θ= 45◦) and [-110]( θ= 135◦).\nForI/bardbl[1−10], the symmetric and asymmetric relations\nwith respect to crystalline direction do not change; sym-\nmetricMRcurvesareobtainedalong H/bardbl[1−10](θ= 0◦)\nand [110] ( θ= 90◦), and asymmetric MR curves are ob-\ntained along H/bardbl[100] (θ= 45◦) and [0-10] ( θ=−45◦).\nAlthough FC-direction dependence was performed, no\nchange was observed. These results indicate that the\nsymmetric and asymmetric curves are not determined by4\n-4 -2 0 2 4 -4 -2 0 2 4 360 270 180900\n[110] [-110] [100] // \nI\n[010] // HFC \n[110]\n[-110][100]\n[010]\n[110]\n[1-10][0-10][100]H (kOe) H (kOe) Angle θ (deg.)\nH (kOe) H (kOe) Angle θ (deg.)Resistance (arb. units) \nResistance (arb. units) (a)\n(b)\n(c)270 18090 0 -90 -4 -2 0 2 4 -4 -2 0 2 4[0-10] [100] [-1-10] \n[1-10] // \nI // HFC H // \nHEasy axisEasy axis\nHard axis\nHard axis(d)\nHEasy axis[100] [100][010] [010]\nEasy axis\nHard axis\n(i)\n(ii)\n(iii)\nHard axis\nFIG. 4. (Color online) (a, b) Magnetotransport properties o f\nthe configurations I/bardbl[100] and /bardbl[1−10], respectively. (c, d)\nFCS magnetization switching process where the field sweeps\nalong the easy and hard axes, respectively.\nthe relative angle θbetween HandI, indicating that\nthe obtained angular dependences are not typical AMR\nof FM. Then, because the symmetric and asymmetric re-\nlations are not changed by the FC directions, the factor\ngoverning the angular dependence of the bilayers is not\nthe sensing current or FC directions but the FCS/RMG\ncrystalline direction.\nOne possible cause of the crystalline-direction-\ndependent AFM rotation is the FM magnetization\nswitching process. FCS has four-fold magnetocrystalline\nanisotropy, where the easy axes are oriented along /angbracketleft100/angbracketright\nand/angbracketleft010/angbracketright, and the hard axes are oriented along /angbracketleft110/angbracketright\nand/angbracketleft1−10/angbracketright[29]. This property indicates that symmetric\nMR curvesappearalongthe hardaxesofFCS,andasym-\nmetric MR curves appear along the easy axes of FCS. It\nis well known that the FM magnetization switching pro-\ncesses with four-fold magnetocrystalline anisotropydiffer\nalong the easy and hard axes. As presented in Fig. 4(c),\nthe magnetization rotates by 180◦along the easy axes.On the other hand, as presented in Fig. 4(d), the mag-\nnetization rotates in 3 steps along the hard axes; (i) the\nmagnetization rotates toward the nearest easy axis, (ii)\nthe magnetization jumps in a direction close to the other\neasy axis, and (iii) the magnetization finally rotates to-\nward the applied field direction [41]. Therefore, the mag-\nnetization rotates by up to 90◦along the hard axis. A\npossible explanation for the full and partial AFM rota-\ntionsalongFCS/RMG /angbracketleft100/angbracketrightand/angbracketleft110/angbracketrightarethat the AFM\ncan fully follow the FM magnetization switching via the\nFM rotation when the field sweep is along the hard axis\nof FM because the magnetization rotates slightly (up to\n90◦). On the other hand, the AFM cannot fully follow\nwhen the field sweep is along the easy axis of FM be-\ncause the magnetization rotates a lot (180◦). These re-\nsults could provide a route to understanding the AFM\nrotation behavior.\nSince the MR curves suggest the importance of AFM\nspin configuration, as discussed above, the larger ∆ Rra-\ntio compared with that of only single-layer FCS films\ncan be considered to be due to AFM moments. To\ndate, there have been several studies of AFM AMR due\nto spin flop [6], crystalline AMR originating from large\nanisotropies in the relativistic electronic structure [16]\nand AFM spin configurations with respect to current di-\nrection [12]. In the crystalline AMR study, the AFM\nspins were reoriented by applying magnetic fields via the\nexchangespringeffect[14], whichisthesameconditionas\nthat in our study. In addition,the ∆ Rratio was obtained\nunderH= 4 kOe with small angle steps ( ∼15◦), indi-\ncating that the AFM moments can fully follow the FM\nmagnetization rotation as mentioned above. The AFM\nspinreorientationisreinforcedbythedelayofangularde-\npendence due to the exchange-springeffect [15], asshown\nin Fig. 4(b). Then, the resistance is higher for the con-\nfiguration of AFM moments aligned along [110] and [-1-\n10] than for the configuration of AFM moments aligned\nalong [1-10] and [-110]. These resistance changes due to\nthe AFM spin direction have been reported in FM/AFM\nbilayers [16]. Moreover, the asymmetric MR curves were\ntransformed into symmetric MR curves with increasing\ntemperature. Therefore, we conclude that the obtained\nlarger ∆Rratio compared with that of only single-layer\nFCS films originates from the reorientation of the AFM\nmoments by applying magnetic fields via FM rotation.\nThen, the unusual tRMG-dependent HexandHcmight be\ncaused by rotating AFM and/or exchange-spring effect.\nAs mentioned in the introduction, rotating AFM can be\nlinked to HexandHc[18]. As similar to HexandHc,\ntRMG= 20 nm shows much larger ∆ Rratio than that of\ntRMG= 30 nm. Since ∆ Rratio originates from the reori-\nentationofthe AFM momentsviaexchange-springeffect,\ntheseresults mightindicate the tRMG-dependent rotating\nAFM and/or exchange-spring effect. The obtained ∆ R\nratio of approximately 5.9 % is much larger than other\nAFM AMR ratios for Sr 2IrO4of approximately 1 % and5\nMnTe of approximately 1.6 %. This result might be re-\nlated to either the nearly half-metallic RMG electronic\nstructure or the electronic structures of both RMG and\nFCS.\nCONCLUSION\nWeperformedamagnetotransportstudyofFCS/RMG\nbilayers to clarify the AFM rotation behavior. In ad-\ndition to the same tRMGthickness dependence of the\nmagnitude of the ∆ Rratio and exchange bias, the MR\ncurves changed from symmetric to asymmetric based on\ntRMGin response to the direction of FCS magnetocrys-\ntalline anisotroy. The maximum ∆ Rratio of the bilay-\ners was more than an order of magnitude larger than\nthat of single-layer FCS films due to the reorientation of\nAFM moments via the FM rotation. We also observed\nthat the AFM moments could fully rotate when the field\nsweep was along the hard axes of FCS but could not fully\nrotate along the easy axes of FCS, demonstrating the im-\npact of FCS magnetocrystalline anisotropy to govern the\nAFM rotation. These results provide profound insights\ninto the control of the AFM moments and promote the\napplication of AFM in spintronics.\nACKNOWLEDGMENTS\nThe authors gretefully acknowledge M. Kuwahara and\nK. Saitoh for their invaluable support. Part of this work\nwas supported by the Japan Society for the Promotionof\nScience(JSPS)ProgramforAdvancingStrategicInterna-\ntionalNetworkstoAcceleratethe CirculationofTalented\nResearchers.\n∗Electronic mail: t.hajiri@numse.nagoya-u.ac.jp\n[1] H. V. Gomonay and V. M. Loktev, Low Temp. Phys. 40,\n17 (2014).\n[2]T. 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Phys. 73, 6393 (1993).\n[41] T. Hajiri, T. Yoshida, S. Jaiswal, M. Filianina, B. Bori e,\nH. Ando, H. Asano, H. Zabel, and M. Kl¨ aui, Phys. Rev.\nB94, 184412 (2016)."    },    {        "title": "1703.01849v1.Spindynamics_in_the_antiferromagnetic_phases_of_the_Dirac_metals__A_MnBi__2____A___Sr__Ca_.pdf",        "content": "Spin dynamics in the antiferromagnetic phases of the\nDirac metals AMnBi 2(A=Sr, Ca)\nM. C. Rahn,1,\u0003A. J. Princep,1A. Piovano,2J. Kulda,2Y. F. Guo,3, 4Y. G. Shi,5and A. T. Boothroyd1,y\n1Clarendon Laboratory, Department of Physics, University of Oxford, Oxford, OX1 3PU, United Kingdom\n2Institut Laue-Langevin, 6 rue Jules Horowitz, 38042 Grenoble Cedex 9, France\n3School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China\n4CAS Center for Excellence in Superconducting Electronics (CENSE), Shanghai 200050, China\n5Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n(Dated: March 7, 2017)\nThe square Bi layers in AMnBi 2(A= Sr, Ca) host Dirac fermions which coexist with antiferro-\nmagnetic order on the Mn sublattice below TN= 290 K (Sr) and 270 K (Ca). We have measured\nthe spin-wave dispersion in these materials by triple-axis neutron spectroscopy. The spectra show\npronounced spin gaps of 10 :2(2) meV (Sr) and 8 :3(8) meV (Ca) and extend to a maximum energy\ntransfer of 61{63 meV. The observed spectra can be accurately reproduced by linear spin-wave theory\nfrom an Heisenberg e\u000bective spin Hamiltonian. Detailed global \fts of the full magnon dispersion are\nused to determine the in-plane and inter-layer exchange parameters as well as on the magnetocrys-\ntalline anisotropy constant. To within experimental error we \fnd no evidence that the magnetic\ndynamics are in\ruenced by the Dirac fermions.\nPACS numbers: 75.30.Ds, 75.25.-j, 75.30.Gw, 74.70.Xa\nI. INTRODUCTION\nFollowing the observation of the topological proper-\nties of electrons on the honeycomb layers of graphene,\nthe universal characteristics of massless dispersing low-\nenergy quasiparticles have been realized across a variety\nof condensed matter systems. The ternary bismuthides\nAMnBi 2(A= Ca, Sr)1{4are a recent addition to this\nfamily of so-called Dirac materials. The Bi square lay-\ners of AMnBi 2have been found to show the same un-\nusual transport characteristics as graphene or topolog-\nical insulators5{8. Due to the suppression of backscat-\ntering processes, the electronic and thermal conductivity\nare enhanced, and the large separation of Landau levels\nproduces a large linear magnetoresistance. Indeed, angle-\nresolved photoemission spectroscopy (ARPES) has pro-\nvided direct evidence of the linear band crossings in both\nSrMnBi 2and CaMnBi 29,10, with a highly anisotropic\nDirac cone.\nAmong other Dirac materials, these bismuthides at-\ntract special interest because their Dirac fermions may\ncouple to transition-metal states, promising an indirect\nexperimental handle to tune the topological bands. Be-\nlowTSr\nN'290 K and TCa\nN'270 K, the large diva-\nlent Mn (3d5;S= 5=2) magnetic moments of magnitude\n\u00193:7\u0016Bin these materials align parallel to the c-axis\nand form antiferromagnetic structures11. The two com-\npounds were found to di\u000ber in the sign of their interlayer\ncoupling, resulting in ferro- and antiferromagnetic stack-\ning of N\u0013 eel-ordered layers in CaMnBi 2and SrMnBi 2,\nrespectively11. An interpretation based on \frst princi-\nples calculations suggests that in the ferromagnetically\nstacked case (CaMnBi 2), the Dirac bands may provide\nan itinerant inter-layer exchange path and thus directly\ncouple to the magnetic ground state11. This appeared tobe supported by a weak resistivity anomaly observed at\nTNin CaMnBi 2, but not in SrMnBi 211. Earlier transport\nstudies, however, had not registered such an anomaly in\neither SrMnBi 2(Ref. 4) or CaMnBi 2(Refs. 12 and 13).\nIn metallic magnets a coupling between the ordered\nmagnetic moments and conduction electron states can\nreveal itself in the magnetic excitation spectrum. For ex-\nample, there can be damping due to spin-wave decay into\nthe Stoner continuum, anomalies in the magnon disper-\nsion due to modi\fcations of the exchange interactions by\nconduction electron states, or gap formation due to an\nadditional Kondo energy scale.\nHere we report on a single-crystal neutron inelastic\nscattering study of SrMnBi 2and CaMnBi 2in the mag-\nnetically ordered state. Our analysis shows that the\nmagnon spectrum in both materials can be accurately\nreproduced from a Heisenberg model describing a local-\nmoment, quasi-two-dimensional (2D) antiferromagnet.\nThe model includes nearest- and next-nearest-neighbor\nin-plane exchange interactions and a weak inter-layer ex-\nchange interaction, together with an easy-axis anisotropy.\nWe did not \fnd any anomalies that would suggest sig-\nni\fcant coupling between the magnons and conduction\nelectron states. The interlayer coupling is smaller than\nfound in the reference compound BaMn 2Bi2, consistent\nwith the larger separation of the Mn spins along the c-\naxis inAMnBi 2.\nII. EXPERIMENTAL DETAILS\nThe preparation and characterisation of the single\ncrystals used in the experiments has been reported\npreviously11. Polycrystalline AMnBi 2was \frst synthe-\nsized by solid-state reaction of the elements. SinglearXiv:1703.01849v1  [cond-mat.str-el]  6 Mar 20172\nFIG. 1. (color online). The crystal and magnetic struc-\ntures of SrMnBi 2(a\u00194:58\u0017A,c\u001923:14\u0017A, magnetic space\ngroupI40=m0m0m) and CaMnBi 2(a\u00194:50\u0017A,c\u001911:07\u0017A,\nmagnetic space group P40=n0m0m)11. The magnetic propaga-\ntion vectors kindicated in this \fgure describe the magnetic\nstructures m(rj) (in lattice coordinates rj) by the relation\nm(rj) =m(0) exp(2\u0019ik\u0001rj), where m(0) is the magnetic\nmoment at an arbitrary origin located on a Mn site. For clar-\nity, the origin of the SrMnBi 2unit cell has been shifted by\n(1\n2;0;1\n4) relative to the conventional cell. The exchange paths\nJ1,J2andJcare indicated by red lines.\ncrystals were then grown from self-\rux in an alumina\ncrucible. Electron-probe microanalysis con\frmed near-\nideal stoichiometry, with a small ( \u00192%) Bi de\fciency\nin the Sr compound (for details, see Ref. 11). Lab-\noratory x-ray di\u000braction measurements con\frmed the\ntetragonal crystal structures reported previously,1,2space\ngroupsI4=mmm (SrMnBi 2) andP4=nmm (CaMnBi 2),\nsee Fig. 1. Magnetization measurements on the batch of\ncrystals used here were consistent with previous studies\n(see supplemental material14).\nNeutron inelastic measurements were performed at the\nInstitut Laue{Langevin on the triple-axis neutron spec-\ntrometer IN8 (Ref. 15) with the FlatCone detector.16By\nkeeping the outgoing energy \fxed and recording rock-\ning scans at various incident energies, this setup allows\nan e\u000ecient collection of constant energy-transfer maps\ncovering a wide range of reciprocal space. The Flat-\nCone array of analyzer crystals and helium tube detec-\ntors consists of 31 channels spaced by 2.5o, thus covering\na 75orange of scattering angle. Throughout the study,\nthe FlatCone was used with its Si (111) analyzer crys-\ntals selecting a \fxed outgoing wavevector of kf= 3\u0017A\n0 50 100 150 200 250 300\nT (K)-20246810normalized intensity\nI ( T ) - I ( 315 K ) CaMnBi2\n(cooling)(b)\nforbidden nuclear Bragg peak (100)\npower law fit\ndiffuse magnetic scattering at (10½)\n( scale factor ×20 )-20246810normalized intensity\nI ( T ) - I ( 315 K ) SrMnBi2\n(cooling)(a)\nweak nuclear Bragg peak (101)\npower law fit\ndiffuse magnetic scattering at (102)\n( scale factor ×100 )\nstrong nuclear Bragg peak (103)FIG. 2. (color online). Temperature dependence of the\ndi\u000berence intensity I(T)\u0000I(315 K) at selected wavevectors\n(see legend), recorded while cooling (a) SrMnBi 2and (b)\nCaMnBi 2. Power law \fts to the Bragg peaks yield transi-\ntion temperatures of TSr\nN= 287(5) K and TCa\nN= 264(2) K.\nAboveTN, incipient in-plane correlations contribute di\u000buse\nrods of magnetic scattering along (10 L). These \ructuations\nare enhanced towards TN(critical scattering) and then freeze\nout with the onset of inter-plane order (blue symbols). The\ndecrease in intensity of the (100) re\rection of CaMnBi 2be-\nlow 200 K is not consistent with previous data and should be\ndisregarded (see main text).\n(Ef= 18:6 meV). For energy transfers below and above\n40 meV (incoming energies Ei?58:6 meV), the double-\nfocusing Si (111) and pyrolitic graphite (002) monochro-\nmators were used, respectively. In four separate exper-\niments, the scattering from the SrMnBi 2and CaMnBi 2\nsingle crystals (of mass 3.3 g and 1.6 g, respectively) was\ninvestigated in the ( HK0) (aborientation) and ( H0L)\n(acorientation) scattering planes. Throughout this pa-\nper we give wavevectors in reciprocal lattice units (r.l.u.)\nq= (H;K;L )\u0011(H\u00022\u0019=a;K\u00022\u0019=b;L\u00022\u0019=c). The\nsamples were mounted in a standard top-loading liquid\nhelium cryostat. All spectra were recorded at a sample\ntemperature of approximately 5 K.\nIII. RESULTS AND ANALYSIS\nWhile cooling the samples in the acorientation, we\ntracked the intensities at selected positions in the ( H0L)\nplane of reciprocal space. Figures 2(a) and (b) show\nthe resulting temperature dependences for SrMnBi 2and3\nFIG. 3. (color online). Magnon spectrum of SrMnBi 2in the (a) ( H0L) and (b) (HK0) planes in reciprocal space. The data\nare illustrated by a stacking plot of constant-energy slices (left panels), and the best-\ft spin-wave model is represented by the\ncorresponding dispersion surface (right panels). A more quantitative comparison between data and simulation is provided in\nthe Supplemental Material.14\nCaMnBi 2, respectively. This includes the magnetic\nBragg contribution at (101) and (103) (for Sr) and (100)\n(for Ca), as well as the di\u000buse magnetic scattering at an-\nother position along the (10 L) direction away from the\nBragg condition. The data, here represented as the rela-\ntive change in the square root of the intensity, demon-\nstrate the order parameter characteristics of magnetic\nBragg scattering at the antiferromagnetic transitions.\nWe note that the decrease of the CaMnBi 2(100) mag-\nnetic scattering below 200 K is not consistent with our\nprevious powder neutron di\u000braction data,11and could be\ndue to a shift of the peak between two detector channels\nas the lattice contracts.\nAbove the ordering temperature, incipient in-plane\nmagnetic correlations form di\u000buse rods of magnetic scat-\ntering along the c\u0003direction of reciprocal space, as re-\nvealed at the (10 L) non-Bragg positions. When cooling\ntowardsTN, this di\u000buse scattering initially intensi\fes and\nthen subsides when the weaker inter-layer correlations set\nin and neutron spectral weight is con\fned to the Bragg\npeaks. Fitting a power law to the thermal variation of\nthe (101) (Sr) and (100) (Ca) peaks yields N\u0013 eel tempera-\ntures ofTSr\nN= 287(5) K and TCa\nN= 264(2) K. These val-\nues are consistent with previous single crystal bulk mea-\nsurements of transport and ARPES samples4,9,12,13, butdi\u000ber slightly from the values found in our earlier neutron\npowder di\u000braction study11. This di\u000berence is likely due\nto small structural or compositional variations among\nthe samples. The critical exponents \fSr= 0:15(3) and\n\fCa= 0:11(2) obtained from the power law \ft are much\nsmaller than the value \f= 0:365 of the three-dimensional\nHeisenberg model, indicating the reduced dimensionality\nof the magnetism in these systems. Due to the additional\nbismuth layers in the unit cells, the magnetism is more\ntwo-dimensional in AMnBi 2than in the related (122)\nmanganese arsenide BaMn 2As2,\f= 0:35(2) (Ref. 17).\nInstead, the inter-layer correlations compare well to the\nparent compounds of (122) iron-based superconductors,\ne.g.\f= 0:098(1) for SrFe 2As2(Ref. 18) and \f= 0:125\nfor BaFe 2As2(Ref. 19). A detailed description of the\npower law \ft to this data is provided in the Supplemen-\ntal Material.14\nThe measured neutron spectra are summarized in\nFig. 3 (SrMnBi 2) and Fig. 4 (CaMnBi 2). A more quanti-\ntative presentation of the data is provided in \fgures S2{\nS5 of the Supplemental Material.14Due to the periodicity\nof the antiferromagnetic order, the magnetic zone centers\nare located at ( HKL ) positions with ( H+K) andLboth\nodd integers for SrMnBi 2, and at positions with ( H+K)\nodd andLany integer for CaMnBi 2. For both com-4\nFIG. 4. (color online). Magnon spectrum of CaMnBi 2presented in the same way as in Fig. 3.\nM\n(1\n21\n20)Γ\n(000)X\n(1\n200)Γ\n(100)0102030405060E(meV)SrMnBi 2(a)\nM\n(1\n21\n20)Γ\n(000)X\n(1\n200)Γ\n(100)CaMnBi 2(b)\nFIG. 5. (color online). Magnon dispersion along high sym-\nmetry directions in the ( HK0) plane, for (a) SrMnBi 2and\n(b) CaMnBi 2. The black line indicates the best \ft from the\nlinear spin-wave model. Red markers represent the position\n(vertical bars) and full-width at half-maximum (horizontal\nlines) of gaussian \fts to cuts through the raw data along the\ncorresponding directions.\npounds, the spectra reveal a well-de\fned magnon disper-\nsion above spin gaps of approximately 10 meV (Sr) and\n8 meV (Ca). The magnons are highly dispersive parallel\nto the layers, but only weakly dispersive perpendicular tothe layers. For both samples the magnon bandwidth is\naround 50 meV for spin waves propagating in the ( HK0)\nplane and 3{4 meV along (10 L). Figure 5 represents more\nquantitatively the magnon dispersion in the ( HK0) plane\nas obtained from gaussian \fts to constant-energy cuts,\nand the left-hand panels of Figs. 6(a) and (b) illustrate\nthe out-of-plane dispersion by energy-wavevector slices of\nthe data along the (10 L) direction.\nTo obtain quantitative information on the magnetic\ncouplings we have compared the data with the linear\nspin-wave spectrum calculated from an e\u000bective spin\nHamiltonian that includes a Heisenberg coupling term\nand an Ising-like single-ion anisotropy,\n^H=X\nhi;jiJij^Si\u0001^Sj\u0000X\niD(^Sz\ni)2; (1)\nwhere we include nearest-neighbor ( J1) and next-nearest-\nneighbour ( J2) exchange constants, an inter-layer ex-\nchange interaction Jc, and the anisotropy constant D.\nThe exchange paths are shown in Fig. 1. Using the\nHolstein{Primako\u000b transformation of two interacting\nBose \felds, corresponding to the two collinear antifer-\nromagnetic sublattices, we obtain the dispersion relation\nE(q) =p\nA(q)2\u0000B(q)2 (2)5\n0 5 10 15\nL along (10L)05101520Energy (meV)\nSrMnBi2 (data)\n0 5 10 15\nL along (10L)SrMnBi2 (model)\n0 50 100 150 200Intensity (arb. units)\n0 2 4 6\nL along (10L)05101520\nCaMnBi2 (data)\n0 2 4 6\nL along (10L)CaMnBi2 (model)(b) (a)\n0 20 40 60 80Intensity (arb. units)\nFIG. 6. (color online). Out-of-plane dispersion of the magnon spectra of (a) SrMnBi 2and (b) CaMnBi 2, illustrated by slices\nalong the (10 L) direction in reciprocal space (intensity averaged over the range H= 0:95{1.05 r.l.u.). The left-hand panels show\ninterpolated plots of the data, and the right-hand panels give the corresponding best-\ft spin-wave spectra convoluted with the\ninstrumental resolution (for details, see text and Supplemental Material14). The superimposed red dashed line indicates the\ntheoretical dispersion using best-\ft parameters.\nwhere qis the magnon wavevector,\nA(q) =S[JAF(0)\u0000J F(0) +JF(q) + 2D]\nB(q) =SJAF(q)\nand\nJ(q) =X\nnJne2\u0019iq\u0001rn(3)\nis the Fourier transform of the exchange interac-\ntions. The subscripts F and AF refer to summa-\ntion over ferromagnetically- and antiferromagnetically-\naligned spins, respectively. The resulting di\u000berential\nscattering cross-section for single-magnon creation is\nd\u001b\nd\nd!=kf\nki\u0010\rr0\n2\u00112\nS(q;!) (4)\nS(q;!) =g2NSA(q)\u0000B(q)\nE(q)fn(!) + 1g\u000ef~!\u0000E(q)g\n(5)\nwhere ~!is the neutron energy transfer, kfandkiare the\noutgoing and incoming neutron wave vectors, \r= 1:913,\nr0is the classical electron radius, gthe Land\u0013 eg-factor,\nNthe number of magnetic ions per sublattice, Sthe spin\nquantum number, and n(!) = (e~!=kBT\u00001)\u00001the bo-\nson occupation number. Given the magnetic structures\nand exchange paths de\fned in Fig. 1, the explicit Fourier\nexchange terms for the case of SrMnBi 2are\nJSr\nAF(q) = 2J1[cos (\u0019H+\u0019K) + cos (\u0019H\u0000\u0019K)] +\n+ 2Jccos(\u0019L)\nJSr\nF(q) = 2J2[cos(2\u0019H) + cos(2\u0019K)]and, in the case of CaMnBi 2,\nJCa\nAF(q) = 2J1[cos (\u0019H+\u0019K) + cos (\u0019H\u0000\u0019K)]\nJCa\nF(q) = 2J2[cos(2\u0019H) + cos(2\u0019K)] + 2Jccos(2\u0019L):\nThis allows an analytical description of the spin gaps:\n\u0001Sr\u0019\u0001Ca\u00194p\nSJ1p\nSD; (6)\nwhere we have applied the appropriate approximations\nfor the present case ( J1\u001dJc,J1\u001dD; for full expres-\nsions see the Supplemental Material14). Similarly, the\nband width Wof the dispersion along (10 L) is given by\nWSr\u0019WCa\u00194p\nSJ1\u0010p\nSD+ 2jSJcj\u0000p\nSD\u0011\n:\n(7)\nIfJ1is the dominant exchange, as is found to be the\ncase here, then the maximum of the in-plane dispersion\nis\u00184SJ1. GivenJ1, we see from Eqs. (6) and (7) that\nin the relevant parameter regime the parameters Dand\nJcare determined by the size of the gap \u0001Sr=Caand\nband width of the out-of-plane modulation WSr=Ca, re-\nspectively. On the other hand, the balance between the\nparameters J1andJ2determines details of the dispersion\nat higher energies in the ( HK0) plane. For example, a\nlocal minimum of the dispersion at the M point, (1\n2;1\n2;0),\nas observed in both materials, will only occur for posi-\ntiveJ2, indicating a competition (frustration) between\nnearest- and next-nearest-neighbor exchange.\nWe \fnd that the above model is able to reproduce very\nwell all features in the data. For quantitative analysis we\nfolded and averaged the raw constant-energy maps of re-\nciprocal space into tiles of 2 \u00022 r.l.u. With the data in\nthis reduced form we could compare it to the model af-\nter convolution of the theoretical spectrum, Eqs. (4){(5),6\nwith an energy- and wavevector-dependent broadening\nfunction to take into account the instrumental resolution.\nA phenomenological gaussian broadening of the ana-\nlytical dispersion proved insu\u000ecient to achieve a con-\nsistent global \ft to the data, particularly for the low-\nenergy part of the magnetic dispersion in the acplane.\nInstead, it was necessary to take into account the res-\nolution of the triple-axis spectrometer, which was cal-\nculated with the RESTRAX ray-tracing algorithm20,21.\nOur procedure to determine the parameters of the spin\nHamiltonian J1,J2,JcandDwas carried out in three\nsteps: First, a global \ft of all data, using phenomeno-\nlogical gaussian broadening of the dispersion, produced\nrough estimates of all parameters. Using these as start-\ning values and \fxing the in-plane exchange interactions\nJ1andJ2, we obtained precise bounds on the inter-layer\nexchangeJcand anisotropy Dby \ftting the resolution-\nconvoluted spectrum for low energies (0{20 meV) to an\nenergy{wavevector slice with wavevector along (10 L), as\nillustrated in Fig. 6. Finally, J1andJ2were re\fned by\n\ftting the in-plane ( ab) dispersion at high energies (30{\n44 meV) using gaussian broadening.\nFigure 5 provides a quantitative plot of the \fts of the\ndispersions in the ab-plane. A more detailed description\nof the data processing, \ftting and error estimation, and\nan explicit comparison of the data and best \fts is pro-\nvided in the Supplemental Material.14\nIV. DISCUSSION\nThe exchange parameters for SrMnBi 2and CaMnBi 2\nobtained from the \fts are summarized in Table I. Apart\nfrom the opposite sign of the inter-layer exchange Jc,\nthere are no signi\fcant di\u000berences between the parame-\nters of SrMnBi 2and CaMnBi 2. The absolute values of\nJ1andJ2are slightly larger in the case of CaMnBi 2, con-\nsistent with the smaller nearest-neighbor spacing ( dNN).\nThe magnitudes of Jcfor the two compounds, which are\nthe same to within the error, are much smaller than J1\nandJ2, con\frming the quasi-2D character of the mag-\nnetism in these materials. Notably, these results are in\ngood agreement with previous estimations based on \frst\nprinciples calculations of the electronic structure, which\ngave an average in-plane exchange of SJab\u001930 meV and\njSJcj\u00190:3 meV11.\nRegarding the magnetocrystalline anisotropy, we ob-\nserve thatDis enhanced by a factor 1.8 in SrMnBi 2com-\npared with CaMnBi 2. According to the initial structure\ndeterminations at room temperature1,2, the local envi-\nronment of the Mn ion is similar in both compounds: The\nMnBi 4tetrahedra are elongated by \u001914 % along c and the\nligand distances are dCa\nMn\u0000Bi= 2:87(1) \u0017A anddSr\nMn\u0000Bi=\n2:89(1) \u0017A. The signi\fcant di\u000berence in anisotropy may\ntherefore point to unknown structural distortions at 5 K\n(at present, no full re\fnement of crystallographic param-\neters at low temperatures is available). The anisotropy\nin good agreement with the result of our earlier densityfunctional prediction ( SDCa\nDFT= 0:3 meV11), as was also\nthe case with the exchange constants.\nIt is instructive to compare the present results to two\navailable inelastic neutron studies of the related com-\npounds BaMn 2Bi222and BaMn 2As223. The correspond-\ning parameters for these materials are quoted in Ta-\nble I. The pnictide-coordinated magnetic Mn2+layers in\nBaMn 2Bi2and BaMn 2As2(\\122 materials\") are analo-\ngous to those in the 112 materials investigated in the\npresent study. However, the I4=mmm 122 compounds\ndo not feature additional pnictide layers (which carry\nthe Dirac bands in the present case). Hence, while the\nin-plane Mn{Mn spacing is very similar, the spacing of\nthe magnetic layers in the 122 compounds is only 58{\n66% of that in CaMnBi 2and SrMnBi 2. Both BaMn 2Bi2\nand BaMn 2As2form antiferromagnetically stacked layers\nof N\u0013 eel type order, in analogy to SrMnBi 2. As may be\nexpected from these circumstances, we \fnd that the in-\nplane exchange interactions in 122 compounds are simi-\nlar or identical to those in 112 compounds. On the other\nhand, in the present 112 materials the inter-plane ex-\nchange is signi\fcantly reduced. This is consistent with\nthe much higher transition temperatures and the smaller\nseparation of the Mn layers in the 122 materials com-\npared with the 112 compounds.\nWe \fnd no evidence that the additional Bi layers in 112\nmaterials, which host the Dirac fermions, cause any qual-\nitative changes in the magnon spectrum, such as anoma-\nlous broadening or dispersion. The instrument's simu-\nlated energy resolution provides an upper bound on the\nin\ruence of such e\u000bects. The characteristics of the Bragg\n(0.5{1:0 meV) and vanadium (1{4 meV) widths of energy\nresolution are illustrated in the Supplemental Material.14\nBy contrast, neutron inelastic measurements of many\niron-based superconductors show obvious signatures of a\nstrong hybridization of magnetic and itinerant states. A\ntypical example is SrFe 2As225, which shows a crossover\ninto the regime of itinerant ( Stoner ) spin \ructuations.\nThis manifests itself as an increased dampening of spin\n\ructuations (i.e. a broadening of the neutron spectrum)\nabove a characteristic energy of approximately 80 meV.\nAs in the 122 compounds, both J1andJ2are posi-\ntive (antiferromagnetic) in SrMnBi 2and CaMnBi 2, re-\nsulting in frustration between nearest- and next-nearest-\nneighbor interactions. The theoretical phase diagram\nof the frustrated J1{J2model on a square lattice has\nbeen investigated extensively in the context of iron-based\nsuperconductors.26{28There is special interest in this\nphase diagram owing to a possible quantum critical point\nand spin liquid phase around J2=J1\u00191\n2. This regime\nseparates two distinct ordered magnetic phases, with\nN\u0013 eel type order for J2=J1<1\n2and stripe antiferromag-\nnetic order for J2=J1>1\n2. Both 112 and 122 Mn-based\ncompounds exhibit dominant nearest-neighbor exchange,\nwithJ2=J1\u00190:3. According to one study the exchange\nand anisotropy parameters for AMnBi 2places these ma-\nterials close to the phase boundary between N\u0013 eel-ordered\nand frustrated paramagnetic phases.28The resulting7\nTABLE I. Exchange parameters, magnetocrystalline anisotropy constants and spin gaps of SrMnBi 2and CaMnBi 2obtained\nfrom a \ft of the linear spin-wave model, as described in the text. The parameters can be related to the nearest-neighbor\n(dNN) and interlayer ( dc) Mn{Mn atomic spacings, the ordered magnetic moment \u0016, and the ordering temperature TN.11The\ncorresponding values for two related Mn pnictides are reproduced below.22,23\nSJ1(meV)SJ2(meV)SJc(meV)SD(meV) \u0001 (meV) dNN(\u0017A)dc(\u0017A)\u0016(\u0016B)TN(K)\nSrMnBi 2 21.3(2) 6.39(15) 0.11(2) 0.31(2) 10.2(2) 3.24 11.57 3.75(5) 290.2(3)\nCaMnBi 2 23.4(6) 7.9(5) -0.10(5) 0.18(3) 8.3(8) 3.18 11.07 3.73(5) 267.0(1.6)\nBaMn 2Bi2[22] 21.7(1.5) 7.85(1.4) 1.26(2) 0.87(15)a16.29(26) 3.18 7.34 3.83(4) 387.2(4)\nBaMn 2As2[23] 33(3) 9.5(1.3) 3.0(6) - - 2.95 6.73 3.88(4) 625(1)\naThe value of SDfor BaMn 2Bi2was misquoted in Ref. 22. In this table we give the correct value24.\nquantum \ructuations could explain some of the observed\nreduction in ordered magnetic moment ( '3.7\u0016B) com-\npared to the ideal local-moment value of 5 \u0016B23. By con-\ntrast, in parent compounds of iron-based superconduc-\ntors such as BaFe 2As2and SrFe 2As2,J1andJ2are of\nsimilar magnitude, resulting in stripe-antiferromagnetic\norder.\nV. CONCLUSIONS\nIn summary, we have performed a comprehensive\ntriple-axis neutron scattering study of the anisotropic\nDirac materials SrMnBi 2and CaMnBi 2, with the aim\nof searching for possible in\ruences of the unusual band\ntopology at their Fermi surfaces on their magnetism. In\nparticular, for CaMnBi 2our previous \fndings had indi-\ncated that the Bi 6 px;ybands may play a role in mediat-\ning the magnetic exchange between Mn layers.\nIn both compounds, we observed well-de\fned magnon\nspectra consistent with local-moment, semi-classical an-\ntiferromagnetism. Using linear spin-wave theory to de-\nscribe the neutron spectra we have identi\fed and quan-\nti\fed all relevant exchange and anisotropy parameters\nof a Heisenberg model for the two compounds. In both\ncases, all details of the dispersion are well reproduced\nby the model and there is no indication of anomalous\nbroadening or dispersion to within experimental preci-\nsion. The absolute values of the exchange parameters in-\ndicate no substantive di\u000berences between the compounds\n(aside from opposite interlayer coupling).\nThese results suggest that di\u000berent routes have to be\nfound to achieve an entanglement of magnetic order and\nnon-trivial band topology. One very promising option\nis the substitution of magnetic rare earth ions on the A\nsite, providing a more direct interaction with the relevant\nBi layers. In particular, a strong response of the trans-\nport properties to rare earth magnetic order has recently\nbeen observed in EuMnBi 229, along with the trademark\nsignatures of Dirac transport30. Furthermore, recent\nhigh-resolution ARPES results and \frst principles cal-\nculations identify YbMnBi 2as a type-2 Weyl semimetal\nwith canted antiferromagnetic order31. The latter study\nfurther suggests that this state would be tuned to a Diracmetal by spin alignment. Naturally, it would be of great\ninterest to perform analogous inelastic neutron studies of\nthe magnetic ground states in those materials.\nNote added in proof: After submission of this\nmanuscript, a Raman spectroscopic study of SrMnBi 2\nand CaMnBi 2was reported by Zhang et al.32. Raman\nspectroscopy probes the spin dynamics through a small\nnumber of characteristic frequencies which are associated\nwith van-Hove singularities in the two-magnon density of\nstates. The authors of Ref. 32 interpret their data us-\ning a similar spin Hamiltonian as in the present study\nbut without the magnetocrystalline anisotropy term ( D\nin our study). Their analysis yields values for the spin\nexchange parameters J1andJ2that are similar to our re-\nsults, but produces anomalously large values of the inter-\nlayer exchange Jcfor both materials (one order of mag-\nnitude larger than in our study or in other related mate-\nrials). The authors of Ref. 32 suggest that this enhanced\ncoupling is caused by the Bi Dirac bands. We would like\nto draw attention to the fact that the parameters J1,Jc\nandDare strongly correlated in modelling key features of\nthe magnon dispersion (see Eqs. 6 and 7), so the neglect\nofDin the Raman analysis could signi\fcantly a\u000bect the\nobtained values of Jc. We note that the Raman value\nofJcwould imply an inter-layer dispersion of the one-\nmagnon spectrum at a factor of 11 (Sr) or 7 (Ca) larger\nthan that found here directly by neutron spectroscopy\n(Fig. 6, Eq. 7).\nACKNOWLEDGMENTS\nWe would like to thank Dr. Paul Ste\u000bens (ILL) for\nproviding the FLATCONE data treatment suite, and\nDr Stuart Calder for helpful clari\fcations on the re-\nsults reported in Ref. 22. This work was supported\nby the U.K. Engineering and Physical Sciences Re-\nsearch Council (grant no. EP/J017124/1), the Chi-\nnese National Key Research and Development Program\n(2016YFA0300604) and the Strategic Priority Research\nProgram (B) of the Chinese Academy of Sciences (Grant\nNo. XDB07020100). MCR is grateful to the Oxford Uni-\nversity Clarendon Fund for provision of a scholarship.8\n\u0003marein.rahn@physics.ox.ac.uk\nya.boothroyd@physics.ox.ac.uk\n1G. Cordier and H. Sch afer, Z. Naturforsch. 32, 383 (1977).\n2E. Brechtel, G. Cordier, and H. Sch afer, Z. 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Shi,5and A. T. Boothroyd1\n1Clarendon Laboratory, Department of Physics, University of Oxford, Oxford, OX1 3PU, United Kingdom\n2Institut Laue-Langevin, 6 rue Jules Horowitz, 38042 Grenoble Cedex 9, France\n3School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China\n4CAS Center for Excellence in Superconducting Electronics (CENSE), Shanghai 200050, China\n5Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences,\nBeijing 100190, China\n1. Power law \ft of magnetic Bragg peaks\nAs described in the manuscript, the thermal varia-\ntion of the intensities of the (101) (SrMnBi 2) and (100)\n(CaMnBi 2) magnetic peaks were \ftted with a power law.\nTo reproduce the incipient \ructuations above TN, the\npower law \ftting function was convoluted by a Lorentzian\ndistribution of the ordering temperature:\nI/A(TN\u0000T)2\f\u00031\n\r2+ (T\u0000TN)2:\nThe \fve (unconstrained) \ftting parameters where then\n(1) a constant background, (2) an overall scale factor A,\n(3) the critical exponent \f, (4) the N\u0013 eel temperature TN\nand (5) the Lorentzian full width at half maximum \r\n(see Fig. S1). While TNalways converges to the results\nquoted in the main text, the critical exponents \fweakly\ndepend on the selected \ftting range. In the main text\nwe therefore quote the mean \ft results and standard de-\nviations of separate \fts to data ranges varying between\n(TN\u000065 K) to 313 K and ( TN\u00005 K) to 313 K.\n240 260 280 300\nT ( K )0246810normalized intensity\nI ( T ) - I ( 315 K )CaMnBi 2(100)\ndata\nbestfit (236K – 313 K)\nspread of TN\nFig. S1 (color online). Lorentzian-convoluted power law \ft to\nthe relative thermal variation of the scattering amplitude at\nthe magnetic Bragg re\rection (100) of CaMnBi 2.2. Magnetic susceptibility\nTo check for consistency with our previous study11, we\nmeasured the thermal variation of the magnetic proper-\nties of the batch of samples probed by neutron inelastic\nscattering (see Fig. S2). Single crystals were \frst aligned\non an x-ray di\u000bractometer. The magnetic susceptibility\n02468χ ( 10-3 emu mol-1 Oe-1 )SrMnBi2   µ0H=1 TH || a, FC\nH || a, ZFC\nH || c, FC\nH || c, ZFC\n0 50 100 150 200 250 300 350\nT (K)0123456χ ( 10-3 emu mol-1 Oe-1 )CaMnBi2   µ0H=1 T\nFig. S2 (color online). Magnetic susceptibility of SrMnBi 2\n(top) and CaMnBi 2(bottom), measured on the batch of single\ncrystals probed by neutron spectroscopy.10\nwas then measured on a Magnetic Properties Measure-\nment System (Quantum Design) with a magnetic \feld\nof \rux density 1 Tesla applied either in-plane or out-of-\nplane. The key characteristics are qualitatively consis-\ntent with our earlier results11. The present samples have\na larger Curie contribution which may be attributed to\nparamagnetic impurities induced by sample decay. Since\nthe magnon dispersion is a coherent response of the main\ncrystal phase, such impurities are not of relevance to the\npresent study, apart from a small contribution to the dif-\nfuse background scattering.\n3. Analytical expressions for features in the\nmagnon dispersion\nThe linear spin-wave model yields the following ana-\nlytical expressions for the spin gaps:\n\u0001Sr=S\u0002\n(4J1+ 2Jc+ 2D)2\u0000(4J1+ 2Jc)2\u00031\n2\n\u00194p\nSJ1p\nSD\n\u0001Ca=S\u0002\n(4J1+ 2D)2\u0000(4J1)2\u00031\n2\n\u00194p\nSJ1p\nSD;\nwhere we have applied the appropriate approximations\nfor the present case ( J1\u001dJc;J1\u001dD). Similarly, for\nthe bandwidth Wof the dispersion along (10 L) we obtain\nWSr=S\u0002\n(4J1+ 2Jc+ 2D)2\u0000(4J1\u00002Jc)2\u00031\n2\u0000\n\u0000S\u0002\n(4J1+ 2Jc+ 2D)2\u0000(4J1+ 2Jc)2\u00031\n2\n\u00194p\nSJ1\u0010p\nSD+ 2SJc\u0000p\nSD\u0011\nWCa=S\u0002\n(4J1\u00004Jc+ 2D)2\u0000(4J1)2\u00031\n2\u0000\n\u0000S\u0002\n(4J1+ 2D)2\u0000(4J1)2\u00031\n2\n\u00194p\nSJ1\u0010p\nSD\u00002SJc\u0000p\nSD\u0011\n:\n4. Analysis of neutron spectra\nFor both materials, and for each in two crystal orien-\ntations (momentum transfers in the ( HK0) and (H0L)\nplanes of reciprocal space), rocking scans were recorded\nat a number of energy transfers up to 60 meV. The scat-\ntering was recorded in the FlatCone analyser{detector\nassembly16producing a set of constant-energy maps.\nIn each rocking scan the wide FlatCone detector bank\nis mapped onto an arc in reciprocal space intercepting\nseveral Brillouin zones. The spectral weight in these\nmaps shows contributions from Bragg di\u000braction (at\nE= 0 meV), acoustic phonons (below E\u001910 meV)\nand magnons (above E= 8{10 meV). In addition, forE6= 0 meV, the data contain accidental Bragg re\rections\n(i.e. the strong elastic signal passes through the analyzer\nby parasitic scattering), as well as powder rings corre-\nsponding to Bragg di\u000braction by the sample environment.\nThe magnon spectral weight can clearly be distinguished\nfrom the phonon- and accidental Bragg scattering in ev-\nery dataset from the form of the scattering and from the\nreduction in the magnetic signal with jqj.\nTo allow a direct comparison with the model, each raw\ndataset was corrected in several steps. First, all intensity\nother than a di\u000buse background and aluminium powder\nrings were masked. This masked data was then radially\naveraged (along the rocking angle). Next, both the raw\ndataset and the radially averaged powder/background\ndataset were divided by the magnetic form factor of\nMn2+. After subtraction, the trajectories of intense spu-\nrious (parasitic) Bragg scattering were deleted from the\ndata manually. The resulting corrected constant-energy\nmaps were then interpolated to a regular grid and di-\nvided into equivalent tiles of 2 \u00022 reciprocal lattice units\n(r.l.u.). Finally, the arithmetic mean of these cells was\ncalculated and all relevant symmetries (mirror planes and\n4- or 2-fold rotation of the ( HK0) and (H0L) reciprocal\nlattice planes, respectively) were applied in order to distil\nthe full statistical signi\fcance of the data. The resulting\nconstant-energy intensity maps for both compounds and\nboth sample orientations are shown in Figs. S4{S7.\n5. Fitting procedure\nThe constant-energy maps thus obtained are in a con-\nvenient form for \ftting to the spin-wave model. However,\nin particular for the low energy part of the magnetic dis-\npersion, a phenomenological gaussian broadening of the\nanalytical dispersion (along the reciprocal space and en-\nergy dimensions) proved insu\u000ecient. Instead, it was nec-\nessary to take into account the resolution of the triple-\naxis spectrometer. This was done by numerically sim-\nulating four-dimensional resolution matrices using the\nRESTRAX ray tracing algorithm20,21. This algorithm\ntakes into account a full physical model of all relevant\ncomponents (including collimators, monochromator, an-\nalyzer, detectors) of the IN8 instrument. To calculate the\nresolution matrix for a particular instrument con\fgura-\ntion, the program was set to trace 5000 random neutron\nevents. The resulting cloud of ( H;K;L;E ) positions was\n\ftted to a four-dimensional ellipsoid.\nThe energy resolution of the triple-axis spectrometer\ndepends only weakly on the momentum transfer, but in-\ncreases with energy transfer. While the energy Bragg-\nwidth of the resolution ellipsoid remains smaller than\n1 meV (standard deviation) up to \u0001 E= 60 meV, the\nvanadium width increases up to \u00194 meV in this range.\nFigure S3 illustrates these characteristics, which rep-\nresent the upper bounds within which we can exclude\nanomalies in the dispersion.11\n0 20 40 60\nenergy transfer (meV)01234energy resolution (meV)Bragg width\nVanadium width\nFig. S3 (color online). Simulated energy resolution of the\nIN8 spectrometer (RESTRAX algorithm) for the experimen-\ntal conditions of the present study.\nIn order to simulate the e\u000bect of the resolution on the\ndistribution of magnetic intensity, the resolution ellip-\nsoids were calculated for every pixel of a full dataset.\nFor a particular set of model parameters, the contribu-\ntion to the intensity at a pixel was calculated by per-\nforming a four-dimensional convolution of the ellipsoid\nwith the analytical dispersion. Finally, the so-obtained\nresolution-corrected intensity corresponding to the full\nexperimental dataset was folded into 2 \u00022 r.l.u. tiles as\ndescribed for the raw data above.\nWhile this procedure produces a satisfactory simu-\nlation of the data, it is unfortunately computationallytoo intensive to include in an e\u000ecient global \ftting rou-\ntine. The best-\ft parameters were therefore determined\nin three stages: First, a global \ft (with phenomenological\nbroadening) of all available datasets (all energies, both\norientations) was used to estimate rough starting values\nfor all parameters. Secondly, for an appropriate grid of\nJcandDvalues, resolution-convoluted constant-energy\nmaps corresponding to the low energy ( E\u001420 meV) data\nsets were calculated. Momentum transfer slices along the\n(10L) direction were then calculated for both data and\nmodel. The \u001f2maps resulting from this comparison are\nshown in Figs. S8(a) and (c) for SrMnBi 2and CaMnBi 2,\nrespectively, with the contour of one standard deviation\nindicated by a red line. The best \ft (10 L) slices thus\ndetermined are illustrated in Fig. 6 of the main article.\nFinally, with JcandD\fxed, global \fts of the high en-\nergy dispersion (30 \u0014E\u001445 meV, both HK0 andH0L\norientations) were performed for an appropriate grid of\nJ1andJ2values, using a phenomenological broadening\nof the dispersion. The resulting \u001f2maps are shown in\nFig. S8(b) and (d) for SrMnBi 2and CaMnBi 2, respec-\ntively. The J1andJ2parameters are seen to be strongly\ncorrelated in both cases, with a minimum in \u001f2extending\nalong a straight line. However, although \u001f2does not vary\nsigni\fcantly along this line, there are weak features in the\nhigh energy ( HK0) maps which do depend sensitively on\n(J1;J2) along this line. These features enabled us to es-\ntablish empirically the errors on J1andJ2quoted in the\nmain article and indicated by dashed lines in Fig. S8.\nFigures S4{S7 give a comparison of all the data with\nthe corresponding simulations from the best-\ft model.\n\u0003marein.rahn@physics.ox.ac.uk\nya.boothroyd@physics.ox.ac.uk\n1G. Cordier and H. Sch afer, Z. Naturforsch. 32, 383 (1977).\n2E. Brechtel, G. Cordier, and H. Sch afer, Z. Naturforsch.\n35, 1 (1980).\n3J. H. Shim, K. Haule, and G. Kotliar, Phys. Rev. B 79,\n060501 (2009).\n4J. K. Wang, L. L. Zhao, Q. Yin, G. Kotliar, M. S. Kim,\nM. C. Aronson, and E. Morosan, Phys. Rev. B 84, 064428\n(2011).\n5Y. Ran, F. Wang, H. Zhai, A. Vishwanath, and D.-H. Lee,\nPhys. Rev. B 79, 014505 (2009).\n6P. Richard, K. Nakayama, T. Sato, M. Neupane, Y.-M. Xu,\nJ. H. Bowen, G. F. Chen, J. L. Luo, N. L. Wang, X. Dai,\nZ. Fang, H. Ding, and T. Takahashi, Phys. Rev. Lett. 104,\n137001 (2010).\n7T. Morinari, E. Kaneshita, and T. Tohyama, Phys. 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Commun.\n7, 13833 (2016).13\n2\n1\n0\n8.5meV\n1\n0            L (r.l.u.)\n9meV\n 9.5meV\n 10meV\n2\n1\n0\n10.5meV\n1\n0            L (r.l.u.)\n11meV\n 11.5meV\n 12meV\n2\n1\n0\n12.5meV\n1\n0            L (r.l.u.)\n13meV\n 13.5meV\n 14meV\n2\n1\n0\n14.5meV\n1\n0            L (r.l.u.)\n15meV\n 16meV\n 18meV\n2\n1\n0\n20meV\n1\n0            L (r.l.u.)\n22meV\n 26meV\n 30meV\n2\n1\n0\n34meV\n1\n0            L (r.l.u.)\n38meV\n 42meV\n 46meV\n2\n1\n0\n50meV\n0 0.5 1.0 1.5 2\nH (r.l.u.)1\n0            L (r.l.u.)\n54meV\n0 0.5 1.0 1.5 2\nH (r.l.u.)\n58meV\n0 0.5 1.0 1.5 2\nH (r.l.u.)\n62meVModel\nData\n0 0.5 1.0 1.5 2\nH (r.l.u.)\n0 100 200 300 400 500 600\nintensity (arb. units)\nFig. S4 (color online). SrMnBi 2: Constant-energy maps of a 2 \u00022 r.l.u. area of the ( H0L) plane of reciprocal space. Each\ndouble panel shows the processed data (lower panel, explanation see text) as well as the best \ft with a phenomenologial gaussian\nbroadening (upper panel).14\n2\n1.5\n1\n2 meV\n 10 meV\n 16 meV\n2\n1.5\n1\n20 meV\n 25 meV\n 30 meV\n2\n1.5\n1\n34 meV\n 38 meV\n 42 meV\n0 0.5 1 1.5\nH (r.l.u.)2\n1.5\n1\n46 meV\n0 0.5 1 1.5 2\nH (r.l.u.)\n0 0.5 1 1.5\nH (r.l.u.)\n50 meV\n0 0.5 1 1.5 2\nH (r.l.u.)\n0 0.5 1 1.5\nH (r.l.u.)\n54 meVModel Data\n0 0.5 1 1.5 2\nH (r.l.u.)\n0 50 100 150\nintensity (arb. units)\n2\n1.5\n1\n2 meV\n 10 meV\n 16 meV\n2\n1.5\n1\n20 meV\n 25 meV\n 30 meV\n2\n1.5\n1\n34 meV\n 38 meV\n 42 meV\n0 0.5 1 1.5\nH (r.l.u.)2\n1.5\n1\n46 meV\n0 0.5 1 1.5 2\nH (r.l.u.)\n0 0.5 1 1.5\nH (r.l.u.)\n50 meV\n0 0.5 1 1.5 2\nH (r.l.u.)\n0 0.5 1 1.5\nH (r.l.u.)\n54 meVModel Data\n0 0.5 1 1.5 2\nH (r.l.u.)\n0 50 100 150\nintensity (arb. units)\nFig. S5 (color online). SrMnBi 2: Constant-energy maps of a 2 \u00022 r.l.u. area of the ( HK0) plane of reciprocal space. Each\ndouble panel shows the processed data (right panel, explanation see text) as well as the best \ft with a phenomenologial gaussian\nbroadening (left panel).15\n2\n1\n0\n6meV\n1\n0            L (r.l.u.)\n6.5meV\n 7meV\n 7.5meV\n2\n1\n0\n8meV\n1\n0            L (r.l.u.)\n8.5meV\n 9meV\n 9.5meV\n2\n1\n0\n10meV\n1\n0            L (r.l.u.)\n10.5meV\n 11meV\n 11.5meV\n2\n1\n0\n12meV\n1\n0            L (r.l.u.)\n12.5meV\n 13meV\n 13.5meV\n2\n1\n0\n14meV\n1\n0            L (r.l.u.)\n14.5meV\n 15meV\n 15.5meV\n2\n1\n0\n16meV\n1\n0            L (r.l.u.)\n18meV\n 20meV\n 24meV\n2\n1\n0\n28meV\n1\n0            L (r.l.u.)\n32meV\n 36meV\n 40meV\n2\n1\n0\n44meV\n0 0.5 1.0 1.5 2\nH (r.l.u.)1\n0            L (r.l.u.)\n48meV\n0 0.5 1.0 1.5 2\nH (r.l.u.)\n52meV\n0 0.5 1.0 1.5 2\nH (r.l.u.)\n56meVModel\nData\n0 0.5 1.0 1.5 2\nH (r.l.u.)\n0 20 40 60 80 100 120 140 160 180\nintensity (arb. units)\nFig. S6 (color online). CaMnBi 2: Constant-energy maps of a 2 \u00022 r.l.u. area of the ( H0L) plane of reciprocal space. Each\ndouble panel shows the processed data (lower panel, explanation see text) as well as the best \ft with a phenomenologial gaussian\nbroadening (upper panel).16\n2\n1.5\n1\n2 meV\n 10 meV\n 16 meV\n2\n1.5\n1\n20 meV\n 25 meV\n 30 meV\n2\n1.5\n1\n34 meV\n 38 meV\n 42 meV\n0 0.5 1 1.5\nH (r.l.u.)2\n1.5\n1\n46 meV\n0 0.5 1 1.5 2\nH (r.l.u.)\n0 0.5 1 1.5\nH (r.l.u.)\n50 meV\n0 0.5 1 1.5 2\nH (r.l.u.)\n0 0.5 1 1.5\nH (r.l.u.)\n54 meVModel Data\n0 0.5 1 1.5 2\nH (r.l.u.)\n0 5 10 15 20 25 30 35 40 45 50\nintensity (arb. units)\nFig. S7 (color online). CaMnBi 2: Constant-energy maps of a 2 \u00022 r.l.u. area of the ( HK0) plane of reciprocal space. Each\ndouble panel shows the processed data (right panel, explanation see text) as well as the best \ft with a phenomenologial gaussian\nbroadening (left panel).17\nFig. S8 (color online). \u001f2maps of least-squares \fts of the linear spin-wave model for SrMnBi 2(a,b) and CaMnBi 2(c,d). Panels\n(a) and (c) re\rect \fts of a low energy ( E\u001420 meV) cut along the (10 L) direction of reciprocal space, using the resolution\nproperties calculated by the RESTRAX ray-tracing routine20,21(best-\ft result shown in Fig. 6 of the main article). Panels (b)\nand (d) illustrate \ft results of the high-energy (30 \u0014E\u001445 meV) dispersion in the ( HK0) plane, used to determine J1and\nJ2. The red lines represent the one- \u001bcontour. Dashed lines indicate the best-\ft values and error margins quoted in the main\narticle. Note that while J1andJ2appear to be strongly correlated, weak details of the dispersion in the ( HK0) plane are not\nadequately re\rected in the \u001f2value and constrain the values further than suggested by the one- \u001bcontour."    },    {        "title": "1703.05753v1.Discovery_of_intrinsic_ferromagnetism_in_2D_van_der_Waals_crystals.pdf",        "content": "Discovery of intrinsic ferromagnetism in 2D van der Waals crystal s  \nCheng Gong,1+Lin Li,2+ Zhenglu Li,3,4+ Huiwen Ji,5 Alex Stern,2 Yang Xia,1 Ting Cao,3,4 Wei \nBao,1 Chenzhe Wang,1 Yuan Wang,1,4 Z. Q.  Qiu,3 R. J. Cava,5 Steven G. Louie ,3,4* Jing Xia,2* \nXiang Zhang1,4* \n \n1NSF Nano -scale Science and Engineering Center (NSEC), 3112 Etcheverry Hall, University of \nCalifornia, Berkeley, California 94720, USA  \n2Department of Physics and Astronomy, University of California, Irvine, Irvine, California  \n92697, USA  \n3Department of Physics, University of California, Berkeley, California 94720, USA  \n4Material Sciences Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, \nBerkeley, California 94720, USA  \n5Department of Chemistry, Princeton University, Princeton, New Jersey 08540, USA  \n+These authors  contribute d equally to this work.  \n*Emails: xiang@berkeley.edu , xia.jing@uci.edu , sglouie@berkeley.edu   \n  It has been long  hoped  that the realization  of long-range ferromagnetic order in two-\ndimensional  (2D) van der Waals (vdW)  crystals , combined with the ir rich electronic and \noptical properties,  would open up new possibilities for magnetic, magnetoelect ric and \nmagneto -optic  applications.1-4 However , in 2D systems , the long-range magnetic order is \nstrongly hampered by thermal fluctuations which may be counteracted  by magnetic \nanisotropy , according to  the Mermin -Wagner theorem .5 Prior efforts via  defect and \ncomposition engineering ,6-11 and proximity effect only  locally or extrinsically introduce  \nmagnetic responses . Here we report the first experimen tal discovery  of intrinsic long-range \nferroma gneti c order  in pristine  Cr2Ge2Te6 atomic layers  by scanning magneto -optic Kerr \nmicroscopy. In such a  2D vdW soft ferromagnet , for the first time,  an unprecedented \ncontrol  of transition temperature of ~ 35% – 57% enhancement  is realized via surprisingly \nsmall fields  (≤ 0.3 Tesla in this work) , in stark contrast to the stiffness  of the transition \ntemperature to magnetic fields in the three -dimensional regime . We found that t he small  \napplied  field enables  an effective anisotropy far surpassing  the tiny magnetocrystalline \nanisotropy , opening up a sizable spin wave excitation gap . Confirmed by renormalized spin \nwave theory , we explain the phenomenon and  conclude that the unusual field dependence \nof transition temperature constitutes  a hallmark of 2D soft ferro magnetic vdW crystals . \nOur discovery  of 2D soft ferromagneti c Cr2Ge2Te6 presents a close -to-ideal  2D Heisenberg \nferro magnet  for studying fundamental spin behaviors, and opens the door for exploring \nnew applications such as  ultra -compact spintronics.  \n \n  Atomic layer ed vdW crystals provide  ideal 2D material systems  hosting e xceptional physical  \nproperties . 12-14 Emerging  functional  devices14 (e.g., ultrafast photodetector, broadband optical \nmodulator, excitonic semiconductor laser , etc. ) have  been  derive d primarily  from the electron  \ncharge  degree of freedom , whereas 2D spintronic s1-3,15 based on vdW crystals is still in its \ninfancy , severely  hindered  by the lack of long-range ferro magnetic order that is crucial for \nmacroscopic magnetic effects .4,7 The emergence of ferromagnetism  in 2D vdW crystals , if \npossible,  combined  with their rich electronics and optics , could open up numerous opportunities \nfor 2D magnetic, magnetoelectric, and magneto -optic applications .2,3  \nThe absence of ferromagneti c order in many intensively studied  2D vdW crystals  (e.g., graphene  \nor MoS 2) has motivated tremendous effort s to extrinsically induc e magneti sm by defect \nengineering via vacancies, adatoms, grain boundaries, and edges ,6-11,16 -18  by adding magnetic \nspecies via intercalation or substitution ,19 or by magnetic proximity effect.  In these schemes , \nextrinsically introduce d local magnetic moments are difficult to be correlated in long range via \nrobust exchange interaction,7 or substrate -induced magnetic responses of 2D materials are rather \nlimited . Theoretical proposals for inducing  ferromagnetism by band  structure  engieering18,20,21 \nstill await  experimental realization.  In contrast , if realizable, intrinsic  ferromagnetism  originating \nfrom the parent 2D lattice  itself is fundamental for both understanding the underlying  physics of \nelectronic and spin processes , and device  applications .  \nWhether or not long -range ferromagnetic order  that exist s in bulk can  persist in 2D regime is a \nfundamental question , because the strong thermal fluctuation s may easily destroy  the 2D \nferro magnetism , according to the Mermin -Wagner theorem.5 As illustrated in Fig . 1, down to the \nroot of 2D ferromagnetism, the presence of a spin wave excitation gap  – as a direct result from  \nmagnetic anisotropy – is a must  for long-range ferromagnetic order  at finite temperature . Most of \nthe reported  ferromagnetic vdW bulk crystals  so far  are magnetically soft with small  easy-axis \nanisotropy. The challenge of harnessing the long -range ferromagnetic order in 2D vdW crystals \nhinges on the strength of magnetic  anisotropy retained in the 2D regim e. In this work, we report \nthe first experimental discovery of long -range ferromagnetic order in pristine Cr 2Ge2Te6 atomic \nlayers by temperature - and magnetic field -dependent Kerr effect via scanning magneto -optic \nKerr microscopy.  In our 2D soft ferromagnetic  vdW crystal, a n unprecedented magnetic field control of ferromagnetic transition temperature  of ~ 35% – 57% enhancement  is achieved  by \nsurpri singly small field s (≤ 0.3 T ). \nMechanical exfoliation via scotch tape was applied to prepare Cr 2Ge2Te6 atomic layers on 260 \nnm SiO 2/Si chips. Bulk Cr 2Ge2Te6 was reported22,23 to be ferromagnetic below 61 K , with an \nout-of-plane magnetic easy axis and negligible coercivity ( Fig. S1b,c in supplementary \ninformation (SI)). The monolayer (1L) was found to degrade rapidly and become invisible under \nan optical microscope (Fig . S2), and the peeling off of bilayer  (2L) flakes was repeatable and \nshown to be robust without traceable degradation within 90 minutes in ambient atmosphere (Fig s. \nS4 and S 5). Therefore, the thinnest structure we studied in this work is a bilayer.  The thickness \nof the atomic layers is determined by the combination of optical contrast and atomic force \nmicroscopy ( Fig. S3 ). \nThe scanning Kerr microscope used in this work was constr ucted with a fiber -optic Sagnac \ninterferometer24 with 10-8 rad DC Kerr sensitivity and micrometer spatial resolution, which is an \nideal  nondestructive optical means  for imaging and measuring  the magnetism of nanometer thick \nand micrometer -size flakes . The geometry of the fiber -based zero -area loop Sagnac \ninterferometer  guarantees the rejection of  reciprocal effects and sense s only nonreciprocal effects \nwith high precision, thus affording  a static measurement of the absolute Kerr rotation angle \nwithout s ample modulations . These  merits make it ideal for detecting small signals from  as-\nexfoliated 2D flake s without involving any device fabrication process , which in this work might  \nperturb the magnetism. The detailed e xperimental setup can be referred  to Fig. S6. It is important \nto note , in such a sensitive experiment , ~3.5 μm light spot is focused on nanometer thick and \nmicrometer size flakes, any horizontal drifting and vertical defocusing would possibly induce \nlarge signal fluctuations . In order to avoid the drifting  induced invariance , every Kerr rotation \ndata point throughout this paper is a summary based on a scanned image  including the target \nflake and the surrounding substrate,  except the bulk measurement in Fig . 2j. Magnetic properties \nof bulk crysta l have also been examined by superconducting quantum interference device \n(SQUID) in this work.  \nIn order to observe  ferromagnetism in  few layer s, we monitor ed the temperature dependent Kerr \nimage  of the 2L sample in Fig. 2a with the help of a small perpendicular field of 0.075 T to \nstabilize the magnetic moments . Strictly speaking, upon applying an external magnetic field, we may no longer have a well -defined  ferromagnetic  phase transition  (at temperature 𝑇𝐶 with zero \nfield), therefore we call it a transition between a ferromagnetic -like state and a paramagnetic -like \nstate, separated by a transition temperature , 𝑇𝐶∗. Fig. 2b-e clearly  show s the emergence of \nferromagnetic  order  in bilayer  Cr2Ge2Te6 as temperature decreases : at 40 K, the Kerr intensity of \nthe whole scanning area except for the region of thicker flakes (≥ 3 layers ) is hardly discernable ; \nas temperature continues to drop , the recognition of the long strip is gradually enhanced;  while \napproaching the  liquid helium temperature , the bilayer  long strip is clearly  distinguishable from \nthe surrounding bare substrate, in terms of  the Kerr rotation angle intensity. The clear visibility \nof thicker (≥ 3 layers ) flakes at 40  K implies a higher 𝑇𝐶∗ for thicker crystals , which will be \nfurther discussed .  \nThe non -zero transition temperature  𝑇𝐶∗ in the bilayer sample reflects a true 2D ferromagnetic \norder in our system.  Firstly, the observed ferro magnetism is not due to substrate polarization \nbecause silicon dioxide is nonmagnetic and does not strongly bond to Cr 2Ge2Te6 layers . \nSecondly, the exposure -to-air time from the flake exfoliation to the specimen loading in a \nvacuum  chamber (10-6 torr) is strictly controlled to be under 15 minutes. The ambient effect s, \neven if there were some, would generally reduce  𝑇𝐶∗. Therefore, the observation of finite 𝑇𝐶∗ is an \nunambiguous evidence of 2D ferromagnetism originating from Cr 2Ge2Te6 atomic layers.  \nA strong  dimensionality effect  is revealed by a thickness -dependent study  under 0.075 T field . \nFig. 2f-k display the prominent monotonic uplifting of 𝑇𝐶∗ with increas ed thickn ess, from  a \nbilayer value ~3 0 K to a bulk limit ~68 K. The behavior of 𝑇𝐶∗ from 2D to 3D regime s is similar \nto the universal trend  of many magnetic transition metal thin films,25-27 although the  interlayer \nbonding strength  in vdW crystals is two-to-three  orders of magnitude  weaker than that  of \ntraditional metals.  The strong thickness dependence of  𝑇𝐶∗ suggests the essential role of interlayer \nmagnetic coupling in establishing  the ferromagnetic order  in Cr 2Ge2Te6 crystals . In other  word s, \na ferromagnetic order in bulk vdW crystals does not guarantee a ferromagnetic order in 2D \nsheets , which  highlights the necessity of a  measure ment direct ly on atomic layers  as in the \npresent  work.  \nTo gain a deeper level of understanding of the observed 2D ferro magnetic behavior in Cr 2Ge2Te6, \nwe study a Heisenberg Hamiltonian with magnetic anisotropies, 𝐻=1\n2∑𝐽𝑖𝑗𝑺𝑖⋅𝑺𝑗 𝑖,𝑗 +∑𝐴(𝑆𝑖𝑧)2\n𝑖−𝑔𝜇𝐵∑𝐵𝑆𝑖𝑧\n𝑖  (𝑺𝑖 : spin operator on site i, 𝐽𝑖𝑗 : exchange interaction between site  i and \nj, A: the single -ion anisotropy, 𝑔: the Land é g-factor, 𝜇𝐵: Bohr magneton, and B: external \nmagnetic field). To solve this spin Hamiltonian, w e apply the renormalized spin wave theory \n(RSWT)28 which includes magnon -magnon scattering s at the Hartree -Fock level  self-consistently  \n(SI). The input interaction parameters  (three intralayer 𝐽∥’s, three interlayer 𝐽⊥’s and A) are \ncalculated  based on ab initio  density functional method29 at the local spin density approximation \nplus U level  (SI). We find small U values ( U = 0.5 eV  in this work) reproduce the magnetic \nground state properties of bulk Cr2Ge2Te6 (SI). The J’s directly mapped out from first -principles \ndensity functional calculations  appear  to be  overestimated within RSWT, and are rescaled by a  \nfixed and uniform  factor of 0.72 that pins to the bulk TC (SI). We perform layer -dependent \ncalculations including Zeeman effect from the 0.075 T external field ( Fig. 2k) assuming A = 0 in \nfew layers and A = -0.05 meV in bulk  (SI), and observe a strong  dimensionality effect  in line \nwith experiment al observations . At finite temperatures, s pontaneous symmetry breaking  \n(ferromagnetic order)  takes place in 3D for the rotationally invariant isotropic Heisenberg system \n(A = 0  and B = 0), but is completely suppressed by the thermal fluctuations of the long -\nwavelength gapless Nambu -Goldstone modes in 2D.  Nevertheless , magnetic anisotropy  (A ≠ 0 \nor B ≠ 0) could establish ferromagnetic order  in 2D  at finite temperatures by  breaking the \ncontinuous  rotational  symmetry in Hamiltonian, therefore  giving rise to a nonzero excitation gap \nto the lowest energy mode of the  acoustic magnon branch. Thermal energy at finite  𝑇𝐶∗ excites a \nlarge number of low -energy  but finite -frequency magnon modes, flattening the expectation value \nof the collective spins. As the number of layer increases (from 2D to 3D ), the increased thickn ess \nrapidly reduces the density of states per spin  for the magnon modes near excitation gap  (Fig. 1c-\ne), requiring a higher 𝑇𝐶∗ to create enough population of excitations to destroy the long -range \nmagnetic order, thus exhibiting a strong dimensionality effect. This picture clearly distinguishes \nour observed 2D ferromagnetism in a real 2D material from the quasi -2D ferromagnetism \nembedded in bulk materials ,30,31 in which there is still weak out -of-plane ferromagnetic coupling \nand the magnon  density of states is different  (see Fig. 1). \nThe agreement between experiments and the theoretical calculations  including Zeeman effect in \nthe Hamiltonian reveals a strong  role of the external field in the observed transition temperature . \nIn order to pin down the effects of external field and intrinsic anisotropy, we did the experimental study of the hysteresis loop on the 6L sample at 4. 7 K. The extreme softness (about  \n2% zero-field remanence of the maximum Kerr signal at 0.6  T) is observed , suggesting a very \nsmall intrinsic anisotropy (< 1 μeV , SI). Two zero -field Kerr images (Fig . 3b,c) after \nwithdrawing  the magnetic field  from 0.6  T and -0.6 T show tiny yet definitive remanence with \nopposite signs (see line cuts in Fig . 3d,e). Kerr images under descending fields down  to zero  (Fig. \n3f-k,b) show the flake is a single  ferromagnetic  domain through  out. In ultrathin flakes, dipolar \ninteraction is expected to be very small, and the equilibrium domain size which diverges \nexponentially  with thickness would likely be very large.32 The observation of about  2% single -\ndomain remanence is a n unambiguous evidence of the strong thermal fluctuations in 2D regime , \nin contrast to 3D ferromagnets where fractional remanence  (if any)  is usually  caused by multi -\ndomains.  We also scan ned the magnetic field on 2L and 3L flakes and did not  see remanence \nwithin the detection limit.   \nThe vanishing remanence in 3L at 4. 7 K, together with the observation of 𝑇𝐶∗ = 41 K under 0.075  \nT field, suggests a strong  magnetic field control of transition temperatures  in our 2D magnetic \nsystems . A field that is usually deemed to be too small (e.g., < 0.5 T) to affect transition \ntemperatures of 3D systems can drastically affect the behavior of 2D soft fe rromagnetism by \nopening the spin wave excitation gap , as sketched in Fig . 1c-e. In order to further examine this \nscenario , we did temperature dependent Kerr rotation study on 2L, 3L, and 6L samples under \ntwo contrasting fields: 0.065  T and 0.3  T. From 0.065  T to 0.3  T, 𝑇𝐶∗ of 2L increases from 28 K \nto 44 K  (57% enhancement) , 𝑇𝐶∗ of 3L increases from 35 K to 49 K  (40% enhancement) , and 𝑇𝐶∗ \nof 6L increases from 48 K to 65 K  (35% enhancement)  approaching the bulk limit.  In st ark \ncontrast, the 𝑇𝐶∗ in the bulk from SQUID measurements under 0.025 T to 0.03 T does not show \nclear change . Under the two fields  (0.065 T and 0.3 T) , the overall  shift of  magnetization -\ntemperature  curve s in 2D layers  (Fig. 4a-c) is obviously distinguished  from the tail effect of a  \nmagnetic field on the critical region above  𝑇𝐶∗ in bulk (Fig. 4d). The possibility of \nsuperparamagnetism can be ruled out because of t he integrity of the crystalline flake from high -\nquality single -phase crystals, evidenced by X -ray diffraction23 and transmission electron \nmicroscopy measurements .33  \nThe observed field effect on transition temperature can be explained well within the picture of \nspin wave excitations as illustrated in Fig . 1. In 2D,  ferromagnetic transition temperature critically depends on the size of the spin wave excitation gap. In a 2D ferromagnet  with \nnegligible single -ion anisotropy , a magnetic field help s to increase the magnetic stiffness \nlogarithmically, leading to a rapid increment of 𝑇𝐶∗, particularly in the range of small fields. \nHowever , in 3D, 𝑇𝐶∗ is predominantly  determined by exchange interactions, and is insensitive to  \nsmall single -ion anisotropies and small fields  that are usually orders -of-magnitude smaller than \nJ’s.  \nBy means o f RSWT method, we calculated the magnetic field dependence of 𝑇𝐶∗ in few-layer s \nand bulk. The calculation results are  quantitatively  consistent with experimental values, as shown \nin Fig . 4. In all the 2L, 3L and 6L samples, a remarkable change of 𝑇𝐶∗ can be obtained in the \nrange of the magnetic fields used in experiments  (Fig. 4f-h), but not in the bulk  (Fig. 4i). \nConsidering the experimental challenge to determine the value of the tiny intrinsic anisotropy  in \na quantitative accuracy , we further investigated  theoretically  the effects of reasonably small but \nfinite intrinsic anisotropies ( see SI and Fig . S10), and find that the effect of field control on \ntransition temperatures is still robust.  Interestingly, for 6L under  0.3 T, the calculated 𝑇𝐶∗ \napproaches  the bulk TC, agreeing well with experiments. It thus provides a n intriguing platform \nin which  magnetic properties can be easily modulated between 2D and 3D regimes (at least in \nview of the transition temperature ).  \nWe report  the first experimental  observation of  intrinsic ferromagnetism in  2D ferromagnetic \nvdW crystal  Cr2Ge2Te6, where  a strong dimensionality effect  arises from the low -energy \nexcitations of magnons . Through the effective engineering of the magnetic anisotropy by small \nmagnetic fields , we discover  the unprecedented  magnetic field control of  transition temperatures  \nin 2D soft ferromagnetic vdW crystal . Our experimental observations are c onfirmed by  the \nrigorous renormaliz ed spin wave theory analysis and calculations , which can provide  generic \nunderstanding of the ferromagnetic behaviors of many  2D vdW  soft ferro magnet s such as \nCr2Si2Te6 and CrI 3. 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Kosterlitz -Thouless transition in two -dimensional planar ferromagnet K 2CuF 4. \nJ. Appl. Phys.  53, 1893 (1982).  \n32. Kaplan, B. & Gehring, G. A. The domain structure in ultrathin magnetic films. J. Mag. Mag. \nMater.  128, 111 (1993).  \n33. Alegria, L. D. et al. Large anomalous Hall effect in ferromagnetic insulator -topological \ninsulator heterostructures . Appl. Phys. Lett.  105, 053512 (2014)  \n  Figure  legends:  \nFigure 1. Schematics of spin wave excitations in two dimensions (2D) and three dimensions \n(3D). a, Ferromagnetic spin wave (magnon) excitations in 2D, with intralayer exchange \ninteraction 𝐽∥, single ion anisotropy 𝐴, and magnetic field 𝐵. b, Ferromagnetic spin wave \nexcitations in 3D, with extra  interlayer exchange interaction 𝐽⊥. c-e, Magnon density of states \n(DOS) around the  low-energy  magnon band edge of monolayer, multi -layer and bulk \nferromagnetic materials , respectively . The low -energy excitations from the ferromagnetic ground \nstate follow parabolic dispersions, and accordingly the DOS is a step function in 2D, and is ~ √𝐸 \nin 3D, where 𝐸 is excitation energy . Therefore, more magnons are excited  for a given thermal \nenergy  in 2D than in 3D due to the reduced dimension s. In 2D, the ferromagnetic transition \ntemperature 𝑇𝐶 is primarily determined by the excitation gap, which result s from the magnetic \nanisotropy , whereas in 3D, 𝑇𝐶 is primarily det ermined by exchange interactions.  f, Crystal \nstructure (side view and top view) of Cr 2Ge2Te6. Bulk Cr2Ge2Te6 has layered structure, and the \nlayers are coupled through vdW interaction with a 3.4 Å interlayer spacing.  \nFigure  2. Observation of ferromagnetism  in bilayer  (2L) Cr2Ge2Te6 and temperature \ndependent Kerr rotation of few -layer and bulk  Cr2Ge2Te6. a, The optical image of exfoliated \nCr2Ge2Te6 atomic layers on 260 nm SiO 2/Si, consisting of a 31 μm long bilayer strip attached to \na trilayer flake with an even thicker end. The bar is 10 μm. b-e, The emergence of a Kerr rotation \nsignal for the bilayer flake  upon the application of a small magnetic field of 0.075 T , as the  \ntemp erature decreases from 40  K to 4.7  K. As the temperature drops, the Kerr rotation signal \nfrom the bilayer flake increases, making the bilayer region in sharp contrast with the nearby bare \nSiO 2 substrate.  The clear visibility of thicker flakes (≥ 3L) indica tes a higher TC. The average \nbackground signal is subtracted (Figure S6) and the signals of higher intensities from the thicker \nflakes (≥ 3L) are truncated at 30 μrad. f-j, The temperature dependent Kerr rotation intensities of \n2L, 3L, 4L, 5L and bulk samples under 0.075 T field. Error bar represents the standard deviation \nof sample signals. k, Transition temperature 𝑇𝐶∗ (defined in the presence of external magnetic \nfield)  between a ferromagnetic -like state and a paramagnetic -like state in sample with different \nthickness, obtained from Kerr measurements and renormalized spin wave theory calculations . A \nstrong dimensionality effect is seen, owing to the rapid reduction of on set magnon DOS as the \nthickness increases. The transition temperature 𝑇𝐶∗ is determined  experimentally  in the range approximating the paramagnetic tail of the effective Kerr signal , and theoretically by the \nvanishing net magnetization. The statistics yield a finite uncertainty in the deterministic  \nexperimental  value s.  \nFigure 3. Ferromagnetic hysteresis loop with single domain remanence in a six -layer (6L) \nCr2Ge2Te6. a, Hysteresis loop of a six -layer flake at 4.7 K showing saturating trend at 0.6 T and \ntiny non -vanishing remanence at zero field. The solid red (blue) arrow represents the descending \n(ascending) field from the positive (negative) maximum. Loop starts from p ositive maximum 0.6 \nT. Inset is an optical image of the flake and the bar is 10 μm.  b, c, Scanned Kerr rotation images \nof the flake at zero field after withdrawing  the field  from 0.6 T and -0.6 T, respectively. d, e,  \nLine scanning crossing the flake with long acquisition time  (each physical spot with 100 data \nacquisitions) . The approximate line positions are indicated by black dashed lines in b and c, \nrespectively, but extending further  out of the window b and c. b-e show a clear signature of \nsmall but definitive remanent Kerr rotation angle s (about 2%  of the “saturated” Kerr rotation at \n0.6 T) with opposite signs on ascending and descending branches. f-k, Kerr images of the \nhighlighted area in a (within dashed rectangle ), at different descending fields, showing the \npersistence of a magnetic single domain through out. The tiny remanence in a single domain is \nstrong evidence that the effect of thermal fluctuations rather than the formation of multi -domains \n(usually in the b ulk) is the reason of the reduced magnetization in our few -layer samples.  \nFigure 4. Magnetic field control of transition temperature of few -layer Cr 2Ge2Te6. a-c, \nNormali zed Kerr rotation angle as a function of temperature, under two different magnetic fields \nof 0.065 T and 0.3 T, for 2L, 3L and 6L, respectively. The 0.3 T field shifts the curve drastically, \nwith respect to curve under 0.065 T field, indicating a strong renormalization of t he transition \ntemperature 𝑇𝐶∗ in atomic layered Cr 2Ge2Te6. Experimental error bars are smaller than the plotted \ndot size if not shown.  d, Temperature dependent magnetization curves of bulk crystals measured \nby SQUID under 0.025 T and 0.3 T. Compared with the curve under 0.025 T, the 0.3 T field only \nintroduces a s lightly distorted tail above 𝑇𝐶∗. The different behavior below 𝑇𝐶∗ is a possible result \nfrom domains: under 0.025 T field, multi -domains are likely formed (suggested by reduced \nabsolute magnetization); but under 0.3 T, single -domain has been approached  (see Fig. S1). e-i, \nExperimental and theoretical field dependence of 𝑇𝐶∗ in samples of various thickness.  In f and g, \nexperimental results at 0.075 T (Fig. 2) are plotted as well. In the 2D limit, if the single ion anisotropy is negligibly small, the tr ansition temperature will be very low, and can be easily \ntuned with a small magnetic field ( e.g., B < 0.5 T). In the bulk limit, due to the 3D nature, such \ntuning ability does not hold . Due to the large signal and the continuous temperature sweeping, all \n𝑇𝐶∗ values of the bulk crystal measured by SQUID are determined at the steepest slop of the \nmagnetization -temperature characteristic curve.  \n  Acknowledgements:  \nWe thank Dr. J. -G. Zheng for helping with AFM measurement at UC Irvine. This work was \nprimarily supported by the Director, Office of Science, Office of Basic Energy Sciences, \nMaterials Sciences and Engineering Division of the US Department of Energy under Contract \nNo. DE -AC02 -05-CH11231 (van der Waals heterostructures program,  KCWF16). The bulk \ncrystal growth at Princeton University was supported by the NSF MRSEC program, grant NSF \nDMR -1420541, and part of measurement at UC Irvine was supported by NSF DMR -1350122.  \n \nAuthor  Contribution s: \nC.G. and X.Z. conceived and initiated thi s resear ch and designed the experiments;  C.G. \nperformed the Sagnac MOKE measurements with assistance from L.L., A.S. and Y.X., under the \nguidance of J.X. ; Z.L. and T.C.  performed theoretical calculations under the guidance of S.G.L. ; \nH.J. synthesized Cr 2Ge2Te6 bulk crystals under the guidance of R.J.C. ; C.G. prepared and \ncharacterized few -layer samples with assistance from Y.X., W.B. and C.W. ; C.G., Z.L., T.C., \nY.W., Z.Q.Q., S.G.L., J.X. and X.Z. analyzed the data and wrote the paper.  \n \n \n \n \n \n \n \n   \n \n \nFigure 1  \n  \n \nFigure 2  \n \n  \n \nFigure 3  \n \n \n  \n \nFigure 4  \n \n"    },    {        "title": "1704.07371v2.Geometrical_dependence_of_domain_wall_propagation_and_nucleation_fields_in_magnetic_domain_wall_sensor_devices.pdf",        "content": "arXiv:1704.07371v2  [physics.app-ph]  15 Aug 2017APS/123-QED\nGeometrical dependence of domain wall propagation and nucl eation fields in magnetic\ndomain wall sensor devices.\nB. Borie\nInstitut f¨ ur Physik, Johannes Gutenberg-Universit¨ at Ma inz, Staudinger Weg 7, 55128 Mainz, Germany and\nSensitec GmbH, Hechtsheimer Str. 2, Mainz D-55131, Germany\nA. Kehlberger, J. Wahrhusen, and H. Grimm\nSensitec GmbH, Hechtsheimer Str. 2, Mainz D-55131, Germany\nM. Kl¨ aui\nInstitut f¨ ur Physik, Johannes Gutenberg-Universit¨ at Ma inz, Staudinger Weg 7, 55128 Mainz, Germany\n(Dated: August 16, 2017)\nWe studythekeydomain wall properties in segmentednanowir es loop-based structures usedin do-\nmain wall based sensors. The two reasons for device failure, namely the distribution of domain wall\npropagation field (depinning) and the nucleation field are de termined with Magneto-Optical Kerr\nEffect(MOKE)andGiantMagnetoresistance (GMR)measuremen ts for thousandsofelementstoob-\ntain significant statistics. Single layers of Ni 81Fe19, a complete GMR stack with Co 90Fe10/Ni81Fe19\nas a free layer and a single layer of Co 90Fe10are deposited and industrially patterned to determine\nthe influence of the shape anisotropy, the magneto-crystall ine anisotropy, and the fabrication pro-\ncesses. We show that the propagation field is little influence d by the geometry but significantly\nby material parameters. Simulations for a realistic wire sh ape yield a curling mode type of the\nmagnetization configuration close to the nucleation field. N onetheless, we find that the domain wall\nnucleation fields can be described by a typical Stoner-Wohlf arth model related to the measured\ngeometrical parameters of the wires and fitted by considerin g the process parameters. The GMR\neffect is subsequently measured in a substantial number of de vices (3000), in order to accurately\ngauge the variation between devices. This reveals a correct ed upper limit to the nucleation fields of\nthe sensors that can be exploited for fast characterization of working elements.\nI.Introduction\nMagnetic domain walls in soft thin films nanostruc-\ntures[1]areinterestingobjectsfornumerousapplications\nsuch as logic [2] and memory devices [3]. Furthermore,\nseveral sensor ideas based on domain walls have been de-\nveloped and reported in the literature [4–6].\nThe potential of magnetic domain wall based sensors\nis due to their many attractive attributes compared to\nother technologies. Magnetic domain walls can be stable\nwell above room temperature, making it a potential can-\ndidate to store data and be used for non-volatile sensing\nand they can be displaced rapidly in application’s rele-\nvant geometries [7]. The second point of such technolo-\ngies relies on the fact that no external power is required\nto create, stabilize or manipulate the storing element,\nmeaning that a power failure in the system does not af-\nfectthefunctionalityofthedevice. Thisensuresnon-stop\nsensing even in cases where power is lost. Moreover, the\nonly necessary power is that associated with the injected\ncurrent that reads the state of the device, which can be\napplied during only a small part of the total operating\ntime. Finally, the integration of the technology is rela-\ntively cheap allowing for an abundance of sensors in the\ndesired system.\nThe first such device to be implemented [6] was de-\nsigned a few years ago by Novotechnik [8] and is now\nproduced by Sensitec [9]. The purpose of this sensoris to count the number of rotations of a magnetic field.\nThe sensorrelieson the Giant Magnetoresistance(GMR)\n[10, 11] effect to generate the signal. The information of\nabsolute rotation is stored by the use of a combination of\ndetermining the domain wall positions and their number\nwithin the device. This simple method provides an ele-\ngant and reliable solution, which enables a versatile sen-\nsor design as required by the application. The structure,\nunderarotatingappliedmagneticfield, nucleatesonedo-\nmain wall every180◦rotation, and hence by counting the\nnumber of domain walls, the number of 360◦turns is ex-\ntracted (see Supplemental Material for the details on the\nfunctionality of the device). These magnetic domain wall\ndevices exhibit two types of failure events, the pinning of\ndomain walls if a particular propagation field threshold\nis not reached and the undesired nucleation of domain\nwalls at too high fields. Those two critical fields ought to\nbe as separated as possible to allow an extensive range\nof operating applied fields and thus of industrial applica-\ntions.\nIn this paper, we study the field operating window by\nmeasuring the successful operations and errors occurring\nin the propagation and the nucleation field for several\nmaterials and different geometries. We employ Magneto-\nOptical Kerr Effect [12] microscopy for the precise inves-\ntigation of the position of domain walls in the devices\nunder various conditions and measure the GMR effect\nfor high-statistics device characterization to obtain rele-\nvant information for the industry. The propagation field2\nis identified as being affected by the edge roughness and\nthecrystallinestructureofthewires, whilethenucleation\nfield isdependent ontheshapeofthe sampleandthe pro-\ncessing parameters. Simulations are used to identify the\n2-dimensionalvariationsofthemagnetizationclosetothe\nnucleation field and the expected nucleation field value.\nFurthermore, comprehensive statistics are obtained with\nthe use of an automatized measurement using the GMR\neffect tocomparewiththe MOKEmicroscopyresultsand\ndetermine a precise nucleation field for a significant num-\nber of devices.\nII.The investigated sys-\ntems\nNi81Fe19(28 nm) Ni81Fe19(32 nm) Co90Fe10(17 nm)\nNominal Width (nm) Average Width (nm)\n200 205±9 240±15 211±6\n250 264±10 288±15 266±6\n300 328±6 342±9 323±9\n350 372±18 391±9 357±19\nTABLE I. Summary of the measured samples. The average\nwidth is determined from the average of 15 width measure-\nments from Scanning Electron Microscopy micrographs of the\nwires.\nThe samples were deposited in a magnetron sputter-\ning system employing a seed layer. The investigated sys-\ntems are two single layers of Ni 81Fe19(28 nm and 32 nm),\nand one single layer of Co 90Fe10(17 nm) (referenced in\nthe Table I). All the samples were capped with a 4 nm\nTa layer. For the GMR measurement, a complete GMR\nstack with a free layer of Co 90Fe10(1 nm)/Ni 81Fe19(32\nnm) was used. The GMR ratio was measured with a\nfour-point probe technique under an applied magnetic\nfield. A resist was spin-coated on the as-deposited wafers\nand patterned with photolithography in the shape of the\nstructures (see Fig. 1). After the development of the\nresist, the material was then etched away by an Ar+ion\netching process.\nTo fabricate electrical connections, part of the batch\nwent through a second lithography followed by Au de-\nposition and lift-off processing. The latter allows for the\nmeasurement of the GMR effect in the devices (see mea-\nsurement scheme detailed in Appendix [13] in the sup-\nplemental material).\nThe materials used are magnetically soft and exhibit\na full film coercivity of 2 Oe for the Ni 81Fe19films and 4\nOe for the Co 90Fe10films. Both materials were selected\nfor their softness. Nonetheless, Co 90Fe10also exhibits an\nincreased saturation magnetization ( M s= 1334 kA/m)\ncompared with Ni 81Fe19(Ms= 795 kA/m). These val-\nues were measured using a BH-looper set-up [14] which\ncan detect the strayfield ofan entirewafer. The films are\npolycrystalline, with an expected crystallitesize of10nm[15]. The individual crystallites of Co 90Fe10can be ex-\npected to exhibit a large magneto-crystalline anisotropy\nconstant (K = 45 ·104J/m3for pure Co [16]) compared\nwith Ni 81Fe19.\nA scheme ofthe used architecture is depicted in Fig. 1.\nThe structure starts with a nucleation pad to introduce\ndomain walls in the looping wires [17] and finishes with\na tapered tip to prevent nucleation on that end. The\ndevices are designed with 16 loops (i.e. the complete\nlength of the device is 31 mm from the nucleation pad\nto the tapered tip end). This structure can contain a\nmaximum of 33 domain walls (i.e. 2 domain walls per\nloop plus 1 in the wire following the nucleation pad),\nthis allows for the sensing of 16 360◦-turns of the applied\nfield.\nThis type of structure is very large compared to the\ntypical dimensions encountered for domain wall-based\ndevices in the literature [18–21]. The large size is advan-\ntageous for the assessment of devices reliability since the\ndomain wall needs to cover a sizeable distance. The do-\nmain wall has thus a high probability of encountering the\nfull distribution of geometrical and structural variations\ninduced by the fabrication. Furthermore, the inner loops\nof the devices are far away from the starting nucleation\npad and the end of the sample which could otherwise\nprovide an unwanted extra source of device variability\n(reduction of the nucleation field due to flux closure at\nthe edges). In order to investigate the influence of vari-\nations of the cross section, four different devices widths\nwere compared (200, 250, 300 and 350 nm).\nIII.Characterization\nAfter processing, a subset of the devices was charac-\nterized under Scanning Electron Microscopy (SEM) and\nAtomic Force Microscopy (AFM) to study the topogra-\nphy and assess the pattern transfer of the geometrical\nshape.3\nFIG. 1. a) Schematic of the structure, 3 loops are represente d.\nb) Atomic Force Microscopy image of a Ni 81Fe19wire of 300\nnm nominal width c) Scanning Electron Microscopy image of\na Ni81Fe19wire of 300 nm nominal width.\nFrom the acquired images (see example in Fig. 1),\nwe observe a variation of the widths compared to the\nnominal widths. Furthermore, the AFM profiles reveal\na trapezoidal shape instead of a square shape which is a\nconsequence ofthe etching processbeing unable to trans-\nfer precisely the photolithographically defined structures\nfor all depths. Such variations are caused by shadow-\ning effects during the ion milling and might affect the\ndynamics of the domain wall propagation as well as the\nnucleation process [20–22]. Material redeposition is seen\non the images as side bumps on top of the wire. The re-\ndepositionisexpectedtobe composedofallthe materials\nconstituting the stack. It, therefore, remains difficult to\ndetermine the effective ’magnetic’ width of the effective\nmagnetic layer. In our case, we define the width as the\ndistance between the two bright lines in the SEM image,\ncorresponding to the bump regions of the AFM profile.\nThe width can vary up to 60 nm between devices and a\nfurther60nmbetweenthetopandthebottomofthewire\ndue to the shape of the wire profile. This effect results\nfrom the inhomogeneity of the resist thickness over the\nwafer and the difficulty in controlling photolithography\ndimensions on lengthscales below the diffraction length.\nIV.Sensor operation condi-\ntions\nA.MOKE investigation\nTo operate the device, the processes necessary for the\nsensing must work flawlessly. There are two main fail-ure events ofmagnetic devices, namely unwanted domain\nwallnucleationanddomainwallpinningasaresultoftoo\nlow fields being applied to allow for domain wall prop-\nagation. The Magneto-Optical Kerr Effect microscope\nset-up used is the following: The objective used is a x50\nmagnification, the source is a white linearly polarized\nlight from an incandescent bulb source, and the micro-\nscope is operated in the longitudinal configuration. The\nwires of the sensor are positioned parallel to the cam-\nera’s field of view to provide a reference for the angle of\nthe applied field. A vector magnet is utilized for the ap-\nplication of a rotating field up to 100 mT. To detect a\nswitching event, a differential contrastmethod is used. A\nbackground image is saved and subtracted to the current\nfield of view yielding a clear contrast observable even for\nwidths assmall as200nm which is lowerthan the diffrac-\ntion limit of our light source.\n1.Propagation field\nThe propagationfield is the lowestfield value, at which\nthe domain wall freely propagates and is not pinned at\nany point in the whole structure. Since domain wall pin-\nning is a highly stochastic phenomenon [23, 24] a signif-\nicant amount of statistics is required to reliably charac-\nterize it. In our scheme, for a single measurement to be\nsuccessful, all the 33 possible domain walls must propa-\ngate along the entire device without anyone experiencing\na failure event since strong pinning of one wall necessar-\nily leads to it being annihilated with the subsequent wall\nwhen it also reaches the pinning site. At least 33 com-\nplete rotations of the field were performed, and every\ndomain wall probed 31 mm of wire for a pinning event.\nIn total, 10 structures were probed for a combined length\nof more than 10 m of magnetic materials. The data are\nshown by the discs in Figure 3.4\nFIG. 2. Schematic representation of the field sequences for\nthemeasurementofthepropagation andnucleationfieldsfor a\nsimplified device based on twoturns. a)-d) Propagation field -\na) The sensor is in the initial state with a pair of domain wall s\nin it. The propagation field measurement can be carried out\nfor any state of the sensor (filled with DWs or empty). b) The\nfield is rotated clockwise and increased to inject domain wal ls\nfrom the nucleation pad at each 180◦turn. c) The nucleated\ndomain wall should continue to propagate around the corners\nand further into the device as the field rotation continues.\nAs long as the wall continues to propagate, the field is kept\nconstant, however, if pinning is detected the field is increa sed\nto sustain the propagation. d) The device is entirely filled u p\nwith domain walls at all alternate corners in the structure.\ne)-g) Nucleation field measurement. e) Empty state for the\ninitial configuration is obtained by rotating the field count er-\nclockwise with a field higher than the propagation field value .\nf) The field is increased and rotated counter-clockwise unti l a\nnucleation eventoccurs. g) The final state with afilled senso r.\nAll the propagation field values for the Ni 81Fe19sam-\nples havebeen found to be between 5 and25 mT. Despite\nthe variations in width and thickness, all the Ni 81Fe19\nsamples, exhibit similar propagation fields. The shape of\nthe wire is thus not entirely governing the domain wall\npropagation field values. The propagation field is mainly\naffected by the irregularities of the shape and the ma-\nterial. However, due to the redeposition on the wires,\nit is difficult to directly ascertain the relevant magnetic\nroughness of the wires, which might be different from the\ntopographical roughness. A priori, this is not very sur-\nprising as for perfect wires the propagation field would\nbe zero and independent of the wire geometry. How-\never, in real wires, defects and edge roughness play the\nrole of governing mechanisms for the propagation field,\nand these effects are not strongly geometry dependent\n(width, thickness). The small increase with decreasing\nwire width can be explained by the edge roughness be-\ncoming relatively more important for narrowerwires (the\nedge roughness is largely wire width independent). The\nCo90Fe10samples have propagation fields that are two\ntimeshigherthanthe onesforthe Ni 81Fe19. Weattribute\nthe latter effect to the magneto-crystalline anisotropy of\neach Co 90Fe10crystallite generating an energy landscape\nwhich increases the pinning of the domain walls. Fur-\nthermore, samples with a thin layer of Co 90Fe10(1 nm)\nbelow the Ni 81Fe19layer were investigated, and we did\nnot observe significant differences between the sampleswith and without the thin layer, which thus plays only a\nminor role for the magnetic properties.\nIn summary, for devices, it is therefore desirable\nto avoid materials with strong magneto-crystalline\nanisotropy and limit processing variability to ensure the\nreliability of domain wall propagation. The propagation\nfield appears then as a characteristic materials param-\neter that is not strongly dependent on the wire width\nand thus cannot be easily tailored by the geometry to\nimprove the operational reliability. Since the propaga-\ntion field does not provide for an easy handle for the im-\nprovement of magnetic sensors operation conditions as\nthe minimum propagation field cannot be easily reduced\nby for instance changing the geometry, we next investi-\ngate the nucleation field of the devices.\nFIG. 3. Propagation and nucleation fields for the investi-\ngated samples. The boxes associated with the data points in\nthe plot represent 25% (first quartile) to 75 % (third quartil e)\nof the distribution. The whiskers or dashed lines represent 5%\nto 25% and 75% to 95%. In this manner, the plot represents\nthe key features of the entire distribution. The round point s\nrepresent the average value of the propagation while the dia -\nmonds represent thenucleation field. a) Plot of the nucleati on\nand propagation field values as a function of the width of the\nwire for the Ni 81Fe19(28 nm) and Ni 81Fe19(32 nm). The black\nline is the pure Stoner-Wohlfarth behaviour for Ni 81Fe19(32\nnm). The blue and green lines are the adapted model fitted\nwith a scaling constant C = 0.7. b) Similar plot as a) for\nCo90Fe10(17 nm).\n2.Nucleation field\nAt high fields, instead of domain wall nucleation only\noccurring in the pad region of the device, undesired do-\nmain wall nucleation takes place in the wires. For the\ntapered end of the device, it is expected that the shape\nanisotropy, in this part, is too high for the nucleation\nof a new domain wall for the probed field range. Hav-\ning established the propagation field, we can rotate the\napplied field counter-clockwise to empty the device com-\npletely of all the domain walls by annihilating them in\nthe nucleation pad. The resulting magnetic state is the\ninitial state, which we term the vortex state due to its re-\nsemblance to the one for ring structures as seen in Fig. 2\ne). By continuing to rotate the field in this direction5\nwith incrementing field strength, we selectively detect\ndomain wall generation through spontaneous nucleation\nsomewhere within the wire, as depicted in Fig. 2f). The\nfield at which such an event is detected is termed the nu-\ncleation field and is of interest since it yields the field at\nwhich the measured information can potentially be lost\nin a failure scenario.\nWe find that the scaling of the nucleation field follows\na power law as a function of the width. The nucleation\nfieldfordomainwallsinnanopatternedsoftmagneticma-\nterials is mainly determined by the cross-sectional shape\nof the system (width and thickness). For a closed sys-\ntem such as a ring or a loop, the only boundaries are\nthe two side edges thus which can be approximated as\ninfinitely long. Furthermore, if the radius of curvature is\nmuch larger than the width of the wire, then no lower-\ning of the nucleation field value is expected at corners.\nSuch effects are usually provoked by a flux closure spin\nconfiguration at the ends of the wire. This reduction of\nthe nucleation field has been observed in the case where\na wire relaxes into an S or C state [25]. Since our ma-\nterials are relatively thick and soft, the magnetization is\nlying in the plane in the direction of the wire length due\nto a dominant magnetostatic energy contribution. The\nexpectation is that the shape anisotropy is playing a pri-\nmary role in the determination of the nucleation field\nvalue. Within the framework of the Stoner-Wohlfarth\nmodel, a particle with dimensions smaller than the ex-\nchange length (5 nm in Ni 81Fe19), the magnetization is\napproximatedwith a macrospinand is expected to rotate\ncoherently during the switching. In the most simplistic\nversion of the model, this particle is only subject for in-\nstance to a uniaxial anisotropy of the form K·sin2(θ),\nin our case, K being mainly the shape anisotropy.\nFor larger systems that are not fully described as a\nmacrospin, one can expect an activation volume, located\nfor instance at the point of lowest anisotropy, to rotate\ncoherently. Mathematically, this can be described as fol-\nlows:\nE=EZeeman+EDemag=−µ0HMsV+1\n2µ0NyM2\nsV\n(1)\nwithMsbeing the saturation magnetization, Vthe\nactivation volume and Nythe demagnetizing factor de-\nscribed in [26] for an infinitely long wire:\nHn=1\n2t\nt+wMs (2)\nWithtthe thickness and wthe width of the sample.\nIn this simple formula, the nucleation field is then deter-\nmined by the geometry as well as the saturation mag-\nnetization, which is a materials constant. As plotted, in\nFig.3in black and Fig. 4full lines, such a calculated\ncurve for the pure Stoner-Wohlfarth behaviour does not\nreproduce the obtained data quantitatively.To understand the magnetization behavior close but\nbelow the nucleation field, some simulations are per-\nformed with the software Mumax3 [27]. AFM profiles\nare used as the simulated shape, and periodic boundary\ncondition serves to extend the length of the wire. The\nframework is discretized with a cell volume of 2x2x2 nm3\nto permit a good representation of the realistic shape.\nThe magnetization is initialized upward and left to relax.\nA field is then applied for 20 ns following an equation of\nthe form B= (−√\n2\n2Bext(1−exp/parenleftbig−t\n4e−9/parenrightbig\n),−√\n2\n2Bext(1−\nexp/parenleftbig−t\n4e−9/parenrightbig\n),0) to avoid artefacts due to an instantaneous\napplied field. A bisection method was then used to de-\ntermine the nucleation field within 1 mT precision.\nThe results are presented in Fig. 4. A snapshot of the\nmagnetization in Fig. 4represents the relaxed state of\nthe stripe at an applied field value that is just 1 mT be-\nlowthe nucleation field value. The spin structureand the\nsubsquent dynamics show that the reversal mode of our\nstripes ressembles an in-plane curling mode. The latter\nis expected for large systems with inhomogeneities in the\nanisotropies [28]. As compared to the Stoner-Wohlfarth\nmodel, the rotation of magnetization is not coherent in\nthe whole structure and exhibits a 2-dimensional varia-\ntion along the wire [29]. A pure curling mode would not\nyield nucleation field values significantly different from\nthe Stoner-Wohlfarth model [30]. Furthermore, our data\nalso show that despite the trapezoidal shape and the\nincluded edge roughness, the nucleation field at 0 K is\non average 90 % of the expected one from the Stoner-\nWohlfarth model.\nFIG. 4. Simulation results of the nucleation field as a functi on\nof the minimum width of the wire. The green and blue full\nline represent the SW model for the Ni 81Fe19(28 nm) and\nNi81Fe19(32 nm), respectively. The dashed lines are the SW\nmodel with a scaling factor C = 0.9. A simulation snapshot\nof the magnetization was taken at a field just 1 mT below the\nnucleation field value of a 200 nm wide wire of Ni 81Fe19(32\nnm).\nHowever,wireirregularitiesareexpectedtoyieldalow-6\nerednucleationfieldasseenfromtheinhomogeneousspin\nconfigurations shown in the simulation results in Fig. 4.\nIn the real system, further effects can lead to a reduc-\ntion of the nucleation field. Damage from the ion milling\ncausing a change of crystallization at the edges and a\ndecrease of the saturation magnetization or a doping of\nthe wire due to material implantation can lead to a lo-\ncally reduced shape anisotropy due to a reduced satu-\nration magnetization. Finally, thermal activation is also\ngoing to have an impact not considered in the Stoner-\nWohlfarth model. In order to account for such effects in\nthe simplest manner, we include a scaling factor C to the\ndemagnetization factor as follows:\nHn=C1\n2t\nt+wMs (3)\nBy fitting the data in Fig. 3, we find that a value of\nC = 0.7 provides good agreement with the results of the\nsamples. Overall, the clearly observable variations of the\nnucleation field with the geometry provide a handle to\ntailor the characteristics of the device thus the operation\ncondition and the reliability.\nB.GMR investigation\nTo investigate this further and obtain better statis-\ntics, we measure a very large number (3200) of 16 loop\ndevices. The majority of the possible geometrical vari-\nations are going to be encountered, generating a clear\nidea of the maximum deviations still enabling a working\ndevice. Furthermore, by taking the absolute resistance\nvalue of the sensor based on the geometry (width, thick-\nness, and length), we can compare it to the nucleation\nfield to check for possible connections.\nAn entire waferwas prepared with the previouslymen-\ntioned GMR stack structures and contacted to measure\nthe resistance of the wires. The resistance in the initial\nstate (no domain walls) is probed and compared to the\ncase where a domain wall is inside. An algorithm using\nalso a bisection method is used to define the exact nucle-\nationfield value. The appliedfield sequenceis made of17\nrotationscounter-clockwisefollowedbyanelectricalmea-\nsurement. After the 17 turns, the device is expected to\nbe empty if the nucleation field is higher than the tested\nvalue. We describe the standard measurement scheme\nfor a working element, any failure in the steps result in\nthe device being counted as defective. A starting value of\n30 mT is used since the propagation limit was previously\nmeasured lower than this. After the sequence, if the sen-\nsor is measured empty then the field is increased to 80\nmT, and the sensor should be entirely filled with domain\nwalls as we expect the nucleation field of all the struc-\ntures to be lower. The next field value is taken as half\nof the difference of the previous ones added to the low-\nest value giving 55 mT. If the device is measured empty,then the following field is half of the difference between\nthe highest (80 mT) and the middle value (55 mT) added\ntothemiddlevalue. Thelowestvalue(30mT)isreplaced\nby the middle one (55 mT). If the sensor is filled, then\nhalf of the difference between the lowest (30 mT) and\nthe middle value (55 mT) is added to the lowest value.\nThe highest value (80 mT) is replaced by the middle one\n(55mT). The algorithmcontinuesuntil the difference be-\ntween the lowest and the highest field values is smaller\nthan 1 mT.\nThe latter processservedto measurethe specific nucle-\nation field of 3200 structures. Approximately 800 mea-\nsurementsperwirewidthwereperformed. Theresistance\nis measured across the device between the V ccand GND\n(as seen in the Appendix [13]). Due to the connection\nlayout, the number of wire connected is 33, and their\nlengthis400 µm. Beforetheresistancemeasurement, the\nsensors were initialized with domain walls in the whole\ndevice. Due to the looping configuration and the domain\nwalls present, half of a wire is in a ‘high’ resistive state\nwith the GMR and the other half is in a ‘low’ resistive\nstate. Thus the measured resistance does not contain a\nGMR component(the DWhasanegligiblecontribution).\nThe resistance of every wire is then similar. We then ap-\nplythe formula r=RsL\nwforthe resistanceofawire, with\nRs= 4.09 Ω/sq being the sheet resistance, L the length\nof a wire and w the width. The wires are connected in\nparallel thus the resistance of the device is R=r\n33. We\nplug the latter in equation 3to obtain:\nHn=1\n2t\nRsL+33Rt33MsR (4)\nwithRbeing the resistance of the sensor, tthe thick-\nness,Msthe saturation magnetization and Lthe length\nof the wire. The previously described model with a\nconstant C= 0.7 is plotted in brown while the Stoner-\nWohlfarth nucleation field is in black.\nFIG. 5. Nucleation field of the device as a function of the\nresistance. The different nominal widths are represented by\ndifferent colors: 350 nm (blue), 300 nm (red), 250 nm (green)\nand 200 nm (purple).7\nIn the Fig. 5, a fixed reference value for propagation\nis used (30 mT). Indeed, for the algorithm to work a\nstarting propagation field value is needed.\nThe expected theoretical resistances are 247 Ω, 198 Ω,\n165 Ω, 141 Ω for respectively 200 nm (purple), 250 nm\n(red), 300 nm (green), 350 nm (blue). The measurement\nof the resistance is realized over the whole structure with\nthe 33 wires in a parallel configuration. The resistance of\nthe device is then plotted as a function of the nucleation\nfield. The previously described model is shown as well as\nthe Stoner-Wohlfarth model.\nOnly a few points are visible in the figure for the 350\nnmwirewidthduetoanaveragenucleationlowerthan30\nmT. The nucleation field values are observed to lie below\nthe adapted model. Furthermore, we observethat the re-\nsistance is always higher than the one expected from the\nnominalwidth. Thelatterisaconfirmationthatthemea-\nsured width with the SEM is probably not the effective\nelectrical width contributing to the magnetic and elec-\ntrical signal and that most of the sides are covered with\nredeposited material thus generating an effectively larger\ntopographical width. Note that taking into account the\nmultilayernatureofthe GMRstackbycalculatingthere-\nsistance using the Fuchs-Sondheimer model [31, 32] does\nnot significantly change the values as the conduction is\ndominated by the thick Ni 81Fe19free layer which carries\nmost of the current. The limit shown by the adapted\nStoner-Wohlfarth model demonstrate the impossibility\nfor our current architecture (3 vertices and 16 loops of\nthe geometries used) to reach the ideal Stoner-Wohlfarth\nmodel behaviour. The fitting constant of 0.7 sets the\nmaximum average nucleation field obtainable for these\nindustrially produced devices in these geometries. These\nresults are of major importance for applications due to\nthe fact that a simple resistance measurement allows to\nthe identification of a non-working device. As an exam-\nple, if the requirements are that devices should not ex-\nhibit a nucleation field lowerthan 40 mT then any device\nwitharesistancebelow200Ωcanbediscarded. Thispro-\nvides a tremendous gain of time since the measurement\nduration for the characterization of a single device can\nbe 1 minute.\nV.Conclusion\nTo conclude, we have determined the critical fields for\nsensor operation based on domain wall propagation al-\nlowing us to gauge the limitations for the operation of\nthe devices. We find that both the materials and the\ngeometry play a key role. Firstly, the origin of the ge-\nometrical dependence of the propagation is difficult to\npinpoint, since a variety of factors contribute to the vari-\nation in propagation field and these are hard to pre-\ncisely characterize and quantify. Compared to Ni 81Fe19,\nthe Co 90Fe10samples yield a drastically increased prop-\nagation field. This can be attributed to the enhancedmagneto-crystalline anisotropy in each individual crys-\ntallite compared to Ni 81Fe19.\nFor the nucleation field, the dependence on the ge-\nometry exhibits a geometrical scaling of the form that\none expects if the shape anisotropy dominates. It can\nbe described by the Stoner-Wohlfarth model, despite the\nsimulation not showing a coherent rotation of the mag-\nnetization in the complete stripe. Thus the nucleation\nfield geometry dependence provides a handle for the im-\nprovement of magnetic domain wall sensors. For all the\nmeasured materials, the maximum expected nucleation\nvalue can be fitted by a corrected uniaxial anisotropy\nrectified by a constant accounting for the processing, the\nangle segmentation, and the geometrical scale. A mea-\nsurement of a large number of devices is performed to\nallow for an accurate assessment of the fitting constant\nused for the MOKE measurement results. We can ascer-\ntain a definite limit by the previously mentioned maxi-\nmum nucleation field value and find that this is related\nto the resistance of the sensor. This limit can be used as\na tool for future fast analysis of magnetic sensors.\nAcknowledgements\nThe authors would like to acknowledge the company\nSensitec GmbH for providing the samples, the company\nNovotechnik for providing the designs and the WALL\nproject for financial support. The work and results re-\nported in this publication were obtained with research\nfunding from the European Community under the Sev-\nenth Framework Programme - The people Programme,\nMulti-ITN “WALL” Contract Number Grant agreement\nno.: 608031, and a European Research Council Proof-of-\nConcept grant (MultiRev ERC-2014-PoC (665672)).8\n[1]Mathias Kl¨ aui, “Head-to-head domain walls in magnetic\nnanostructures,” J. Phys. Condens. Matter 20, 313001\n(2008).\n[2]D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson,\nD. Petit, and R. P. Cowburn, “Magnetic domain-wall\nlogic,”Science309, 1688–1692 (2003) .\n[3]Stuart S. P. Parkin, Masamitsu Hayashi, and Luc\nThomas, “Magnetic domain-wall racetrack memory,”\nScience320, 190–194 (2008) .\n[4]H Corte-Le´ on, P Krzysteczko, H. W. Schumacher,\nA. Manzin, D. Cox, V. Antonov, and O. Kaza-\nkova, “Magnetic bead detection using domain wall-based\nnanosensor,” J. App. Phys. 117, 17E313 (2015) .\n[5]M. A. Bashir, M. T. Bryan, D. A. Allwood, T. 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Phys. 1, 1 (1952)."    },    {        "title": "1705.00700v2.One_dimensional_in_plane_edge_domain_walls_in_ultrathin_ferromagnetic_films.pdf",        "content": "One-dimensional in-plane edge domain walls in\nultrathin ferromagnetic \flms\nRoss G. Lund1, Cyrill B. Muratov1, and Valeriy V. Slastikov2\n1Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ\n07102, USA\n2School of Mathematics, University of Bristol, Bristol BS8 1TW, UK\nFebruary 7, 2018\nAbstract\nWe study existence and properties of one-dimensional edge domain walls in ultrathin ferro-\nmagnetic \flms with uniaxial in-plane magnetic anisotropy. In these materials, the magnetization\nvector is constrained to lie entirely in the \flm plane, with the preferred directions dictated by\nthe magnetocrystalline easy axis. We consider magnetization pro\fles in the vicinity of a straight\n\flm edge oriented at an arbitrary angle with respect to the easy axis. To minimize the micro-\nmagnetic energy, these pro\fles form transition layers in which the magnetization vector rotates\naway from the direction of the easy axis to align with the \flm edge. We prove existence of edge\ndomain walls as minimizers of the appropriate one-dimensional micromagnetic energy functional\nand show that they are classical solutions of the associated Euler-Lagrange equation with Dirich-\nlet boundary condition at the edge. We also perform a numerical study of these one-dimensional\ndomain walls and uncover further properties of these domain wall pro\fles.\n1 Introduction\nThe \feld of ferromagnetism of thin \flms is currently undergoing a renaissance driven by advances\nin theory, experiment and technology [7, 11, 14, 24, 31, 42, 44]. Study of magnetic domain walls is\nremaining at the forefront of this activity and attracts a lot of attention from engineering, physical\nand mathematical communities. These studies are in part motivated by a new \feld of applied\nphysics { spintronics { o\u000bering a great promise for creating the next generation of data storage and\nlogic devices combining spin-dependent e\u000bects with conventional charge-based electronics [2,3].\nThere are two most common types of domain walls connecting the distinct preferred directions\nof magnetization in uniaxial materials: Bloch and N\u0013 eel walls [22]. Bloch walls appear in bulk\nferromagnets, where the magnetization pro\fle connecting the two opposite easy axis directions\nprefers an out-of-plane rotation with respect to the plane spanned by the wall direction and the\neasy axis. In ultrathin ferromagnetic \flms, on the other hand, the stray \feld energy penalizes\nout-of-plane rotations, and as a result the magnetization pro\fle is constrained to the \flm plane. A\ndomain wall pro\fle connecting the two distinct preferred directions of magnetization via an in-plane\nrotation is called a N\u0013 eel wall .\nN\u0013 eel walls have been thoroughly investigated theoretically since their discovery, and their inter-\nnal structure is currently fairly well understood. The main characteristics of the one-dimensional\nN\u0013 eel wall pro\fle typically include an inner core, logarithmically decaying intermediate regions\n1arXiv:1705.00700v2  [math.AP]  5 Feb 2018and algebraic tails. These features have been predicted theoretically, using micromagnetic treat-\nments [16, 19, 22, 36, 39], and veri\fed experimentally [5]. Recent rigorous mathematical studies of\nN\u0013 eel walls con\frmed these predictions and provided more re\fned information about the pro\fle of\nthe N\u0013 eel wall, including uniqueness, regularity, monotonicity, symmetry, stability and precise rate\nof decay [9,10,12,20,33,38].\nAnother type of a domain wall has been recently observed in ultrathin ferromagnetic \flms with\nperpendicular anisotropy and strong antisymmetric exchange referred to as Dzyaloshinskii-Moriya\ninteraction (DMI). The presence of DMI signi\fcantly alters the structure of domain walls, leading\nto formation of chiral domain walls in the interior and chiral edge domain walls at the boundary\nof the ferromagnetic sample [37, 40, 43]. These chiral domain walls and chiral edge domain walls\nplay a crucial role in producing new types of magnetization patterns inside a ferromagnet and have\nbeen rigorously analyzed in [37].\nIt is well known that magnetization con\fgurations in ferromagnets are signi\fcantly a\u000bected\nby the presence of material boundaries [11, 14, 22]. To reduce the stray \feld, the magnetization\nvector tries to stay tangential to the material boundary, thus minimizing the presence of boundary\nmagnetic charges. In ultrathin \flms, this forces the magnetization vector to lie almost entirely in\nthe \flm plane and align tangentially along the \flm's lateral edges [27]. At the same time, these\ngeometrically preferred directions may disagree with the intrinsic directions in the bulk \flm, deter-\nmined by either a strong in-plane uniaxial crystalline anisotropy or an external in-plane magnetic\n\feld. The result of this incompatibility is another type of magnetic domain walls { edge domain\nwalls . These domain walls have been observed experimentally in magnetically coupled bilayers\nin the shape of strips with an easy axis normal to the strip [41], and in single-layer strips with\nnegligible crystalline anisotropy and varying in-plane uniform magnetic \feld [32].\nThe origin of edge domain walls is due to the competition between magnetostatic, exchange and\nanisotropy energies. Strong uniaxial anisotropy de\fnes the two preferred magnetization directions\nwithin the ferromagnetic \flm plane. On the other hand, at the \flm edge the magnetostatic energy\npenalizes the magnetization component normal to the edge, and consequently the magnetization\nprefers to lie in-plane and tangentially to the edge of the ferromagnetic \flm. The exchange en-\nergy allows for a continuous transition between these states and as a result an edge domain wall\nconnecting the direction tangent to the \flm edge and the anisotropy easy axis direction is created.\nExperimental observations in soft ferromagnetic thin \flms and bilayers [32,41] indicate that edge\ndomain walls, formed near the boundary of the sample due to a misalignement of the tangential\nand applied \feld directions, have an essentially one-dimensional character. This is con\frmed by\nmicromagnetic simulations in extended ferromagnetic strips performed in several regimes, including\nstrong in-plane uniaxial anisotropy with no applied \feld (see Fig. 1) and no crystalline anisotropy\nwith strong in-plane applied \feld (results not shown). Numerical simulations suggest that away\nfrom the side edges the domain walls have essentially one-dimensional pro\fles. Therefore, in order\nto investigate these pro\fles it is enough to model their behavior, employing a simpli\fed one-\ndimensional micromagnetic energy capturing the essential features of the wall pro\fles. Such a\ndescription is expected to be appropriate for strips of soft ferromagnetic materials whose thickness\ndoes not exceed signi\fcantly the exchange length and whose width is much larger than the N\u0013 eel\nwall width.\nThe goal of this paper is to understand the formation of edge domain walls viewed as global\nenergy minimizers of a reduced one-dimensional micromagnetic energy. We begin our analysis by\nderiving a one-dimensional energy functional describing edge domain walls (see (15)). Since we\nare speci\fcally interested in one-dimensional domain wall pro\fles, we consider the problem on an\nunbounded domain consisting of a ferromagnetic \flm occupying a half-plane times a \fxed interval\nwith small thickness. However, this setup makes the energy of the wall in\fnite due to inconsistency\n2Figure 1: A magnetization con\fguration containing edge domain walls in a strip obtained from\nmicromagnetic simulations of a 20 :7\u0016m\u00025:2\u0016m\u00024nm permalloy sample with vertical uniaxial\nanisotropy and no applied \feld (for further details, see section 8). The colormap corresponds\nto the angle between the magnetization vector (also shown by arrows) and the y-axis. Inside the\ndashed box (i.e., far from the side edges) the edge wall pro\fles are essentially one-dimensional.\nbetween the preferred magnetization directions at the \flm edge and inside the \flm (see section 3).\nTherefore, in order to have a well de\fned minimization problem, we need to renormalize the one-\ndimensional energy per unit edge length in a suitable way (see (24)). We show existence of a\nminimizer for this energy, using standard methods of the calculus of variations; see Theorem 1.\nThe main di\u000eculty lies in dealing with nonlocal magnetostatic energy term and identifying the\nproper space where the minimization problem makes sense. We continue our analysis by deriving\nthe Euler-Lagrange equation characterizing the pro\fle of the edge domain wall; see Theorem 2.\nThis seemingly straightforward task, however, requires a rather careful and proper dealing with the\nnonlocal energy term. The main di\u000eculty is related to the fact that we only have rather limited\nregularity of the energy minimizing solutions a priori. The information about further regularity is\nusually recovered through the use of the Euler-Lagrange equation and a bootstrap argument. As\nthis information is not yet available, we need to carefully analyze the nonlocal term using methods\nfrom fractional Sobolev spaces and recover a weak form of the Euler-Lagrange equation. After\nderiving the Euler-Lagrange equation, we can prove higher regularity of the solutions using an\nadaptation of the standard elliptic regularity techniques. However, due to the di\u000eculties arising in\ndealing with nonlocality we can only show the C2regularity of solutions. Further application of the\nbootstrap argument is then hindered by the lack of integrability of the contribution of the nonlocal\nterm to the Euler-Lagrange equation when higher derivatives of the solution are considered, and\nthe second derivative of the solution indeed blows up at the \flm edge.\nAfter establishing existence and regularity of the edge domain wall pro\fles, we investigate\ntwo speci\fc regimes where we can provide re\fned information about the properties of the energy\nminimizing solutions of the Euler-Lagrange equation; see Theorem 3 and Theorem 4. The \frst\nregime that we consider is the regime of relatively small magnetostatic energy, which corresponds\nto very thin \flms. In this regime we show that all minimizers of the energy (24) are close to the\nstandard local N\u0013 eel wall-type pro\fle. The second regime is the regime in which the boundary\ntangent and the easy axis directions are nearly parallel. In this case we show that there is a unique\nminimizer of the energy in (24), and this minimizer is close to the uniform state. We corroborate\nour analytical \fndings and provide more information about the pro\fles of edge domain walls, using\none-dimensional numerical simulations that employ the method from [36].\n3Our paper is organized as follows. In section 2, starting from the full three-dimensional micro-\nmagnetic model we derive a variational model for edge domain walls that we intend to investigate\nin this paper. Section 3 is devoted to a rigorous formulation of the problem and includes the\nstatements of the main results about existence, regularity and the qualitative features of edge do-\nmain walls. In sections 4, 5, 6 and 7 we prove the main theorems formulated before in section 3.\nIn section 8, we present the results of numerical simulations, compare them with our analytical\n\fndings and discuss open problems. Finally, in Appendix A we provide a rigorous derivation of\nthe one-dimensional micromagnetic energy in magnetic strips under natural assumptions on the\none-dimensional magnetization pro\fle.\n2 Model\nConsider a uniaxial ferromagnet occupying a domain \n \u001aR3, with the easy axis oriented along the\nsecond coordinate direction. Then the micromagnetic energy associated with the magnetization\nstate of the sample reads, in the SI units [22,29]:\nE(M) =A\nM2sZ\n\njrMj2d3r+K\nM2sZ\n\n(M2\n1+M2\n3)d3r\n\u0000\u00160Z\n\nM\u0001Hd3r+\u00160Z\nR3Z\nR3r\u0001M(r)r\u0001M(r0)\n8\u0019jr\u0000r0jd3rd3r0:(1)\nHere M= (M1;M2;M3) is the magnetization vector that satis\fes jMj=Msin \n and M= 0 in\nR3n\n, the positive constants Ms,AandKare the saturation magnetization, exchange constant\nand the anisotropy constant, respectively, His an applied external \feld, and \u00160is the permeability\nof vacuum. In (1), the terms in the order of appearance are the exchange, crystalline anisotropy,\nZeeman and stray \feld terms, respectively, and r\u0001Mis understood distributionally.\nIn this paper, we are interested in the situation in which \n is a \rat ultra-thin \flm domain, i.e.,\nwe have \n = D\u0002(0;d), whereD\u001aR2is a planar domain specifying the \flm shape and dis the\n\flm thickness of a few nanometers. In this case the magnetization is expected to be essentially\nindependent from the third coordinate, and the full three-dimensional micromagnetic energy admits\na reduction to an energy functional that depends only on the average of the magnetization over\nthe \flm thickness (see, e.g., [27, Lemma 3]; for an analytical treatment in a closely related context,\nsee [26, 35]). Therefore, we introduce an ansatz M(x1;x2;x3) =Ms(m(x1;x2);0)\u001f(0;d)(x3), where\nm:R2!R2is a two-dimensional in-plane magnetization vector satisfying jmj= 1 inDand\njmj= 0 outside D, and\u001f(0;d)is the characteristic function of (0 ;d). Next, we de\fne the exchange\nlength`, the Bloch wall width L, and the thin \flm parameter \u0017measuring the relative strength of\nthe magnetostatic energy [36]:\n`=s\n2A\n\u00160M2s; L =r\nA\nK; \u0017 =\u00160M2\nsd\n2p\nAK; (2)\nand note that the above ansatz is relevant when d.`[13, 17, 18, 27, 36]. Then, measuring the\nenergy in the units of 2 Adand lengths in the units of L, we obtain the following expression for the\nenergy as a function of m[19]:\nE(m) =1\n2Z\nD\u0000\njrmj2+m2\n1\u00002h\u0001m\u0001\nd2r+\u0017\n2Z\nR2Z\nR2K\u000e(jr\u0000r0j)r\u0001m(r)r\u0001m(r)d2rd2r0;(3)\n4where\u000e=d=Lis the dimensionless \flm thickness,\nK\u000e(r) =1\n2\u0019\u000e(\nln \n\u000e+p\n\u000e2+r2\nr!\n\u0000r\n1 +r2\n\u000e2+r\n\u000e)\n; (4)\nand we set H=K=(\u00160Ms)(h;0) for h:R2!R2, assuming that the applied \feld lies in the \flm\nplane. More explicitly, assuming that @Dis of classC2, we have\nE(m) =1\n2Z\nD\u0000\njrmj2+m2\n1\u00002h\u0001m\u0001\nd2r+\u0017\n2Z\nDZ\nDK\u000e(jr\u0000r0j)r\u0001m(r)r\u0001m(r)d2rd2r0\n\u0000\u0017Z\nDZ\n@DK\u000e(jr\u0000r0j)r\u0001m(r)(m(r0)\u0001n(r0))dH1(r0)d2r\n+\u0017\n2Z\n@DZ\n@DK\u000e(jr\u0000r0j)(m(r)\u0001n(r))(m(r0)\u0001n(r0))dH1(r0)dH1(r);(5)\nwhere nis the outward unit normal vector to @D, and we took into account that the distributional\ndivergence of mis the sum of the absolutely continuous part in Dand a jump part on @D.\nWe now consider the thin \flm limit introduced in [36] by sending \u000eto zero with \u0017andD\fxed.\nObserve that when \u000eis small, we have\nK\u000e(r)'1\n4\u0019randZ\n@DK\u000e(jr\u0000r0j)dH1(r0)'1\n2\u0019ln\u000e\u00001: (6)\nTherefore, when mdoes not vary appreciably on the scale of \u000e, to the leading order we have\nE(m)'E\u000e(m), where\nE\u000e(m) =1\n2Z\nD\u0000\njrmj2+m2\n1\u00002h\u0001m\u0001\nd2r+\u0017\n8\u0019Z\nDZ\nDr\u0001m(r)r\u0001m(r)\njr\u0000r0jd2rd2r0\n\u0000\u0017\n4\u0019Z\nDZ\n@Dr\u0001m(r)(m(r0)\u0001n(r0))\njr\u0000r0jdH1(r0)d2r+\u0017ln\u000e\u00001\n4\u0019Z\n@D(m(r)\u0001n(r))2dH1(r):(7)\nSince the last term in (7) blows up as \u000e!0, unless m\u0001n= 0 a.e. on @D, in the limit we recover\nE0(m) =1\n2Z\nD\u0000\njrmj2+m2\n1\u00002h\u0001m\u0001\nd2r+\u0017\n8\u0019Z\nDZ\nDr\u0001m(r)r\u0001m(r)\njr\u0000r0jd2rd2r0; (8)\nwith admissible con\fgurations m2H1(D;S1) satisfying Dirichlet boundary condition m=st\non@D, where tis the positively oriented unit tangent vector to @Dands:@D!f\u0000 1;1g. In\nfact, since the trace of mbelongs toH1=2(@D;R2), the function sis necessarily constant on each\nconnected component of @D. Note that this creates a topological obstruction in the case when Dis\nsimply connected, giving rise to boundary vortices at the level of E\u000e[27,28,34]. At the same time, it\nis clear that for suitable multiply connected domains the considered admissible class is non-empty.\nA canonical example of the latter is an annulus (for a physics overview, see [25]). In the absence\nof crystalline anisotropy and applied \feld, the ground state of the magnetization in an annulus is\neasily seen to be a vortex state. However, this result no longer holds in the presence of crystalline\nanisotropy, since the latter does not favor alignment of mwith the boundaries. In large annuli,\nthis would lead to the formation of a boundary layer, in which the magnetization rotates from the\ndirection tangential to the boundary to the direction of the easy axis. We call such magnetization\ncon\fgurations edge domain walls (for similar objects in a di\u000berent micromagnetic context, see [37]).\n52xaxiseasy\nβ\nx1xFigure 2: Illustration of the strip geometry.\nFocusing on one-dimensional transition pro\fles in the vicinity of the boundary, we now consider\nDto be a strip of width woriented at an angle \f2[0;\u0019=2] with respect to the easy axis (see Fig.\n2). We de\fne\nx=x1cos\f+x2sin\f (9)\nto be the variable in the direction normal to the strip axis. Then, with the applied \feld hset to\nzero, the energy of a magnetization con\fguration m=m(x) per unit length of the strip is equal\nto (see Appendix A)\nE\f;w(m) =1\n2Zw\n0\u0000\njm0\n1j2+jm0\n2j2+m2\n1\u0001\ndx+\u0017\n4\u0019Zw\n0Zw\n0lnjx\u0000yj\u00001m0\n\f(x)m0\n\f(y)dxdy; (10)\nwherem\f=e\f\u0001m, with e\f= (cos\f;sin\f), provided that\nm\f(0) =m\f(w) = 0: (11)\nThe energy in (10) may also be rewritten using the operator\u0010\n\u0000d2\ndx2\u00111=2\n(acting from H1(R) to\nL2(R) and understood via Fourier space, see Appendix A)\nE\f;w(m) =1\n2Zw\n0\u0000\njm0\n1j2+jm0\n2j2+m2\n1\u0001\ndx+\u0017\n4Z1\n\u00001m\f\u0012\n\u0000d2\ndx2\u00131=2\nm\fdx: (12)\nAnother useful representation of the energy in (10) that expresses the double integral in terms of\nm\frather than its derivative is (see Appendix A)\nE\f;w(m) =1\n2Zw\n0\u0000\njm0\n1j2+jm0\n2j2+m2\n1\u0001\ndx+\u0017\n8\u0019Z1\n\u00001Z1\n\u00001(m\f(x)\u0000m\f(y))2\n(x\u0000y)2dxdy: (13)\n6Lastly, we express the energy in (13) in terms of the angle \u0012between mand the easy axis in the\ncounter-clockwise direction:\nm= (\u0000sin\u0012;cos\u0012): (14)\nWith a slight abuse of notation, we get that the energy associated with mis given by\nE\f;w(\u0012) =1\n2Zw\n0\u0000\nj\u00120j2+ sin2\u0012\u0001\ndx+\u0017\n8\u0019Z1\n\u00001Z1\n\u00001(sin(\u0012(x)\u0000\f)\u0000sin(\u0012(y)\u0000\f))2\n(x\u0000y)2dxdy; (15)\nwhere we set \u0012(x) =\f, for allx62(0;w). The energy functional in (15) is the starting point of\nour analysis throughout the rest of this paper. In particular, it is straightforward to show that\nminimizers of (15) exist among all \u0012\u0000\f2H1\n0(0;w), are smooth in the interior and satisfy the\nEuler-Lagrange equation\n0 =d2\u0012\ndx2\u0000sin\u0012cos\u0012\u0000\u0017\n2cos(\u0012\u0000\f)\u0012\n\u0000d2\ndx2\u00131=2\nsin(\u0012\u0000\f)x2(0;w); (16)\nwhere [15]\n\u0012\n\u0000d2\ndx2\u00131=2\nu(x) =1\n\u0019\u0000Z1\n\u00001u(x)\u0000u(y)\n(x\u0000y)2dy; (17)\nand here and everywhere below \u0000R\ndenotes the principal value of the integral. Notice that the Euler-\nLagrange equation in (16) coincides with the one for the classical problem of the N\u0013 eel wall [10].\n3 Statement of results\nWe now turn to the problem of our main interest in this paper, which is to characterize a single\nedge domain wall. For this purpose, we would like to send the parameter wto in\fnity and obtain\nan energy minimizing pro\fle \u0012(x) solving (16) for all x >0 and satisfying \u0012(0) =\f(also setting\n\u0012(x) =\ffor allx<0 in the de\fnition of the last term in (16)). We note that for the problem on\nthe semi-in\fnite domain with \f2[0;\u0019=2] the boundary condition at x= 0 is equivalent to that\nin (11) because of the re\rection symmetry, which makes the energy invariant with respect to the\ntransformation\n\u0012!\u0000\u0012; \f!\u0000\f: (18)\nWe also note that for \f= 0 we clearly have \u0012= 0 as the unique global minimizer for the energy in\n(15). Therefore, in the following we always assume that \f >0.\nAs can be seen from standard phase plane analysis, in the absence of the nonlocal term due to\nstray \feld, i.e., when \u0017= 0, the edge domain wall solution is explicitly\n\u0012(x) = 2 arctan\u0012\ne\u0000xtan\f\n2\u0013\nforx>0; (19)\nnoting that for \f=\u0019=2 there is also another solution which is obtained from the one in (19) by a\nre\rection with respect to \u0012=\u0019=2. Furthermore, for all \f2(0;\u0019=2) and\u0017= 0 this is the unique\nsolution of (16) satisfying \u0012(0) =\fand approaching a constant as x!+1. The pro\fle \u0012(x) is\ndecreasing monotonically from \u0012=\fatx= 0 to\u0012= 0 atx= +1and decays exponentially at\n7in\fnity. It also minimizes the energy in (15) with \u0017= 0 andw=1among all\u0012\u0000\f2\u0017H1\n0(R+). By\n\u0017H1\n0(R+) we mean the Hilbert space obtained as the completion of the space C1\nc(R+) with respect\nto the homogeneous Sobolev norm\nkuk2\n\u0017H1\n0(R+):=Z1\n0ju0j2dx: (20)\nNote that by Sobolev embedding the elements of \u0017H1\n0(R+) may be identi\fed with continuous func-\ntions vanishing at x= 0 (cf. [8, Section 8.3]). The minimizing property of \u0012(x) in (19) may be seen\ndirectly from the Modica-Mortola type inequality for the energy with \u0017= 0 andw=1:\nE0\n\f(\u0012) :=1\n2Z1\n0\u0000\nj\u00120j2+ sin2\u0012\u0001\ndx=Z1\n0j(cos\u0012)0jdx+1\n2Z1\n0\u0000\nj\u00120j\u0000jsin\u0012j\u00012dx\n\u0015\f\f\f\fZ1\n0(cos\u0012)0dx\f\f\f\f=jcos\f\u0000cos\u00121j\u00151\u0000cos\f: (21)\nIn writing (21), we used weak chain rule [8, Corollary 8.11] and the fact that sin \u00122H1(R+)\nwheneverE0\n\f(\u0012)<+1and, hence, sin \u0012(x)!0 asx!+1, implying that \u0012(x)!\u001212\u0019Z[8,\nCorollary 8.9]. Furthermore, by inspection the case of equality holds if and only if \u0012is given by\n(19).\nA natural question is whether this type of boundary layer solution also exists for \u0017 >0. We\npoint out from the outset that if one formally sets w=1in (15), one runs into a di\u000eculty\nthat the nonlocal term in the energy evaluated on the function in (19) is in\fnite. Indeed, by\npositivity of the nonlocal and anisotropy terms, for any con\fguration with bounded energy we\nwould have sin \u00122H1(R+) and, therefore, lim x!1\u0012(x) =\u001212\u0019Z, as before. On the other hand,\nif\u00121\u0000\f622\u0019Z, the nonlocal part of the energy becomes in\fnite:\nZ1\n\u00001Z1\n\u00001(sin(\u0012(x)\u0000\f)\u0000sin(\u0012(y)\u0000\f))2\n(x\u0000y)2dxdy\u0015Z1\nRZ0\n\u00001sin2(\u0012(y)\u0000\f)\n(x\u0000y)2dxdy\n\u0015Z1\nRsin2(\u0012(y)\u0000\f)\nydy\u00151\n2sin2\fZ1\nRdy\ny= +1; (22)\nwhere we chose a su\u000eciently large R> 0, such that sin2(\u0012(y)\u0000\f)\u00151\n2sin2(\u00121\u0000\f) =1\n2sin2\f >0\nfor ally > R . This phenomenon has to do with the divergence of the energy of a pair of edge\ndomain walls minimizing E\f;win (15) asw!1 . Indeed, for \f6= 0 an edge domain wall carries\na net magnetic charge spread over a region of width of order 1 near the edge. Therefore, the\nself-interaction energy per unit length of a single edge domain wall diverges logarithmically with\nw, as can be seen by examining the argument in (22). Thus, in order to concentrate on a single\nedge domain wall, we need to appropriately renormalize the wall energy by \\subtracting\" the\nin\fnite self-interaction energy of a single wall. To this end, we introduce a smooth cuto\u000b function\n\u0011\f:R![0;\f] that satis\fes \u0011\f(x) =\fwhenx\u00140,\u0011\f(x) = 0 when x\u00151, and\u00110\n\f(x)\u00140 for\nallx2R, and formally subtract its contribution from the integrand in the last term in (15). This\nproduces the following renormalized energy\nE\f(\u0012) :=1\n2Z1\n0\u0000\nj\u00120j2+ sin2\u0012\u0001\ndx\n+\u0017\n8\u0019Z1\n\u00001Z1\n\u00001(sin(\u0012(x)\u0000\f)\u0000sin(\u0012(y)\u0000\f))2\u0000(sin(\u0011\f(x)\u0000\f)\u0000sin(\u0011\f(y)\u0000\f))2\n(x\u0000y)2dxdy; (23)\n8which is clearly \fnite when \u0012=\u0011\f. Notice that \u0011\f(x) mimics the edge domain wall pro\fle near\nthe edge and thus has the same leading order self-energy as the edge wall, which is what motivates\nits introduction in (23).\nCare is needed in de\fning a suitable admissible class of functions \u0012(x) in order to make the\nlast term in (23) meaningful, as the integrand there may not be in L1(R2). The latter is related to\nthe logarithmic divergence at in\fnity of the respective integrals mentioned earlier. Therefore, we\nrewrite the energy E\f(\u0012) in an equivalent form for smooth functions \u0012(x) satisfying \u0012(x) =\ffor\nallx<0 and\u0012(x) =\u0019nfor somen2Zand allx>R\u001d1:\nE\f(\u0012) =Z1\n0\u00121\n2j\u00120j2+1\n2sin2\u0012+\u0017\n4\u0019\u0001sin2(\u0012\u0000\f)\u0000sin2(\u0011\f\u0000\f)\nx\u0013\ndx\n+\u0017\n8\u0019Z1\n0Z1\n0(sin(\u0012(x)\u0000\f)\u0000sin(\u0012(y)\u0000\f))2\n(x\u0000y)2dxdy\n\u0000\u0017\n8\u0019Z1\n0Z1\n0(sin(\u0011\f(x)\u0000\f)\u0000sin(\u0011\f(y)\u0000\f))2\n(x\u0000y)2dxdy; (24)\nas can be veri\fed by a direct computation. We observe that this energy is well-de\fned, possibly\ntaking value +1, on the admissible class\nA:=n\n\u00122C\u0000\nR+\u0001\n:\u0012\u0000\f2\u0017H1\n0(R+)o\n: (25)\nIndeed, the last term in (24) is independent of \u0012and \fnite (see Lemma 5). Therefore, the main\ndi\u000eculty with the de\fnition of E\fcomes from the last term in the \frst line of (24). Nevertheless,\nas we show in Lemma 6, the integrand in the \frst line of (24) may be bounded from below by an\nintegrable function that does not depend on \u0012. This makes the de\fnition of E\fin (24) meaningful.\nWe are now in the position to state our existence result for the edge domain walls, viewed as\nminimizers of the one-dimensional energy E\fin (24) over the admissible class Ain (25).\nTheorem 1. For each\f2(0;\u0019=2]and each\u0017 > 0, there exists \u00122 A such thatE\f(\u0012) =\ninf~\u00122AE(~\u0012). Furthermore, we have \u00122L1(R+)andlimx!1\u0012(x) =\u00121for some\u001212\u0019Z.\nWe remark that the minimizers obtained in Theorem 1 do not depend on the speci\fc choice of \u0011\f.\nIndeed, denoting by E\f(\u0012;\u0011\f) the value of the energy for a given \u0012and\u0011\f, we have for any \u0012and\n\u0011(1;2)\n\fsuch thatE(\u0012;\u0011(1;2)\n\f)<+1:\nE\f(\u0012;\u0011(2)\n\f)\u0000E\f(\u0012;\u0011(1)\n\f) =\u0017\n4\u0019Z1\n0sin2(\u0011(2)\n\f\u0000\f)\u0000sin2(\u0011(1)\n\f\u0000\f)\nxdx\n+\u0017\n8\u0019Z1\n0Z1\n0(sin(\u0011(1)\n\f(x)\u0000\f)\u0000sin(\u0011(1)\n\f(y)\u0000\f))2\u0000(sin(\u0011(2)\n\f(x)\u0000\f)\u0000sin(\u0011(2)\n\f(y)\u0000\f))2\n(x\u0000y)2dxdy;\n(26)\nwhich is a constant independent of \u0012. In particular, minimizers of E\f(\u0001;\u0011(1)\n\f) overAcoincide with\nthose ofE\f(\u0001;\u0011(2)\n\f).\nThe result in Theorem 1 should be contrasted with that for the case \u0017= 0 discussed at the\nbeginning of this section. For the latter, for all \f2(0;\u0019=2) we have existence of a unique minimizer\ninA, which is monotone decreasing and converging to zero exponentially at in\fnity. In the case\n\u0017 >0, on the other hand, our result does not exclude a possibility of winding near the \flm edge,\n9expressed in the fact that one may have \u0012!\u0019nfor somen6= 0 asx!+1. Similarly, neither\nuniqueness nor monotonicity of the energy minimizing pro\fle are guaranteed a priori, and the decay\nat in\fnity is expected to follow a power law (cf. [10] and section 8 below).\nWe now turn to the questions of further regularity and the Euler-Lagrange equation satis\fed by\nthe minimizers obtained in Theorem 1. Formally, the Euler-Lagrange equation associated with (24)\ncoincides with (16) for all x2R+(again, extending \u0012to\u0012(x) =\fforx<0). However, care needs\nto be exercised once again, since the function sin( \u0012\u0000\f) no longer belongs to L2(R), so the standard\napproach to the de\fnition of\u0010\n\u0000d2\ndx2\u00111=2\nvia Fourier space no longer applies directly. Nevertheless,\nwe show that (16) still holds for the minimizers, provided that one uses the integral representation\nin (17) as the de\fnition for\u0010\n\u0000d2\ndx2\u00111=2\n. The latter is meaningful whenever \u0012is smooth, and we\nhave explicitly\n\u001200(x) = sin\u0012(x) cos\u0012(x) +\u0017\n2\u0019\u0001sin(\u0012(x)\u0000\f) cos(\u0012(x)\u0000\f)\nx\n+\u0017\n2\u0019cos(\u0012(x)\u0000\f)\u0012\n\u0000Z1\n0sin(\u0012(x)\u0000\f)\u0000sin(\u0012(y)\u0000\f)\n(x\u0000y)2dy\u0013\n8x>0:(27)\nThis picture is made precise in the following theorem.\nTheorem 2. For each\f2(0;\u0019=2]and each\u0017 >0, let\u0012be a minimizer from Theorem 1. Then\n\u00122C2(R+)\\C1(R+)\\W1;1(R+)and (27) holds. In addition, we have j\u00120(0)j= sin\fand\nlimx!0+j\u001200(x)j=1.\nWe remark that the last statement in Theorem 2 prevents the minimizer in Theorem 1 to be\nsmooth up to x= 0, in contrast with the case \u0017= 0 (see (19)). In turn, because of the presence of\nthe nonlocal term in (27) further regularity of the minimizer for x>0 cannot be established by a\nstandard bootstrap argument. A further study of higher regularity of the domain wall pro\fles in\nthe \flm interior would require a \fner simultaneous treatment of the exchange and stray \feld terms\nand goes beyond the scope of the present paper.\nWe end with a consideration of two parameter regimes in which further information can be\nobtained about the detailed structure of the energy minimizing pro\fles. In both these regimes\nthe nonlocal term in the equation may be viewed as a perturbation. The \frst is the regime when\n\f2(0;\u0019=2) is arbitrary, but \u0017is su\u000eciently small depending on \f. Then we have the following\nresult about the behavior of minimizers.\nTheorem 3. Let\f2(0;\u0019=2)and let\u0012\u0017be a minimizer of (24) for a given \u0017 >0. Then, as\u0017!0\nthe minimizers \u0012\u0017converge uniformly on [0;+1)to the minimizer \u00120of(21), de\fned in (19).\nWe note that we need to avoid the value of \f=\u0019=2 in the statement of Theorem 3, because even\nin the case \u0017= 0 there are two possible minimizers: one is given by (19) and the other by its\nre\rection with respect to \u0012=\u0019=2. On the other hand, it is easy to see by an inspection of the\nproof of Theorem 3 that convergence in its statement is uniform in \ffor all 0< \f\u0014\f0< \u0019= 2.\nNote also that the statement of Theorem 3 implies that \u0012\u0017(x)!0 asx!+1for all\u0017 > 0\nsu\u000eciently small depending on \f. In other words, the domain wall pro\fles cannot exhibit winding\nin this parameter range.\nThe second regime is for \fxed values of \u0017 >0 and su\u000eciently small \f >0. Here we have the\nfollowing result.\nTheorem 4. Let\u0017 >0, let 0< \f\u0014\f0for some\f0(\u0017)>0, and let\u0012\fbe a minimizer of (24).\nThen\u0012\fis unique, and \u0012\f!0uniformly on [0;+1)as\f!0.\n10Again, the statement of Theorem 4 implies that \u0012\f(x)!0 asx!+1for all\f > 0 su\u000eciently\nsmall depending on \u0017.\n4 Proof of Theorem 1\nWe begin by de\fning\nJ\f(\u0012) :=Z1\n0Z1\n0(sin(\u0012(x)\u0000\f)\u0000sin(\u0012(y)\u0000\f))2\n(x\u0000y)2dxdy; (28)\nand making the following basic observation.\nLemma 5. We have\nJ\f(\u0011\f)<+1: (29)\nProof. Since\u0011\f(x) = 0 for all x\u00151, we may write\nJ\f(\u0011\f) =Z1\n0Z1\n0(sin(\u0011\f(x)\u0000\f)\u0000sin(\u0011\f(y)\u0000\f))2\n(x\u0000y)2dxdy\n+ 2Z1\n0Z1\n1(sin(\u0011\f(y)\u0000\f)\u0000sin\f)2\n(x\u0000y)2dxdy\n=Z1\n0Z1\n0(sin(\u0011\f(x)\u0000\f)\u0000sin(\u0011\f(y)\u0000\f))2\n(x\u0000y)2dxdy\n+ 2Z1\n0(sin(\u0011\f(y)\u0000\f)\u0000sin\f)2\n1\u0000ydy: (30)\nBy smoothness of \u0011\fwe have\njsin(\u0011\f(x)\u0000\f)\u0000sin(\u0011\f(y)\u0000\f)j\u0014Cjx\u0000yj; (31)\nfor someC > 0 depending only on \u0011\f. Therefore, the integrands in the right-hand side of (30) are\nessentially bounded, yielding the result.\nWith the result of Lemma 5 in hand, we can write the energy in (24) as\nE\f(\u0012) =F\f(\u0012) +\u0017\n8\u0019J\f(\u0012)\u0000\u0017\n8\u0019J\f(\u0011\f); (32)\nwhere\nF\f(\u0012) :=Z1\n0\u00121\n2j\u00120j2+1\n2sin2\u0012+\u0017\n4\u0019\u0001sin2(\u0012\u0000\f)\u0000sin2(\u0011\f\u0000\f)\nx\u0013\ndx: (33)\nWhileJ\f(\u0012)\u00150 by de\fnition, we also have the following result concerning F\f(\u0012).\nLemma 6. For every\u00122A we have\n1\n2sin2\u0012(x) +\u0017\n4\u0019\u0001sin2(\u0012(x)\u0000\f)\u0000sin2(\u0011\f(x)\u0000\f)\nx\u0015\u0000C\n1 +x28x>0; (34)\nfor someC > 0depending only on \f,\u0017and\u0011\f. Furthermore, F\f(\u0012)is bounded below independently\nof\u0012.\n11Proof. Since by the de\fnition of \u0011\fwe havej\u0011\f(x)\u0000\fj\u0014Cxfor someC > 0 and allx2(0;1),\nthe left-hand side of (34) is bounded below on this interval. At the same time, by trigonometric\nidentities and Young's inequality we have for all x\u00151 and any\">0\nsin2(\u0012\u0000\f)\u0000sin2(\u0011\f\u0000\f)\nx=sin(\u0012\u00002\f) sin\u0012\nx\u0015\u0000\"sin2\u0012\u00001\n4\"x2: (35)\nHence, choosing \"su\u000eciently small depending only on \u0017, we can control the left-hand side of (34)\nfrom below by\u0000C=x2for allx2(1;1), where the constant C > 0 depends only on \u0017. Combining\nthe estimates on the two intervals then yields (34). Finally, the last statement follows from (34)\nand the fact that the integrand in (33) is measurable on R+.\nProof of Theorem 1. Since inf ~\u00122AE\f(~\u0012)\u0014E\f(\u0011\f)<+1, and since E\f(\u0012)\u0015\u0000Cfor someC > 0\nand all\u00122Aby Lemmas 5 and 6, there exists a sequence of \u0012n2Asuch that\n\u00001<inf\n~\u00122AE\f(~\u0012) = lim inf\nn!1E\f(\u0012n)<+1: (36)\nFurthermore, by Lemma 6 and positivity of J\fwe havek\u00120\nnkL2(R+)\u0014Cfor someC > 0. Therefore,\nupon extraction of subsequences we have \u00120\nn* \u00120inL2(R+) for some \u00122H1\nloc(R+) and\u0012n!\u0012\ninC([0;R]) for anyR > 0 \fxed by Sobolev embedding [8, Theorem 8.8 and Proposition 8.13].\nBy a diagonal argument we then conclude that upon further extraction of a subsequence we have\n\u0012n(x)!\u0012(x) for every x>0.\nNow, by the lower semicontinuity of the norm we have lim inf n!1k\u00120\nnkL2(R+)\u0015k\u00120kL2(R+).\nTherefore, by Lemma 6 and Fatou's lemma we get lim inf n!1F\f(\u0012n)\u0015F\f(\u0012). Again, by Fatou's\nlemma we also get lim inf n!1J\f(\u0012n)\u0015J\f(\u0012). In fact, since ( \u0012n) is a minimizing sequence, the\ninequalities above are equalities. This implies that \u0012n\u0000\f!\u0012\u0000\fstrongly in \u0017H1\n0(R+), and, hence,\nwe have\u00122A. Thus,\u0012is a minimizer of E\foverA.\nFinally, observe that by weak chain rule [8, Corollary 8.11] and Lemma 6 we have sin \u00122\nH1(R+). Therefore, by [8, Corollary 8.9] we also have sin \u00122C\u0000\nR+\u0001\nand sin\u0012(x)!0 asx!+1.\nOn the other hand, from the Modica-Mortola type inequality and Lemmas 5 and 6 we obtain\nZ1\n0jsin\u0012jj\u00120jdx\u00141\n2Z1\n0\u0000\nj\u00120j2+ sin2\u0012\u0001\ndx< +1; (37)\nwhich implies that \u0012(x)!\u00121for some\u001212\u0019Zasx!+1. This concludes the proof.\nRemark 7. We have shown existence of a minimizer for the energy in (24) since a priori the\nenergy in (23) did not make sense for all \u00122A. However, if \u00122A satis\fesE\f(\u0012)< C (in the\nsense of (24)) then it is not di\u000ecult to see that the energy in (23) also makes sense and coincides\nwith that in (24).\n5 Proof of Theorem 2\nWe now proceed to the proof of Theorem 2, where we have to compute a variation of the energy\nin (24). It is clear how to deal with all the terms except J\f(\u0012). In order to \fnd the variation of\nJ\f(\u0012), for notational convenience we de\fne\nu(x) := sin(\u0012(x)\u0000\f); (38)\n12for\u00122Aand notice that\nZ1\n0Z1\n0(u(x)\u0000u(y))2\n(x\u0000y)2dxdy =J\f(\u0012): (39)\nWe are taking a variation with respect to \u0012as follows: \u0012(x)7!\u0012(x) +\"\u001e(x), with\u001e2C1\nc(R+)\nand\"2R. It is convenient to introduce the corresponding variation in u, de\fned as u(x)7!\nu(x) +\" \"(x), where\n \"(x) :=(sin(\u0012(x)+\"\u001e(x)\u0000\f)\u0000sin(\u0012(x)\u0000\f)\n\"\"6= 0;\ncos(\u0012(x)\u0000\f)\u001e(x) \"= 0:(40)\nNote that for every x2R+we have\n \"(x)! 0(x) =: (x) as\"!0: (41)\nBefore computing the variation of J\f(\u0012) we will need two technical lemmas concerning the\nproperties of  \".\nLemma 8. Let\"2R,\u00122A,\u001e2C1\nc(R+), and let \"be de\fned in (40). Then \"2H1(R+),\nand for almost every x2R+we have\nj \"(x)j\u0014j\u001e(x)j andj 0\n\"(x)j\u0014j\u001e0(x)j+j\u00120(x)jj\u001e(x)j: (42)\nProof. By mean value theorem, we have\n \"(x) = cos(\u0012(x) +\"\u0015\"(x)\u001e(x)\u0000\f)\u001e(x); (43)\nfor some\u0015\"(x)2(0;1), which yields the \frst inequality in (42). Next, applying the weak chain\nrule [8, Corollary 8.11], we obtain  \"2H1\nloc(R+), and for almost every x2R+we have\n 0\n\"(x) = cos(\u0012(x) +\"\u001e(x)\u0000\f)\u001e0(x) +cos(\u0012(x) +\"\u001e(x)\u0000\f)\u0000cos(\u0012(x)\u0000\f)\n\"\u00120(x): (44)\nIn particular,  0\n\"has compact support and, hence,  \"2H1(R+). Therefore, again by mean value\ntheorem and triangle inequality we have for some \u0015\"(x)2(0;1)\nj 0\n\"(x)j\u0014jcos(\u0012(x) +\"\u001e(x)\u0000\f)jj\u001e0(x)j+jsin(\u0012(x) +\"\u0015\"(x)\u001e(x)\u0000\f)jj\u001e(x)jj\u00120(x)j; (45)\nyielding the result.\nLemma 9. Let\"2R,\u00122A,\u001e2C1\nc(R+), and let \"be de\fned in (40). Then there exists C > 0\nindependent of \"such that for every \u000e>0there holds\nZZ\nfjx\u0000yj\u0014\u000eg( \"(x)\u0000 \"(y))2\n(x\u0000y)2dxdy\u0014C\u000e andZZ\nfjx\u0000yj\u0015\u000eg( \"(x)\u0000 \"(y))2\n(x\u0000y)2dxdy\u0014C\u000e\u00001:\n(46)\nIn particular\nZ1\n0Z1\n0( \"(x)\u0000 \"(y))2\n(x\u0000y)2dxdy\u00142C: (47)\n13Proof. First of all, since  \"has compact support lying in R+, we can extend  \"by zero to the\nwhole real line. By Lemma 8,  \"is absolutely continuous and, hence, for all x6=ywe have\n \"(x)\u0000 \"(y)\nx\u0000y=Z1\n0 0\n\"(y+t(x\u0000y))dt: (48)\nTherefore\nZZ\nfjx\u0000yj\u0014\u000eg( \"(x)\u0000 \"(y))2\n(x\u0000y)2dxdy\n=ZZ\nfjx\u0000yj\u0014\u000egZ1\n0Z1\n0 0\n\"(y+t(x\u0000y)) 0\n\"(y+s(x\u0000y))dtdsdxdy: (49)\nInterchanging the order of integration and applying Cauchy-Schwarz inequality, from (49) we obtain\nZZ\nfjx\u0000yj\u0014\u000eg( \"(x)\u0000 \"(y))2\n(x\u0000y)2dxdy\u0014Z1\n0ZZ\nfjx\u0000yj\u0014\u000egj 0\n\"(y+s(x\u0000y))j2dxdyds: (50)\nFinally, using the new variable z=x\u0000yin place ofxyields\nZZ\nfjx\u0000yj\u0014\u000eg( \"(x)\u0000 \"(y))2\n(x\u0000y)2dxdy\u0014Z1\n0Z+\u000e\n\u0000\u000eZ1\n\u00001j 0\n\"(y+sz)j2dydzds = 2\u000eZ1\n0j 0\n\"(y)j2dy;\n(51)\nwhich in view of Lemma 8 gives the \frst estimate in (46).\nTo obtain the second estimate in (46), simply note that\nZZ\nfjx\u0000yj\u0015\u000eg( \"(x)\u0000 \"(y))2\n(x\u0000y)2dxdy\u00142ZZ\nfjx\u0000yj\u0015\u000egj \"(x)j2+j \"(y)j2\n(x\u0000y)2dxdy\n= 4ZZ\nfjx\u0000yj\u0015\u000egj \"(y)j2\n(x\u0000y)2dxdy =8\n\u000eZ1\n0j \"(y)j2dy; (52)\nyielding the claim, once again, by Lemma 8.\nLastly, (47) is an immediate corollary to (46) with \u000e= 1.\nWe now establish G^ ateaux di\u000berentiability of J\f(\u0012) with respect to compactly supported smooth\nperturbations of \u0012.\nLemma 10. Let\u00122A be such that J\f(\u0012)<1, let\u001e2C1\nc(R+), and letuand be de\fned in\n(38) and(41), respectively. Then\nlim\n\"!0J\f(\u0012+\"\u001e)\u0000J\f(\u0012)\n\"= 2Z1\n0Z1\n0(u(x)\u0000u(y))( (x)\u0000 (y))\n(x\u0000y)2dxdy: (53)\nProof. Observe that, using  \"de\fned in (40), we can write\nJ\f(\u0012+\"\u001e)\u0000J\f(\u0012)\n\"= 2Z1\n0Z1\n0(u(x)\u0000u(y))( \"(x)\u0000 \"(y))\n(x\u0000y)2dxdy\n+\"Z1\n0Z1\n0( \"(x)\u0000 \"(y))2\n(x\u0000y)2dxdy: (54)\n14Next, for\u000e2(0;1) we split the integrals above into those over fjx\u0000yj\u0014\u000egand those over\nfjx\u0000yj> \u000eg. SinceJ\f(\u0012)<1, by Cauchy-Schwarz inequality and Lemma 9 the former are\nbounded by Cp\n\u000ewithC\u00150 independent of \"2(\u00001;1)nf0g. To compute the latter, we use\nLebesgue dominated convergence theorem to pass to the limit as \"!0 with\u000e\fxed. Observe that\nsince\n2ZZ\nfjx\u0000yj>\u000egju(x)\u0000u(y)jj \"(x)\u0000 \"(y)j\n(x\u0000y)2dxdy\u0014ZZ\nfjx\u0000yj>\u000eg(u(x)\u0000u(y))2\n(x\u0000y)2dxdy\n+ZZ\nfjx\u0000yj>\u000eg( \"(x)\u0000 \"(y))2\n(x\u0000y)2dxdy; (55)\nand since by our assumption and (39) the \frst integral is bounded, it is su\u000ecient to dominate the\nintegrand in the second term of the right-hand-side of (55) by an integrable function independent\nof\".\nUsing Lemma 8, we can write for all jx\u0000yj>\u000e:\n( \"(x)\u0000 \"(y))2\n(x\u0000y)2\u0014(j \"(x)j+j \"(y)j)2\n(x\u0000y)2\u00142j\u001e(x)j2+j\u001e(y)j2\n(x\u0000y)2\u001ffjx\u0000yj>\u000eg(x;y) =:G\u000e(x;y);(56)\nwhere\u001ffjx\u0000yj>\u000egis the characteristic function of the set fjx\u0000yj> \u000eg. Since\u001eis bounded and\nhas compact support, we have G\u000e2L1(R+\u0002R+). Therefore, since  \"(x)! (x) as\"!0 for all\nx2R+, by Lebesgue dominated convergence theorem we have\nlim\n\"!0ZZ\nfjx\u0000yj>\u000eg(u(x)\u0000u(y))( \"(x)\u0000 \"(y))\n(x\u0000y)2dxdy =ZZ\nfjx\u0000yj>\u000eg(u(x)\u0000u(y))( (x)\u0000 (y))\n(x\u0000y)2dxdy\n(57)\nlim\n\"!0ZZ\nfjx\u0000yj>\u000eg( \"(x)\u0000 \"(y))2\n(x\u0000y)2dxdy =ZZ\nfjx\u0000yj>\u000eg( (x)\u0000 (y))2\n(x\u0000y)2dxdy<1: (58)\nLastly, combining this result with the estimates of the integrals over fjx\u0000yj\u0014\u000egand sending\n\u000e!0 completes the proof, once again, by Lebesgue dominated convergence theorem, Lemma 9\nand Cauchy-Schwarz inequality.\nWith the di\u000berentiability of J\festablished in Lemma 10, di\u000berentiability of E\fthen follows\nby a standard argument. Thus, we arrive at the following result that yields the Euler-Lagrange\nequation for a minimizer of E\foverAin weak form.\nProposition 11. Let\u0012be a minimizer of E\foverA. Then\n\u0017\n4\u0019Z1\n0Z1\n0(sin(\u0012(x)\u0000\f)\u0000sin(\u0012(y)\u0000\f))(cos(\u0012(x)\u0000\f)\u001e(x)\u0000cos(\u0012(y)\u0000\f)\u001e(y))\n(x\u0000y)2dxdy\n+Z1\n0\u0000\n\u00120\u001e0+\u001esin\u0012cos\u0012\u0001\ndx+\u0017\n2\u0019Z1\n0\u001esin(\u0012\u0000\f) cos(\u0012\u0000\f)\nxdx= 08\u001e2C1\nc\u0000\nR+\u0001\n:(59)\nOur next goal is to \fnd an alternative representation of the nonlocal term in (59) that would\nallow us to proceed with establishing higher regularity of the minimizers of E\f, ultimately obtaining\nthe classical form of the Euler-Lagrange equation in (27).\n15Lemma 12. Let\u00122A be such that J\f(\u0012)<1. Then for every \u001e2C1\nc\u0000\nR+\u0001\nwe have\nZ1\n0Z1\n0(sin(\u0012(x)\u0000\f)\u0000sin(\u0012(y)\u0000\f))(cos(\u0012(x)\u0000\f)\u001e(x)\u0000cos(\u0012(y)\u0000\f)\u001e(y))\n(x\u0000y)2dxdy\n+2Z1\n0sin(\u0012(x)\u0000\f) cos(\u0012(x)\u0000\f)\nx\u001e(x)dx\n= 2Z1\n0\u0012\n\u0000Z1\n0cos(\u0012(y)\u0000\f)\u00120(y)\nx\u0000ydy\u0013\ncos(\u0012(x)\u0000\f)\u001e(x)dx: (60)\nProof. We begin by de\fning uand as in (38) and (40), respectively, and extending them by zero\nto the whole of R. We also similarly extend \u001e. To simplify the notations, we still denote those\nextensions as u, and\u001e, respectively.\nNext, we de\fne\nI:=Z1\n\u00001Z1\n\u00001(u(x)\u0000u(y))( (x)\u0000 (y))\n(x\u0000y)2dxdy; (61)\nand for\u000e>0 we writeI=I\u000e\n1+I\u000e\n2, where\nI\u000e\n1:=ZZ\nfjx\u0000yj>\u000eg(u(x)\u0000u(y))( (x)\u0000 (y))\n(x\u0000y)2dxdy; (62)\nI\u000e\n2:=ZZ\nfjx\u0000yj\u0014\u000eg(u(x)\u0000u(y))( (x)\u0000 (y))\n(x\u0000y)2dxdy: (63)\nNote that by our assumptions (61) and, hence, (62) and (63), de\fne absolutely convergent integrals.\nIndeed, by Lemma 8 and Cauchy-Schwarz inequality we have\njIj\u0014J1=2\n\f(\u0012)\u0012Z1\n\u00001Z1\n\u00001( (x)\u0000 (y))2\n(x\u0000y)2dxdy\u00131=2\n=J1=2\n\f(\u0012)\u0012Z1\n0Z1\n0( (x)\u0000 (y))2\n(x\u0000y)2dxdy + 2Z1\n0j (x)j2\nxdx\u00131=2\n; (64)\nwhich is \fnite by Lemma 9.\nLet nowwn2C1\nc(R+) be such that wn!\u0012\u0000\fin\u0017H1\n0(R+), and extend wnby zero for x<0.\nWe claim that if un:= sinwn, then we have In!Iasn!1 , where\nIn:=Z1\n\u00001Z1\n\u00001(un(x)\u0000un(y))( (x)\u0000 (y))\n(x\u0000y)2dxdy: (65)\nIndeed, de\fne I\u000e\n1;nandI\u000e\n2;nas in (62) and (63) with ureplaced by un. Arguing as in the proof of\nLemma 10, in view of the pointwise convergence of untouwe haveI\u000e\n1;n!I\u000e\n1. At the same time,\nby the argument in the proof of Lemma 9 and boundedness of u0\nninL2(R), we also havejI\u000e\n2;nj\u0014C\u000e\nfor someC > 0 independent of n. Thus, the claim follows by arbitrariness of \u000e.\nNow, since both unand belong toH1(R) by Lemma 8 and [8, Corollary 8.10 and Corollary\n8.11], we may proceed by expressing Inin Fourier space [30, Theorem 7.12]:\nIn=Z1\n\u00001jkjF\u0003[un]F[ ]dk; (66)\n16where \\\u0003\" stands for complex-conjugate and\nF[u] :=Z1\n\u00001e\u0000ikxu(x)dx: (67)\nAlternatively, we can write (66) as\nIn=Z1\n\u00001(\u0000isgn(k))\u0003F\u0003[u0\nn]F[ ]dk; (68)\nWe now pass to the limit n!1 by taking advantage of the fact that u0\nn!u0inL2(R) and,\ntherefore, we have\nI=Z1\n\u00001(\u0000isgn(k))\u0003F\u0003[u0]F[ ]dk= 2Z1\n\u00001\u0012\n\u0000Z1\n\u00001u0(y)\nx\u0000ydy\u0013\n (x)dx; (69)\nwhere we inserted the de\fnition of Hilbert transform [21, Section 5.1.1].\nFinally, to obtain the desired formula, we separate out the contribution of the negative real axis\nin the integral over yin (61). We have\nI=Z1\n0Z1\n0(u(x)\u0000u(y))( (x)\u0000 (y))\n(x\u0000y)2dxdy + 2Z0\n\u00001Z1\n0u(x) (x)\n(x\u0000y)2dxdy\n=Z1\n0Z1\n0(u(x)\u0000u(y))( (x)\u0000 (y))\n(x\u0000y)2dxdy + 2Z1\n0u(x) (x)\nxdx; (70)\nwhere we took into account that since \u001e2C1\nc\u0000\nR+\u0001\n(see (41) for the de\fnition of  ), there is no\nsingularity in the integral near x= 0. At the same time, from (69) we get\nI= 2Z1\n0\u0012\n\u0000Z1\n0u0(y)\nx\u0000ydy\u0013\n (x)dx; (71)\nwhich concludes the proof.\nProof of Theorem 2. By Proposition 11, \u0012solves (59). At the same time, by Lemma 12 we have\nZ1\n0\u0012\u001e00dx=Z1\n0g\u001edx8\u001e2C1\nc(R+); (72)\nwhere\ng(x) := sin\u0012(x) cos\u0012(x) +\u0017\n2\u0019cos(\u0012(x)\u0000\f)\u0000Z1\n0cos(\u0012(y)\u0000\f)\u00120(y)\nx\u0000ydy: (73)\nNotice that since sin \u00122L2(R+) and\u001202L2(R+), by the properties of Hilbert transform [21,\nTheorem 5.1.7 and Theorem 5.1.12] we have that g2L2(R+) as well. Hence \u0012002L2(R+), which\nby Sobolev embedding [8, Theorem 8.8] implies that \u00122C1(R+) and\u001202L1(R+).\nFocusing now on the nonlocal term, let u(x) be de\fned by (38), extended, as usual, by zero\ntox <0, and leth(x) denote the integral in (73), with x2R. Note that by chain rule we have\nu2C1(R+), andu0(x) experiences a jump discontinuity at x= 0 whenever u0(0+)6= 0. Also, by\nweak chain rule [8, Corollary 8.11] we have u002L2(R+). Passing to the Fourier space as in the\n17proof of Lemma 12 (but treating has a member ofS0(R) now), for any \u001e2C1\nc(R+), extended by\nzero tox<0, we can write\n\u0000Z1\n\u00001h\u001e0dx=1\n2Z1\n\u00001jkjF\u0003[\u001e]F[u0]dk=1\n2Z1\n\u00001(\u0000isgn(k))F\u0003[\u001e]F[u00]dk\n+u0(0+)\n2Z1\n\u00001(\u0000isgn(k))F\u0003[\u001e]dk=Z1\n0\u0012\n\u0000Z1\n0u00(y)\nx\u0000ydy\u0013\n\u001e(x)dx+u0(0+)Z1\n0\u001e(x)\nxdx; (74)\nwhere byu00we mean the absolutely continuous part of the distributional derivative of u0onR.\nThus, we have\nh0(x) =\u0000Z1\n0u00(y)\nx\u0000ydy+u0(0)\nxinD0(R+): (75)\nIn this expression, the \frst term is still in L2(R). Similarly, the second term is in L2\nloc(R+). By\n(72) and weak product and chain rules [8, Corollary 8.9 and Corollary 8.11], this then yields that\n\u00120002L2\nloc(R+), implying, in particular, that \u00122C2(R+). Furthermore, by (72) we have\n\u001200(x) =g(x)8x>0: (76)\nFinally, again by passing to Fourier space [15, Section 3] we have\n\u0000Z1\n0u0(y)\nx\u0000ydy=\u0000Z1\n\u00001u(x)\u0000u(y)\n(x\u0000y)2dy=\u0000Z1\n0u(x)\u0000u(y)\n(x\u0000y)2dy+u(x)\nx; (77)\nwhere we took into account that u(0) = 0. Substituting this expression to (76) yields (27).\nTo prove the remaining statements about the behavior of \u0012(x) nearx= 0, we multiply (76) by\n\u00120(x) and integrate over R+. Since the Hilbert transform is an anti-hermitian operator from L2(R)\ntoL2(R) [21, Section 5.1.1], and since cos( \u0012(x)\u0000\f)\u00120(x), extended by zero for x <0, belongs to\nL2(R), the contribution of the last term in the right-hand side of (73) vanishes. At the same time,\nsince\u001202H1(R+) we have\u00120(x)!0 asx!+1[8, Corollary 8.9]. Since also sin \u0012(x)!0 as\nx!+1, by [8, Theorem 8.2] we have\nj\u00120(0)j2= sin2\u0012(0): (78)\nThe boundary condition then implies that j\u00120(0)j= sin\f. In particular, \u00120(0)6= 0. Thus, the\nfunctionu0(x) de\fned above experiences a jump discontinuity at x= 0, leading to a logarithmic\ndivergence of the integral in (73) as x!0+. This concludes the proof.\n6 Proof of Theorem 3\nWith a slight abuse of notation we denote the energy in (24) by E\u0017\n\f. We \frst show that \u0012\u0017\u0000\f!\n\u00120\u0000\fin\u0017H1\n0(R+) as\u0017!0. Let us assume that E\u0017\n\f(\u0012\u0017)\u0014Cand 0< \u0017 < 1. Using the same\narguments as in Lemma 6, we obtain\n1\n4sin2\u0012(x) +\u0017\n4\u0019\u0001sin2(\u0012(x)\u0000\f)\u0000sin2(\u0011\f(x)\u0000\f)\nx\u0015\u0000C\n1 +x28x>0; (79)\nwhereC > 0 depends only on \fand\u0011\f. Therefore we have\nE0\n\f(\u0012\u0017) =1\n2Z1\n0\u0000\nj\u00120\n\u0017j2+ sin2\u0012\u0017\u0001\ndx\u0014C; (80)\n18which immediately implies (see the proof of Theorem 1) that \u0012\u0017\u0000\f *\u0012\u0000\fin\u0017H1\n0(R+) and\u00122A.\nNow we prove \u0000-convergence of energies with respect to the weak convergence in \u0017H1\n0(R+) (for a\ngeneral introduction to \u0000-convergence, see, e.g., [6]). Let us assume that \u0017n!0 and\u0012n\u0000\f *\u0012\u0000\f\nin\u0017H1\n0(R+). Then by Sobolev embedding [8, Theorem 8.8], upon extraction of a subsequence we\nalso have\u0012n(x)!\u0012(x) for allx > 0. Therefore, by lower semicontinuity of the norm, Fatou's\nlemma and positivity of J\fwe have\nlim inf\nn!1E\u0017n\n\f(\u0012n)\u0015E0\n\f(\u0012): (81)\nNow, taking any \u0012\u0000\f2\u0017H1\n0(R+) such that E\u0017n\n\f(\u0012)<+1, we can construct a sequence \u0012n\u0011\u0012\nthat trivially satis\fes\nlim sup\nn!1E\u0017n\n\f(\u0012n) =E0\n\f(\u0012); (82)\nestablishing the \u0000-limit sought.\nUsing the properties of \u0000-convergence [6], we then have lim \u0017!0E\u0017\n\f(\u0012\u0017) =E0\n\f(\u00120), where\u00120is\ngiven by the right-hand side of (19), which implies \u0012\u0017\u0000\f!\u00120\u0000\fin\u0017H1\n0(R+) and sin\u0012\u0017!sin\u00120\ninH1(R+)\\C(R+). However, from the Modica-Mortola type inequality in (37) we also have for\nsomeC > 0 depending only on \fand\u0011\f:\nZ1\n0jsin\u0012\u0017jj\u00120jdx\u0014E\u0017\n\f(\u0012\u0017) +C\n2\u0017\u0014E0\n\f(\u00120) +C\u0017= 1\u0000cos\f+C\u0017: (83)\nThis implies that \u0012\u0017(x)2(\u0000C\u0017;\f +C\u0017) for allx>0 and some C > 0 depending only on \fand\n\u0011\f. Hence for any \fxed \f2(0;1\n2\u0019) we can always choose \u00170>0 such that for all \u0017 < \u0017 0we\nhave\u0012\u0017(x)2[\u00001\n2\f\u00001\n4\u0019;1\n2\f+1\n4\u0019]\u001a(\u00001\n2\u0019;1\n2\u0019) for allx > 0. Recall also that \u00120(x)2(0;\f)\u001a\n[\u00001\n2\f\u00001\n4\u0019;1\n2\f+1\n4\u0019] for allx>0. Thus, using mean value theorem, with some ~\u0012(x) between\u0012\u0017(x)\nand\u00120(x), we arrive at\njsin\u0012\u0017(x)\u0000sin\u00120(x)j=jcos~\u0012(x)jj\u0012\u0017(x)\u0000\u00120(x)j\u0015Cj\u0012\u0017(x)\u0000\u00120(x)j 8x>0; (84)\nfor someC > 0 depending only on \f. In particular, in view of the uniform convergence of sin \u0012\u0017to\nsin\u00120, for any\">0 and all\u0017su\u000eciently small we have\nsup\nx2Rj\u0012\u0017(x)\u0000\u00120(x)j\u0014Csup\nx2Rjsin\u0012\u0017(x)\u0000sin\u00120(x)j<\": (85)\nThis concludes the proof.\n7 Proof of Theorem 4\nLet us \frst show that \u0012\f!0 as\f!0 uniformly in C(R+). We de\fne \u001e\f(x) = maxf\f(1\u0000x);0gfor\n0<\f <\u0019\n4and allx2R. It is clear that E\f(\u0012\f)\u0014E\f(\u001e\f) and, therefore, after a straightforward\n19computation,\n1\n2Z1\n0\u0000\nj\u00120\n\fj2+ sin2\u0012\f\u0001\ndx\u00141\n2Z1\n0\u0000\nj\u001e0\n\fj2+ sin2\u001e\f\u0001\ndx+\u0017\n4\u0019Z1\n0sin2(\u001e\f\u0000\f)\u0000sin2(\u0012\f\u0000\f)\nxdx\n+\u0017\n8\u0019Z1\n0Z1\n0(sin(\u001e\f(x)\u0000\f)\u0000sin(\u001e\f(y)\u0000\f))2\n(x\u0000y)2dxdy\n\u0014\f2+\u0017\n4\u0019Z1\n0sin2(\u001e\f\u0000\f)\u0000sin2(\u0012\f\u0000\f)\nxdx\n+\u0017\n8\u0019Z1\n0Z1\n0(\u001e\f(x)\u0000\u001e\f(y))2\n(x\u0000y)2dxdy\n\u0014\u0010\u0017\n4\u0019+ 1\u0011\n\f2+\u0017\n4\u0019Z1\n0sin2(\u001e\f\u0000\f)\u0000sin2(\u0012\f\u0000\f)\nxdx: (86)\nNow, with the help of trigonometric identities and Young's inequality we estimate the last integral\nin (86):\n\u0017\n4\u0019Z1\n0sin2(\u001e\f\u0000\f)\u0000sin2(\u0012\f\u0000\f)\nxdx\n\u0014\u0017\n4\u0019Z1\n0sin2(\u001e\f\u0000\f)\nxdx+\u0017\n4\u0019Z1\n1sin(2\f\u0000\u0012\f) sin\u0012\f\nxdx\n\u0014\u0017\f2\n4\u0019+\u0017\n4\u0019Z1\n1sin 2\fcos\u0012\fsin\u0012\f\u0000cos 2\fsin2\u0012\f\nxdx\n\u0014\u0017\f2\n4\u0019+\u0017\f\n2\u0019Z1\n1jsin\u0012\fj\nxdx\u0014\u0017(\u0017+\u0019)\f2\n4\u00192+1\n4Z1\n1sin2\u0012\fdx; (87)\nrecalling that 0 <\f <\u0019\n4and, therefore, cos 2 \f >0. Combining the above inequalities, we obtain\n1\n4Z1\n0\u0000\nj\u00120\n\fj2+ sin2\u0012\f\u0001\ndx\u0014C\f2; (88)\nfor someC > 0 depending only on \u0017. Hence the left-hand side of (88) vanishes as \f!0, and by\n(37) this implies \u0012\f!0 uniformly in C(R+) in this limit.\nNow we prove uniqueness of minimizers when \fis small enough. From the arguments above\nit is clear that for all \u000e > 0 and all\fsu\u000eciently small we have \u0012\f2(\u0000\u000e;\u000e). Therefore, if\nu\f:= sin(\u0012\f\u0000\f), then\u0012\f=\f+ arcsinu\f2C1(R+) andu\f2(\u00002\u000e;2\u000e) for all su\u000eciently small \u000e.\nWe rewrite the energy E\f(\u0012\f) in terms of u\f:\nE\f(\u0012\f) =1\n2Z1\n0 \nju0\n\fj2\n1\u0000u2\n\f+ sin2(\f+ arcsinu\f) +\u0017\n4\u0019\u0001u2\n\f\u0000sin2(\u0011\f\u0000\f)\nx!\ndx\n+\u0017\n8\u0019Z1\n0Z1\n0(u\f(x)\u0000u\f(y))2\n(x\u0000y)2dydx\n\u0000\u0017\n8\u0019Z1\n0Z1\n0(sin(\u0011\f(x)\u0000\f)\u0000sin(\u0011\f(y)\u0000\f))2\n(x\u0000y)2dydx: (89)\nIt is straightforward to show that the right-hand side of (89) is a strictly convex functional of u\f\nfor all\u000esu\u000eciently small and, therefore, the minimizer of E\fis unique when \fis small enough.\n208 Numerics and discussion\nWe conclude this paper by presenting the results of some numerical simulations that exhibit edge\ndomain walls and discussing some of their distinctive characteristics. We \frst perform a micromag-\nnetic simulation of the remanent magnetization con\fguration in a ferromagnetic strip, using the\nsimpli\fed two-dimensional thin \flm model from section 2 (for details of the numerical algorithm,\nsee [36]). The result of the simulation is presented in Figure 1 and shows the long-time asymptotic\nstationary magnetization con\fguration formed as the result of solving the overdamped Landau-\nLifshitz-Gilbert equation with the energy from (7) in a strip of lateral dimensions 128 :25\u000232:25\n(in the units of L). The initial condition was taken in the form of the magnetization saturated to\nthe direction slightly away from vertical. The thin \flm parameter was \fxed at \u0017= 20. The dimen-\nsionless parameters above correspond, for example, to those of a permalloy strip ( Ms= 8\u0002105\nA/m,A= 1:3\u000210\u000011J/m andK= 5\u0002102J/m3[1]) with dimensions 20 :7\u0016m\u00025:2\u0016m\u00024nm, for\nwhich`= 5:69 nm and L= 161 nm, giving an edge domain wall width of order 1 \u0016m.\nTo obtain the one-dimensional domain wall pro\fles numerically, we solve a parabolic version of\n(27) for\u0012=\u0012(x;t) inR+\u0002R+:\n\u0012t=\u0012xx\u0000sin\u0012cos\u0012\u0000\u0017\n2cos(\u0012\u0000\f)\u0012\n\u0000d2\ndx2\u00131=2\nsin(\u0012\u0000\f); (90)\nwith Dirichlet boundary condition\n\u0012(0;t) =\f; (91)\nand initial data\n\u0012(x;0) =2\f\n1 +ex=2; (92)\nwhich is a monotonically decreasing function that asymptotes to zero as x!+1. To solve the\nabove problem, we employ a \fnite-di\u000berence discretization and an optimal-grid-based method to\ncompute the stray \feld, extending \u0012by its boundary value to x<0. The details of the numerical\nmethod can be found, once again, in [36]. The domain wall pro\fles are then identi\fed with the\nsteady states of (90) reached as t!1 .\nWe collect the results of our numerical solution of the above problem for a range of physically\nrelevant values of \fand\u0017in Figs. 3 and 4. The upper panels (a)-(c) of Fig. 3 show the pro\fles\nof edge domain walls with \fequal to\u0019=2 (red curves), \u0019=4 (green curves) and \u0019=8 (blue curves).\nThe lower panels show the same pro\fles in log-log coordinates, with the dashed lines indicating\n1=xdecay. Each pair of panels corresponds to a di\u000berent value of \u0017:\u0017= 1 in panels (a) and (d);\n\u0017= 10 in panels (b) and (e); \u0017= 50 in panels (c) and (f). In all the simulations, a discretization\nstep \u0001x= 0:125 was used near the edge on a non-uniform grid with a stretch factor b= 20 [36] and\nterminating at x'6\u0002103. A 16-node optimal geometric grid was used in the transverse direction\nto compute the stray \feld [23,36].\nFor 0<\f\u0014\u0019=2 the obtained domain wall pro\fles exhibit monotonic decay from \u0012=\fatx= 0\nto\u0012= 0 atx= +1. Thus, qualitatively the pro\fles are similar to those in (19) corresponding to\nthe case\u0017= 0. This is in agreement with the predictions of Theorem 3 and Theorem 4 for the\ncases\u0017.1 and\f.\u0019\n2, respectively. At the same time, one can see from Figs. 3(b) and 3(c) that as\nthe value of \u0017is increased, the pro\fles develop a multiscale structure, whereby an inner core forms\nnear the edge on the O(\u0017) length scale for \u0017\u001d1, followed by either an exponential ( \f=\u0019=2) or\nan algebraic ( \f6=\u0019=2) tail. Heuristically, this scaling may be derived by balancing the anisotropy\n21(a)\n(d)(b)\n(e)(c)\n(f)Figure 3: Computed boundary wall pro\fles for \f=\u0019=2;\u0019=4;\u0019=8. In panels (a) and (d), \u0017= 1;\nin panels (b) and (e), \u0017= 10; in panels (c) and (f), \u0017= 50. The upper panels (a)-(c) show the\ncomputed pro\fles for the given values of \fand\u0017. The lower panels (d)-(f) show the same pro\fles\nin log-log coordinates, with the dashed lines indicating the 1 =xdecay.\nand the stray \feld terms in (27). A structure similar to this was reported previously for N\u0013 eel walls\nat large values of \u0017[19, 20, 22, 33, 36]. Notice that all the pro\fles are regular and exhibit a \fnite\nslope near the edge, in agreement with Theorem 2.\nFocusing on the decay of the domain wall pro\fles, we observe that even though the overall\nshape of the pro\fle may be qualitatively similar to that in (19), they exhibit slow algebraic decay\nfor all\f6=\u0019=2, even for small values of \u0017, see Figs. 3(d){(f). In all those cases, the pro\fles exhibit\na decay rate proportional to 1 =x, which can be explained by the fact that there is a build-up of\nmagnetic charge near the material edge, which results in a stray \feld decaying like 1 =xaway from\nthe edge. This should be contrasted to the asymptotic 1 =x2decay observed in N\u0013 eel walls [10]. At\nthe same time, for the special value of \f=\u0019=2 the decay becomes exponential, which can also be\nseen directly from (27). Indeed, when \f=\u0019=2, the cos(\u0012(x)\u0000\f) factor multiplying the contribution\nof the stray \feld vanishes as x!+1, making the anisotropy term dominate at large values of x\nand, therefore, resulting in exponential decay.\nWe note that the domain wall pro\fles obtained by us numerically in Fig. 3 are not guaranteed\nto be those corresponding to the global energy minimizers in Theorem 1. Instead, they may\ncorrespond only to local energy minimizers. In fact, neither monotonicity, nor uniqueness of the\nenergy minimizing edge domain wall pro\fles are known a priori, in contrast to the N\u0013 eel walls in\nthe bulk of the material [10, 38]. In order to assess whether other types of local minimizers may\nexist in the problem under consideration, we performed further numerical studies of solutions of\n(90){(92) by considering values of \foutside the interval (0 ;\u0019=2]. The obtained stationary solutions\nfor\u0017= 10 are shown in Fig. 4. All these solutions decay to zero as x!1 , indicating a nontrivial\nwinding (i.e., a variation of \u0012by more than \u0019=2) in each one for \f62[0;\u0019=2]. We emphasize that\nthese domain wall pro\fles are stabilized by nonlocal e\u000bects, since in the absence of stray \feld, i.e.,\nwhen\u0017= 0, such solutions do not exist. At the same time, the solutions with winding appear\nto have higher energy than the corresponding ones in Fig. 3 for the same value of \u0017, indicating\n22(a) (b)\n(c)Figure 4: Edge domain walls exhibiting winding and lack of monotonicity obtained by solv-\ning (90) for \u0017= 10 and di\u000berent values of \f. In (a), \f=\u0000\u0019=2;\u0019=2;3\u0019=2;5\u0019=2. In (b),\n\f=\u00003\u0019=4;\u0019=4;5\u0019=4;9\u0019=4. In (c), the non-monotone decay in the tails of the solutions for\n\f=\u00005\u0019=8 (red),\f=\u00003\u0019=4 (green) and \f=\u00007\u0019=8 (blue) at large xis emphasized.\nthat the global energy minimizers do not exhibit winding. Furthermore, for \u0000\u0019 < \f <\u0000\u0019=2 the\nsolutions exhibit overshoot and a non-monotone decay to zero as x!+1, see Fig. 4(c). Thus,\nmonotonicity of the initial data in (92) is not preserved under (90). Let us also mention that we\ntried di\u000berent non-monotone initial conditions, but did not obtain any other solutions than those\nshown in Fig. 4. However, monotone solutions with larger winding were observed numerically for\nstill larger values of \f. According to our numerics, it appears that edge domain wall solutions with\narbitrarily large winding are possible.\nTo conclude, we note that an a priori lack of monotonicity is an obstacle for proving the precise\nasymptotic decay of the pro\fles, using the methods of [10]. A broader question of interest is whether\nthe one-dimensional domain wall pro\fles in Theorem 1 are also minimizers, in some suitable sense,\nof the two-dimensional micromagnetic energy in (8). It is well known that magnetic domains often\nexhibit spatially modulated exit structures near the material boundary [22], and spatially periodic\nand more complicated two-dimensional edge domains have been observed experimentally in thin\n\flms with strong in-plane crystalline anisotropy [11].\nA One-dimensional energy\nFor the reader's convenience, below we present a derivation of the one-dimensional energy in (10)\nfrom the two-dimensional energy in (8) and then collect some rather well-known facts about the\none-dimensional fractional Sobolev norm appearing throughout our paper.\nThe energy E0(m) in (8) with hset to zero may be equivalently written as\nE0(m) =1\n2Z\nD\u0000\njrmj2+m2\n1\u0000'r\u0001m\u0001\nd2r; ' (r) =\u0000\u0017\n4\u0019Z\nDr\u0001m(r0)\njr\u0000r0jd2r0; (93)\nwhere'is the stray \feld potential. Since we are interested in one-dimensional transition pro\fles in\nthe vicinity of the material edge, we assume Dto be an in\fnite strip of width woriented at an angle\n\f2[0;\u0019=2] with respect to the easy axis (see Fig. 2) and consider one-dimensional magnetization\ncon\fgurations. Setting m=m(x) and'='(x), wherexis the normal coordinate to the strip\n23axis de\fned in (9), the energy in (93) per unit length becomes\nE\f;w(m) =1\n2Zw\n0\u0000\njm0\n1(x)j2+jm0\n2(x)j2+m2\n1(x)\u0001\ndx\n+\u0017\n8\u0019lim\nL!1ZL\n\u0000LZw\n0Zw\n0m0\n\f(x)m0\n\f(y)\np\njx\u0000yj2+jsj2dxdyds; (94)\nwherem\f=e\f\u0001m, with e\f= (cos\f;sin\f) andm\f(0) =m\f(w) = 0, and we noted that the\ncontributions of the magnetic dipoles to the stray \feld potential at distances jsj\u001dwareO(jsj\u00002),\nmaking the last integral in (94) convergent as L!1 . We compute\nZL\n\u0000L1p\njx\u0000yj2+jsj2ds= 2 log p\nL2+x2+L\nx!\n= 2 lnjx\u0000yj\u00001+ 2 ln(2L) +O(a2); (95)\nwitha=jx\u0000yj\nL. We know thatjx\u0000yj\u0014wfor allx;y2(0;w) and therefore a!0 uniformly in\nx;yasL!1 . Using the fact that m\f(0) =m\f(w) = 0, we then obtain (10) for an arbitrary\nm2H1((0;w);R2) satisfying (11).\nNow we derive several other representations of the one-dimensional energy in (10). We can\nextendm\fby zero outside the interval (0 ;w) and rewrite the energy as\nE\f;w(m) =1\n2Zw\n0\u0000\njm0\n1j2+jm0\n2j2+m2\n1\u0001\ndx+\u0017\n4\u0019Z1\n\u00001Z1\n\u00001lnjx\u0000yj\u00001m0\n\f(x)m0\n\f(y)dxdy: (96)\nNext we want to rewrite the nonlocal part of the energy in Fourier space. It is clear that m\f2\nH1(R)\\L1(R) but lnjxj\u00001diverges at in\fnity. We introduce the following function approximating\nlnjxj\u00001:\nKa(x) :=e\u0000ajxjlnjxj\u00001; a> 0: (97)\nIt is clear that Ka(x)2Lp(R) for allp<1. Moreover, we have as a!0\n\u0017\n4\u0019Z1\n\u00001Z1\n\u00001Ka(x\u0000y)m0\n\f(x)m0\n\f(y)dxdy!\u0017\n4\u0019Z1\n\u00001Z1\n\u00001lnjx\u0000yj\u00001m0\n\f(x)m0\n\f(y)dxdy: (98)\nHere we used dominated convergence theorem, with the help of Young's inequality and the fact that\nm\fhas compact support. Now we can use L2Fourier transform (as de\fned in (67)) to obtain [30]\nZ1\n\u00001Z1\n\u00001Ka(x\u0000y)m0\n\f(x)m0\n\f(y)dxdy =Z1\n\u00001k2bKa(k)jbm\f(k)j2dk\n2\u0019; (99)\nwhere [4, Eq. (6) on p. 18]\nbKa(k) =2a\r+aln(k2+a2) + 2karctan(k=a)\nk2+a2; (100)\nand\r\u00190:5772 is the Euler constant.\nWe can split the integral in the right hand-side of (99) as\nZ1\n\u00001k2bKa(k)jbm\f(k)j2dk\n2\u0019=Z\njkj<1k2bKa(k)jbm\f(k)j2dk\n2\u0019+Z\njkj>1k2bKa(k)jbm\f(k)j2dk\n2\u0019: (101)\n24Using the facts that bm\f(k)2L2(R),kbm\f(k)2L2(R) and\nbKa(k)!\u0019\njkjasa!0 uniformly in kfor alljkj\u00151; (102)\nk2bKa(k)!\u0019jkjasa!0 uniformly in kfor alljkj\u00141; (103)\nwe obtain\nZ1\n\u00001Z1\n\u00001Ka(x\u0000y)m0\n\f(x)m0\n\f(y)dxdy!1\n2Z1\n\u00001jkjjbm\f(k)j2dk: (104)\nCombining (98), (99) and (104) we arrive at the following representation for the energy in (96):\nE\f;w(m) =1\n2Zw\n0\u0000\njm0\n1j2+jm0\n2j2+m2\n1\u0001\ndx+\u0017\n4Z1\n\u00001jkjjbm\f(k)j2dk\n2\u0019; (105)\nwhich is equivalent to (12).\nWe now want to rewrite the nonlocal term in the real space involving only m\f, but notm0\n\f. In\norder to do this we note that\n1\n2\u0019Z1\n\u00001j1\u0000eikzj2\nz2dz=1\n\u0019Z1\n\u000011\u0000cos(kz)\nz2dz=jkj; (106)\nand obtain\nZ1\n\u00001jkjjbm(k)j2dk=Z1\n\u00001\u0012Z1\n\u00001j1\u0000eikzj2\nz2dz\u0013\njbm\f(k)j2dk\n2\u0019\n=Z1\n\u000011\nz2\u0012Z1\n\u00001j(1\u0000eikz)bm\f(k)j2dk\n2\u0019\u0013\ndz\n=Z1\n\u00001Z1\n\u00001jm\f(x)\u0000m\f(x+z)j2\nz2dzdx\n=Z1\n\u00001Z1\n\u00001jm\f(x)\u0000m\f(y)j2\njx\u0000yj2dydx: (107)\nTherefore we can rewrite the original energy in the following way:\nE\f;w(m) =1\n2Zw\n0\u0000\njm0\n1j2+jm0\n2j2+m2\n1\u0001\ndx+\u0017\n8\u0019Z1\n\u00001Z1\n\u00001(m\f(x)\u0000m\f(y))2\n(x\u0000y)2dxdy; (108)\nwhich coincides with (13).\nAcknowledgements. The work of RGL and CBM was supported, in part, by NSF via grants\nDMS-1313687 and DMS-1614948. 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Phys. { Condensed Matter , 26:394002, 2014.\n28"    },    {        "title": "1705.03371v2.Ab_initio_study_of_magnetocrystalline_anisotropy__magnetostriction__and_Fermi_surface_of_L10_FeNi__tetrataenite_.pdf",        "content": "arXiv:1705.03371v2  [cond-mat.mtrl-sci]  17 Aug 2017Ab initio study of magnetocrystalline anisotropy, magneto striction, and Fermi surface\nof L1 0FeNi (tetrataenite)\nMiros/suppress law Werwi´ nski∗\nInstitute of Molecular Physics Polish Academy of Sciences,\nM. Smoluchowskiego 17, 60-179 Pozna´ n, Poland\nWojciech Marciniak\nFaculty of Technical Physics, Pozna´ n University of Techno logy,\nPl. M. Sk/suppress lodowskiej-Curie 5, 60-965 Pozna´ n, Poland\n(Dated: August 18, 2017)\nThe ordered L1 0FeNi phase (tetrataenite) is recently considered as a promi sing candidate for the\nrare-earth free permanent magnets applications. In this wo rk we calculate several characteristics of\nthe L1 0FeNi, where most of the results come form the fully relativis tic full potential FPLO method\nwith the generalized gradient approximation (GGA). A speci al attention deserves the summary of\nthe magnetocrystalline anisotropy energies (MAE’s), the f ull potential calculations of the anisotropy\nconstant K3, and the combined analysis of the Fermi surface and three-di mensional k-resolved\nMAE. Other calculated parameters presented in this article are the magnetic moments msandml,\nmagnetostrictive coefficient λ001, bulk modulus B 0, and lattice parameters. The MAE’s summary\nshows rather big discrepancies between the experimental MA E’s from literature and also between\nthe calculated MAE’s. The MAE’s calculated in this work with the full potential and GGA are equal\nto 0.47 MJ m−3from WIEN2k, 0.34 MJ m−3from FPLO, and 0.23 MJ m−3from FP-SPR-KKR\ncode. These last results strongly suggest that the value of M AE in GGA is below 0.5 MJ m−3. It\nis also expected that this value is significantly underestim ated due to the limitations of the GGA.\nUnfortunately, as other authors suggest, even the MAE equal 1.3 MJ m−3would be insufficient\nto raise the L1 0FeNi from the category of semi-hard magnets. However the L1 0FeNi has still a\npotential to improve its MAE by modifications, like e.g. tetr agonal strain or alloying. The presented\nthree-dimensional k-resolved map of the MAE combined with the Fermi surface give s a complete\npicture of the MAE contributions in the Brillouin zone. The c alculated Fermi surface consists of\nclosed hole pockets and open sheets. It reflects a four-fold s ymmetry of the crystal and is closely\nrelated to the MAE( k). The analysis of the effects of external factors, like strai n, on the k-resolved\nMAE and Fermi surface should be beneficial in engineering of t he hard magnetic properties. The\nobtained from full potential FP-SPR-KKR method magnetocry stalline anisotropy constants K2\nandK3are several orders of magnitude smaller than the MAE/ K1and equal to -2.0 kJ m−3and\n110 J m−3, respectively. The calculated partial spin and orbital mag netic moments of the L1 0\nFeNi are equal to 2.72 and 0.054 µBfor Fe and 0.53 and 0.039 µBfor Ni atoms, respectively. The\ncalculations of geometry optimization lead to a c/aratio equal to 1.0036, B 0equal to 194 GPa, and\nλ001equal to 9.4 ×10−6.\nI. INTRODUCTION\nThe electric power generators, motors, and transform-\ners are just a few examples where the magnetic materials\nfind an application in the modern technology. The hard\nmagnetic materials used the most are the alnicos, hexa-\nferrites, and Nd-Fe-B alloys. The economical event from\n2011 called The Rare-Earth Crisis1destabilized i.a. the\nprices of neodymium motivating efforts to find new rare-\nearth free permanent magnets. The ongoing search for\nhard magnetic materials free from rare-earth elements\nis summarized in several review articles.2–7Some of the\npromising candidates studied recently are e.g. Fe/Co\nnanowires8,9, Fe-Co alloys doped with B and C10–13,\n(Fe/Co) 2B14–17, (Fe/Co) 5XB218–20, MnBi21, and the\nL10phases such as FePt, CoNi, MnAl, and FeNi.22–24\n∗Corresponding author: werwinski@ifmpan.poznan.plThe last candidate on the list is a subject of this work.\nExistence of an ordered L1 0-type FeNi phase (tetrataen-\nite) was confirmed in the sixties by N´ eel et al.25in the\nstudy of a single crystal ordered by neutron bombard-\nment and once again in the seventies by Paterson et al.26\nin the study of taenite lamellae from the iron meteorite.\nThe L1 0is the strukturbericht designation of the CuAu\nI-type ordered tetragonal phase. The L1 0unit cell con-\nfined by solid lines in Fig. 1consists of two formula units.\nTwo faces of its are occupied by one type of atoms. The\nthird face and corners are occupied by the second type\nof atoms. A detailed study of crystallographic aspects\nof L1 0magnetic materials can be found in a paper of\nLaughlin et al.27\nIn this work we investigate theoretically the magne-\ntocrystalline anisotropy constants K2andK3, Fermi sur-\nface, and bulk modulus of the tetrataenite. This ef-\nforts are followed by the theoretical reinvestigation of th e\nmagnetostrictive coefficient λ001and magnetocrystalline\nanisotropy energy (MAE) treated with a full potential2\nFIG. 1. The L1 0crystallographic structure. The solid lines\ndesignate a unit cell containing two formula units and the\ndashed lines confine a unit cell with a single formula.\nfully relativistic method based on the full four component\nrepresentation of the Bloch states.28Also the experimen-\ntal29–32and theoretical24,29,33results from literature of\nthe magnetocrystalline anisotropy of the L1 0FeNi are\nconsidered. One of the most important parameters from\nthe perspective of permanent magnets application is a\nmagnetocrystalline anisotropy constant K1. The mea-\nsured values of K1of the L1 0FeNi are relatively high32,34\nand equal up to 1.0–1.3 MJ m−3. Skomski and Coey6\nsuggest that even such high values of K1are insufficient\nto raise FeNi off the category of semi-hard magnets. How-\never it has been shown for the L1 0FeNi films, that their\nintrinsic magnetic properties can be altered e.g. by en-\ngineering larger strains.35,36Furthermore, the composi-\ntion and microstructure of the L1 0FeNi may be tailored\nas well to improve the FeNi potential for rare-earth-free\npermanent magnet application.32Skomski points out the\nbeneficial self-organized microstructure of the L1 0FeNi\nbeing reflected in a relatively high coercivity of about\n120 mT.37Some other characteristics indicating the L1 0\nFeNi as a good candidate for hard magnets are magne-\ntization approaching that of the Nd-Fe-B and relatively\nhigh Curie point near 550° C38, however preceded by\nthe critical temperature of the ordered state of about\n320° C.25Considering from the application point of view,\na serious weakness of the FeNi remains achieving and re-\ntaining the L1 0atomic order.36,39\nII. COMPUTATIONAL DETAILS\nThe determination from the first principles of the\nmagnetocrystalline anisotropy constants and the magne-\ntostrictive coefficient λ001requires the fully relativistic\nelectronic band structure calculations. The calculations\nare carried out by using the full-potential local-orbital\nminimum-basis scheme FPLO-14.0-4928,40with the gen-\neralized gradient approximation (GGA) in the Perdew-\nBurke-Ernzerhof (PBE) form.41The calculations are per-\nformed up to a 803k-mesh with tetrahedron method for\nintegration, an energy convergence criterion 10−8Ha,\nand a charge density convergence criterion 10−6. The\ntwo-dimensional maps of MAE( k) are constructed on the1000 × 1000 k-mesh and the three-dimensional plot of\nMAE( k) is based on 250 × 250 × 170 k-mesh within a\nselected one-eighth part of the full Brillouin zone. An\ninitial spin splitting is applied assuming the ferromag-\nnetic structure. The bct representation of the L1 0unit\ncell is used (as described e.g. by Edstr¨ om et al.24) with a\nspace group P4/mmm and atomic coordinates Fe (0, 0,\n0) and Ni (0.5, 0.5, 0.5). Both the volume and c/aratio\nare optimized. The calculated structural parameters are\nin good agreement with the corresponding experimental\nand theoretical values, see Table I. For the visualization\nof crystal structure the VESTA code42is used. The mag-\nnetostriction is calculated with the same scheme as used\nby Wu et al.29,43–45In the latter references one can find\nthe detailed description of the method but also the mag-\nnetostrictive coefficients calculated for several material s\nthat stay in a good agreement with the experiment. In\norder to calculate the magnetostrictive coefficient of the\nL10FeNi we use the approach developed for cubic geom-\netry as the tetragonal distortion in this system is very\nsmall (c/a = 1.0036). To determine the magnetostric-\ntive coefficient for a cubic material one has to calculate\nthe strain dependences of the total energy ( E) and the\nmagnetocrystalline anisotropy energy (MAE), see Fig. 3.\nThe dependence of the fractional change in length can be\nwritten based on the direction cosines of the magnetiza-\ntion ( α) and of the strain measurement ( β):\n∆l\nl0=3\n2λ001/bracketleftBigg3/summationdisplay\ni=1α2\niβ2\ni−1\n3/bracketrightBigg\n+ 3λ1113/summationdisplay\ni/negationslash=jαiαjβiβj.(1)\nIf the measurement is made along the [001] direction the\nequation simplifies to:\n∆l\nl0=3\n2λ001/bracketleftbigg\nα2\nz−1\n3/bracketrightbigg\n(2)\nand for a single domain system it takes a form:\nλ001=1\n3l0(θ= 0◦)−l0(θ= 90◦)\nl0(θ= 0◦) +l0(θ= 90◦), (3)\nwhere θis the angle between the magnetization direction\nand the caxis. If the ab initio calculated total energies\nare fitted in a quadratic form:\nE(θ= 0◦) =al2+bl+c;\nE(θ= 90◦) =al2+bl+c+ MAE ,(4)\nthe equation for magnetostrictive coefficient can be writ-\nten as:\nλ001=−2\n3d(MAE)\ndl\nb. (5)\nIn this work the MAE’s are evaluated based on the total\nenergies calculated self-consistently for two perpendicu lar\nquantization axes:\nMAE = E(θ= 90◦)−E(θ= 0◦). (6)3\nIn addition to the FPLO calculations, the Korringa-\nKohn-Rostoker (KKR) approach as implemented in the\nMunich SPR-KKR package (non-public full potential\nversion 7.6.0) is used to calculate the magnetocrystalline\nanisotropy constant K3.46,47The advantage of using full\npotential method for MAE calculations has been dis-\ncussed before.14,48The ab initio calculations of K3are\ntoday still a numerically demanding task. But what dis-\ntinguishes the SPR-KKR among the other ab initio codes\nis a numerical accuracy on the level of about 0.1 µeV in\ncalculating total energy, which is the same order of mag-\nnitude as expected for the K3value of the L1 0FeNi.\nAnother argument in favor of the SPR-KKR is that it\nhas been successfully applied before to calculate K3for\nthe magnetic shape memory Fe-Pd alloys.49In order to\nget a converged values of K3for the L1 0FeNi, the FP-\nSPR-KKR parameters of 10−10Ry energy convergence\ncriterion and up to 225 × 225 × 158 k-points (about 8\nmillion) are necessary. For Brillouin zone integration the\nspecial point method with a regular k-point grid is used.\nKhan et al. have shown for L1 0FePt phase22that the\nKKR calculations of total energies are also quite sensitive\nto the angular momentum expansion lmaxcutoff used for\nthe multipole expansion of the Green function. Khan et\nal.concluded that the angular momentum lmax= 3 cut-\ntoff yields to a qualitatively correct value of the MAE,\nhowever even for lmax= 7 a full convergence is still dif-\nficult to reach. Taking this conclusion into account we\nchoose for our calculations a maximum angular momen-\ntum value lmax = 4 (NL = 5 in the KKR configura-\ntion file). Our decision is further motivated by results\nof a convergence test of K1with respect to the angular\nmomentum performed up to lmax = 6. The test indi-\ncated that problems with convergence occur above the\nlmax = 4 leading to divergence of the K1value. This\nbehavior may come from numerical problems in evalu-\nating the Madelung potential and near-field corrections,\npointed out by Khan et al.22The energy integrals are\nevaluated by contour integration on a circular energy\npath in complex plane (GRID = 5), using 40 points of\nthe E-mesh. The calculations within the FP-SPR-KKR\nare carried out with the PBE exchange-correlation po-\ntential and with the same crystallographic parameters of\nFeNi as used for FPLO MAE calculations. The magne-\ntocrystalline anisotropy energy in tetragonal crystal can\nbe described by the following equation50:\nMAE = K1sin2θ+K2sin4θ+K3sin4θcos4φ, (7)\nwhere K iare the anisotropy constants, θis the angle\nbetween the magnetization direction and the caxis, and\nφis the angle between the magnetization and the aaxis\nwithin the basal plane of a tetragonal lattice. For θ= 90◦\nthe Eq. 7takes the following form:\nMAE −K1−K2=K3cos4φ. (8)\nThe K3is then evaluated based on the total energies\nE100andE110calculated self-consistently for φequal to0◦and 45◦(directions [100] and [110] in the bct unit cell)\nfrom equation:\n2K3=E100−E110. (9)\nThe computational parameters used to obtain K 1and K 2\nwith FP-SPR-KKR are the same as presented above for\nthe K 3calculations, with an exception that lower num-\nber of k-points, about 1.5 million, is used in the whole\nBrillouin zone.\nFurthermore the full-potential augmented plane-wave\nmethod FP-LAPW as implemented in the WIEN2k\ncode51is used to calculate a reference value of MAE.\nMuffin-tin radii RMT are 2.29 a 0for Fe and 2.29 a 0for\nNi atoms, where a 0is Bohr radius. The PBE exchange-\ncorrelation potential is used. Plane wave cut-off param-\neterRK maxis set to 10, which leads to above 245 basis\nfunctions. Relativistic effects are included with the sec-\nond variational treatment of spin-orbit coupling. The\ntotal energy convergence criterion is set to 10−8Ry. The\n55 × 55 × 39 k-points (about 120 thousand) are used in\nthe whole Brillouin zone.\nAlthough, the reproducibility of the results in den-\nsity functional theory calculations of solids has been well\nestablished on the level of scalar relativistic estimation\nof the lattice parameters52, the accurate calculations of\nthe magnetocrystalline anisotropy energy MAE and its\nderivatives, like magnetostrictive coefficient, remain a\nchallenge.22It is even harder to calculate the values of\nmagnetocrystalline anisotropy constant K3which hap-\npens to be two orders of magnitude smaller than MAE.49\nIn this work we managed to calculate the K3of the L1 0\nFeNi with the full potential thanks to the very accurate\nconvergence tests.\nIII. DENSITY FUNCTIONAL THEORY\nCALCULATIONS\nThe L1 0FeNi phase has been studied ab inito several\ntimes before.24,29,33,53In this work we reinvestigate the\nmagnetocrystalline anisotropy energy MAE and the mag-\nnetostrictive coefficient λ001of the L1 0FeNi phase and\ncalculate its magnetocrystalline anisotropy constant K2,\nK3, Fermi surface, and bulk modulus. Our results to-\ngether with the literature data are summarized in Tab. I.\nA. Bulk Modulus\nWe start from the calculations of the equilibrium vol-\nume (22.686 ˚A3), followed by the calculations of equi-\nlibrium c/aratio (1.0036), see Fig. 2. The optimization\nleads to a tetragonal structure with the lattice parame-\ntersa= 3.56˚A and c= 3.58˚A. The energy-volume data,\nsee Fig. 2(a), allows to calculate the bulk modulus B 0for\nthe L1 0FeNi by fitting the third-order Birch-Murnaghan\nequation of state.56It leads to a B 0equal to 194 GPa at4\nTABLE I. The lattice parameters ( aandc), spin ( MS) and orbital ( ML) magnetic moments, magnetocrystalline anisotropy\nenergies (MAE), magnetostrictive coefficients ( λ001), and bulk moduli (B 0) of the L1 0FeNi phase.\nquantity a c MS(Fe) M L(Fe) M S(Ni) M L(Ni) MAE or K1MAE or K1λ001 B0\nunit ˚A ˚A µB µB µB µB µeV formula−1MJ m−310−6GPa\nexperiment 3.57363.57362.54±0.1654∼0.05310.73±0.04540.1031— 0.58–1.330,32∼955—\nFLAPW-GGA293.58 3.58 2.71 0.052 0.69 0.038 32 0.22 9.7 —\nVASP-GGA32— — — — — — 110 0.78 — —\nVASP-GGA333.556 3.584 2.65 — 0.61 — 78 0.56 — —\nWIEN2k-GGA33— — — — — — 69 0.48 — —\nWIEN2k-GGA243.56 3.58 2.69 — 0.67 — 69 0.48 — —\nWIEN2k-GGA (this work) — — 2.69 0.052 0.66 0.036 67 0.47 — —\nASA-SPR-KKR-GGA24— — 2.73 — 0.62 — 110 0.77 — —\nFP-SPR-KKR-GGA (this work) — — 2.69 0.053 0.60 0.036 32 0.23 — —\nFPLO14-GGA (this work) 3.56 3.58 2.72 0.054 0.53 0.039 48 0.34 9.4 194\n22.4 22.6 22.8 23.00.01.02.03.0\n1.00 1.010.00.20.40.6\nvolume (Å3)                                    c/aE - E0 (meV formula-1)\nE - E0 (meV formula-1)(a) (b)\nFIG. 2. (a) The volume dependence of the total energy and\n(b) total energy versus the lattice parameters c/aratio as cal-\nculated for the L1 0FeNi with the FPLO14 PBE+so method.\nTheV(E) is calculated with a fixed c/a∼1.0056 as obtained\npreviously by Edst¨ om et al.24The equilibrium volume equals\n22.686 ˚A3. The equilibrium c/aratio calculated with a fixed\nvolume 22.686 ˚A3equals 1.0036. The energy scale is shifted\nso that the energy minimum equals zero.\n0 K. The previous calculations with coherent potential\napproximation CPA, of the Fe 0.5Ni0.5random alloy have\ngiven the B 0value of about 220 GPa.57For comparison,\nthe experimental values of B 0for the Fe 0.5Ni0.5alloy\nmeasured at room temperature vary between 165 GPa\nand 177 GPa58and the experimental B 0values for Fe\nand Ni are about 170 GPa and 180 GPa, respectively.\nB. Magnetostrictive Coefficient λ001\nThe total and magnetocrystalline anisotropy energies\nversus the length of the lattice parameter care presented\nin Fig. 3as necessary to evaluate the magnetostrictive\ncoefficient λ001from Eq. 5. The total energy and MAE\nare calculated based on the optimized crystal structure\nin a constant volume mode adopted for the distortion.\nThe MAE calculated for the equilibrium L1 0FeNi struc-\nture equals 48 µeV formula−1(0.34 MJ m−3). Initial\ntests performed with volume relaxation for 503k-points\nhave shown that the constant volume mode underesti-3.55 3.56 3.57 3.58 3.59 3.600100200300400500600700\n454647484950E - E0  (µeV formula-1)\nMAE (µeV formula-1)\nlattice parameter c (Å)\nFIG. 3. The calculated total energy and magnetocrystalline\nanisotropy energy of the L1 0FeNi versus the length of the lat-\ntice parameter c. The results are obtained within the FPLO14\nPBE+so method. A constant volume mode is adopted for the\ndistortion. The dotted line indicates the equilibrium latt ice\nparameter.\nmates the λ001of the L1 0FeNi by about 10%. For the\nprice of this inaccuracy, in constant volume mode we can\nperform calculations up to 803k-points. The resultant\nλ001obtained for 603, 703, and 803k-points are equal to\n11.3 ×10−6, 8.7 ×10−6, and 9.4 ×10−6, respectively.\nThe variation of the estimated λ001with number of k-\npoints comes from the numerical inaccuracy in the eval-\nuation of a very small quantity as MAE. Nevertheless,\nthe calculated here λ001= 9.4 ×10−6stays in a good\nagreement with the previous theoretical result by Wu and\nFreeman29,λ001= 9.7 ×10−6, and with the experimen-\ntal value by Bozorth55λ100∼9×10−6.λ001∼10×10−6\nfor the L1 0FeNi is rather small value. It is of the same\norder of magnitude as for elements Fe and Ni and three\norders of magnitude smaller than for the magnetostric-\ntive material Terfenol-D.5\nC. Magnetic Moments\nThe spin magnetic moments calculated with the FPLO\nfor the L1 0FeNi are equal 2.72 µBfor the Fe atom\nand 0.53 µBfor the Ni atom, see Tab. I. The calculated\nspin magnetic moments on Fe and Ni stay in relatively\ngood agreement with the values measured on Fe equal to\n2.54±0.16 µBand on Ni equal to 0.73±0.04 µBfor the\nL10FeNi phase.54The calculated orbital magnetic mo-\nments are equal to 0.054 µBfor the Fe atom and 0.039 µB\nfor the Ni atom. They are also close to the experimen-\ntal values on Fe equal to ∼0.05 µBand on Ni equal to\n0.10 µBas measured for the L1 0FeNi phase.31The cal-\nculated and measured orbital magnetic moments on Fe\nin the L1 0FeNi are reduced in comparison to the exper-\nimental value of 0.086 µBfor the bcc iron.59The calcu-\nlated orbital magnetic moments on Ni (0.039 µB) in the\nL10FeNi are also reduced in respect to the experimen-\ntal value of 0.055 µBfor the fcc nickel.60The calculated\ntotal magnetic moment is equal to 3.34 µBformula−1\n(1.67 µBatom−1) which is not particularly high value\nfor 3 d-based magnetic materials. From the perspective\nof hard magnetic materials this reduction of magnetic\nmoment in comparison to e.g. pure bcc Fe is beneficial\nfor the magnetic hardness but adversely affects the en-\nergy product.\nD. Magnetocrystalline Anisotropy Energy\nGetting consistent MAE results from different first\nprinciples codes is still a challenge. The difficulties come\nfrom such factors as a complex shape of the valence band\nstructure or a demand of very high numerical accuracy.22\nThe differences between the results from various codes\nmay come from application of different approximations\nlike the atomic sphere approximation, the lack of crys-\ntal structure optimization, or the insufficient number of\nk-points.14,22,61Some recent papers discuss however the\nreproducibility of the MAE between WIEN2k and KKR\nmethods22and between FPLO and WIEN2k.14\nThe calculated in GGA MAE’s of the L1 0FeNi taken\nfrom literature are equal to 0.2229, 0.4824, 0.5633, and\n0.7832MJ m−3. The experimental determination of the\nL10FeNi magnetocrystalline anisotropy constant K1is\nambiguous as well, with a spread in the K1values from\n0.5830, trough 0.6725, up to 1.332MJ m−3. The MAE’s\nof the L1 0FeNi calculated in GGA in this work are equal\nto 0.23, 0.34, and 0.47 MJ m−3from FP-SPR-KKR,\nFPLO, and WIEN2k, respectively.\nBased on the above theoretical results we expect that\nthe accurate GGA value of the MAE for the L1 0FeNi\nis below 0.5 MJ m−3. Such value could be an argument\nfor removing the L1 0FeNi from a list of candidates for\nrare-earth free permanent magnets. Some authors sug-\ngest however that bare GGA is insufficient for describ-\ning the L1 0FeNi and a consideration of orbital polar-\nization corrections is necessary.33,53It has been shownthat inclusion of the orbital polarization causes a signif-\nicant increase of the MAE of the L1 0FeNi (from ∼0.55\nto∼1.23 MJ m−3)53and (from 0.48 to 0.84 MJ m−3)33.\nUnfortunately, even the MAE of about 1 MJ m−3may\nbe insufficient to raise the L1 0FeNi from the category\nof semi-hard magnets.6To reach that goal further efforts\non increasing the magnetocrystalline anisotropy have to\nbe made.\nE. Magnetocrystalline Anisotropy Constant K3\nThe magnetocrystalline anisotropy constant K3of the\nL10FeNi is particularly interesting from a perspective of\nspintronic applications of the material, e.g. for the mag-\nnetic tunnel junction.33In a tetragonal crystal the K3\ncan be defined by Eq. 7. Unfortunately we cannot calcu-\nlate the K3with the FPLO code, which was applied to\nobtain previous results. To get the K3we use then the\nFP-SPR-KKR package which produces the value of K3\nequal to 110 J m−3. It is four orders of magnitude smaller\nfrom the value of MAE or K1for the L1 0FeNi. For com-\nparison, the values of K3calculated in ASA-SPR-KKR\nfor magnetic shape memory alloys Fe-Pd49have similar\norder of magnitude ( ∼103J m−3) as the one presented\nabove. The details of calculation method and the motiva-\ntions for using FP-SPR-KKR package for evaluating K3\nare presented in the introductory section. As the K3isk-\n0 2 4 6 8 10 12-50005001000\nmillions of k-pointsK3 (J m-3)\nFIG. 4. The magnetocrystalline anisotropy constant K3of\nthe L1 0FeNi phase versus the number of k-points. The full\npotential FP-SPR-KKR code with the PBE approximation is\nused.\nmesh sensitive the convergence test is made, see Fig. 4. It\ncan be noticed that large number of k-points is necessary\nto get a satisfactory convergence of K3. To complete the\nFP-SPR-KKR analysis also the K 1and K 2magnetocrys-\ntalline anisotropy constants are calculated. In Fig. 5we\npresent the energy dependence as a function of the polar\nangle θbetween the magnetization direction and the c\naxis. Parameters from a fit to K1sin2θ+K2sin4θare\nK1= 0.23 MJ m−3andK2= -2.0 kJ m−3, see Eq. 8.\nThe lowest energy in Fig. 5corresponds to [001] quanti-\nzation axis and the highest energy to [100] axis.6\n0.000.050.100.150.200.25\nCalculated data\nK1 sin2θ + K2 sin4θEnergy (MJ m-3)\nθ0                   π/4                   π/2                   3π/4                    π\nFIG. 5. Energy as a function of the polar angle θbetween\nthe magnetization direction and the caxis for the L1 0FeNi.\nK1= 0.23 MJ m−3,K2= -2.0 kJ m−3. The full potential\nFP-SPR-KKR code with the PBE approximation is used.\nF. Fermi Surface\nFIG. 6. The Fermi surface of the L1 0FeNi obtained within\nthe FPLO14 PBE+so method.\nFigure 6presents the Fermi surface of the L1 0FeNi.\nThe tetragonal crystal structure of the L1 0FeNi is re-\nflected in a uniaxial anisotropy and a four-fold symme-\ntry of Fermi surface sheets. The seven sheets are divided\ninto two groups of closed hole pockets and open sheets.\nThe nested hole pockets (a), (b), and (c) are centered at\nthe high symmetry point M. One another hole pocket is\ncentered at point Γ, see panel (e). When the hole pockets\npermit for only closed orbits the mainly open sheets (d),\n(e), and (f) allow for both the opened and closed orbits.\nWe have calculated the Fermi surface of the L1 0FeNiboth as the basic characteristic of the material and as the\nbasis for a k-resolved MAE analysis which will be pre-\nsented in the next paragraph. Basic knowledge on the\nFermi surface of the crystal lets us think on manipulat-\ning of its properties using the emerging method called\nthe Fermi surface engineering. The method makes rela-\ntions between a shape of the Fermi surface and external\nfactors like doping or strain. The Fermi surface engi-\nneering technique has been successfully applied e.g. to\ntransparent conductors62and superconductors.63\nG. MAE Analysis in Reciprocal Space\nMost often the anisotropic magnetic properties of the\nmaterials are analyzed in the real space. The spin and\norbital moments and magnetocrystalline anisotropy con-\nstants are presented as functions of the atomic position or\nangle.21,33,64The magnetocrystalline anisotropy energy\n(MAE) is one of the most important intrinsic properties\nof the hard magnetic crystals which can be analyzed in\nreciprocal space as a k-resolved quantity. The MAE can\nbe determined with the magnetic force theorem29,65,66\nfrom a formula:\nMAE = E(θ= 90◦)−E(θ= 0◦) =\n=/summationdisplay\nocc′ǫi(θ= 90◦)−/summationdisplay\nocc′′ǫi(θ= 0◦) +O(δρn),(10)\nwhere ǫiis the band energy of the ith state and θis the\nangle between the magnetization direction and the caxis.\nThe modern computer technique allows to visualize the\nMAE( k) structure in a three-dimensional (3D) Brillouin\nzone. The 3D maps of magnetocrystalline anisotropy\nhave been recently calculated for (Fe/Co) 2B alloys16–\nanother candidates for the rare-earth free permanent\nmagnets. However usually the MAE( k) dependences\nare presented along one-dimensional k-path.14,29,45Al-\nthough various authors attempt to interpret the MAE( k)\nwithout a 3D-resolution, in our opinion the k-path or sin-\nglek-point analyses do not cover properly the complex-\nity of magnetocrystalline anisotropy. We think that in\norder to get a full picture of it the entire Brillouin zone\nshould be considered.14For a hard magnetic material\nthe calculated 3D-MAE( k) plot is a unique characteristic\nas for example the Fermi surface is for a metal. Inter-\nesting is that the connections between the 3D-MAE( k)\nand the Fermi surface go beyond the above comparison\nand a fact is a close relation between these two. Both,\nthe 3D-MAE( k) and Fermi surface, are reciprocal space\nelectronic structure characteristic. Furthermore, as the\nFermi level is an upper integration boundary of the MAE,\nthe sheets of the Fermi surface indicate the borders be-\ntween the distinct regions of the MAE. Because of that\nthe Fermi surface is a link between the calculated 3D-\nMAE( k) and experiment. The 3D-MAE( k) analysis is\nrelevant for hard magnetic materials also because it can\ndetermine directions to improve the MAE. In relation to7\nthe Fermi surface engineering an improvement technique\nbased on the MAE( k) analysis could be called magne-\ntocrystalline anisotropy engineering.\nFIG. 7. (c), (d): The Fermi surface of the L1 0FeNi view\nfrom two perspectives, together with corresponding (a), (b ):\nk-resolved contributions to MAE. The MAE( k) results are\nobtained by the magnetic force theorem within the FPLO14\nPBE+so method. Only a reducible one-eighth part of the full\nBrillouin zone is presented. The magnitude of negative and\npositive values is indicated by the intensity of blue and red\ncolor, respectively.\nIn Fig. 7we present the calculated 3D-MAE( k) plots\nof the L1 0FeNi. We can see that the whole Brillouin\nzone is filled by the positive (red) and negative (blue)\ncontributions. The magnitude of the k-resolved contribu-\ntions is in order of 10−3eV per k-point, where the value\nof the total MAE is three orders of magnitude smaller\n(48µeV formula−1). A comparison between the MAE\ncontributions and total value indicates a fine compensa-\ntion of the relatively large negative and positive compo-\nnents. From Fig. 7it is easy to notice that the over-\nall shape of the MAE( k) is dictated by the Fermi sur-\nface. The Fermi surface sheets divide the Brillouin zone\ninto several mainly positive or negative regions. The two\nmost prominent positive contributions come from (1) the\nvicinity of the R-Z line along the [100] quantization axis\nand (2) from the spherical region around the M point.\nActually, around the M point we observe a sequence of\nnegative, positive, and again negative regions filling the\nvolumes between the corresponding nested hole pockets\nof the Fermi surface. A similar alternating ordering of\nthek-resolved contributions to anisotropy constant K\nhas been presented for the (Fe/Co) 2B alloys.16\nIn Fig. 8the cross-sections of the MAE( k) for planes\nΓ-X-M and R-Z-A are shown together with correspond-\nM\nX\u0001(a) (b)\n(c) (d)\nR ZA\n-0.004   K-point resolved MAE (eV)   0.004X\u0000\nR Z\nFIG. 8. (a), (c): The Γ-X-M and R-Z-A cross-sections of\nthe Fermi surface of the L1 0FeNi, together with the cor-\nresponding (b), (d): cross-sections of MAE contributions o f\neach k-point. The results are obtained by the magnetic force\ntheorem within the FPLO14 PBE+so method. The different\ncolors of Fermi surface contours are used as the guide for the\neye only.\ning cross-sections of the Fermi surface. At these plots\nit is even easier to notice how the Fermi surface splits\nthe regions with differing MAE( k) components. The\nMAE cross-sections also confirm that the first hole pocket\naround the M point contains mainly negative k–resolved\ncontributions and the regions between the next two hole\npockets are respectively positive and negative. Although\nthe MAE( k) structure reflects the uniaxial anisotropy of\nthe crystal, it is easy to notice how the four-fold symme-\ntry is broken. A reason for this is that the [100] direction\nhas been distinguished as a quantization axis.\nThe above-described MAE( k) analysis reveals details\nof the structure of magnetocrystalline anisotropy of the\nL10FeNi which are the distribution of positive and neg-\native components, the magnitude of these shares, or the\ndetailed relationship between the MAE( k) and the Fermi\nsurface. Unfortunately, an extra fine compensation of the\nlarge MAE( k) contributions makes it difficult to predict\nthe roads to increase the MAE of the L1 0FeNi. Be-\ncause of the compensation effect, even large changes of\nthe MAE( k) structure will finish as hard to predict small\nchanges to the MAE. Based on the combined 3D-MAE\nand Fermi surface analysis we are able however to point8\nout the sheets of Fermi surface which should be extended\nor reduced to enlarge the regions of positive MAE con-\ntributions. Unfortunately, this simple line of reasoning\nis difficult to realize in practice. Taking into account the\ninnovative nature of the combined 3D-MAE and Fermi\nsurface analysis, we do not exclude that this approach\nwill be more fruitful for another hard magnetic material.\nIV. SUMMARY AND CONCLUSIONS\nThe magnetocrystalline anisotropy energy MAE of the\nL10FeNi calculated in this work within GGA goes below\n0.5 MJ m−3. It has been shown in literature that the\nmore reliable model including orbital polarization cor-\nrections gives about twice this value, whereas the experi-\nmental values of the anisotropy constant K1from litera-\nture oscillate around 1.0 MJ m−3. Regarding application\nof the L1 0FeNi as rare-earth free permanent magnets,\nthe above numbers of MAE/ K1are rather discouraging.\nOther known limitations of tetrataenite are practical dif-\nficulties with synthesis of the ordered phase and relatively\nlow critical temperature of the ordered state. However\nthe L1 0FeNi has still a potential to improve its MAE by\nmodifications, like e.g. a tetragonal strain or an intersti-\ntial or substitutional alloying. In favor of the L1 0FeNi\nspeak also the high saturation magnetization and Curie\ntemperature. The L1 0FeNi with well defined anisotropic\nparameters may also find applications in modern elec-\ntronics. From point of view of computations it can be\nalso used as a reference standard for advanced ab initio\ncalculations due to relatively simple crystal structure.\nIn the reciprocal space analysis of the hard magnetic\nmaterials it is common to present the MAE( k) contribu-\ntions together with the bandstructure along the k-path.\nIn this work we have shown the three-dimensional k-\nresolved analysis of the MAE (3D-MAE) which gives a\ncomplete picture of the MAE contributions in the Bril-\nlouin zone. Unfortunately, in case of the L1 0FeNi we\ncannot use the k-resolved analysis to give specific sug-\ngestions. In return we show the close interconnection\nbetween the 3D-MAE and the Fermi surface. We expectthat analysis of the effect of strain or alloying on the\nk-resolved MAE may allow to improve the properties of\nhard magnetic materials.\nThe calculated spin and orbital magnetic moments and\nmagnetostrictive coefficient λ001stay in an acceptable\nagreement with the previous theoretical results and mea-\nsurements. The calculated bulk modulus B0, magne-\ntocrystalline anisotropy constants K2andK3, and Fermi\nsurface require experimental confirmation.\nThe conducted full relativistic calculations can be also\nused to formulate several general conclusions regarding\nthe computational parameters. The fact that that spin-\norbit coupling is the origin of the orbital magnetic mo-\nment, magnetocrystalline anisotropy, and magnetostric-\ntion is well known. In order to accurately calculate the\nmagnetic parameters it is also crucial to consider the full\npotential. The significant discrepancies between the cal-\nculated MAE’s even from various full potential GGA im-\nplementations show how sensitive the calculations are.\nTo improve the reliability of the results we suggest to\nchoose the computational parameters very carefully and\nuse a second code for cross-checking. Though it is well\nknown that the orbital magnetic moment on Fe is un-\nderestimated in GGA, it is not obvious that it leads to\nunderestimation of the MAE. For better description of\nthe magnetic moments in considered system the orbital\npolarization corrections and dynamical mean field theory\ncan be used. In this work we have also shown that it is\ntechnically possible to calculate the anisotropy constant\nK3with full potential. The calculated value of K3is four\norders of magnitude smaller than the MAE and a huge\nnumber of k-points is indispensable to get a consistent\nvalue.\nV. ACKNOWLEDGEMENTS\nWe thank A. Edstr¨ om and B. Wasilewski for discus-\nsions. 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Inosov1,\u0003\n1Institut für Festkörper- und Materialphysik, Technische Universität Dresden, D-01069 Dresden, Germany\n2Institut für Theoretische Physik, Universität Würzburg, 97074 Würzburg, Germany\n3Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany\n4Department of Physics, Princeton University, Princeton, New Jersey 08544, USA\n5School of Physics, University of Melbourne, Parkville, VIC 3010, Australia\n6Institute for Theoretical Solid State Physics, IFW Dresden, Helmholtzstraße 20, D-01069 Dresden, Germany\n7Experimental Physics V , Center for Electronic Correlations and Magnetism,\nInstitute of Physics, University of Augsburg, D-86135 Augsburg, Germany\n8Institute of Applied Physics, Academy of Sciences of Moldova, Chisinau MD-2028, Republic of Moldova\n9Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA\n10Institut Laue-Langevin, 71 avenue des Martyrs, CS 20156, F-38042 Grenoble cedex 9, France\n11Jülich Center for Neutron Science (JCNS), Forschungszentrum Jülich GmbH,\nOutstation at Heinz Maier – Leibnitz Zentrum (MLZ), Lichtenbergaße 1, D-85747 Garching, Germany\nLow-energy spin excitations in any long-range ordered magnetic system in the absence of magnetocrystalline\nanisotropy are gapless Goldstone modes emanating from the ordering wave vectors. In helimagnets, these\nmodes hybridize into the so-called helimagnon excitations. Here we employ neutron spectroscopy sup-\nported by theoretical calculations to investigate the magnetic excitation spectrum of the isotropic Heisenberg\nhelimagnet ZnCr2Se4with a cubic spinel structure, in which spin-3 /2 magnetic Cr3+ions are arranged in a geo-\nmetrically frustrated pyrochlore sublattice. Apart from the conventional Goldstone mode emanating from the\n(0 0qh)ordering vector, low-energy magnetic excitations in the single-domain proper-screw spiral phase show\nsoft helimagnon modes with a small energy gap of \u00180.17 meV , emerging from two orthogonal wave vectors\n(qh0 0)and(0qh0)where no magnetic Bragg peaks are present. We term them pseudo-Goldstone magnons,\nas they appear gapless within linear spin-wave theory and only acquire a finite gap due to higher-order\nquantum-fluctuation corrections. Our results are likely universal for a broad class of symmetric helimagnets,\nopening up a new way of studying weak magnon-magnon interactions with accessible spectroscopic methods.\nPACS numbers: 75.30.Ds 75.10.Hk 78.70.Nx\nI. INTRODUCTION\nA. Classical Heisenberg model on the pyrochlore lattice\nThe Heisenberg model on the pyrochlore lattice attracts a\nlot of interest as various spin models on this lattice give rise\nto the simplest three-dimensional frustrated spin systems.\nEven for classical spins, this model hosts a wide range of dif-\nferent ground states. Considering only the antiferromagnetic\nnearest-neighbor (NN) interactions results in a classical spin\nliquid [1], exhibiting no long-range magnetic order down\nto zero temperature. This is explained by strong geometric\nfrustration that leads to a highly degenerate classical ground\nstate. However, inclusion of further-neighbor interactions re-\nlieves this frustration and stabilizes various ordered ground\nstates, among them ferromagnetism, single- or multi- qspin\nspirals, nematic order, and other exotic phases [2–6].\nChromium spinels provide great opportunities to inves-\ntigate magnetic interactions between classical spins on the\nstructurally ideal pyrochlore lattice. These compounds\nhave the general formula ACr2X4, where AandXare non-\nmagnetic ions and Cr3+is the magnetic cation in the 3d3\nconfiguration [7]. Its magnetic sublattice has the pyrochlore\nstructure with spins S=3=2at the Cr sites. Using the classi-\ncal Heisenberg model,\nH=X\ni jJi jSi\u0001Sj, (1)\n\u0003Corresponding author: dmytro.inosov@tu-dresden.deis justified by the negligibly small magneto-crystalline\nanisotropy [8–10]. Thus we consider throughout the pa-\nperJi j\u0011Jnif sites iandjarenthneighbors [see Fig. 1 (a) ].\nDepending on the chemical composition, chromium spinels\nexhibit different mechanisms of frustration, such as geomet-\nric frustration that occurs if dominant NN interactions are\nantiferromagnetic, or bond frustration which originates from\ncompetition between ferromagnetic NN and antiferromag-\nnetic further-neighbor exchange.\nTo estimate the range and relative strengths of coupling\nconstants Jnin chromium spinels, Yaresko [7]performed ab\ninitio calculations to extract exchange parameters up to the\nfourth nearest neighbor for various compounds of this fam-\nily. Calculations showed that the NN interaction J1changes\ngradually from antiferromagnetic in some oxides to ferro-\nmagnetic in sulfides and selenides, while the next-nearest-\nneighbor (NNN) interaction J2is noticeably weaker than the\nantiferromagnetic J3exchange parameter (see Table I). For\nthe HgCr2O4system, J1can be even weaker than J2(or com-\nparable, depending on the effective Coulomb repulsion U),\nso that the third-nearest-neighbor interaction J3may become\ndominant. Therefore, the existing theoretical phase diagram\nrestricted to only NN and NNN interactions [3]appears in-\nsufficient for a realistic description of these materials. The\nimportance of the two 3rd-nearest-neighbor exchange paths\non the pyrochlore lattice has been also emphasized for the\nspin-1\n2molybdate Heisenberg antiferromagnet Lu2Mo2O5N2\n[11], where J0\n3andJ00\n3have opposite signs and dominate\nover J2. It was recently conjectured that this may lead to an\nunusual “gearwheel” type of a quantum spin liquid [12].arXiv:1705.04642v3  [cond-mat.str-el]  5 Oct 2017B. Classical phase diagram relevant for chromium spinels\nAn inspection of the theoretically predicted exchange pa-\nrameters for various ACr2X4spinels presented in Ref. 7re-\nveals non-negligible exchange integrals up to the fourth-\nnearest neighbor, J4, which is ferromagnetic in all the studied\ncompounds. The third-nearest-neighbor exchange constants\nare, on the other hand, antiferromagnetic. Note that the\npyrochlore lattice exhibits two inequivalent sets of third-\nnearest neighbors. Following Ref. 7, we treat their corre-\nsponding exchange couplings as equal, J0\n3=J00\n3\u0011J3, since\ndeviations are considered to be small. The classical ground\nstate depends only on the ratios between the exchange con-\nstants, which can be much more accurately predicted in first-\nprinciples calculations than their absolute values. Here we\nrestrict our attention to the sulfides, selenides, and HgCr2O4\nwith ferromagnetic NN interactions, J1<0. In Table I, we\nsummarize the calculated ratios J2=jJ1j,J3=jJ1j, and J4=jJ1j\nfor all seven compounds considered in Ref. 7(to be specific,\nwe list the ratios for LSDA +Uwith U=2eV), along with\ntheir experimentally determined ground states.\nWe determine the classical phase diagram by means of\na direct energy minimization scheme [3]. We consider a\npyrochlore lattice (periodic boundary conditions imposed)\nwith typically L=8unit cells (containing 16 spins) in each\nspatial direction, resulting in N=16\u0001L3=213spins in\ntotal. Starting from a random initial spin configuration, one\nrandomly picks a lattice site iand rotates its spin Siantipar-\nallel to its local field defined as hi=@H=@Si=P\nj6=iJi jSj,\nthereby minimizing the energy and simultaneously keeping\nthe spins normalized. Once an energetically minimal spin\nconfiguration has been found, the object of interest is the\nspin structure factor S(q) =1\nN\f\fP\njSjexp(iq\u0001rj)\f\f2with its\nmagnetic Bragg peaks.\nThe pure J1–J2phase diagram already has been com-\nputed [3]. As mentioned before, we consider only ferro-\nmagnetic J1. Table I reveals that the couplings J4of all\nconsidered spinels are ferromagnetic and roughly equal,\nJ4=jJ1j\u0019\u0000 0.014(which also corresponds to the value de-\nFig. 1. (a) Pyrochlore-lattice structure, representative of the mag-\nnetic Cr sublattice in ACr2X4spinels. Small spheres represent Cr3+\nions. The pyrochlore lattice can be described either as an fcc ar-\nrangement of separated Cr4tetrahedra formed by NN bonds (large\ncubic unit cell, solid lines) or as a half-filled fcc lattice of Cr3+ions\nwith a twice smaller lattice parameter (small cubic unit cell, dashed\nlines). We also show different exchange paths corresponding to\nJ1...J4interactions that are discussed in the text. (b) The simple-\ncubic Brillouin zone with dimensions 2\u0019=a(central cube), and\nthe two unfolded Brillouin zones corresponding to fcc lattices with\nparameters aanda=2(truncated octahedra). The high-symmetry\npoints are marked according to the large unfolded zone.\nFig. 2. Classical J2–J3phase diagram for J1=\u00001andJ4=jJ1j=\n\u00000.014containing five phases: ferromagnetic (FM), (00q)proper-\nscrew spiral with the corresonding q-value shown in color code, the\ncuboctahedral stack (CS), the multiply modulated commensurate\nspiral (MMCS), and the\u00001\n81\n27\n8\u0001\nphase. In addidtion, the location\nof the seven spinel compounds with ferromagnetic J1is indicated\nby the symbols. For details see the text and Table I.\ntermined by fitting the experimental data to spin wave ex-\ncitation spectra, see Sec. II.B). Thus we restrict our inves-\ntigation of the phase diagram to J4=jJ1j=\u00000.014, but we\nchecked for all spinels that the ground state remains un-\nchanged when taking into account the couplings listed in\nTable I. Therefore we map out the classical phase diagram\nfor\u00000.05\u0014J2=jJ1j\u00140.2and0\u0014J3=jJ1j\u00140.5as pre-\nsented in Fig. 2. We find five different phases: ferromag-\nnetic (FM), (00q)proper-screw spiral, cuboctahedral stack\n(CS), multiply modulated commensurate spiral (MMCS),\nand the\u00001\n81\n27\n8\u0001\nphase. Quite generally, we have observed for\nJ2=jJ1j>0.1andJ3=jJ1j>0.1that the energy landscape\nbecomes extremely flat. In order to yield reliable results we\nhad to increase system sizes drastically, for some parame-\nter settings we even used L=32corresponding to more\nthan half a million spins. For dominant J1<0we find FM\norder, but both antiferromagnetic J2andJ3destabilize the\nphase. For sufficiently strong J3>0(and not too large J2)\nthere is a phase transition from the FM to the (00q)heli-\nCompound J2=jJ1jJ3=jJ1jJ4=jJ1jground state Refs.\nHgCr2O40.1714 0.471 0 AFM [13]\nZnCr2S4 0.0395 0.198 –0.014 (00q)spiral [14]\nCdCr2S4 0.0065 0.116 –0.015 FM [15]\nHgCr2S4 0.0222 0.111 –0.013 (00q)spiral [16, 17 ]\nZnCr2Se40.0102 0.169 –0.018 (00q)spiral [18, 19 ]\nCdCr2Se4–0.0071 0.101 –0.013 FM [15]\nHgCr2Se4–0.0014 0.109 –0.013 FM [20]\nExperimental values:\nZnCr2Se40.0118 0.170 –0.014 [this work ]\nTable I. Ratios of the exchange constants in different chromium\nspinels after Ref. 7, taken for a moderate effective Coulomb repul-\nsion U=2eV . For each compound, its experimentally measured\nground state is also given: antiferromagnetic (AFM), ferromag-\nnetic (FM), or proper-screw spin spiral. The bottom row shows\nexperimental values for ZnCr2Se4estimated from our INS data in\nSection II.B of this work.\n– 2 –cal phase. The wave vector qchanges continuously from\nthe phase transition line ( q=0) up to a maximal value of\nq=7=8for large J3(color-coded in increments of q=1=8\nin Fig. 2). For J3=0andJ2=jJ1j\u00150.2we find the CS phase\nof Ref. [3]characterized by Bragg peaks at three of the four\b1\n21\n21\n2\t\n-type qvectors. Finite J3stabilizes the phase down to\nsmaller values of J2. Further increase of J3drives the system\ninto the multi- qMMCS phase which is also present in a very\nsmall parameter range of the J1–J2phase diagram [3]. This\nstate is characterized by four main Bragg peaks at\u00001\n43\n41\n2\u0001\n,\u00003\n41\n41\n2\u0001\n,\u00001\n43\n43\n2\u0001\nand\u00003\n41\n43\n2\u0001\n, and a subdominant peak at\u00003\n43\n40\u0001\n(or symmetry related combinations of these vectors, respec-\ntively). Making J3even larger eventually turns the system’s\nground state into an exotic phase with four dominant Bragg\npeaks at q=\u0000\n\u00061\n81\n2\u00067\n8\u0001\nand\u0000\n\u00067\n81\n2\u00061\n8\u0001\n. To our best\nknowledge, this phase has not been mentioned before in the\nliterature, and we simply call it the\u00001\n81\n27\n8\u0001\nphase.\nThe phase diagram in Fig. 2 contains all spinels listed\nin Table I, and it is interesting to compare the results of\nour calculations with available experimental data. CdCr2S4,\nHgCr2Se4, and CdCr2Se4have been reported to possess fer-\nromagnetic ground states [15,20]in agreement with our\nsimulations. HgCr2S4develops a (00q)spiral configuration\nwith a temperature-dependent pitch [16], but the experi-\nmental results emphasize a strong tendency to ferromagnetic\ncorrelations [17]; in our phase diagram it is located in the\nFM phase close to the spiral phase. The material of highest\ninterest is, of course, ZnCr2Se4where both experiment and\nsimulation consistently find a (00q)helix with q=0.468\n(or 0.28 Å\u00001). ZnCr2S4has been reported to host a sim-\nilar proper-screw spin structure in the temperature range\n8K<T<15K with q=0.787[14], which agrees fairly well\nwith our simulated value q=0.625based on the coupling\nconstants from Table I. This spinel undergoes a structural\nphase transition at 8 K with a change of the magnetic order.\nThe only oxide in our list is HgCr2O4, where Bragg peaks\ncorresponding to q1= (100)andq2= (101\n2)were mea-\nsured [13]. The compound undergoes, however, a structural\nphase transition into an orthorhombic phase [21]for which\nour pyrochlore description is of course inadequate. Other-\nwise, according to its exchange couplings, HgCr2O4would\nbe located in the\u00001\n81\n27\n8\u0001\nphase.\nWe emphasize that our phase diagram, based on the ab\ninitio parameters of Ref. [7], agrees fairly well with the avail-\nable experimental results. The major discrepancies are as-\nsociated with structural transitions of the spinels ZnCr2S4\nand HgCr2O4, leading to distortions or even different crys-\ntal structure. Since both effects cannot be captured by our\nsimulations, the discrepancies between experiment and the-\noretical modeling in these particular cases are plausibly un-\nderstood.\nC. Crystal structure and spin-spiral order in ZnCr2Se4\nAmong the materials listed in Table I, characterized by\nferromagnetic NN interactions, only ZnCr2Se4and HgCr2S4\nhost an incommensurate spin-spiral ground state resulting\nfrom the bond frustration imposed by competition with\nfurther-neighbor interactions. The proper-screw spin struc-\nture of ZnCr2Se4has a short helical pitch, \u0015h, of only\n22.4 Å [18,22], as compared to that of 42 Å in HgCr2S4\n(which in addition increases with temperature to \u001890 Å at\nT=30K)[16]. This results in magnetic Bragg peaks inZnCr2Se4that are sufficiently remote from the commensu-\nrate structural reflection, with a propagation vector (0 0qh),\nwhere qh\u00190.28Å\u00001[18,22]. Therefore, low-energy mag-\nnetic excitations emerging from different Bragg peaks are\neasy to resolve, which makes ZnCr2Se4a perfect model\nmaterial for investigations of low-energy spin dynamics in\nsymmetric (Yoshimori-type) [23,24]helimagnets that are\ncommon among multiferroics [25,26]. Moreover, from the\ncomparison of the Néel temperature, TN=21K, with the\nCurie-Weiss temperature \u0002CW=90K, the frustration ratio\nof\u0002CW=TN=4.3can be estimated, suggesting a consid-\nerable degree of frustration in this system [27]. The small\nsingle-ion anisotropy energy in ZnCr2Se4is evidenced by the\nmagnetic resonance data [9], by the gapless high-resolution\nneutron powder spectra down to at least 0.05 meV in energy\n[28], as well as by the absence of any anisotropy-induced\ndeformation of the spin spiral. According to a recent study\n[29], such a deformation would lead to the appearance of\nodd-integer higher-order magnetic Bragg peaks in neutron\ndiffraction that are absent in the neutron-diffraction patterns\nof ZnCr2Se4[18, 22 ].\nZnCr2Se4crystallizes in the normal spinel (Fd¯3m)struc-\nture at room temperature having 8 formula units per simple-\ncubic unit cell with a lattice constant a=10.497Å[18,30].\nAt low temperatures, the lattice undergoes a tiny distor-\ntion to an orthorhombic structure with c=a=0.9999 that\ncould not be resolved in neutron diffraction measurements\n[14], but was suggested from earlier x-ray data [30–32].\nFor the purposes of our study, we neglect this distortion\nand describe the lattice as cubic at all temperatures. Cr3+\nions form a pyrochlore magnetic sublattice which consists\nof corner-sharing tetrahedra [Fig. 1 (a) ]. The sublattice can\nbe described as a face-centered-cubic (fcc) arrangement of\nequally oriented Cr4tetrahedra (shown in dark color) or,\nalternatively, as a half-filled fcc lattice of individual Cr3+\nions with a twice smaller unit cell, as shown in Fig. 1 with\ndashed lines. This allows us to introduce a larger unfolded\nBrillouin zone in reciprocal space, as shown in Fig. 1 (b), by\nanalogy with the procedure described for the structurally\nrelated compound Cu2OSeO3by Portnichenko et al. [33].\nBelow the ordering temperature, Cr3+spins form an in-\ncommensurate helical structure with a propagation vector\npointing along the [001]direction. In the absence of an\nexternal magnetic field, there are three possible domains\nwith mutually perpendicular directions of the spin spirals,\ncorresponding to the cubic crystal symmetry. Application\nof an external magnetic field leads to a domain selection\ntransition, which stabilizes domains with the smallest angle\nbetween the propagation vector of the spiral and the mag-\nnetic field direction [19,34]. This offers the possibility to\nprepare a single-domain state either by cooling the sample\nin magnetic field or by applying and removing the field at\nbase temperature.\nIn this paper we present inelastic neutron scattering (INS)\nmeasurements of magnetic excitations in ZnCr2Se4over a\nwide range of energies in the entire Brillouin zone. Compar-\ning our data with spin-dynamical calculations, we have ex-\ntracted exchange parameters up to the fourth nearest neigh-\nbor and found a good agreement with the isotropic Heisen-\nberg model. Furthermore, measurements of low-energy\nhelimagnon excitations performed in the single-domain spin-\nspiral state revealed two distinct modes: the Goldstone mode\nemanating from incommensurate magnetic Bragg peaks and\n– 3 –Energy (meV)\nEi = 20 meV\n0 1 2 3 −3−2−1 0 1 2 1 2 0 1 2 1 2 0 1 2 3 −3−2−10246810(a)\n010-3\nL in (11L ) (r. l. u.) L in (22L ) (r. l. u.) H in (HH1) (r. l. u.) H in (HH2) (r. l. u.)\n0 1 2 3 −3−2−1 0 1 2 3 −3−2−1 0 1 2 1 2 0 1 2 1 205101520253035Energy (meV)\nL in (11L ) (r. l. u.) L in (22L ) (r. l. u.) H in (HH1) (r. l. u.) H in (HH2) (r. l. u.)Ei = 70 meV(b)\n05×10-4INS intensity\n(arb. units)INS intensity(arb. units)Fig. 3. Energy-momentum cuts through the TOF data measured at the ARCS spectrometer (setup 1), plotted along high-symmetry\ndirections after symmetrization respecting the cubic crystal symmetry. All the presented data were measured at the base temperature\nof 5 K with incident neutron energies of 20 meV (top row) and 70 meV (bottom row). The momentum integration range in directions\northogonal to the image was set to \u00060.14 r.l.u.\na soft pseudo-Goldstone mode with a small spin gap of\n\u00180.17 meV that emerges from the structurally equivalent\northogonal wave vectors (qh0 0)and(0qh0).\nII. EXPERIMENTAL RESULTS\nA. Instrumental conditions for the experiments\nWe have used thermal- and cold-neutron time-of-flight\n(TOF) and triple-axis spectroscopy (TAS) techniques to\nmap out dispersions of magnetic excitations in ZnCr2Se4\nin a broad range of energies. The advantage of the TOF\nmethod is the possibility to obtain data covering the whole\n4-dimensional (4D) energy-momentum space (Q,}h!)in a\nsingle measurement. Further, the combination of data col-\nlected with high-energy ( Ei=20and 70 meV) neutrons\nat the ARCS spectrometer [35]at ORNL, Oak Ridge, and\nlow-energy ( Ei=3.27meV or\u0015i=5Å) neutrons at the IN5\nspectrometer [36]at ILL, Grenoble, provides an overview of\nthe whole magnon spectrum together with high-resolution\nimaging of low-energy excitations.\nExperiments were performed on a mosaic of co-aligned sin-\ngle crystals of ZnCr2Se4with a total mass\u00181 g, prepared by\nchemical transport reactions and characterized as described\nelsewhere [34]. For the ARCS experiment (setup 1), the\nsample was mounted in such a way that the (HH L )plane\nwas horizontal. The experiment was performed without\nmagnetic field in the multi-domain state at the base temper-\nature of T=5K. We performed rocking scans for incoming\nneutron energies of 20 and 70 meV , which correspond to the\nenergy resolution of 0.87 and 3.4 meV , respectively, defined\nas the full width at half maximum of the incoherent elasticline. We then processed the data using the HORACE software\npackage [37,38]. The 4D datasets were symmetrized during\ndata reduction by combining statistics from all symmetry-\nequivalent directions in the same Brillouin zone.\nTo confirm the positions of high-energy excitations at\ncertain high-symmetry points of the Brillouin zone, we sup-\nplemented our TOF data with thermal-neutron TAS measure-\nments carried out using the IN8 thermal-neutron spectrome-\nter at ILL (setup 2). Here the sample was mounted inside an\n“orange”-type cryostat in the (HK0)scattering plane, so that\ntheW(320)point could be reached. The spectrometer was\noperated with the vertically focused pyrolytic graphite (PG)\nmonochromator and analyzer, with the final neutron wave\nvector fixed to kf=4.1Å\u00001, and a PG filter installed between\nthe sample and the analyzer. The collimation before and\nafter the sample was set to 300and400, respectively. The\nresulting energy resolution, calculated for this spectrometer\nconfiguration at 25 meV energy transfer, was about 4 meV .\nThe cold-neutron TOF experiment at IN5, ILL (setup 3),\nwas carried out using the same sample placed in the “orange”-\ntype 2.5 T cryomagnet. This time the sample was rotated\nabout the (111)axis so that its (112)direction was point-\ning vertically (parallel to the direction of the magnetic\nfield). Correspondingly, the equatorial scattering plane was\nspanned by the mutually orthogonal (110)and(111)vectors.\nIn order to stabilize one helimagnetic domain, we cooled\ndown the sample in a vertical magnetic field of 1.5 T , with\nBk(112). In this geometry, the (00qh)ordering vector\nforms an angle of 35.3\u000ewith the field direction, leading to a\ntwice larger projection of the field on this axis as compared\nto the ordering vectors of the two other domains, (0qh0)\nand (qh00), that are inclined at 65.9\u000ewith respect to the\n– 4 –L in (HHL) (r. l. u.)\nH in (HHL ) (r. l. u.) H in (HHL ) (r. l. u.) H in (HHL ) (r. l. u.) H in (HHL ) (r. l. u.)\n010-3ħω = 1.7±0.5 meV ħω = 3.5±0.5 meV ħω = 6.0±0.5 meV ħω = 8.5±1.5 meV (a) (b) (c) (d)L in (HHL) (r. l. u\nH in (HHL ) (r. l. u.) H in (HHL ) (r. l. u.) H in (HHL ) (r. l. u.) H in (HHL ) (r. l. u.)\n04×10-4ħω = 11±2 meV ħω = 21±1 meV ħω = 27±2 meV ħω = 30±2 meV (e) (f) (g) (h)0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3               4\n0 1 2 3 0 1 2 3\n0 1 2 3 0 1 2 3              40124\n3\n0124\n3\nINS intensity\n (arb. units)INS intensity (arb. units).)Fig. 4. Constant-energy slices through the TOF data (setup 1) in the (HH L )plane, integrated within different energy windows as indicated\nabove each panel. The top and bottom rows of panels were measured at the base temperature of 5 K with incident neutron energies of 20\nand 70 meV . The out-of-plane momentum integration range was set to \u00060.1,\u00060.14, and\u00060.15 r.l.u. for panels (a), (b – d), and (e – h),\nrespectively. The white stripes in some panels originate in regions with no data coverage.\nmagnetic field. As a consequence, out of the three possible\nhelimagnetic domains, only the one with the propagation\nvector along the (001)direction is selected by the applied\nfield. When the base temperature of 1.5 K was reached, the\nmagnetic field was switched off, and the measurement was\nperformed in the single-domain state in zero field with the\nincident neutron wavelength of 5 Å and the elastic energy\nresolution of 0.084 meV .\nIn this scattering geometry, neither of the three propa-\ngation vectors lies within the horizontal scattering plane.\nHowever, they are sufficiently short to be covered by the out-\nof-plane acceptance range of the position-sensitive TOF de-\ntector bank despite the limitation from the vertical opening\nangle of the cryomagnet. Therefore, magnetic satellites of\nthe(000),(111), and (222)zone centers could be accessed\nin this configuration, which would be unfeasible on most\ninstruments restricted to the horizontal scattering plane.\nDuring data reduction, the complete dataset was then con-\nverted into energy-momentum space using the conventional\nbasis vectors of the simple-cubic crystallographic unit cell,\nso that the presentation of the data is not affected by the\nrotated sample mounting. We visualize the obtained re-\nsults by presenting two-dimensional (2D) cuts through the\n4D dataset that are taken along high-symmetry momentum\ndirections. To designate points in Qspace, we will use re-\nciprocal lattice units corresponding to the simple-cubic unit\ncell ( 1r.l.u. =2\u0019=a), whereas high-symmetry points will be\nmarked according to the unfolded Brillouin zone following\nthe standard notation for an fcc lattice [33].\nFinally, we performed detailed spin-gap measurements\nat the cold-neutron TAS PANDA [39]operated by JCNS at\nMLZ, Garching (setup 4). They were also carried out at\nzero field in the single-domain state, which was prepared\nby field-cooling the sample in a field of 2.5 T using the 5 T\nvertical-field cryomagnet. The same mosaic sample was re-\nmounted with its [001]axis pointing vertically, along themagnetic field direction, which resulted in the selection of\nthe domain with the (00qh)ordering vector, whereas the\naccessible range of Qspace was restricted to the (H K0)\nscattering plane spanned by the propagation vectors of the\ntwo suppressed domains, (qh00)and(0qh0). Their Bragg\nintensities were reduced by a factor of \u0018200 as a result\nof domain selection. To resolve these wave vectors at low\nscattering angles (in the vicinity of the direct beam), we oper-\nated the spectrometer with the horizontally flat but vertically\nfocused monochromator and analyzer. The final neutron\nenergy was fixed at Ef=3.0meV ( kf=1.2Å\u00001), providing\nan energy resolution of about 0.1 meV . A cold beryllium filter\nwas used to suppress higher-order contamination from the\nmonochromator.\nB. Experimental determination of the exchange constants\nWe will start the presentation of our results with the\nhigher-energy data measured at the ARCS spectrometer\nin the multi-domain state. Figure 3 shows typical energy-\nmomentum cuts along several high-symmetry lines in Q\nspace that are parallel to the (001) and (110) directions. Pan-\nels (a) and (b) show low- and high-energy datasets ( Ei=20\nand 70 meV), respectively. In the (22L)and (HH2)cuts,\nwe clearly see intense magnon branches emanating from\nincommensurate magnetic satellites of the (222)Bragg peak,\nwhile similar modes near the (111)and (113)points are\nnoticeably weaker. This behavior of the dynamic structure\nfactors can be explained using the approach of unfolded\nBrillouin zones, which was detailed in Ref. 33for another\ncompound with a structurally similar magnetic sublattice.\nOne characteristic energy scale revealed in Fig. 3 (a) is given\nby the crossing point between equivalent magnon branches\nin the (11L)direction, located at \u00186.8 meV .\nIn Fig. 3 (b), we additionally observe high-energy modes\nnear the top of the magnon band, between 25 and 35 meV .\n– 5 –20253035\n15\n10\n5\n0\nMomentumEnergy (meV)\nX'(220) (211)L(111)Γ'(222)W(120)L(111)Γ(000) Γ(000)X(020)W(120)\n3\n2_3\n2_(0) K05×10-4\nINS intensity\n(arb. units)Ei = 70 meV\nT = 5 K(b)\n810\n6\n4\n2\n0\nMomentumEnergy (meV)\n1\n2_1\n2_K( 2) X'(220) (211)L(111)Γ'(222)W(120)Γ(000) Γ(000)X(020)W(120)\n3\n2_3\n2_(0) K(a)\n05×10-3\nINS intensity\n(arb. units)Ei = 20 meV\nT = 5 K\n20253035\n15\n10\n5\n0\nMomentumEnergy (meV)\nX'(220) (211)L(111)Γ'(222)W(120)L(111)Γ(000) Γ(000)X(020)W(120)\n3\n2_3\n2_(0) K010\nINS intensity\n(arb. units)(c)1\n2_1\n2_K( 2)\n1\n2_1\n2_K( 2)L(111)Fig. 5. (a,b) Energy-momentum cuts thought the TOF data, taken\nwith Ei=20and 70 meV at the ARCS spectrometer (setup 1),\ncombined into a “spaghetti”-type plot that covers all high-symmetry\ndirections of the unfolded Brillouin zone. (c) Results of our spin-\ndynamical calculations performed using the SPINWsoftware for\nthe exchange constants given in Eq. 4, plotted along the same\npolygonal path in momentum space as the data in panels (a) and\n(b). The calculations were averaged over three possible domain\norientations to mimic the multi-domain spin-spiral state realized\nin a zero-field experiment.\nWhile there is no energy gap separating the high- and low-\nenergy modes, unlike in Cu2OSeO3[33], the intensity is\ncertainly enhanced near the top of the magnon band, as\nbest seen along the (HH2)direction (rightmost panel). This\nis reminiscent of the situation in another recently studied\nhelimagnet with a similar pitch of the spin spiral, Sr3Fe2O7,\nwhere an intense and well separated high-energy magnon\nbranch was observed at a similar energy of 25 meV [40].\nHowever, the low-energy behavior of the spin waves in\nSr3Fe2O7is considerably different. There, apart from the\nsteeply dispersing outward branches, we observed\n -shaped\ninner branches of helimagnetic spin waves connecting the\nincommensurate ordering vectors, which extended up to\n\u00184 meV in energy. This form of spin-wave dispersion, bridg-\ning the incommensurate magnetic satellites, is a common\nfeature of helimagnets with symmetric exchange interac-\ntions [24]. In the case of ZnCr2Se4, however, this branch\noccurs at 10 times lower energies, so it cannot be seen in\nFig. 3 (a) above the elastic line. We will return to the discus-\nsion of this low-energy magnon branch when presenting the\ncold-neutron data in Section II.C.\nTo visualize the magnon spectrum in Qspace, we are pre-\nsenting constant-energy cuts in Fig. 4. All of them show in-\ntensity distributions within the (HH L )plane at different en-ergies, covering wave vectors up to (444). Here one can ob-\nserve the complex hierarchy of energy scales in the magnon\ndispersion. The low-energy cut at 1.7 meV in Fig. 4 (a) dis-\nplays clearly separated rings of scattering intensity around\nthe zone centers. Stronger modes are found near \u0000points of\nthe unfolded Brillouin zones, such as \u0000(000),\u0000(222), and\n\u0000(004), whereas weaker modes appear as replicas shifted\nby the (111)wave vector to the centers of the folded fcc Bril-\nlouin zone [33]. Around 2.5 meV , these two modes start to\nhybridize, leading to a saddle point in the spin-wave disper-\nsion that can be most clearly seen along the (HHH )direction\nat the right-hand side of Fig. 5 (a). Above the saddle-point\nenergy, the rings of scattering break into separated segments\nas shown in Figs. 4 (b – e). Above 6.8 meV , which corre-\nsponds to the crossing of the bands in the (11L)direction\nin Fig. 3 (a), the weaker rings of scattering surrounding\nthe(111)and(113)points cross each other, resulting in a\nstreak of intensity centered at (112)in Figs. 4 (c,d). Going\neven higher in energy, we observe an elliptical feature sur-\nrounding the X(002)point in Fig. 4 (f) that finally shrinks\nin Figs. 4 (g,h) into a set of broad peaks corresponding to\nthe top of the magnon band at the unfolded-zone boundary\nat an energy of\u001830 meV . What is not seen in this figure is\nthe bottom of the high-energy magnon branch that results\nin an intense peak centered at 24 meV at the W(120)point,\nwhich lies outside of the (HH L )plane, but can be well seen\nin Fig. 5 (b). A similar feature was also seen in the spec-\ntrum of Cu2OSeO3at the same Qpoint and nearly the same\nenergy [33].\nFor a more comprehensive representation of magnetic\nexcitations in ZnCr2Se4, we put together energy-momentum\ncuts along a polygonal path which contains all high symme-\ntry directions in Qspace, creating a “spaghetti”-type plot\nshown in Figs. 5 (a,b) for both incident energies. Kinematic\nconstraints lead to the lack of data at high energy transfer\nin the first Brillouin zone, however these data have better\nsignal-to-noise ratio at low energies. Therefore, in Fig. 5 (b)\nwe have overlayed data from the first Brillouin zone on top\nof those obtained from equivalent points at higher jQjto\ncomplete the missing parts of the dataset. The stitching lines\n0 5 10 15 20 25 30 35\nEnergy transfer (meV)150200250300350Intensity (arb. units)W(320)\nfit\npeak 1\npeak 2\nbackground\nFig. 6. A typical energy scan measured at the W(320)point with\nthe IN8 spectrometer (setup 2) in the multi-domain state, revealing\ntwo marginally resolved peaks centered at 23.7meV (dash-dotted\nline) and 30.8meV (dotted line), in perfect agreement with our\nspin-dynamical calculations.\n– 6 –Fig. 7. Constant-energy cuts through the low-energy TOF data measured at IN5 (setup 3) in the single-domain state. (a) Elastic intensity\nmap in the (H1L)plane, demonstrating the selection of a single magnetic domain with the propagation vector parallel to the caxis.\n(b – d) Constant-energy cuts at 0.2, 0.3, and 0.4 meV , which show magnetic Goldstone modes emanating from the (0 0\u0006qh)ordering\nvectors and pseudo-Goldstone modes at the orthogonal (\u0006qh00)vectors, which are seen to merge at 0.4 meV . Note different intensity\nscales in all images.\nMomentum (r. l. u.)Energy (meV)\n00.20.40.60.81\n(110) (111) (112)\nEnergy (meV)\n00.20.40.60.81\n(011) (111) (211)\nMomentum (r. l. u.)\nEnergy (meV)\n00.20.40.60.81\n(111)\nMomentum (r. l. u.)\n00.055INS intensity\n(arb. units)(a) (b) (c)\n( )11\n2_1\n2_( )11\n2_3\n2_\nFig. 8. Energy-momentum cuts through the IN5 data (setup 3), centered at the (111)wave vector and taken along different directions:\n(a) along the (11L)line passing through the (111\u0006qh)ordering vectors of the field-selected domain; (b) average of the equivalent\n(1\u0006h1 1)and(1 1\u0006h1)directions passing through the suppressed (1\u0006qh1 1)and(1 1\u0006qh1)ordering vectors; (c) along two orthogonal\ndiagonal directions: (1\u0000h1\u0000h1)(left) and (1\u0006h1\u0007h1)(right), which pass through the 0.35 meV inflection point in the dispersion in\nthe plane orthogonal to the ordering vector. In the right part of panel (c), datasets for positive and negative hhave been averaged.\nbetween high- and low- jQjdata can be recognized at the\nleft-hand side of the figure.\nTo extract the experimental exchange constants from our\ndata, we compared them with spin-dynamical calculations\nperformed in the framework of linear spin-wave theory\n(LSWT) using the SPINWsoftware package [41,42]. The\nmagnon spectrum was calculated using the classical Heisen-\nberg model with interactions up to the 4thshell of Cr neigh-\nbors[7]:\nH=1\n44X\ni=14X\nn=1znX\nj=1JnSi\u0001Sj, (2)\nwhere inumbers Cr sites in the unit cell, and jruns over zn\nneighbors in the nthcoordination shell around the site i. In\nour notation, positive Jnstands for AFM coupling between\nCrS=3=2spins. The model does not differentiate between\ntwo inequivalent exchange paths for third-nearest neighbors,\nJ0\n3andJ00\n3, hence these two exchange parameters were as-\nsumed equal. The pitch angle \rbetween the neighboring\nmagnetic moments of the spin helix can be found by solving\nthe equation that minimizes the energy for classical spins:\nJ1+J2+4J4+4(J2+2J3)cos(\r) +6J4cos(2\r) =0. (3)\nTo enable direct comparison with our ARCS data, we mod-\neled the multi-domain state by averaging the calculated spec-\ntra for three possible orientations of the magnetic domains.The calculations were first carried out with the theoretically\npredicted exchange parameters from Ref. 7, which were\nthen iteratively adjusted to achieve the best agreement with\nthe measured spectrum. The resulting spectrum is shown\nin Fig. 5 (c) for the following optimized values of exchange\nconstants:\nJ1=\u00002.876 meV , J2=0.034 meV ,\nJ0\n3=J00\n3=0.490 meV , J4=\u00000.041 meV .(4)\nThe absolute value of the experimental NN exchange con-\nstant J1is approximately 50% lower than the theoretical\nprediction. However, the signs of all interactions are well\ncaptured by the calculation. Their ratios, listed in Table I,\nshow fair agreement (within 20%) with the results obtained\nin Ref. 7 for an effective Coulomb repulsion U=2 eV .\nAs an additional verification of the exchange parameters\nand the overall correctness of our spin-wave model, in Fig. 6\nwe show a typical energy scan measured at the IN8 thermal-\nneutron spectrometer at the W(320)wave vector. It reveals\nan intense peak corresponding to the bottom of the high-\nenergy magnon branch (cf. Fig. 5). A closer analysis of\nthe peak shape reveals that it consists of a stronger exci-\ntation centered at (23.7\u00060.2)meV and a weaker one at\n(30.8\u00061.2)meV . The strong lower-energy peak originates\nfrom the sum of two magnetic domains whose propagation\n– 7 –Fig. 9. Schematic illustration of the low-energy magnon dispersion:\n(a) in the (HK0)plane orthogonal to the ordering vector and (b) in\nthe(H0L)plane passing through the ordering vector. Here \u0001PG\ndenotes the gap of the pseudo-Goldstone magnon branch, \u001eand\u0012\nare spherical angles. The filled and open dots represent Bragg peaks\nof the selected and suppressed magnetic domains, respectively.\nvectors are directed along the yandzcubic axes, whereas\nthe weaker upper peak presumably consists of two unre-\nsolved excitations originating from the x-oriented domains.\nBoth the positions and the relative intensities of the two\npeaks match fairly well with the spin-dynamical calculation\nresults in Fig. 5 (c).\nC. TOF measurements in the single-domain state\nWe proceed with presenting low-energy data measured at\nthe IN5 spectrometer in the single-domain state prepared\nby cooling the sample in magnetic field as described in Sec-\ntion II.A. To confirm that our domain-selection procedure\nwas successful, in Fig. 7 (a) we show the elastic-scattering\nintensity map in the (h1l)plane in the vicinity of the (111)\nstructural Bragg reflection. This plane contains both the\n(1 1 1\u0006qh)ordering vectors, where we observe intense mag-\nnetic Bragg peaks, and the (1\u0006qh1 1)wave vectors, where\nthe corresponding peaks are suppressed by two orders of\nmagnitude in intensity. We therefore conclude that all the\ninelastic spectra obtained from the same dataset originate\nfrom a single magnetic domain oriented along the caxis.\nThe inelastic data shown in Fig. 7 (b – d) represent equiv-\nalent cuts taken in the vicinity of Q=0, where the signal-\nto-background ratio is maximized. The constant-energy\nmaps exhibit two inequivalent types of low-energy spin-wave\nmodes seen as elliptical features emanating from the selected\nand suppressed ordering vectors. From different sizes of the\nellipses it is clear that the dispersion of the two modes is\nnot identical. Moreover, the intensity of the second mode\ncorresponding to the suppressed Bragg reflection appears to\nbe even higher compared to the true Goldstone mode. The\ntwo modes merge at 0.4 meV [Fig. 7 (d) ], resulting in an\ninflection point (flattening) of the dispersion seen as bright\nintensity spots along the diagonal directions of the image.\nThe dispersion of the same low-energy excitations can be\nseen in Fig. 8, where we compare energy-momentum cuts\npassing through the (1 1 1\u0006qh)ordering vectors with orthog-\nonal cuts through the suppressed magnetic Bragg positions\n[panels (a) and (b), respectively ]. In both directions, we\nobserve Goldstone-like magnon modes that appear gapless\nwithin the experimental energy resolution. Their disper-\nsion resembles the\n -shaped low-energy magnon branch in\nSr3Fe2O7[40], with the top of the inner branch reaching\nto only 0.5 meV for the Goldstone modes in Fig. 8 (a) and\n0.28 meV for the pseudo-Goldstone modes in Fig. 8 (b). A\nweak spin-wave band dispersing downward towards (111)can be seen in both directions. The nearly twofold difference\nin the band width of the inner\n -shaped branch connecting\nthe magnetic Bragg peak with the zone center confirms that\nthe two modes seen in panels (a) and (b) are distinct. The\nbehavior of the steeply dispersing outward branch is, on the\nother hand, much more isotropic. We also show in Fig. 8 (c)\ndiagonal cuts that cross the inflection points of the spin-wave\ndispersion in the plane perpendicular to the ordering vector.\nAs can be estimated from the image, these inflection points\nare located at 0.35 meV in energy.\nD. Spin gap of the pseudo-Goldstone magnons\nThe presented IN5 data indicate that a soft magnon mode\npersists practically in all momentum directions around the\nzone center atjQj\u0019qh, by analogy with the result obtained\nby Kataoka [43]for conical spin density waves with sym-\nmetric exchange interactions. This results in a corrugated\nMexican-hat-like dispersion schematically shown in Fig. 9.\nThe locus of dispersion minima along different directions\nforms an approximately spherical surface in Qspace of ra-\ndius qh. On this surface, the spin gap \u0001(\u0012,\u001e)has two\ndistinct minima: at the ordering vector itself ( \u0012=0), form-\ning the true (gapless) Goldstone mode emanating from the\nmagnetic Bragg peaks (filled dots), and pseudo-Goldstone\nmodes at two orthogonal wave vectors ( \u0012=\u0019=2;\u001e=0or\n\u0019=2) corresponding to the would-be Bragg peak positions\nof the suppressed magnetic domains (open dots). The latter\nare characterized by a small spin gap \u0001PGthat can be only\nmarginally resolved in our TOF data. Along all intermediate\n0.1 0.2 0.3 0.4 0.5\nEnergy transfer (meV)50100150200250300Intensity (arb. units)\n(a)φ=0◦\nφ=8◦\nφ=15◦\nφ=26◦\nφ=45◦\n45 55 65 75 85 5 15 25 35 450.150.200.250.300.350.40Energy transfer (meV)\n(b)fit\nIN5\nPanda\nθ(degrees) φ(degrees)\nFig. 10. (a) Several unprocessed energy scans from the triple-axis\nspectrometer PANDA (setup 4), measured at the minimum of the\nspin-wave dispersion in the plane orthogonal to the ordering vector\n(\u0012=\u0019=2) along different directions with \u001e=0\u000e, 8\u000e, 15\u000e, 26\u000e,\nand 45\u000e. (b) Angular dependence of the minimum energy in the\nspin-wave dispersion \u0001(\u0012,0)and\u0001(\u0019=2,\u001e)extracted from the\nIN5 and PANDA measurements. The solid line is an empirical fit\ndescribed in the text.\n– 8 –directions, the spin gap is larger, reaching a maximum along\nthe Brillouin-zone diagonals.\nTo measure the angular dependence of the spin gap di-\nrectly, we performed additional measurements at the PANDA\nspectrometer within the \u0012=\u0019=2plane along several ra-\ndial directions relative to the zone center for a number of\ndifferent\u001eangles. The measurement at each angle was\ndone in three steps. First, we performed an energy scan at\njQj=qh(assuming that the locus of the dispersion minima\nis spherical) to measure the approximate onset energy of\nmagnetic scattering. Then, we did a radial scan at an en-\nergy characterized by the maximal slope of the spectrum, in\norder to find the true location of the dispersion minimum\nthat could slightly deviate from qhfor various angles. Fi-\nnally, we repeated the energy scan at the new wave vector\njQjfor every angle. These final energy scans are shown\nin Fig. 10 (a). It can be seen that the signal at \u001e=0, at\nthe location of the pseudo-Goldstone mode, has a finite en-\nergy that is clearly resolved from the elastic line. The max-\nimum of inelastic scattering intensity is observed around\n0.15 meV . A more robust estimate of the gap energy results\nfrom fitting both the IN5 and PANDA data from different\nradial directions, which resulted in the angular dependence\nof the spin-gap energy shown in Fig. 10 (b). We can see\nthat the results of both TOF and TAS measurements are in\ngood agreement. The accuracy of our spin-gap measure-\nment benefits from the extrapolation of the data taken away\nfrom the ordering vector, where the peak is better resolved,\ntowards the low-energy region. We fitted the combined\ndataset in Fig. 10 (b) with the simple empirical function\n\u0001(\u0012,\u001e) =q\n\u00012\nPG+A2sin22\u001e+B2sin22\u0012, which resulted\nin the experimental spin-gap value \u0001PG=0.166(7)meV .\nOn the other hand, recent high-resolution powder data on\nZnCr2Se4demonstrate a gapless spin-wave spectrum down\nto at least 0.05 meV [28], which indicates that the anisotropy\ngap at the ordering vector (0 0qh)must be negligibly small,\nat least a factor of 3 smaller than \u0001PG.\nIII. DISCUSSION AND CONCLUSIONS\nA remarkable hallmark of the observed pseudo-Goldstone\nmodes is that LSWT predicts them to be gapless in the ab-\nsence of magneto-crystalline anisotropy [43]at wave vectors\nwhere no magnetic Bragg reflections are found below TN.\nThis is in contrast to magnetic soft modes observed, for ex-\nample, in\u000b-CaCr2O4[44], which originate from a proximity\nto another phase with a different magnetic ordering vector\nand therefore have a much larger gap that is well captured\nby LSWT . The existence of gapless modes without an under-\nlying Bragg reflection would violate the Goldstone theorem,\nyet this contradiction can be resolved by taking into account\nquantum fluctuation corrections to the linear spectrum.\nAs long as the low-energy magnon spectrum is concerned,\nspin systems can be analyzed in the frame of the non-linear\nsigma model [45]or mapped onto a one-sublattice Heisen-\nberg model. Therefore, to obtain an estimate of the pseudo-\nGoldstone spin gap, it is sufficient to consider a simplified\nmodel involving J1<0,J2>0, and J4>0interactions on a\nsimple-cubic lattice,\nH=J1X\nhi,jiSiSj+J2X\nhhi,jiiSiSj+J4X\nhhhhi,jiiiiSiSj, (5)\nwhere the summations are taken over the 1st, 2nd, and 4thnearest neighbors, respectively. This minimal model results\nin the correct pitch angle \r\u00190.23\u0019and similar spin-wave\nvelocities as those realized in ZnCr2Se4for the following\nvalues of parameters: J2=4meV , J4=1.8meV , and J1=\n\u00004(J2+J4cos\r).\nTo consider the magnon dispersion in the helical spin\nstructure it is worthwhile to introduce local spin quanti-\nzation axes x,y,z, such that zkhSiand ykQ. The\nHolstein-Primakoff transformation in the lowest 1=Sor-\nder followed by the Bogoliubov transformation yields the\nmagnon Hamiltonian Hm=P\nq!q\u000b†\nq\u000bqwith the disper-\nsion!q=SÆ\n(Jq\u0000JQ)(Jq+Q+Jq\u0000Q\u00002JQ)=2, where Jq=P\nRJRexp(iq\u0001R). This magnon dispersion acquires six zeros\nat the (0 0\u0006qh),(0\u0006qh0), and (\u0006qh0 0)points. Quantum\nfluctuations given by the next orders in the 1=Sexpansion\nof the Holstein-Primakoff transformation open a gap at the\n(0\u0006qh0)and(\u0006qh0 0)points orthogonal to the propagation\nvector [45–48]. The lowest order of terms which gives such\ncorrections is represented by three- and four-magnon pro-\ncesses shown in Fig. 11. Both !(4)\nqandRe\u000611(!,q)terms\ncontribute to the magnon dispersion. The imaginary part\nof the self-energy term Im\u000611(!,q)describes the magnon\ndamping\u0000q:\n˜!q+i\u0000q=!q+!(4)\nq+\u000611(!,q). (6)\nWe calculated the magnon dispersion ˜!qat the helix prop-\nagation vector Q= (0 0qh)and at the orthogonal wave vec-\ntorQ0= (qh0 0). Our calculation shows that in the first case\nthe quantum fluctuation corrections cancel out as expected\nin the 1=Sexpansion theory, whereas at the second vector Q0\na spin-wave gap opens up with a magnitude \u0001PG=0.18meV ,\nleading to a pseudo-Goldstone magnon observed in the ex-\nperiment. The good agreement between the calculated and\nmeasured values of the spin gap suggests that the described\nquantum-fluctuation corrections to the LSWT are fully suffi-\ncient to describe the experimentally observed magnon spec-\ntrum. It is worth noting that the experimentally measured\nspin gap is expected to have an additional contribution dic-\ntated by the anisotropy energy, which is nonzero in any real\nmaterial. However, the quantitative agreement of the calcu-\nlated and measured gap magnitudes indicates that this con-\ntribution in ZnCr2Se4is not dominant. Inclusion of a finite\nsingle-ion anisotropy term in our model would reduce the es-\ntimate of the spin gap from pure quantum-fluctuation effects\neven further, hence the experimental value of \u0001PGshould\nbe seen as an upper estimate for the pseudo-Goldstone spin\ngap in a purely Heisenberg system.\nIt is interesting to discuss the implications of our results for\ncollinear antiferromagnets. Stripe-like magnetic order with a\n(\u0019,0)ordering vector, like the one realized in iron pnictides\n[49,50], can be seen as a limiting case of a spin spiral with\na pitch length of two lattice constants. In a simple frustrated\nJ1–J2Heisenberg model on a 2D square lattice, a collinear\nFig. 11. The diagrams which give 1=Scorrections to the spectrum:\n(a)!(4)\nqand (b)\u000611(!,q).\n– 9 –ground state is selected by the order-by-disorder mechanism\n[51]. The linear spin-wave spectrum exhibits gapless modes\nnot only at the (\u0019,0)ordering vector, but also at the (0,\u0019)\nand(\u0019,\u0019)points. These accidental zero-energy modes de-\nvelop finite spin gaps due to quantum fluctuations and are\nin this sense analogous to the pseudo-Goldstone modes dis-\ncussed in our present work. Similarly, quantum-fluctuation\neffects were also claimed responsible for the reduction of the\nordered moment in iron pnictides and significant modifica-\ntions of their spin-wave spectrum [52]. However, there such\nchanges appear to be so dramatic that they lead to a com-\nplete destruction of the pseudo-Goldstone modes. Thus, a\nrecent experimental study performed on a uniaxially strained\nsingle crystal of BaFe2As2clearly showed the absence of any\nmeasurable magnon intensity at the (0,\u0019)wave vector up\nto at least 40 meV , where the magnetic Bragg reflection was\nfully suppressed by detwinning [53]. It therefore appears\nthat the noncollinear helimagnetic ground state of ZnCr2Se4\nand the large value of the spin ( S=3=2) that reduces quan-\ntum fluctuations are necessary prerequisites for observing\nclear pseudo-Goldstone modes with a small spin gap.\nIn summary, we have analyzed the phase diagram of the\npyrochlore magnetic lattice for various coupling constants.\nThe obtained theoretical phase diagram is in good agreement\nwith the ground states of several B-site magnetic spinels that\nare known from experiments. Using inelastic neutron scat-\ntering, we investigated the complete spin-wave spectrum of\nZnCr2Se4in the whole of reciprocal space and across two\norders of magnitude in energy. We found two distinct types\nof low-energy magnon modes: the Goldstone mode along\nthe helix propagation vector at (00\u0006qh)and two pseudo-\nGoldstone modes at (\u0006qh0 0)and(0\u0006qh0). Our simulation\nin the framework of the LSWT provides a perfect descrip-\ntion of the spectrum except at the pseudo-Goldstone points,\nwhere LSWT predicts a gapless mode, while the experiment\nshows a gap of the order \u0001PG=0.17meV . This gap can\nbe quantitatively reproduced in the calculations by includ-\ning leading-order corrections due to quantum fluctuations\narising from magnon-magnon scattering processes.\nThe scattering channels contributing to the magnon life-\ntime in insulators are usually very weak and require elabo-\nrate time-consuming experiments with a micro-electronvolt\nenergy resolution to be quantified in direct measurements\n[54]. At the same time, here we demonstrated that the\npseudo-Goldstone magnon gap in helimagnets, first mea-\nsured in our work, represents a well-defined and experi-\nmentally accessible quantity of the spin-wave spectrum that\ncarries indirect information about such magnon-magnon\ninteractions. In spite of its relatively simple experimental\ndetermination, the spin-gap magnitude \u0001PGcannot be ex-pressed analytically as a function of the Hamiltonian pa-\nrameters and is therefore useful for testing spin-dynamical\nmodels beyond LSWT .\nThe existence of nearly gapless spin-wave modes prop-\nagating in the direction orthogonal to the ordering vec-\ntor is expected to leave measurable signatures in the low-\ntemperature thermodynamic and transport properties of the\nmaterial, as well as in its response to local probes, at temper-\natures of the order of \u0001PG\u001cTN. In a cubic system that has\ntwo pseudo-Goldstone modes per one Goldstone mode, the\ncontribution of the former would dominate in all processes\ngoverned by low-energy magnon scattering, including but\nnot limited to magnetic contributions to the low-temperature\nspecific heat and magnon heat conduction [55]. In the pres-\nence of weak anisotropy, a two-gap structure in the magnon\ndensity of states is expected. To the best of our knowledge,\nthese effects still await their detection in future experiments\nperformed on samples in the single-domain magnetic state.\nOur present results should apply not only to simple heli-\nmagnets, but also to a much broader class of materials in\nwhich the magnetic propagation vector is spontaneously cho-\nsen from multiple structurally equivalent alternatives upon\ncrossing a transition to the magnetically ordered state.\nACKNOWLEDGMENTS\nWe acknowledge fruitful discussions with A. Yaresko and\nG. Jackeli. This project was funded by the German Research\nFoundation (DFG) through the Collaborative Research Cen-\nter SFB 1143 in Dresden [projects C03 (D. S. I.) and A06\n(S. R.) ], individual research grant IN 209/4-1, and the Trans-\nregional Collaborative Research Center TRR 80 (Augsburg,\nMunich, Stuttgart). T . M. and R. 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B 95, 224407 (2017).\n– 12 –"    },    {        "title": "1706.03521v1.Voltage_Controllable_Colossal_Magnetocrystalline_Anisotropy_in_Single_Layer_Transition_Metal_Dichalcogenides.pdf",        "content": "arXiv:1706.03521v1  [cond-mat.mtrl-sci]  12 Jun 2017Voltage-Controllable Colossal Magnetocrystalline Aniso tropy in Single Layer\nTransition Metal Dichalcogenides\nXuelei Sui,1,2Tao Hu,2,3Jianfeng Wang,1Bing-Lin Gu,1,4,5Wenhui Duan,1,4,5,∗and Mao-sheng Miao3,2,†\n1Department of Physics and State Key Laboratory of Low-Dimen sional Quantum Physics,\nTsinghua University, Beijing 100084, People’s Republic of China\n2Computational Science Research Center, Beijing 100084, Pe ople’s Republic of China\n3Department of Chemistry and Biochemistry, California Stat e\nUniversity Northridge, California, Los Angeles 18111, Uni ted States\n4Institute for Advanced Study, Tsinghua University, Beijin g 100084, People’s Republic of China\n5Collaborative Innovation Center of Quantum Matter, Beijin g 100084, People’s Republic of China\n(Dated: March 14, 2018)\nMaterials with large magnetocrystalline anisotropy and st rong electric field effects are highly\nneeded to develop new types of memory devices based on electr ic field control of spin orientations.\nInstead of using modified transition metal films, we propose t hat certain monolayer transition metal\ndichalcogenides are the ideal candidate materials for this purpose. Using density functional calcu-\nlations, we show that they exhibit not only a large magnetocr ystalline anisotropy (MCA), but also\ncolossal voltage modulation under external field. Notably, in some materials like CrSe 2and FeSe 2,\nwhere spins show a strong preference for in-plane orientati on, they can be switched to out-of-plane\ndirection. This effect is attributed to the large band charac ter alteration that the transition metal d-\nstates undergo around the Fermi energy due to the electric fie ld. We further demonstrate that strain\ncan also greatly change MCA, and can help to improve the modul ation efficiency while combined\nwith an electric field.\nEnormous efforts, both experimental and theoretical,\nhave been spent to improve the efficiency of magne-\ntization control in nanoscale systems[1, 2]. Conven-\ntional techniques like magnetic-field-induced magnetiza-\ntion switch and spin-current induced torque in magnetic\ntunnel junctions[3], both have a complex design and are\nvery power-consuming. In comparison, controlling the\nspin orientation by applying electric fields to materials\nwithlargemagnetocrystallineanisotropy(MCA) isanew\nand promisingapproach. This method has the advantage\nof ultra-low power consumption and strong coherence of\nthe individual spins. Recently, this has been demon-\nstrated experimentally in itinerant magnetic FePt and\nFePd ultrathin films with liquid interfaces[4]. Soon after,\nelectric-field-controlledMCAs werealsoreported forfew-\nmonolayers-thick magnetic metals[5–7] and alloys[8, 9],\nnano-junctions[10, 11], defected graphene[12], and thin\nfilms[13]. In most of these materials, the magnetism\ncomes from the 3 dtransition metals and the MCA arises\nfrom the strong spin-orbit coupling (SOC) due to the al-\nloying with heavy elements. Markedly, the MCA in these\nmaterialsis still low, andthe spin statesarevulnerableto\nthermal fluctuations. The electric field effect also needs\nlarge improvements for these materials to effectively ma-\nnipulate the spin orientation. Furthermore, other prob-\nlems persist in thin metal films and surfaces, including\nthe difficulty of growing high quality samples, the high\nreactivity while exposed to air and liquid, and especially\nthe strong screening to the applied electric field.\nIn contrast to ultra-thin metal films, many two-\ndimensional (2D) materials can be more easily fabri-\ncated in large quantity and high quality[14–16]. Many\nof them are quite stable and their screening effect toelectric field is relatively small. 2D materials, such as\ngraphene, boron-nitride and transition metal dichalco-\ngenides (TMDs) have shown high stability and superior\ntransport properties[17, 18]. Furthermore, mechanical\nexfoliation or chemical synthesis can be used to produce\nmonolayer TMDs flakes of high purity, such as MoS 2[19],\nMoSe2, TaS2, TaSe 2, and NiTe 2[20–22]. Recently, a\nthorough computational study was carried out for more\nthan 30 monolayer TMDs with varying combinations of\ntransition metal and chalcogen atoms (S, Se, or Te)[23].\nDepending on dband filling, TMD monolayers can be\nmagnetic. The examples also include the well studied\nVSe2, TaS2and TaSe 2[24–26]. A couple of recent stud-\nies used TMDs as the supporting materials for the MCA\ncenters[27, 28]. Since many of the magnetic TMDs con-\ntain heavy elements such as Se and Te, which indicates\nstrong SOC, we propose these monolayer materials may\npossess large voltage-controllable MCA.\nIn this letter, we conduct a systematic first-principles\nstudy of the MCA of single-layer TMDs and related ma-\nterials with and without external electric field. The se-\nlected materials include AX 2(A = Sc, Ti, V, Cr, Mn,\nFe, Co, Ni, X = Se, Te), TaS 2, TaSe 2and also FeI 2. The\nresults clearly demonstrate a large MCA for a number of\nAX2monolayers, including CrSe 2, FeSe 2, FeTe 2, TaS2,\nTaSe2, and FeI 2. Especially, the MCA for some AX 2\nshows a very strong electric field dependence. For CrSe 2,\nFeSe2, FeTe 2, and FeI 2, the electric field can change the\nsign of MCA, which can be used to switch preferred spin\norientation. Ourstudyillustratesthepromisingpotential\nof electric-field-switching of magnetization orientation in\nthese 2D materials. Furthermore, we demonstrate that2\nthe strain effect on MCA is also strong.\nθh\nθ\nA X\na/arrowrightnosp\nb/arrowrightnosp\n(c) (d) (a) (b) \nhhoneycomb  H-AX2 centered honeycomb T-AX 2\nX A X\na/arrowrightnosp\nb/arrowrightnosp\nFIG. 1. (Color online) Atomic structures of single layer AX 2.\n(a) and (c) are the top and side views of H phase, (b) and (d)\nare top and side views of T phase. The shaded areas show\nthe primitive unit cells with Bravais lattice vectors /vector aand/vectorb\n(|/vector a|=|/vectorb|).his the vertical distance between the chalcogen\natom and transition metal atom and θis the X-A-X bond\nangle across the top and the bottom layers. Magenta spheres\nrepresent transition-metal atoms (A) while blue and yellow\nspheres represent the top and the bottom layer chalcogen\natoms (X), respectively.\nAll the calculations are performed within the frame-\nwork of density functional theory (DFT)[29] as im-\nplemented in the Vienna ab initio simulation package\n(VASP)[30]. The PAW[31] potentials are used to de-\nscribe the ionic potential of all the atoms. We employed\nthe Perdew-Burke-Ernzerhof (PBE)[32] generalized gra-\ndient approximation (GGA) for the exchange correlation\nfunctional. In order to treat the strong on-site Coulomb\ninteraction of 3 dmetals, we employed the GGA + U[33]\nmethod. The Uvalues are tested for all the compounds\nby comparing the resulted magnetic moment with those\nobtained by Heyd-Scuseria-Ernzerhof (HSE)[34] hybrid\nfunctional. For most of the cases, we find that U= 2 eV\nyield magnetic moments in good comparison with HSE.\nSeveralfactorsareimportant for the selection ofcandi-\ndate 2D TMD systems. 1) The material should be ther-\nmodynamically stable or metastable with a low energy.\nIdeally, the TMD could exist naturally in single layer or\ninlayeredbulkstructurethatcanbeexfoliatedintosingle\nlayer. 2) The material should be spin polarized, there-\nfore the TMD materials containing 3d transition metals\nespecially V, Cr, Mn, Fe, Co are particullarly interest-\ning. 3) The material should have large SOC, therefore\nshould contain heavy elements. Our work will then fo-\ncus on late chalcogens such as Se and Te. Some com-\npounds, although not dichalcogenides, adopt the same\nlayered structure as TMD and satisfy the above criteria.\nOne such example is FeI 2, in which Iodine atoms occupy\nthe same lattice sites as chacogens. We therefore also\ninclude these layered structures in our study.\nWe first investigate the stability of single layer AX 2\nby comparing their energies with existing A nXmcom-\npounds. For an existing A mXncompound, we assume\na reaction between A mXnand elemental solid A, 2 /nAmXn+ [(n−2m)/n] A→AX2. The formation energy\nEfis defined as: Ef=EAX2- 2/n EAmXn- [(n−2m)/n]\nEA, whereEAmXnandEAarethetotalenergiesofA mXn\ncompound and solid A, respectively. For 3D AX 2com-\npounds,Efis actually the energy difference between sin-\ngle layer and bulk AX 2. There are two different types of\nsingle-layerstructuresfor AX 2compounds: ahoneycomb\nH structure with point-group symmetry of D 3h(trigonal\nprismatic coordination, Fig. 1(a) and (c)) and a centered\nhoneycomb T structure with D 3dsymmetry (octahedral\ncoordination, Fig. 1(b) and (d)). Both structures are\nincluded in our study.\nThe calculated results are shown in Table 1. We would\nlike to point out that severalAX 2compounds arealready\nknown and exhibit layered structure, including CrSe 2,\nTaS2, TaSe 2, and FeI 2. Our calculations show that the\nenergy difference between single layer and bulk structure\nis very small, indicating that CrSe 2monolayer can be\nfabricated by mechanical exfoliation. Single layer TaS 2\nand TaSe 2havebeen fabricated and theirpropertieshave\nbeen studied[35]. Furthermore, we also examine the dy-\nnamic stability of these compounds by calculating the\nphonon spectraof optimized TMDs in both Hand T con-\nfigurations (see Supporting information Fig. S1). Our\nresults reveal that among the 36 structures considered,\n31 of them are dynamically stable. Depending on the\ncoordination and oxidation state of the transition metal\natoms, TMDs can be either metallic or semiconducting.\nNow we will focus on the magnetic AX 2monolayers.\nTable 1 gives the calculated magnetic moments for all\nmagnetic AX 2we have considered. The results obtained\nby GGA + U method ( U= 2 eV) compare very well\nwith the HSE results. Magnetic moments of MnX 2and\nFeX2(CrX2) are around 3 µBand 2µB, respectively,\nwhile others are about 1 µB. The magnetic moment is\nan integer number if the material is semiconducting or\nhalf metallic. In latter case, one spin channel is metal-\nlic while the other one has a gap (see Fig. S2 in SI).\nIn order to examine the magnetic ordering, we consider\ntwo spin configurations of ferromagnetic (FM) and anti-\nferromagnetic (AFM). Both of the energies referring to\nnonmagnetic state are listed. By comparing the energies\nof the FM and AFM states, we find that all magnetic 2D\nAX2are in FM state.\nFor the next step, we examine the MCA and its de-\npendence with the electric field for all magnetic AX 2\nmonolayers. For a surface of area S(in units of cm2),\nmagnetocrystalline anisotropy is the total energy differ-\nence between two magnetic states where the magneti-\nzation is aligned along the [100] or [001] direction, i.e.,\nMCA = ( E[100]−E[001])/S. The results for a number\nof selected compounds are presented in Fig. 2 (please\nsee Supporting Information Table S1 for MCA values for\nall studied magnetic AX 2under zero and finite electric\nfields). Asshownin thefigure, many2DAX 2compounds\nshow very large MCAs at zero field. Especially for H-3\nTABLE I. Properties of magnetic AX 2monolayers, including\nformation energies ( Ef), magnetic energies ( EFMandEAFM)\nandmagnetic moments ( M)for free-standingsingle layer AX 2\nin H and T phasesa\nSystem EfEFMEAFMMGGAMU=2MHSE\nH-ScSe 22.29c-0.32 -0.16 1.00 1.00 1.00 S\nH-ScTe 21.85c-0.25 -0.14 1.00 1.00 1.00 S\nH-VSe 2-0.01b-0.36 -0.15 1.00 1.00 1.00 S\nT-VSe 2-0.02b-0.44 -0.29 0.61 1.15 1.26 M\nH-VTe 20.12b-0.41 -0.26 1.00 1.00 1.00 S\nT-VTe 2-0.04b-0.66 -0.21 0.92 1.50 1.52 M\nT-CrSe 20.00b-1.79 -0.09 2.16 2.42 2.69 M\nT-CrTe 20.08b-1.91 -1.88 2.43 2.66 2.69 M\nT-MnSe 2-0.33b-2.38 -0.19 2.81 3.00 3.00 H\nH-MnTe 20.13b-2.39 -0.06 2.57 3.00 3.00 H\nT-MnTe 2-0.10b-2.76 -0.44 2.82 3.17 3.10 M\nH-FeSe 20.26b-0.92 -0.45 1.99 2.00 2.00 H\nH-FeTe 20.20b-1.04 -0.55 1.80 2.00 2.00 H\nH-TaS 20.01b-0.06 -0.02 0.02 0.75 0.87 M\nH-TaSe 20.01b-0.08 -0.02 0.00 0.83 1.00 M\nT-FeI 20.03b-1.79 1.20 4.00 4.00 4.00 S\naEf(in units of eV) is the formation energy of single layer AX 2.\nbAX2compound exists,cAX2compound does not exist. EFM,\nEAFM(in units of eV) are the relative energies of ferromagnetic\nand anti-ferromagnetic state with respect to the non-magne tic\nstate.MGGA,MU=2andMHSE(in units of µB) are the values of\nmagnetic moment calculated by GGA, GGA + U and HSE\nfunctionals, respectively. S, M and H in the last column stan d for\nsemiconducting, metallic and half metallic, respectively .\nH-FeSe 2\nH-FeTe 2\nH-TaS 2\nH-TaSe 2T-CrSe 2\n2\n-10-8-6-4-202468MCA (erg/cm ) \n2\nelectric field (V/Å) 0.00 0.25 0.50 0.75T-FeI \nFIG. 2. (Color online) The calculated MCA (in units of\nerg/cm2) as a function of electric field ( F= 0, 0.25, 0.5\nand 0.75 V/ ˚A) for T-FeI 2(black lines), T-CrSe 2(red lines),\nH-FeSe 2(cyan lines), H-FeTe 2(yellow lines), H-TaS 2(blue\nlines) and H-TaSe 2(green lines) (H-FeTe 2is unstable when\nF= 0.75 V/ ˚A). Inset: Schematic diagrams for two magnetic\norientations, the upper and lower ones represent the magnet i-\nzation aligned along the out-of-plane [001] and in-plane [1 00]\ndirections, respectively.\nFeTe2and H-TaSe 2, their MCAs are as large as −9.58\nand−8.84 erg/cm2, respectively. Negative values mean\nthat spins favor the in-plane ( xory) orientations. These\nvalues are one order of magnitude larger than those of\ntransition metal thin films. For example, a Pd-capped,\nnine monolayers thick FePd film exhibits an MCA value\nof 0.86 erg/cm2[13]. A gold (Au) capped FeCo film hasa MCA value of -0.56 erg/cm2[9]. It is well known that\nlarge MCA is originated from the strong coupling be-\ntween the local spin (magnetic moments) and the orbital\nmoments. Inclusion of heavy elements in the system, like\nPd or Au capping in the metal films, can greatly enhance\nMCA. However, as shown in our current work, forming\ndirect chemical bonds with heavy elements such as Se\nor Te in AX 2is a more effective way to improve MCA,\nlikely due to the stronger coupling between spin and or-\nbital moments.\nAs shown in Fig. 2, 2D AX 2compounds show ex-\nceptionally large voltage modulation with their magne-\ntocrystallineanisotropy. The MCA values ofall five com-\npoundsshownin thefigurevarydramaticallywiththe in-\ncreasingexternal field. For example, the MCA of H-TaS 2\nchangesfrom −5.65erg/cm2to−2.16erg/cm2whileelec-\ntric field increases from 0 to 0.75 V/ ˚A. More interest-\ningly, MCA values increase monotonically and change\nsign from negative to positive for T-CrSe 2(from−0.93\nerg/cm2to 0.65 erg/cm2), H-FeSe 2(from−2.52 erg/cm2\nto 4.19 erg/cm2) and H-FeTe 2(from−9.58 erg/cm2to\n7.11 erg/cm2) under finite external field. These results\nclearlyshowthatwecaneffectivelyswitchthemagnetiza-\ntion orientation from in-plane [100] to out-of-plane [001]\ndirection in some 2D AX 2materials by applying electric\nfield. In comparison, an earlier work showed that MCA\nof monolayer Fe (001) changes from 0.39 erg/cm2(0.2\nmeV/atom) to −0.19 erg/cm2(−0.1 meV/atom) while\nelectric field varies from 0 to 1.35 V/ ˚A[6].\nNow we will discuss the possible microscopic mech-\nanism of electric-field-driven modulation of MCA. It is\ncommonlyknownthatthemodificationofMCAiscaused\nby the changes in the relative occupation of transition\nmetaldorbitals[7]. Assuming the SOC is a perturba-\ntion term of the Hamiltonian, MCA can be expressed by\nthe coupling terms between the occupied and unoccupied\nstates through the orbital angular momentum operators\nˆLzandˆLxand on the energy difference between these\nstates[36, 37], namely\nMCA =ξ2/summationdisplay\no,u/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\nΨ↑(↓)\no/vextendsingle/vextendsingle/vextendsingleˆLz/vextendsingle/vextendsingle/vextendsingleΨ↑(↓)\nu/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle2\n−/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\nΨ↑(↓)\no/vextendsingle/vextendsingle/vextendsingleˆLx/vextendsingle/vextendsingle/vextendsingleΨ↑(↓)\nu/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle2\nE↑(↓)\nu−E↑(↓)\no(1)\n+/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\nΨ↑(↓)\no/vextendsingle/vextendsingle/vextendsingleˆLx/vextendsingle/vextendsingle/vextendsingleΨ↓(↑)\nu/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle2\n−/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\nΨ↑(↓)\no/vextendsingle/vextendsingle/vextendsingleˆLz/vextendsingle/vextendsingle/vextendsingleΨ↓(↑)\nu/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle2\nE↑(↓)\nu−E↓(↑)\no/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nspin-flip term\nwhereξis the SOC constant, Ψ↑(↓)\noand Ψ↑(↓)\nuindicate\nrespectively the occupied and unoccupied majority-spin\n(minority-spin) bands, and E↑(↓)\noandE↑(↓)\nurespectively\nrepresent the corresponding energies. For the same spin,\nSOC between occupied and unoccupied states with the\nsame (different) magnetic quantum number through the\nˆLz(ˆLx) operator gives a positive (negative) contribution\nto MCA. SOC between opposite spin states gives reverse4\nFIG. 3. (Color online) Energy- and k-resolved distribution of\norbital character for electronic structures along high sym me-\ntry lines for Fe d-orbitals in FeSe 2compound under electric\nfield of (a) 0, (b) 0.25, (c) 0.5 and (d) 0.75 V/ ˚A, respectively.\nThe left and right subpanels are majority and minority spin\nbands, respectively. The thickness represents the amplitu de\nof the orbital character. Numerals refer to points on high-\nsymmetrylines(HSL n,n= 1-5)wheretherearelarge changes\nin MCA.\ncontribution to MCA.\nLet us use the H-FeSe 2monolayer as an example to\nshow how the external field modifies the band structure\nand the perturbation terms in the above expression for\nMCA. We have made this choice because the effect is\nmost significant in this material (Fig. 3). Due to the\ncrystal field effect in trigonal prisms, d-orbitals split into\nthree groups, a(dz2) (|m|= 0),e1(dyz,dxz) (|m|= 1)\nande2(dx2−y2,dxy) (|m|= 2), which are denoted by\nblue, green and red colors, respectively. For FeSe 2un-\nder zero field [Fig. 3(a)], the coupling between majority\nspin (left panel) including e1, and including e2, through\ntheˆLxoperatorgive negative contribution to MCA. This\nhappens both along the (Γ-K) high symmetry line (la-\nbeled as HSL1) as well as along the (Γ-M) (labeled as\nHSL2) directions. For the minority spin (in right panel\nand labeled as HSL3), the coupling between occupied a\nand unoccupied e2through the ˆLxoperator also gives\nnegative contribution.\nIn next step, an electric field perpendicular to the\nTMD plane is applied. It causes Fe d-orbitals shift down-\nwards for the majority-spin band, which substantially\nreduces the proportion of d-orbitals around the Fermi\nlevel (left panels), especially for the states around HSLn\n(n= 1 and 2). When electric field reaches 0.75 V/ ˚A,dstates almost vanish around the Fermi level, except for a\nvery small proportion of e1states [Fig. 3(d) left panel];\nhence the negative contribution to MCA around HSL1\nand HSL2 decrease greatly. Considering the spin flip\nterms, the minority bands of e2states below the Fermi\nenergy (occupied) shift upwards [HSL4 and HSL5 in the\nright panels of Fig. 3(a) - (d)], whereas the majority\nbandsoftheunoccupied e1statesalongthe(Γ-K)(HSL4)\nand (M-K) (HSL5) shift downwards [Fig. 3(c) and (d)\nleft panels]. As a consequence, the energy differences\nbetween the minority occupied e2states and majority\nunoccupied e1states decrease. This causes stronger cou-\npling between these two opposite spin states through the\nˆLxoperator, which gives a large positive contribution to\nMCA. Therefore, with the increasing of electric field, the\nabovenegativetermscontributelesstoMCAwhereasthe\npositive terms contribute more, which eventually causes\nMCA change from negative to positive when the field is\nhigh enough.\n0.00 0.25 0.50 0.751.401.451.501.551.601.651.701.75\n76 78 80 82 84 86 88 \n0.00 0.25 0.50 0.75012345678910 \n-3 -2 -1 0123456789\nelectric field (V/Å)\nh\n(Å)\n澳h\nθ\nelectric field (V/Å)\nMCA (erg/cm ) MCA 2BM\n2orb\n(10 )µ−∆Morb∆\n( )\nθ°\nFIG. 4. (Color online) The variation of relevant structural\nparameters verse electric field for FeSe 2system. h(˚A) andθ\n(◦) represent, respectively, the vertical distance between t he\nchalcogen atom and the transition metal atom, and the X-A-\nX bond angles. While ∆ Morb(in units of µB) is the orbital\nmoment anisotropy.\nFigure 4 shows how the geometry parameters and or-\nbital moment anisotropy(∆ Morb=Morb[001]−Morb[100])\nchange under an electric field. It reveals that the ge-\nometry changes under an electric field together with the\nchange of orbital momentum and MCA. The vertical dis-\ntances(h)betweenthe chalcogenatomandthe transition\nmetal atom change non-linearly from 1.51 ˚A to 1.74 ˚A.\nBy increasing the intensity of the electric field, hgrows\nrapidly. The same behavior also happens to X-A-X bond\nanglesθ, which changes from 75 .8◦to 84.0◦. Both of\nthe variation trends are parallel with that of the MCA.\nDespite the applied field breaks the D3hsymmetry, all\nsix A-X bond lengths remain the same. More significant\nchanges happen to the orbital moments. ∆ Morbchanges\nfrom 0.026 µBto 0.092 µBas the electric field increases\nfrom 0 to 0.75 V/ ˚A. And it is commonly known that\nhigher orbital moments as well as their anisotropy usu-\nally indicate larger MCA. The changes in the orbital mo-\nment of the AX 2monolayers under electric field are very\ndifferent from those of thin transition metal films, which5\nare much smaller and often nonmonotonic[5, 13, 38]. It\nis worth noticing that similar geometry changes under\nelectric field can be observed for most of the 2D AX 2\nmonolayers. However, only those with strong SOC and\nlarge MCA show significant variations of the orbital mo-\nmentum as well as the voltage modulation of MCA. Our\nresults suggest that materials with both strong SOC\nand strong covalent bonds are good choices for voltage-\ncontrollable MCA.\nInspired by the above results, that the geometry and\nmagnetic properties including MCA change simultane-\nously for AX 2monolayers under increasing electric field,\nwe also explored how the biaxial strain could change the\nMCA values. Our results reveal a strong strain effect (re-\nfer to Supporting Information Fig. S3). Among all the\nAX2monolayers, T-CrSe 2, H-FeSe 2and H-TaS 2show\nexceedingly large strain effect on MCA. Particularly, the\nMCA of T-CrSe 2increases nonlinearly with an applied\ntensile biaxial strain and changes from in-plane to out-\nof-plane at strain of about 2%. It is well known that\nmonolayer materials may sustain large biaxial strains.\nAlthough, the direct control of MCA by strain is not\neasy to achieve, the combination of strain and voltage\nmay greatly extend our control of MCA in nano-sized\ndevices.\nIn summary, we propose that single layer transition\nmetal dichalcogenides can be ideal candidate materials\nfor voltagecontrolled memory devices. Using first princi-\nples DFT calculations, we demonstrate that 2D AX 2ma-\nterials may exhibit both colossal MCA and exceedingly\nstrong voltage dependence, and their easy-axes change\nfrom in-plane at zero field to out-of-plane under finite\nelectric field. The polarized covalent bonds signify the\nSOC on heavy atoms, causing large MCA. Comparing\nwith thin metal films, these covalently bonded materials\nexhibit large geometry deformation under electric filed\nand much smaller screening effect, resulting at an enor-\nmous voltage modulation to its MCA. Hence, these ma-\nterials can meet the two opposing demands for the new\ntype of magnetic memory, namely maintaining the mem-\nory against thermodynamic fluctuations and writing or\nrewritingwith low powerconsumption. 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Lett. 102, 247203\n(2009)."    },    {        "title": "1706.07368v1.Magnetocrystalline_anisotropy_in_YCo5_and_LaCo5__A_choice_of_correlation_parameters_and_the_relativistic_effects.pdf",        "content": "Magnetocrystalline anisotropy in YCo 5 and LaCo 5: A choice \nof correlation parameters and the relativistic effects  \nManh Cuong Nguyen* , Yongxin Yao, Cai -Zhuang Wang, Kai-Ming Ho  and Vladimir P. \nAntropov  \nAmes Laboratory – U.S. D epartment of Energy , Iowa State  University, Ames, IA 5001 1, USA  \nAbstract  \nThe dependenc e of the magnetocrystalline anisotropy energy (MAE) of MCo 5 (M = Y , La) on \nthe Coulomb correlation s and strength of spin orbit  (SO)  interaction  within the GGA + U scheme \nis investigated. A range of par ameters suitable for the satisfactory description of key magnetic \nproperties is determined.  The origin of MAE in these materials is mostly related to the large \norbital moment anisotropy  of Co atoms  on the 2c crystallographic site. Dependence of relativistic \neffects on Coulomb corr elations, applicability of t he second order perturbation theory for the \ndescription of MAE and effective screening of the SO interaction in these systems are discussed \nusing a generalized virial theorem . \nPACS : 75.30.Gw, 75.50.Vw, 71.15.Mb  \n \nThe MCo 5 interme tallic compound s (M =  rare-earth and Y) with the hexagonal CaCu 5- \nprototype  structure  have been attracted attention primarily due to their applications as permanent \nmagnets . One of the key intrinsic magnetic propert ies determining such applications is the MAE . \nEssentially , all known magnets with the CaCu 5 structure exhibit  a very high value of MAE  with \na maximum observed in SmC o5 family of magnets,  the most famous applied permanent \nmagnet s [1–3]. In general any MCo 5 system shows interesting properties, with even CeCo 5 \ndemonstrating strong and temperature stable magnetic properties with different doping  [4]. \nTheoretical studies of SmC o5 are very limited due to difficulties in the description of localized \n4f-states of Sm. In order to gain insight s into the mechanism of high MAE  in M Co5, it is natural  \nfirst of all  to study the nature of M AE in similar systems where rare -earth atom  is replaced by  d-\nelements  (Y or La ) which do not have  localized  f-electrons but still demonstrate strong magnetic \nanisotropy. However, numerous studies of these systems using traditional band structure \nmethods did not show a sufficient agreement with the e xperiment.  Moreover, different methods \nproduced very different results demonstrating a large sensitivity to the choice of the exchange \nand Coulomb correlations. A large value of anisotropy in a bulk metallic material can also  be \nrelated to unusual ly strong  relativistic effects  in f-electron systems  [3,5]  and a question of the \napplicability of the traditional perturbation theory is naturally appearing . It would thus  be \ndesirable to understan d how sensitive the importan t physics of these materials  to both correlation  \nand relativistic effects . А specific question  which appears here is the influence of correlations on \nSO interaction that has been discussed recently  [6]. There have been many theoretical works on YCo 5 and LaCo 5 system s to study  the MAEs  \nand other magnetic properties   [7–15]. Density functional theory (DFT) calculations within local  \ndensity approximation (LDA) g ave different MAEs for YCo 5 depending on basis set  and \nimplementation . Some LDA  calculations even predicted a wron g magnetization direction  of \nYCo 5 [8]. DFT calculation within the generalized -gradient approximation (GGA) can predict a \ncorrect magnetization direction along the lattice  vector  c for YCo 5 but the obtained value of \nMAE was about a factor of 4 smaller than the experiment al value  [8]. The large anisotropy in \nYCo 5 and LaCo 5 is believed to be due to the large orbital  magnetic  moment s of Co atoms  as \nobserved in  experiments  [7,16,17] . Some works have been performed taking into account the ad \nhoc orbital polarization (OP) correction to  artificially  increase the orbital magnetic \nmoment  [9,11,18,19] . Somewhat better values for  MAE  and orbital  magnetic  moments  on Co \natoms  in YCo 5 have been  obtained by these  DFT + OP calculations  in comparison with \nDFT  [9,11] . However, the original ly proposed  OP correction  due its unphysical atomic limit  \ncreat es some uncontrolled anisotropies in YCo 5. A proper treatment r equires implement ation of  a \nrelativistic LDA+U type of scheme. There have been only a few calculations for MAE of LaCo 5 \nand they also showed very diverse results.  A LDA  + OP calculation  by Steinbeck et al  [9] \nshowed a MAE of LaCo 5 ~ 9.0 meV/f.u ., which i s severely overestimate d in comparison with  the \nexperiment value . While other LDA calculations showed a MAE of 0.38 meV/f.u. and 2.84 \nmeV/f.u without and with OP correction , correspondingly  [12]. \nIn this work, we use standard  DFT  GGA  + U scheme  [20] taking SO interaction into \naccount  to inves tigate the magnetic properties of YCo 5 and LaCo 5. We show that there exists a \n“universal ” set of U and  J parameters which  leads to a good agreement with the experimental \ndata for t he MAEs,  orbital moment anisotropies  (OMAs ) and other magnetic properties for both  \nYCo 5 and LaCo 5. Our calculation show s that the OMA is the key factor inducing the huge MAEs \nin YCo 5 and LaCo 5. We also verify the applicab ility of a second order perturbation theory for \nsystem s with  large MAEs  and discuss the screening of SO inter action  by Coulomb correlations.  \nThe spin -polarized DFT calculations  are performed by the Vienna Ab-initio  Simulation \nPackage (VASP)  [21] with a projector -augmented wave (PAW) pseudo -potential method  [22,23]  \nwithin the generalized -gradient approximation (GGA) parameterized by Perdew, Burke, and \nErnzerhof  [24]. The energy cutoff is  320 eV and the Monkhost -Pack scheme  [25] is used for \nBrillouin zone sampling with a high quality k -point grid of 2  × 1/60 Å-1, which is equivalent to \na 14 × 14 × 15 k -mesh.  The total energy convergence criterion is 10-8 eV/unit cell.  Strong -\ncorrelat ion effects are  taken into account via a GGA + U scheme  [20] for Co 3 d-electron s. All \ncalculations ar e performed  with experimental lattice parameters  from the ASM Alloy Phase \nDiagram Database . These parameters  are a = 4.924, 5.105 , and 2.698  Å, and c = 4.000, 3.966  and \n3.710  Å for YCo 5, LaCo 5 and L10 CoPt, respectively . The MAE  is calculated  by force theory  and \nis defined as the total energy difference for magneti zation  align ed along  lattice a and c: K = E[100] \n– E[001]. Here,  E[100] and E[001] are the total energies with the magnetization aligned along the  \nlattice a and c of the hexagonal or tetragonal unit cells, respectively.  In details, a collinear self -\nconsistent calculation is performed first. Then  non-self-consistent calculations wit h SO interaction are performed  with magnetization aligned along different directions, e.g. [001] and \n[100] to calculate E[001] and E[100], respectively.  The SO anisotropy energy  for an atom  is defined \nin the same way: K so = E so[100] – Eso[001]. Here,  Eso[100] and E so[001] are the SO interaction  energ ies \nwith the magnetization aligned  along the corresponding directions  of that atom. The K so for the \nsystem is a sum of those from  all atom s in the unit cell . \n \nFigure 1.  Dependence s of YCo 5 and LaCo 5 MAE s on U  and J  parameter s within a GGA + U \nscheme.  Colored bands  show the experimental range of MAEs. The blue diamond and red \ntriangle are MAEs of YCo 5 and LaCo 5, respectively, and with U = J = 1.8 eV  \nFirst, we investigate the dependence of MAE s of YCo 5 and LaCo 5 on on-site Coulomb U \nand exchange J parameters  as shown in Fig.1.  Both Y and La does not have 4 f-electrons  and 4 d-\nelectron in Y and 5 d-electron in La are not localized so the strong -correlation correction will be \napplied to Co  3d-electrons  only.  We vary U and J within reasonable ranges for 3 d-transition \nmetals : 1.0 to 2.4 eV for  U and 0.8 to 1.6 eV for J , and with a step value of 0.2 eV . In Fig. 1, the \ngolden and blue bands represent the range of MAEs for YCo 5 and LaCo 5 respectively reported \nby experimenta l measurements . The top panel  of Fig. 1 shows the dependence of MAEs of YCo 5 \nand LaCo 5 on J when  U is set to be  2.0 eV. As the changes in MAE s with J are very small , we \nuse J=0.8 eV  to investigate the dependence of MAE on the U parameter . \n In contrast with a weak dependence on J, MAEs depend strongly  on U . This can be seen \nin Fig. 1. For GGA + U with 1.0 eV ≤ U ≤ 2.4 eV  and J = 0.8 eV , the MAE of YCo 5 varies from \n2.40 to 4.70 meV/f.u. and it is in the experimental ly observed range with 1.4 ≤ U ≤ 1.7 eV. For \nLaCo 5 the MAE value varies from 2.00 to 6.40 meV/f.u. and it is in the experimental observed \nrange with 1.9 ≤ U ≤ 2.2 eV. It is interesting that for U from 1.8 to 2.0 eV, the MAEs of both \nYCo 5 and LaCo 5 are very close to the experimental values. The calculated MAE of YCo 5 varies \nfrom  3.99 to 4.14 meV/f.u. when U changes from 1.8 to 2.0 eV, about 4.8 to 8.9% \noverestimat ion in comparison with  the upper bound of the experimental value. For the same \nrange of U, the MAE of LaCo 5 is from 4.01 to 4. 80 eV, about 8.4%  smaller than  the lower bound \nvalue to within  the experimental value range . These results  show that we can choose a common  \npair of  on-site Coulomb U  and exchange J  parameter s for Co 3 d-electrons  so that GGA + U can \ndescribe  reasonably well the MAE of both YCo 5 and LaCo 5. This is very interesting : a \n“universal ” pair of U and J for Co 3 d-electrons can be chosen for metallic compound s with Co. \nWe will show later that this choice of a universal  set of U and J for Co also work ve ry well for \nL10 CoPt, which is also an important permanent magnet  but has a very different crystal structure \nfrom YCo 5 and LaCo 5. From  the discuss ion above, a universal  U can be chosen as any value \nbetween 1.8 to 2.0 eV . For all following calculations and analys is, we use an on-site Coulomb U \n= 1.9 eV  and exchange J = 0.8 eV . It should be noted that the MAE of YCo 5 and LaCo 5 by GGA  \nwithout U and J are 0.73 and 0.98 meV/f.u., respectively.  \nTable I.  Atomic site resolved contributions to magnetic properties of MCo 5 (M = Y, La) and \nCoPt. MAEs from GGA  + U calculation and experiment  (K) are in meV/f.u., spin ( S) and orbital \n(L) magnetic moments and OMA  (ΔL) are in μ B/atom, K so is in meV/f.u. . ΔL0 and K so0 are ΔL \nand Kso with U = J = 0 eV , respectively.  \nSystem  YCo 5 LaCo 5 CoPt \nK     \n Calc. 4.090  4.402  1.127  \nExpt . 3.400  3.800  4.400  5.600  ~ 1.000  \nM/Pt S/L \nΔL/ΔL0 -0.419  / 0.037 \n-0.020  /-0.009  -0.283 / 0.027  \n-0.014  /-0.009  0.334 / 0.068  \n0.010  / 0.019  \nCo2c S/L \nΔL/ΔL0 1.607 / 0.15 2 \n-0.062  /-0.020  1.628  / 0.15 2 \n-0.058  /-0.027  \n2.014 / 0.11 2 \n-0.043  /-0.034  Co3g1 S/L \nΔL/ΔL0 1.677 / 0.117  \n-0.036  /-0.033  1.663 / 0.139  \n-0.049  /-0.045 \nCo3g2 S/L \nΔL/ΔL0 1.677 / 0.117  \n-0.020  / 0.003  1.663 / 0.139  \n-0.032  /-0.008  \nM/Pt Kso/Kso0 0.447  / 0.073  0.957  / 0.213  2.189  / 1.945  \nCo2c Kso/Kso0 2.106  / 0.250  1.795  / 0.186  \n-0.109  /-0.781  Co3g1 Kso/Kso0 0.867  / 0.622  1.335  / 0.983  \nCo3g2 Kso/Kso0 0.814  / 0.304  1.335  / 0.394  \nTotal  Kso/Kso0 7.156  / 1.803  8.553  / 2.355  2.080  / 1.164  \n \nTable I shows the atomic site resolved contribution s to the magnetic properties of YCo 5 \nand LaCo 5. Note that  when an external field  is applied  along lattice a and the SO interaction  is \nincluded , the 3 g-site of the crystal is split into 2 inequivalent sites . One  site has a multiplicity of 1 (denoted as 3 g1 hereafter ) and the other has a multiplicity of 2 ( denoted as 3 g2 \nhereafter ) [8,26] . The total spin magnetic mo ment s of YCo 5 and LaCo 5 are 7.83 and 7.96 μ B/f.u., \nrespectively . Taking orbital  moment into account,  the total magnetic moment s are 8.52 and 8.71 \nμB for YCo 5 and LaCo 5, respectively.  This agree s very well with  the experimental observation s \nof 8.3 3 μB/f.u. for YCo 5 [7] and 8.46  μB/f.u. for LaCo 5 [27]. From Table I , we can see that the \norbital  moment of Co is considerably  large r than that of Y or La  by about 3 to 4 time s. This is \nconsistent with experiment al values . A recent X -ray magnetic circular dichroism (XMCD) \nexperiment  [15] show ed that the orbital moment s of Co on both site s in YCo 5 is about 0.2 \nμB/atom  with an orbital moment on the 2c-site being slightly larger . By an inelastic spin flip \nneutron scattering experiment , Heidermann et al [28] showed that the orbital moments of Co  on \nthe 2c and 3 g-sites are 0.26 and 0.24 μB for YCo 5 and about 0.29 and 0.25 (by estimation ) for \nLaCo 5. Our results for the  orbital moment of Co in both YCo 5 and LaCo 5 are smaller than \nexperiment al values.  The trend of larger orbital moment in 2 c-site is observed in our calculation.  \nThis trend of orbital moment is also consistent with previous  calculations i ncluding an orbital \npolarization scheme  [9,11]  or an LDA + DMFT calculation  [15] for YCo 5. We note that a  LDA \ncalculation with a linear -augmented plane -wave (LAPW) method  [19] for YCo 5 observed the \nsame orbital moment s for Co on  both the 2c- and 3 g-sites.  \nFrom site resol ved contributions to the SO anisotropy energy  (Kso), we find that for \nYCo 5, the contribution from Co on the 2c-site is ~2.5 times larger than that from Co on the 3g-\nsite and ~5.0 times larger than that from Y.  These results agree with the experiment al \nobservation that the source of  anisotropy of YCo 5 mainly comes from Co atoms with a dominant \ncontribution from the 2c-site [7,29] . Note that GGA calculations show that dominant \ncontributions to MAE com e from Co on the 3g-site for both YCo 5 and LaCo 5.  This is  in contrast \nto our GGA + U calculation and the experiment al results . The huge MAEs in MCo 5 (M = Y, La) \ncompounds are ascribed to the large Co orbital moment and the large Co OMA  [7,16,17,30] . In \na model developed by Streever  [17], the atomic anisotropy energy  is proportional to the OMA.  \nThe OMA s of Co on  2c- and 3 g-sites were measured from a polarized neutron scattering \nexperiment  [7]. Their values are -0.100 and -0.030 μB/atom, respectively . This gives  a total \nOMA ~ -0.30 μB/f.u. for YCo 5, where  the OMA is defined as the difference in orbital moment \nwhen the magnetization is ali gned along lattice a or c. The OMAs from our calculation are -\n0.062, -0.036 , and -0.020 μB/atom for Co on 2c-, 3g1, and 3 g2-sites, respectively. The absolute \nvalues of OMAs from our calculation are somehow smaller than experimental values . They do, \nhowever,  show the same trend that the 2 c-site Co OMA is much  larger than that of 3 g-site Co by \nabout  2 to 3 times . It is interesting that  when we include the correlat ion effects, the  OMAs are \nincreased (Table I) and our results for OMA become  similar to those obtained by LDA + OP [9], \nalthough the orbital moment s of Co atoms  in those calculations   are about twice larger than our s. \nNote that the LDA + OP values for orbital moment s of Co atoms are also larger than the  values \nfrom flip flop neutron scattering  [28] or recent XMCD  [15] experiments.  In addition, t he MAE \nof YCo 5 from LDA + SO + OP is 4.40 meV/f.u., which is ~ 8% higher  than the value from \npresent c alculation and ~ 16% higher than the upper bound of experimental value . The picture is very similar for LaCo 5 but the difference in OMA and K so between  Co on  2c- and 3 g-sites are \nsmaller and  the contribution from La to K so is larger in comparison with that from Y in YCo5. \nThese results and comparisons with experiments and previous calculation s show that the GGA + \nU calculation  with the universal  pair of U and J can describe the magnetic properties of the \nMCo 5 compound s reasonably well . \nIn general , the model of Streever  [17] is a simplified result of the second order \nperturbation theory. It has been known that the total SO anisotropy K so [3,31]  can be p resented as  \nKso= (λΔ Lz +  λ ΔL↑↓)/2.  (1) \nHere, ΔLz and ΔL↑↓ are anisotropies of the longitudinal and transversal components of the orbital \nmoment  operator. Our results indi cate that the second (transversal) part is relatively small and \ndoes not play any significant role in the total magnetic anisotropy. The a nisotropy of the \nlongitudinal component (usual OMA) is domina ting in Eq. (1). We emphasize that  instead of the \nabsolute value of orbital moment, a  large OMA is a key factor in inducing huge MAE s in YCo 5 \nand LaCo 5. Overall , this implication is consistent with the physics of Streever’s model  [17] and \nis a very common behavior of anisotropy in metals . Our calculation also re -confirm s that the \nmain contributions to MAEs of YCo 5 and LaCo 5 are from Co on the 2c-site.  \nAs an example demonstrating  the optim al choice of U and J, we show  in Table I the \natomic resolved magnetic properties of L1 0 CoPt. CoPt was discovered to be a good permanent \nmagnet material since the 1930s . However, the high cost of Pt metal has made this material  a \npractical ly useless  for permanent magnets  applications . But CoPt is still of great interest for \nmagnetic material and properties research.  There have been many DFT calculation s for MAE of \nCoPt. DFT  calculations gave  widely  diverse MAE results depending on calculation and \nimplementation of the computational method . This is similar to the situation of MCo 5 systems . \nFor example, results from linear muffin -tin orbitals method are 2.29, 1.50 , and 2.00  meV/f.u ., \nand those from full-potential  linear muffin -tin orbitals method  are 1.05 and 2.20 meV/f.u.  [32] \nwithin the LDA . Our  LDA and GGA calculated MAEs  are 1.53 and 0.84 meV/f.u., respectively. \nThe experiment MAE of CoPt is ~ 1.00 meV/f.u. at T = 0 K and ~ 0.82 meV/f.u. at T = 300 \nK [32–34].  \nWe perform a GGA + U  calculation for CoPt with the  universal U and J for Co 3d-\nelectrons.  Like in the calculations above , we do not  apply additional correlation effects to Pt 5d-\nelectrons  because they are  not localiz ed. Our GGA + U result for MAE  (1.127 meV/f.u. ) agree s \nwith the experiment.  From Table 1 w e also can see that the main contribution to MAE of CoPt is \nfrom the Pt atom . The  Co atom  show s a negative contribution (favoring in -plane magnetization  \nor easy plane ), but it is small in amplitude in comparison with the contribution from Pt . OMA is \nnegative for Co and positive for Pt.  These result s are qualitatively consistent with recent work  \nanalyzing the constituents to magnetic anisotropy and other magnetic proper ties of CoPt  [31,35] . \nOur value of MAE  agrees well with experiment s.  \nFigure 2 . Projected d ensity of states ( PDOS)  of Co in YCo 5 and Pt in CoPt from  GGA + U \ncalculations without SO interaction.  \nFigure 2 show the projected density of states ( PDOS) on Co of YCo 5 and Pt of CoPt  from \nGGA + U calculation s without SO interaction. The positive and negative values of PDOS are the \nspin up and spin down components of PDOS , respective ly. YCo 5 and LaCo 5 are isostructur al and \nhave very similar lattice parameters  and the PDOS are very similar . We thus show only the \nPDOS from YCo 5. For YCo 5, the PDOS near the Fermi level are mainly contributed from the Co \ndown spin, where the PDOS from the dx2-y2-orbital of Co on the 2c-site are totally dominant . The \nPDOS of Y (not shown in  plot) near the Fermi level is very small in comparison with that of Co. \nFor CoPt, the spin down  PDOS from different d-orbitals of Pt are comparable near the Fermi \nlevel. However, for the spin up channel, the PDOS from the dx2-y2-orbital is very dominant  and \ndetermines the opposite sign of OMA . \nIn all cases , the spin anisotropy of the DOS at the F ermi level is significant  and allows a \nsimple interpretation of large orbital moments in these systems if we use following \nrelation  [36,37] : \nL = λ∑m2(Nm↑(Ef)-N-m↓(Ef)).  (2) \nHere, m is the magnetic quantum number  and Nm↑/↓(Ef) are the spin up/down  DOS  at the Fermi \nlevel.  Clearly, the largest contribution to the orbital moment is produced by states with a large m \nand the largest anisotropy of DOS.  The r elative signs of  orbital moments obtained from e q.(2) \nand Fig.2  confirm a different behavior of  OMA i n CoPt and Y(La)Co 5 systems.  The eq. (2) \ncannot be used for the calculation of OMA because it does not take into account the change of \nwave functions when SO coupling is included.  \nLet us discuss the relativistic effects and their dependence on Coulomb corr elation from a \ngeneral point of view.  First, we stress that the total energy  of a system  when the atomic SO \ncoupling V so is included  is  \nE = T + V + V so,  (3) \nand the resulting relativistic splitting is expected to be screened by a competition of kinetic a nd \npotential energies terms in e q. (3). Using second order  perturbation theory , it has been shown  \n(Ref. [31]) that the change in the  total energy due to SO couplin g addition  is just half of the \ninitial SO interaction energy  \nE(2) = V so /2.           (4) \nCorrespondingly , the total anisotropy of hexagonal or tetragonal systems is K= Kso/2 [31]. This \nresult is obtained in framework of traditional relativistic perturbation theory and is not \nnecessarily valid  in systems where a strength of SO coupling is comparable to the crystal field \neffects.  \nThis result leads to an important statement : the strength of SO coupling in solids is \nalways smaller (  = 0/2) than the original (atomic) SO coupling  0 due to the “screening ” \nreaction of the system and cannot be enhanced in this approach  overall . We would like to \nemphasize that this scre ening is determined by both kinetic and potential energ ies’ actions. To \nillustrate it in more detail , one can write an analog of the virial theorem with SO coupling \nincluded (mass -velocity and Darwin correction are ignored here for simplicity). Taking into  \naccount that SO coupling is a homogeneous function of the third order (in case of pure Coulomb \npotential) , this virial theorem can be written as  \n2T + V + 3V so = 0.    (5) \nUsing e qs. (3) and (5) , the resulting total energy can be presented in several alter native \nforms : \nE = (V - Vso)/2 = -T - 2Vso  =  (T + 2V)/3.  (6) \nWhile the virial theorem in a form  of eq.  (5) is a very approximate relation for the real \ninteraction in the solid, the point is that there is always  a general relation between the three terms \nin eq. (3) and the total energy in eq. (6) can always be presented as a function of only two terms.  If we now take into account the perturbative eq. (4), one can show that the total energy change \nwhen SO interac tion is added can be presented as  \nE(2) = V/4 = -T/5,                                    (7) \nwhere the symbol  stands for the difference of the corresponding potential /kinetic energies with \nand without SO coupling.  The large relativistic change of potential energy V= 2V so is a generic \nresult and is valid for both LDA and LDA+U treatment of potential energy terms.  The discussion \nof the total renormalization of the SO coupling  constant  must  evidently  take into accoun t the \ncorresponding change of the kinetic energy term (with opposite sign) T = -5Vso/2. Therefore, \nthe resulting “ screening ” of SO coupling (determined by a competition of large kinetic and \npotential energies) is relatively small and is expressed by eq. (4). The importance of the kinetic \nenergy screening has often been  ignored  [6,38]  and has led to fictitious enhancement of the SO \ncoupling due to correlation effects (see V = 2V so above).  In addition, one can write the \ncorresponding expressions for the kinetic and potential magn etic anisotropies analogous to e qs. \n(4)-(7).  \nOverall, this connection  between different energy terms l eads to an important conclusio n: \nthe total magnetic anisotropy of a system can be presented  equally well using only kinetic, \npotential , or SO energ y anisotrop y. We also mention t hat with the addition of Hubbard terms , the \ntotal potential and its radial derivative are modified. Thus , the atomic (non -renormalized) λ is \nalso modified. In the majority of DFT + U formalisms , the Hubbard corrections are added \nwithout radial dependence and λ is not changed. However, the SO energy is modified as new \nwave functio ns are affected by Hubbard potential terms. Now we will show how significant this \nmodification becomes . \n \nFigure 3. Ratios of (a) GGA + U and GGA SO energies as function of U and (b) total relativistic \nand SO energies as function of SO coupling strength  within GGA + U for YCo 5. \nWe artificially vary the strength of Coulomb correlations (potential ener gy) and \naccording to eqs. (3)-(7) this leads to the corresponding changes in the kinetic energy and overall \nscreening of SO interaction. In our calculations, we  did not see the enhancement of atomic SO \ncoupling constants  due to r-independent Hubbard terms addition . This was  as expected . \nHowever,  the Coulomb correlation s have effects on the SO energy via the self -consistent \nwavefunction. On Fig. 3(a), the site resolved ratio s of SO atomic energies from GGA + U and \nGGA calculations for YCo 5 are shown.  The SO energ y on all atoms is practically unchanged \n(within 4%) with correlations added and only the Co atoms  on the 3g-site losing nearly 10% of \nthe orig inal SO energy. Fig. 3 (b) show s the ratio of total relativistic and SO energ ies obtained in \nGGA + U. The ratio is almost two for the whole range of scaling  SO coupling  strength  /0, thus \nconfirming that the SO coupling in this system is screened by half . This is  due to both kinetic \nand potential energy competition as discussed above.  \n \nFigure 4 . MAE as function of SO coupling  strength for YCo 5, LaCo 5, and CoPt. Data points are \nfrom GGA + U calculations and the dashed and solid lines are fitting curves.  Inset shows \ndifferent fitting curves to MAE of YCo 5. \nIn Fig. 1 , we also show the K so/2 together with K as functions of U and J parameters in \norder to further verify the perturbation theory in treating SO interaction. Clearly , the K so/K ratio \nin LaCo 5 is almost  two for different values of U . The deviation from the perturbation theory is \nlarger for YCo 5 where  Kso/K is between 1.75 and 1.95. Another way to verify the perturbation \ntheory is to investigate the change of MAE with SO  coupling  strength ( λ). We scale  the SO  \ncoupling  strength  by the factor , which ranges  from 0 to 2 . This corresponds  to SO  interaction  \nthat w as previously not included  to doubly  strengthen . We extend  to 2 instead of 1 to get a \nbetter fitting which is able to predict the MAE if we can enhance the SO  coupling . The results \nfrom these scaled SO  coupling  calculations are shown in Fig. 4. If the second order perturbation \ntheory is obeyed, the MAE vs.  curve should be ideally parabolic : K2() = k 22. In our \ncalculation for all YCo 5, LaCo 5 and CoPt, the MAE vs.  curves are fitted better to the function \nincluding  not only the  2nd but also the  4th order terms  [K4() = k 22 + k 44], or the function \nincluding 2nd, 4th, and 6th order terms [K6() = k 22 + k 44 + k 66], if we fit to the whole range \nof  from 0 to 2 . The MAE vs.  curves can only be fitted very well to the parabolic K2() \nfunction if we limit the fitting to small , i.e., smaller than 0.5. The K 2() function with fitted k 2 \nparameters describes  the MAE with  > 0.5 very poor ly.  \nThe inset of F ig. 4 show s the MAE from GGA + U calculations and 3 fitting functions: \nK2(), K 4(), and K 6() for YCo 5. We can see that K 2() fits quite poor ly to the MAE curv e, \nwhile  K4() and K 6() fit very well to the MAE curve. The fitted parameter of the K6() \nfunction are k 2 = 4.677, k 4 = -0.628 , and k 6 = 0.071. It is as expected that the parameter for \nhigher order term s is smaller.  The results  of MAE dependence on SO coupling  strength for \nLaCo 5 and CoPt systems  are similar  to that of YCo 5. The values of  fitted  k2, k4, and k 6 are 4.533, \n-0.098 , and -0.015  for LaCo 5 and 1.088, 0.064 , and -0.019 for CoPt, respectively. These results  \ntogether with the deviation of K so/K from 2 discussed above  show that the perturbation theory \nmay not be enough to describe the MAE of systems with large MAE. This observation is also \nconsistent with previous works on large MAE systems such as CoPt  [31] and FePt  [39]. The \ncauses of the deviation of MAE from the perturbation theory are the non -negligible contributions \nfrom higher order terms of perturbation theory and/or self -consistent effects  [31]. \nIn summary, t hrough  a systematic  investigation of the dependence of MAE of MCo5 \n(with M = Y and La ) on the Co on -site Coulomb  U and exchange J  within the GGA + U scheme , \nwe determine a  universal  set of U and J  parameters  for Co 3d-electrons.  The results from \ncalculations with this set  of parameters  are consistent with experiment s and previous calculations \nnot only  for MCo 5 systems but also for CoPt system, which has a very different crys talline \nstructure than MCo 5. We show that the key factor for large  MAEs  of MCo 5 system s is the large  \nOMA  and not the amplitude of the orbital  moment  itself.  The second order perturbation theory  is \nable to  capture the main contributions , but the higher order terms need to be included for a better \ndescription of systems with large MAE s. Using the simplified vers ion of  the virial theorem with \nSO interaction , we analyze the nature of the SO interaction screening in solids  with Coulomb \ncorrelations in cluded . \nACKNOWLEDGEMENTS  \nThis work is supported by t he U.S. Department of Energy  (DOE), Office  of Science , Basic \nEnergy Sciences, Materials Science and Engineering  Division , including the  grant of computer \ntime at the National Energy Research Scientific Computing Center (NERSC) in Berkeley, CA . \nM. C. N. is also supported by U.S. DOE, Energy Efficiency and Renewable Energy, Vehicles \nTechnology Office, EDT program . Work of V. P. A. is also supported by  the Critical Materials \nInstitute, an Energy Innovation Hub funded by the U.S. DOE.  The research was performed at \nAmes Laboratory, which is operated for the U.S. DOE by Iowa State University under contract # \nDE-AC02 -07CH11358.  \n*Email: mcnguyen@ameslab.gov  \nREFERNCES  \n[1] G. Y. Chin, Science 208, 888 (1980).  \n[2] K. J. Strnat and R. M. W. Strnat, J. Magn. Magn. Mater. 100, 38 (1991).  [3] J. Stöhr and H. C. Seigmann, Magnetism: From Fundamentals to Nanoscale Dynamics , \n1st ed. (Springer -Verlag Berlin Heidelberg, New York, 2006).  \n[4] A. Palasyuk  et. al. , Cerium, Cobalt and Copper Alloy doped with Tantalum or/and Iron as \na Permanent Magnet Mat erial , ISURF #04624/AL676 (2017).  \n[5] A. V. Postnikov and V. P. Antropov, Comput. Mater. Sci. 17, 438 (2000).  \n[6] G.-Q. Liu, V. N. Antonov, O. Jepsen, and O. K. Andersen, Phys. Rev. Lett. 101, 26408 \n(2008).  \n[7] J. M. Alameda, D. Givord, R. Lemaire, and Q. Lu, J. Appl. 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B 94, 144436 (2016).  \n "    },    {        "title": "1707.02694v3.Giant_interfacial_perpendicular_magnetic_anisotropy_in_Fe_CuIn___1_x__Ga__x_Se__2__beyond_Fe_MgO.pdf",        "content": "arXiv:1707.02694v3  [cond-mat.mtrl-sci]  2 Nov 2017Giant interfacial perpendicular magnetic anisotropy in Fe /CuIn 1−xGaxSe2\nbeyond Fe/MgO\nKeisuke Masuda,1Shinya Kasai,1Yoshio Miura,1,2,3,4and Kazuhiro Hono1\n1Research Center for Magnetic and Spintronic Materials,\nNational Institute for Materials Science (NIMS), 1-2-1 Sen gen, Tsukuba 305-0047, Japan\n2Kyoto Institute of Technology, Electrical Engineering and Electronics, Kyoto 606-8585, Japan\n3Center for Materials Research by Information Integration,\nNational Institute for Materials Science (NIMS), 1-2-1 Sen gen, Tsukuba 305-0047, Japan\n4Center for Spintronics Research Network (CSRN),\nGraduate School of Engineering Science, Osaka University,\nMachikaneyama 1-3, Toyonaka, Osaka 560-8531, Japan\n(Dated: September 14, 2021)\nWe study interfacial magnetocrystalline anisotropies in v arious Fe/semiconductor heterostructures\nby means of first-principles calculations. We find that many o f those systems show perpendicular\nmagnetic anisotropy (PMA) with a positive value of the inter facial anisotropy constant Ki. In\nparticular, the Fe/CuInSe 2interface has a large Kiof∼2.3 mJ/m2, which is about 1.6 times larger\nthan that of Fe/MgO known as a typical system with relatively large PMA. We also find that the\nvalues of Kiin almost all the systems studied in this work follow the well -known Bruno’s relation,\nwhich indicates that minority-spin states around the Fermi level provide dominant contributions\nto the interfacial magnetocrystalline anisotropies. Deta iled analyses of the local density of states\nand wave-vector-resolved anisotropy energy clarify that t he large Kiin Fe/CuInSe 2is attributed\nto the preferable 3 d-orbital configurations around the Fermi level in the minori ty-spin states of the\ninterfacial Fe atoms. Moreover, we have shown that the locat ions of interfacial Se atoms are the key\nfor such orbital configurations of the interfacial Fe atoms.\nI. INTRODUCTION\nMagnetic tunneling junctions (MTJs), in which an in-\nsulator barrier is sandwiched between two ferromagnetic\nelectrodes, are the most important practical spintronic\ndevices. They are currently used in non-volatile mag-\nnetic random access memories (MRAM), read heads of\nhard disk drives (HDD), and other magnetic sensors. For\nall these applications, high magnetoresistance (MR) ra-\ntios are requiredfor high-voltageoutput as magnetic sen-\nsors. Second, low resistance-area products ( RA) are es-\nsential for achieving high recordingdensities in HDD and\nMRAM. Moreover, we have an additional requirement\nfor spin transfer torque MRAM (STT-MRAM) [1] and\nvoltage-controlled MRAM [2] applications that MTJs\nneed to have perpendicular magnetic anisotropy (PMA)\nat interfacesbetween the electrodeand the barrierlayers.\nIn particular, for STT-MRAM, PMA is indispensable to\nreduce the critical current for STT switching with suf-\nficient thermal stability [1]. Such MTJs with interfacial\nPMA [3–6] are referred to as p-MTJs.\nIn the early stages of research on interfacial mag-\nnetocrystalline anisotropy, magnetization measurements\nshowed that the interfaces of Co/Pt and Co/Pd het-\nerostructures exhibit PMA with an interfacial anisotropy\nconstant Kismaller than 1mJ /m2[7–9]. It has also been\nclarifiedfrom x-raymagnetic circulardichroism(XMCD)\nspectroscopythat the PMAis attributed to the enhanced\norbital moment of Co, which is induced by the hybridiza-\ntion between Co 3 dand Pt 5 d(Pd 4d) states [10, 11].\nThus, it has been believed that heavy elements with 5 d\nor 4delectrons are required to obtain interfacial PMA.In the past two decades, PMA has also been observed\nin heterostructures with oxide barriers [12–18]. Several\nstudies have found perpendicular magnetic anisotropy at\nthe interface of Co(Fe)/MO x(M= Al, Mg, Cr, Ta,\nRu, etc.) under appropriate oxidation conditions [12–\n15]. Moreover, large PMA with Ki>1mJ/m2has been\nachieved in Fe/MgO [16, 17] and Fe-rich CoFeB/MgO\n[18]. These results suggest that the coupling between O\nand Co(Fe) atoms at interfaces of heterostructures plays\na crucial role in PMA, which was actually confirmed by\nx-ray photoelectron spectroscopy (XPS) measurements\n[14].\nRecently, Kasai et al.[19] succeeded in fabricat-\ning novel MTJs where semiconductor CuIn 0.8Ga0.2Se2\n(CIGS) is sandwiched between ferromagnetic electrodes.\nThey found that the CIGS-based MTJs have both high\nMR ratios (100% at 8K and 40% at room temperature)\nand low RAvalues (0 .3-3Ωµm2) [19]. Such high MR\noutput was explained theoreticallyas a result ofthe spin-\ndependent coherent tunneling of ∆ 1wave functions [20].\nIn a more recent study, Mukaiyama et al.demonstrated\nlarge voltage outputs under bias voltages, which indi-\ncatedthattheCIGS-basedMTJisparticularlyattractive\nfor read-head applications [21]. To consider potential ap-\nplications to STT-MRAM cells, the possibility of obtain-\ning large PMA on CIGS-based MTJs must be investi-\ngated; however, no experimental and theoretical studies\nhave been done on this issue.\nIn the present work, we investigate interfacial mag-\nnetocrystalline anisotropies of various Fe/semiconductor\nheterostructures including Fe/CuIn 1−xGaxSe2by means\nof first-principles calculations. We found that all of the2\nx\nyz\nx\nyz\nx\nyz(b)\n(c)(a)\nFe In Se \nSe \nFe Cu In Ga \nFe OMg Cu \nFIG. 1. Supercells of (a) Fe(7)/CuInSe 2(17), (b)\nFe(7)/CuIn 0.5Ga0.5Se2(17), and (c) Fe(7)/MgO(13).\nFe/CuIn 1−xGaxSe2heterostructures show PMA. In par-\nticular, the Fe/CuInSe 2system exhibits a quite large\nKi≈2.3mJ/m2, which is approximately 1.6 times as\nlarge as that of the Fe/MgO system that is currently\nused for p-MTJs. We also found that Bruno’s relation\n[22], which states that magnetocrystalline anisotropy is\nproportional to the anisotropy in the orbital magneti-\nzation of a ferromagnet, holds for almost all the systems\nconsideredinthisstudy. Thissuggeststhatmagnetocrys-\ntalline anisotropiescanbe described bysecond-orderper-\nturbation theory with respect to spin-orbit interactions,\nin which electron scatterings only in minority-spin states\nare considered. By analyzing the local density of states\n(LDOS) and wave-vector-resolved interfacial anisotropy,\nwe show that the large Kiin Fe/CuInSe 2can be un-\nderstood naturally from the orbital configurations in the\nminority-spin states near the Fermi level.\nII. CALCULATION METHOD\nWe carried out first-principles calculations on the ba-\nsis of density-functional theory (DFT) including spin-\norbit interactions, which is implemented in the Vi-\nennaab initio simulation program (VASP) [23]. For\nthe exchange-correlation energy, we adopted the spin-\npolarized generalized gradient approximation (GGA)\nproposed by Perdew, Becke, and Ernzerhof [24]. The\nprojector augmented wave (PAW) potential [25, 26]\nwas also used to take into account the effect of core\nelectrons properly. In this work, we considered 14\ntypes ofFe/semiconductor(001)heterostructures, the list\nof which is given in Table I. We also considered an\nFe/MgO(001) heterostructure to obtain a benchmark of\nKiunder the present calculation conditions. First, we\nprepared a supercell Fe(7)/ X(17) for each heterostruc-\nture with semiconductor Xas shown in Figs. 1(a) and1(b), where each number in the parentheses represents\nthe number of layers. For the Fe/MgO(001) heterostruc-\nture, we used a supercell Fe(7)/MgO(13) [see Fig. 1(c)]\nbecause the thickness of MgO(13) is close to those of\nthe semiconductors X(17). Since the barrier is suffi-\nciently thicker than the Fe electrode in all the super-\ncells, the in-plane lattice constant aof each supercell\nwas fixed to the experimental lattice constant of the bar-\nrierabarriershown in Table I. Note here that we can\nseta=abarrier/√\n2 in the cases of ZnSe, ZnS, GaAs,\nand MgO barriers due to the high symmetry of the\nstructures. As experimental lattice constants of ternary\nchalcopyrite semiconductors, we adopted the values in\nRef. [27]. We also used the equation aCuIn1−xGaxSe2=\n(1−x)×aCuInSe 2+x×aCuGaSe 2to set the experimen-\ntal lattice constant of CuIn 1−xGaxSe2. Unfortunately,\nVASP cannot treat disorder between In and Ga atoms\nin CuIn 1−xGaxSe2. Thus, we treated this as a percent-\nage of the numbers of atoms in the supercell. For ex-\nample, in the case of Fe/CuIn 0.5Ga0.5Se2(001), we as-\nsigned the same number of atomic sites for In and Ga in\nthe supercell, where the atomic configurations of these\natoms were chosen as shown in Fig. 1(b). In all the\nsupercells, we optimized each atomic position and the\ndistance between the barrier and the Fe electrode so that\nthe total energy of the supercell is minimized. Such\noptimizations reduced the energy of each supercell by\n1∼3eV. In addition, we confirmed that Se, S, or As\nlayers are energetically favored as the interfacial layers in\nall the Fe/semiconductor(001) heterostructures as shown\nin Figs. 1(a) and 1(b) (see Appendix for details). The\ninterfacial anisotropy constant Kiwas calculated using\nthe force theorem as Ki= (E[100]−E[001])/2S, where\nE[100](E[001]) is the total energy of the supercell for\nthe magnetization along the [100] ([001]) direction, Sis\nthe cross-sectional area of the supercell, and the factor\n2 in the denominator reflects the fact that two inter-\nfaces are included in the supercell. In this definition,\na positive (negative) Kishows a tendency toward per-\npendicular (in-plane) magnetic anisotropy. The list of k\npoints used in the calculations of Kiis provided in Table\nI. We used 10 ×10×1kpoints for the heterostructures\nwith ternary or quaternary chalcopyrite semiconductors.\nWe used more kpoints for other heterostructures due to\ntheir smaller supercell sizes.\nIII. RESULTS AND DISCUSSION\nWe show the obtained values of Kiin Table I. We find\nthatallsystemsexceptFe/CuInS 2(001)havepositiveval-\nues ofKi. In particular, Fe/CuInSe 2(001) has the largest\nKiof 2.305mJ/m2, which is about 1.6 times as large\nas our benchmark value 1 .396mJ/m2in Fe/MgO(001).\nSimilar to our previous study [20], we additionally con-\nsidered the effect of the Coulomb interaction Uin the\nCu 3dstates of Fe/CuInSe 2(001) on Ki. We obtained\nKi= 2.363 and 2 .403mJ/m2forU= 5 and 10eV, re-3\n 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35\n 5.73  5.74  5.75  5.76  5.77  5.78Ki [mJ/m 2]\nin-plane lattice constant a‒[Å]\nFIG. 2. In-plane lattice constant dependence of the interfa cial\nanisotropy constant Kifor Fe/CuInSe 2(001).\nΔMorb,i 0.0625\n0.0500\n0.0375\n0.0250\n0.0125\n0\n-0.0125\nΔMorb,i  [ μB/atom] \n-0.5 0  0.5 1  1.5 2  2.5 Ki [mJ/m 2]\nZnSe ZnS GaAs CuInSe \n2CuIn \n0.75 Ga \n0.25 Se \n2CuIn \n0.5 Ga \n0.5 Se \n2CuIn \n0.25 Ga \n0.75 Se \n2CuGaSe \n2CuInS \n2CuGaS \n2AgInSe \n2AgGaSe \n2AgInS \n2AgGaS \n2MgO Ki\nFIG. 3. Correlation between the interfacial anisotropy con -\nstantKiand anisotropy of the orbital magnetic moment at\nthe interfacial Fe layer ∆ Morb,i(see text for details).\n-0.01 0  0.01 0.02 0.03 0.04 0.05 0.06\n 0.0  1.0  2.0  3.0  4.0Fe/CuInSe /g3671(001)\nFe/MgO(001)ΔMorb  [ μB/atom] \ndistance from the interface [Å] \nFIG. 4. Anisotropy of the orbital magnetic moment ∆ Morb\nresolved into each Fe-layer contribution for Fe/CuInSe 2(001)\nand Fe/MgO(001). The horizontal axis shows the distance of\neach Fe layer from the interface.TABLE I. List of in-plane lattice constants aof the het-\nerostructures, kpoints used in the calculations of Ki, and\nthe obtained values of Ki.\nXin Fe/X(001) a(˚A)kpoints Ki(mJ/m2)\nZnSe 4.013 15 ×15×1 1.701\nZnS 3.823 15 ×15×1 1.151\nGaAs 3.998 15 ×15×1 0.210\nCuInSe 2 5.782 10 ×10×1 2.305\nCuIn0.75Ga0.25Se2 5.726 10 ×10×1 2.069\nCuIn0.5Ga0.5Se2 5.698 10 ×10×1 1.712\nCuIn0.25Ga0.75Se2 5.670 10 ×10×1 1.575\nCuGaSe 2 5.614 10 ×10×1 1.266\nCuInS 2 5.523 10 ×10×1 -0.373\nCuGaS 2 5.356 10 ×10×1 0.776\nAgInSe 2 6.109 10 ×10×1 0.721\nAgGaSe 2 5.985 10 ×10×1 1.257\nAgInS 2 5.872 10 ×10×1 0.841\nAgGaS 2 5.754 10 ×10×1 0.027\nMgO 2.982 20 ×20×1 1.396\nspectively, which indicates that the interaction Uin the\nbarriers does not have significant effects on Ki. We also\nstudied the in-plane lattice constant adependence of Ki\nfor Fe/CuInSe 2(001), as shown in Fig. 2. The possible\nsmallest value of ais considered to be twice the lattice\nconstant of bulk bcc Fe, a= 2aFe= 5.732˚A. Thus, we\nchanged afrom 2aFetoaCuInSe 2. Note that this range\nincludes the value of athat is compatible with bcc Cr\noften used as buffer layers, a= 2aCr= 5.768˚A. We see\nthatKichanges smoothly with the value of aand is over\n2mJ/m2for all values of ain the considered range.\nAs mentioned in Sec. II, a positive Kiindicates a ten-\ndency toward PMA. However, to be more precise, the\nfollowing effective anisotropy should be used for a more\naccurate estimation of PMA: Keffteff=Ki−2πM2\nsteff,\nwhereMsis the saturation magnetization and teffis the\neffective thickness of a ferromagnetic electrode. The sec-\nond term 2 πM2\nsteffis the contribution from magnetic\nshape anisotropy, which always favors in-plane magne-\ntization. In our present situation, since bcc Fe has a\nmagnetization of 2 .262µBper atom and teff=tFe/2≈\n0.425nm, the shape anisotropy term is estimated as\n2πM2\nsteff∼0.85mJ/m2, which does not exceed Kiin\nmany systems considered in this study [28]. Therefore,\nwe can conclude that many Fe/semiconductor(001) het-\nerostructures favor PMA even if we use Kefftefffor an\nestimation of interfacial magnetocrystalline anisotropy.\nIn Fig. 3, we show the correlation between the in-\nterfacial anisotropy constant Kiand the anisotropy of\nthe orbital magnetic moment in the interfacial Fe atom\n∆Morb,ifor heterostructures studied in this work. Here,\nthe anisotropy of the orbital magnetic moment is defined\nby ∆Morb,i=M[001]\norb,i−M[100]\norb,i, where M[001]\norb,i(M[100]\norb,i)\nis the orbital magnetic moment of the interfacial Fe\natom for magnetization along the [001] ([100]) direc-\ntion. We clearly see that the so-called Bruno’s relation\nKi∝∆Morb,i[22] holdsforalmostallsystemsconsidered4\nin this study. A previous theoretical work has confirmed\nthis relation in the Fe(Co)/MgO interface [31]. Note,\nhowever, that values of Kiand ∆Morb,iin Fe-based het-\nerostructures do not always follow the Bruno’s relation,\nas shownin asystematic theoreticalstudy [32]. Laan[33]\nindicated that the Bruno’s relation is satisfied when no\nspin-flip scattering occurs and majority-spin states are\nfully occupied. In such a case, the minority-spin scatter-\ning between unoccupied and occupied states around the\nFermi level ( EF) provides the dominant contribution to\nthe magnetocrystalline anisotropy [34]. As shown later,\nFe/CuInSe 2(001) has a suitable LDOS and band struc-\nture to yield large PMA through such minority-spin scat-\ntering. Figure 4 shows the anisotropy of the orbital mag-\nnetic moment ∆ Morbresolved into each Fe-layer contri-\nbution for Fe/CuInSe 2(001) and Fe/MgO(001). In both\nsystems, the interfacialFe layerhas amuch larger∆ Morb\ncompared to other layers. Since the magnetic anisotropy\nis proportional to ∆ Morbin these systems, as mentioned\nabove, the results in Fig. 4 clearly indicate that the\nanisotropy Kiis mainly due to the interfacial contribu-\ntion.\nBefore discussing Fe/CuInSe 2(001) with the largest\nKi, let us first focus on the Fe/MgO(001) for compar-\nison. Previous theoretical studies [35–38] have shown\nthat this system has relatively large PMA with Ki=\n1∼2mJ/m2, which is consistent with our present re-\nsultsKi= 1.396mJ/m2. Figure 5(a) shows the calcu-\nlated LDOS for 3 dstates of an Fe atom located at the\ninterface between Fe and MgO layers, whose main fea-\ntures around EFare consistent with previous results on\nsimilar systems [39, 40]. To understand the relationship\nbetween the LDOS andthe magneticanisotropyconstant\nKi, we introduce the following expression for Kiderived\nfrom the second-order perturbation expansion with re-\nspect to the spin-orbit interaction [34]:\nKi≈ξ2/summationdisplay\no↓,u↓|∝angbracketlefto↓|Lz|u↓∝angbracketright|2−|∝angbracketlefto↓|Lx|u↓∝angbracketright|2\nǫu↓−ǫo↓,(1)\nwhereξis the coupling constant of the LS spin-orbit in-\nteraction, |o↓∝angbracketright(|u↓∝angbracketright) is an occupied (unoccupied) state\nwith minority spin, ǫo↓(ǫu↓) is the energy of the |o↓∝angbracketright\n(|u↓∝angbracketright) state, and Lα(α=x,z) are the usual angular mo-\nmentum operators. In Eq. (1), we considered excitation\nprocesses between the minority-spin occupied and unoc-\ncupied states, and we neglected small contributions from\nspin-flip scattering processes. This approach was shown\nto be sufficient to understand PMA in Fe/MgO(001) sys-\ntems [35]. We can easily see that the matrix element of\nLz(Lx)providesapositive(negative)contributionto Ki.\nIn addition, the excitation energy ∆ ǫ≡ǫu↓−ǫo↓is also\nan important factor: the excitation process with smaller\n∆ǫcontributes more significantly to Ki. In the minority-\nspin LDOS of an interfacial Fe atom in Fig. 5(a), the dxy\nstate has a peak a little below EF(E−EF≈ −0.4eV),\nand thedyz,dzx, anddx2−y2states havepeaks just above\nEF(E−EF≈0.2eV). These states yield finite values-1 -0.5 0  0.5 1 \nE-EF [eV] Projected LDOS [states/eV/orbital] -1.0-0.8-0.6-0.4-0.2 0  0.2\n-0.4 -0.2  0  0.2  0.4\n-0.1-0.05 0  0.05 0.1\n-4 -3 -2 -1  0  1  2  3  4 Projected LDOS [states/eV/orbital] (a)\n(b)\nFIG. 5. The projected LDOSs for (a) Fe 3 dstates and (b)\nO 2pstates at the interface of the Fe/MgO(001) heterostruc-\nture. In each panel, positive and negative values indicate t he\nmajority- and minority-spin projected LDOSs, respectivel y.\nThe inset of panel (a) shows an enlarged view near the Fermi\nlevel.\nof∝angbracketleftdxy|Lz|dx2−y2∝angbracketrightand∝angbracketleftdxy|Lx|dyz(dzx)∝angbracketright. As a result,\npositive contributions from ∝angbracketleftLz∝angbracketrightexceed negative ones\nfrom∝angbracketleftLx∝angbracketright, leading to PMA with positive Ki. Figure\n5(b) shows the LDOSs of 2 pstates in an interfacial O\natom. Note that finite values of LDOSs occur around EF\nalthough bulk MgO is a band insulator. Such states are\ninduced by metallic states of interfacial Fe atoms and\nare thus called metal-induced gap states (MIGS). The\nconcept of MIGS was first introduced as a way to under-\nstanding metal/semiconductor interfaces [41, 42], and it\nwas later applied to Cu/MgO [43] and Fe/MgO [44] in-\nterfaces. By comparing Figs. 5(a) and 5(b), we see that\nstructuresofLDOSsaround EFarequite similarbetween\nFed3z2−r2and Opzstates, which is due to the strong\nhybridization between these states at the interface. Such\nstrong hybridization comes from the geometry of the in-\nterface, where O atoms are on top of Fe atoms in the\nzdirection, as seen in Fig. 1(c). We also see that in\nthe minority-spin LDOSs, the peak structure of the Fe\ndyz(dzx) state just above EFis almost the same as that\nof the O py(px) state.5\n-1 -0.5 0  0.5 1 \n-1.0-0.8-0.6-0.4-0.2 0.0 0.2\n-0.4 -0.2  0  0.2  0.4\nE-EF [eV]Projected LDOS [states/eV/orbital] \n-0.1-0.05 0  0.05 0.1\n-2 -1.5 -1 -0.5  0  0.5  1  1.5  2 Projected LDOS [states/eV/orbital] (a)\n(b)\nFIG. 6. The projected LDOSs for (a) Fe 3 dstates and (b) Se\n4pstates at the interface of the Fe/CuInSe 2(001) heterostruc-\nture. In each panel, positive and negative values indicate t he\nmajority- and minority-spin projected LDOSs, respectivel y.\nThe inset of panel (a) shows an enlarged view near the Fermi\nlevel.\nLet us now discuss the relationship between the large\npositive Kiand LDOSs in Fe/CuInSe 2(001). Figure\n6(a) shows the LDOSs for 3 dstates of an Fe atom lo-\ncated at the interface between Fe and CuInSe 2layers.\nWe can readily identify some sharp peaks in minority-\nspin LDOSs both just above and just below EF. Such\nLDOSsstructuresenableexcitationswith quitesmall∆ ǫ,\nwhich provide large contributions to Kias mentioned\nabove. Asseenfromtheorbitalconfigurationsaround EF\nshown in the inset of Fig. 6(a), these peaks can yield fi-\nnitevaluesof ∝angbracketleftdxy|Lz|dx2−y2∝angbracketright,∝angbracketleftdzx|Lz|dyz∝angbracketright,∝angbracketleftdyz|Lz|dzx∝angbracketright,\nand∝angbracketleftdxy|Lx|dyz(dzx)∝angbracketright. Thus, Fe/CuInSe 2(001) has more\nexcitation processes with positive contributions to Ki\nthan Fe/MgO(001). Moreover, such positive processes\nhave smaller ∆ ǫthan those of Fe/MgO(001). These\nfeatures in the LDOSs support our results that Kiof\nFe/CuInSe 2(001) is about 1.6 times larger than that\nof Fe/MgO(001). We emphasize that such preferable\nLDOSs of interfacial Fe have a close relationship with\nthe LDOSs of interfacial Se shown in Fig. 6(b). Similar-0.2-0.15-0.1-0.05 0  0.05 0.1 0.15 0.2E-EF [eV] (b)(a)\nFIG. 7. The direct microscopic information on PMA in the\nFe/CuInSe 2(001) heterostructure: (a) the in-plane wave vec-\ntor (k/bardbl) dependence of ∆ E(k/bardbl)≡E[100](k/bardbl)−E[001](k/bardbl) and\n(b) the minority-spin bands along the high symmetry lines in\nthek/bardblBrillouin zone. In panel (b), orbital components of\neach band are indicated by colors.\nto the interfacial O LDOSs in Fe/MgO(001), Se LDOSs\nhave finite values around EFdue to the MIGS. In the\nminority-spin Se LDOSs, we can find some sharp peaks\naroundEFwithpz-orbitalcharacter,whichcoupletovar-\nious minority-spin Fe 3 dstates around EF[see the inset\nof Fig. 6(a)]. This is in contrast to the Fe/MgO(001)\ncase, in which pzstates in interfacial O atoms couple\nmainly to d3z2−r2states in interfacial Fe atoms. Such\na difference mainly comes from the difference in the ge-\nometry at the interface between Fe and barrier layers.\nIn Fe/CuInSe 2(001), interfacial Se atoms are not on top\nof Fe atoms in the zdirection, unlike the Fe/MgO(001)\ncase, as shown in Fig. 1(a). Owing to such locations of\ninterfacial Se atoms, pzwave functions of interfacial Se\ncan hybridize with almost all 3 dstates of interfacial Fe,\nnot only with d3z2−r2states; this yields favorable orbital\nconfigurations around EF.\nWe carried out further analysis to obtain more de-\ntailed information on large Kiin Fe/CuInSe 2(001). Fig-\nure 7(a) shows the in-plane wave vector ( k/bardbl) dependence6\n-0.5 0  0.5 1  1.5 2  2.5\n-0.5 -0.4 -0.3 -0.2 -0.1  0  0.1  0.2  0.3  0.4  0.5\nΔEF [eV]Fe/CuInSe2Fe/MgOKi [mJ/m 2]\nFIG. 8. Calculated Kias a function of the change in the\nFermi energy ∆ EFfor Fe/CuInSe 2(001) and Fe/MgO(001).\nThe origin of the horizontal axis ∆ EF= 0 corresponds to\nthe original Fermi energy in each system. The changes in the\nvalence electron number ∆ nat ∆EF=±1 eV are also shown\nin units of electrons/atom.\nof ∆E(k/bardbl)≡E[100](k/bardbl)−E[001](k/bardbl). Here,Kiis propor-\ntional to the sum of ∆ E(k/bardbl) over all k/bardblin the Brillouin\nzone. We obtained positive values of ∆ E(k/bardbl) in wide\nregions of the Brillouin zone, including high-symmetry\nΓ and M points. To understand the origin of the pos-\nitive ∆E(k/bardbl), we plotted in Fig. 7(b) the band struc-\nture of the Fe/CuInSe 2(001) supercell along the high-\nsymmetry lines. Around the Γ point, both the high-\nest occupied and lowest unoccupied bands are generated\nby the hybridization between dyzanddzxstates of Fe,\nwhich can enhance ∝angbracketleftdzx|Lz|dyz∝angbracketrightand∝angbracketleftdyz|Lz|dzx∝angbracketright, giv-\ning positive contributions to Ki. Around the M point,\nthe highest occupied and lowest unoccupied bands con-\nsist ofdxyanddx2−y2states of Fe, respectively. These\nbands can enhance ∝angbracketleftdxy|Lz|dx2−y2∝angbracketrightwith positive contri-\nbutions to Ki. On the other hand, we have small neg-\native values of ∆ E(k/bardbl) around X point. As shown in\nFig. 7(b), since the highest occupied (lowest unoccupied)\nband around X point comes from dzxanddxy(dyzand\ndxy) states, ∝angbracketleftdzx|Lx|dxy∝angbracketrightand∝angbracketleftdxy|Lx|dyz∝angbracketrightcontribute to\nsuch negative values of ∆ E(k/bardbl≈X). The smallness of\n∆E(k/bardbl≈X) is due to the relatively large energy differ-\nencebetweenthe highestoccupiedandlowestunoccupied\nbands [see the denominator of Eq. (1)]. All these band\nstructures around EFare consistent with the orbital con-\nfigurations in Fe LDOSs shown in the inset of Fig. 6(a).\nFinally, we show in Fig. 8 the calculated Kias a\nfunction of the change in the Fermi energy ∆ EFfor\nFe/CuInSe 2(001) and Fe/MgO(001). These curves were\nobtained by changing the valence electron number nin\neach system. We see that Kiin Fe/MgO(001) increases\nslightly for small hole and electron dopings. On the other\nhand,Kiin Fe/CuInSe 2(001) decreases a great deal for\nboth hole- and electron-doped cases, which is consistentwith our findings that the large Kiin this system is\nclosely related to the sharp peaks in the LDOS around\nEFin Fe minority-spin states [see Fig. 6(a)]. In combi-\nnation with large Ki, the sharp decrease in Kiby dop-\nings in Fe/CuInSe 2(001) is useful for the voltage-assisted\nMRAM applications [1, 2].\nIV. SUMMARY\nWe investigated interfacial magnetocrystalline\nanisotropies of various heterostructures consisting of\nFe and non-oxide semiconductor layers by using first-\nprinciples calculations. We found that most of those\nsystems show PMA with a positive interfacial anisotropy\nconstant Ki. In particular, Fe/CuInSe 2was found to\nhavethe largest Ki≈2.3mJ/m2, which is approximately\n1.6 times as large as that of Fe/MgO, being a benchmark\nsystem currently used in p-MTJs. We also found that\nthe Bruno’s relation holds for almost all systems con-\nsidered in this study, which means that the interfacial\nmagnetocrystalline anisotropies are determined mainly\nby minority-spin states around EF. By analyzing the\nLDOS and wave-vector-resolved anisotropy energy, we\nclarified that the large Kiin Fe/CuInSe 2is due to the\npreferable 3 d-orbital configurations around EFin the\nminority-spin states of interfacial Fe atoms. Moreover,\nwe found that the positions of interfacial Se atoms play a\nkey role in the appearance of such orbital configurations\nof interfacial Fe atoms.\nNote added in proof. In Ref. [28], we com-\nmented on magnetic shape anisotropy estimated from\nthe magnetostatic dipole-dipole interaction. After our\nmanuscript was accepted, we found a minor error\nin our program for calculating the magnetic shape\nanisotropy. By using the revised program, we ob-\ntained the shape anisotropy of around 1.1 mJ/m2\nfor the present Fe/semiconductor(001) superlattices,\nwhich is sufficiently lower than the crystalline magnetic\nanisotropy in many of these systems.\nAppendix: INTERFACIAL LAYERS OF\nSUPERCELLS\nTo determine the energetically favored interfacial layer\nof the supercell, we compared formation energies of the\nsupercell for different termination layers of the semicon-\nductorbarrier. Here,weshowthedetailsoftheprocedure\nin the caseofFe/CuInSe 2(001). The relativestability be-\ntweenSe- and CuIn-terminated interfacesisestimated by\nusing the following formation energy [46, 47]:\nEterm\nform=Eterm\ntot−/summationdisplay\niNiµi, (A.1)\nwhereEterm\ntotis the total energy of the optimized super-\ncell for each termination, Niis the number of atoms of7\nthe element i, andµiis its chemical potential. Note\nhere that the chemical potential of each atom does\nnot exceed the corresponding one in the bulk phase,\ni.e., the upper limit of µiis given by the bulk to-\ntal energy per atom. In the present study, we used\nµCu(bulk),µIn(bulk), andµSe(bulk)derived from bulk en-\nergies of fcc Cu, tetragonal In, and hexagonal Se, respec-\ntively. From the thermodynamic equilibrium condition\nµCuInSe 2=µCu+µIn+ 2µSe, we can obtain the lower\nlimit ofµSeasµSe≥(µCuInSe 2−µCu(bulk)−µIn(bulk))/2.\nHere, we adopt µCuInSe 2derived from the bulk struc-\nture. Weestimatedthedifferenceintheformationenergy\nEdiff\nform= (ESe−term−ECuIn−term)/2 in the thermodynam-\nically allowed range ( µCuInSe 2−µCu(bulk)−µIn(bulk))/2≤\nµSe≤µSe(bulk), where the upper (lower) limit of µSe\ncorresponds to the Se-rich (Se-poor) case. As a result\nof the calculations, we obtained Ediff\nform=−2.43eV and\n−0.77eV for the Se-rich and Se-poor cases, respectively.\nThese results indicate that the Se-terminated interface\nis energetically preferred in Fe/CuInSe 2(001). We car-ried out similar analysis in other Fe/semiconductor(001)\nheterostructures to determine their accurate interfacial\nstructures. As a result, it was found that Se, S, or As\nlayers are energetically favored as interfacial layers in all\nof the present Fe/semiconductor(001) heterostructures.\nACKNOWLEDGMENTS\nThe authors are grateful to H. Sukegawa and S. Mi-\ntani for useful discussions and critical comments. This\nwork was partly supported by Grant-in-Aids for Scien-\ntific Research (S) (Grant No. 16H06332) and (B) (Grant\nNo. 16H03852) from the Ministry of Education, Cul-\nture, Sports, Science and Technology, Japan, by NIMS\nMI2I, and also by the ImPACT Program of Council for\nScience, Technology and Innovation, Japan. 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College of Science, Guilin University of Technology, Guilin 541004, PR China\n2. Consiglio Nazionale delle Ricerche CNR-SP IN, UOS L’Aquila, Sede Temporanea di Chieti,\n66100 Chieti, Italy\n3. ISIR-SANKEN, Osaka University, 8-1 Mihogaoka, Ibaraki, Osaka, 567-0047, Japan\nABSTRACT\nIridates supply fertile grounds for unconvent ional phenomena and exotic electronic phases.\nWith respect to well-studied octahedrally-coordinated iridates, we pay our attention to a ratherunexplored iridate, Na\n4IrO 4, showing an unusual square-planar coordination. The latter is key to\nrationalize the electronic structure and magnetic property of Na 4IrO 4, which is here explored by\nfirst-principles density functional theory and Monte Carlo simulations. Due to the uncommonsquare-planar crystal field, Ir 5 dstates adopt intermediate-spin state with double occupation of\n݀\nz2orbital, leading to a sizable local spin moment, at variance with many other iridates. The\nsquare-planar crystal field splitting is also crucial in opening a robust insulating gap in Na 4IrO 4,\nirrespective of the specific magnetic ordering or treatment of electronic correlations. Spin-orbitcoupling plays a minor role in shaping the electronic structure, but leads to a strongmagnetocrystalline anisotropy. The easy axis perpendicular to the IrO\n4plaquette, well explained\nusing perturbation theory, is again closely related to the square-planar coordination. Finally, thelarge single-ion anisotropy suppresses the spin frustration and stabilizes a collinearantiferromagnetic long-range magnetic ordering, as confirmed by Monte Carlo simulationspredicting a quite low Néel temperature, expected from almost isolated IrO\n4square-planar units as\ncrystalline building blocks.I. INTRODUCTION\nRecently, 5 dIr oxides (iridates) have attracted extensive attentions, due to the delicate\ncompetition between the on-site Coulomb correlation U, Hund’s coupling JH, spin-orbit coupling\n(SOC) and crystal field splitting [ 1,2,3,4]. New phases, emerging phenomena and fascinating\nphysical properties have been uncovered for iridates. For example, Ir-based pyrochlores display astrong enhancement of SOC by correlations, changing from topological band insulator intotopological Mott insulator [1], and orthorhombic perovskite iridates AIrO\n3(A = alkaline-earth\nmetal) is proposed as a new class of topological crystalline metals [4]. Most of these studiesfocused on tetravalent (Ir\n4+,5d5) iridates, sharing IrO 6octahedron as a common crystalline basis\nblock, where the 5 dstates are split into triply degenerate t2gstates and doubly degenerate egstates\nby the octahedral crystal field (see Figure 1). As a result of the interplay between SOC and crystalfield splitting, the sixfold degenerate (including the spin degree of freedom) Ir t\n2gstates are split\ninto completely filled quartet Jeff= 3/2 and half-filled doublet Jeff=1 / 2s t a t e s[ 5,6]. The\nhalf-filled Jeff= 1/2 level with a hole state is proposed to be a key factor in driving exotic\nphenomena in iridates [3].\nTo the best of our knowledge, Ir atoms in irida tes have been almost exclusively bonded to\noxygen atoms in the form of octahedra. On the contrary, Na 4IrO 4features one of the few examples\nof square-planar coordination geometry in iridates (Figure 1), composed by loosely connectedIrO\n4square-planar plaquettes [ 7,8]. The isolated square-planar IrO 4plaquettes locate in the ab\nplane with tiny deviations of the Ir-O bonds from the crystallographic a/baxis. For each Ir atom,\nthere are two nearest-neighbor (NN) Ir atoms along the caxis and eight next-nearest-neighbor\n(NNN) Ir atoms along the lattice diagonal. Na 4IrO 4can therefore be viewed as consisting of rigid\nIrO 4clusters almost separated one from the other, arr anged on a body-centered tetragonal lattice.\nThe most remarkable feature in Na 4IrO 4is therefore the uncommon local geometry of IrO 4\nplaquette, which will lead the dorbitals to further split under a square-planar crystal field (as\nschematically shown in Figure 1). The square-planar geometry is frequently found in 3 dtransition\nmetal compounds, such as the infinite-layer cuprates SrCuO 2and CaCuO 2or iron oxide SrFeO 2\n[10]. However, square-planar units are corner-sharing in 3 dcompounds [10], whereas IrO 4\nsquare-planar units are separated one from the other in Na 4IrO 4[8]. One of the few experimental\ninvestigations on Na 4IrO 4showed a temperature dependence of the magnetic susceptibilityexhibiting clear antiferromagnetic (AFM) ordering at 25 K [8], although the detailed magnetic\nstructure was not reported. First-principles density functional theory (DFT) calculations proposedthe crucial role of effective Coulomb interactions (Hubbard U) in determining the crystal structure\nof Na\n4IrO 4. In contrast, the magnetic ordering and SOC was reported to play almost no role in the\ncrystal field splitting, orbital filling and structural instability of Na 4IrO 4[8].\nFigure 1 (a) Crystal structure and spin exchange paths ( J1,J2andJ3)o fN a 4IrO 4. The large (green),\nmiddle (yellow), and small (grey) spheres represent the Na, Ir, and O ions, respectively. We usexyzfor the local coordinates and abcfor the global orientation. (b) Schematic d-orbital splittings\nunder octahedral (left) and square-planar (middle) crystal field, and the actual (right) orders of theenergy level arrangements and the intermediate-spin state for Ir\n4+(5d5)i o n si nN a 4IrO 4(see\nbelow).\nIn the present work, we explored the electronic structure, magnetocrystalline anisotropy\n(MCA) and spin exchange interactions in Na 4IrO 4by performing DFT calculations, complemented\nby Monte Carlo (MC) simulations to predict the Néel temperature and magnetic ground state. Allthe remarkable properties of Na\n4IrO 4are closely related to the crucial square-planar coordination\nin IrO 4plaquettes. The electronic structure shows an energy level splitting consistent with\nsquare-planar crystal field and with strong hy bridizations (both inter-atomic between Ir 5 dand O\n2pstates as well as intra-atomic between Ir 5 ݀z2and Ir 6 sstates), leading to an intermediate-spin\nstate, quite unusual for iridates but expected from almost isolated square-planar IrO 4units. The\ninsulating band gap originates from the strong crystal field splitting, independently on themagnetic ordering and Coulomb interactions. SOC interactions almost have no effect on theelectronic structure, but result in a large eas y-axis MCA (single-ion anisotropy (SIA)) of Ir\n4+ion\nin the unusual square-planar crystal field. Finally, our MC simulations predict a rather low Néeltemperature and a collinear long-range AFM magnetic ordering in Na 4IrO 4, again expected from\nloosely connected square-planar IrO 4plaquettes.\nII. COMPUTATIONAL METHODS AND STRUCTURAL DETAILS\nElectronic structure calculations were carried out using the Vienna Ab Initio Simulation\nPackage (VASP) code [ 11] within the projector augmented wave (PAW) method [ 12,13]. The\ngeneralized gradient approximation (GGA) exchange-correlation functional as parameterized bythe Perdew-Burke-Ernzerhof (PBE) was used for all spin polarized calculations [ 14]. SOC was\nincluded in the simulations using the noncollinear magnetism settings. The rotationally invariant +Umethod introduced by Liechtenstein et al. was employed to account for correlations effects [ 15].\nThe values of the Coulomb interactions Uand the Hund’s coupling J\nHfor Ir 5 dorbitals were fixed\nto 2 and 0.2 eV, respectively. K-point meshes of 8 × 8 × 12 for the primitive unit cell and 6 × 6 × 6\nfor 22 2 supercell (see below) were used for the Brillouin zone integration. The cutoff\nenergy was set to 520 eV for all DFT calculations. The threshold for self-consistent-field energy\nconvergence was chosen as 10−6eV .\nX-ray crystal structure refinements of Na 4IrO 4show a tetragonal structure (space group I4/m)\nwith two formula units (f. u.) per unit cell. According to the symmetry, Na, Ir, and O atoms can beclassified as three nonequivalent crystallographic sites in the unit cell. They are located at 8 c(x, y,\n0), 2 a( 0 ,0 ,0 ) ,a n d8 h(x, y, 0) sites, respectively. From x-ray diffraction experiments, the lattice\nconstants of Na\n4IrO 4were determined to be a=b= 7.184 Å and c= 4.725 Å [8]. In the IrO 4\nsquare-plane, there are two O-Ir-O bonds, which are mutually perpendicular but slightly deviating\nfrom the global crystallographic a/baxis in the abplane. To monitor the behavior of the\nsquare-planar crystal field, a local coordinate system (x , y , z ) defined in Figure 1 is employedfor Ir atoms, with z being exactly perpendicular to the IrO\n4square-plane, and x, y are defined\nexactly along one of the Ir-O bonds in the square-planar IrO 4plaquette.\nBased on experimental lattice parameters, we optimized all independent atomic internal\ncoordinates and lattice constants. As the detailed magnetic structure is not available [8], the AFMordering has been simulated by considering an antiparallel alignment of the spin magnetic momentof two Ir atoms in the unit cell, found from first-principles to be the lowest-energy magnetic state(see below). We confirmed that a reasonable Uparameter and SOC have only a small impact onthe crystal structure. As listed in Table I, our theoretical calculated lattice parameters were in good\nagreement with available experimental and theoretical results, with errors less than 2% for thelattice constants and 4% for the volume. We noted that smaller errors and similar results to ref. 8can be obtained for a nonmagnetic state setting. Electronic structure calculations were carried outwith the relaxed lattice parameters for the AFM state. First, we performed spin polarizedcalculations within GGA, then took Coulomb interactions Uand SOC into account by GGA + U,\nGGA + SOC and GGA + SOC + Ucalculations. For the SOC calculations, the quantization axis\nwas set along [0 0 1] (the crystallographic caxis, except where specifically noted otherwise).\nTABLE I Theoretical calculated and experimental measured lattice constants (Å), unit cell\nvolume (V, Å\n3), atomic internal coordinates and Ir-O bond length (Å) of Na 4IrO 4.\na=bc VNa O\nIr-O\nxyxy\nExp. [a] 7.167 4.713 242.09 0.1962 0.4059 0.2526 0.0815 1.902\nE x p . [ b ] 7 . 1 8 4 4 . 7 2 5 2 4 3 . 8 5 ---- 1 . 9 4 2\nT h e o . [ b ] 7 . 2 0 7 4 . 7 0 4 2 4 4 . 3 3 ---- 1 . 9 3 8GGA [c] 7.256 4.759 250.53 0.197 40 . 4 0 4 90 . 2 5 3 80 . 0 8 2 6 1 . 9 3 7\nGGA+ U[c] 7.280 4.742 251.33 0.196 60 . 4 0 2 80 . 2 5 2 50 . 0 8 1 9 1 . 9 3 2\nGGA+SOC [c] 7.262 4.755 250.78 0.1972 0.4043 0.2536 0.0824 1.937GGA+SOC+ U[c] 7.287 4.739 251.63 0.196 50 . 4 0 2 40 . 2 5 2 40 . 0 8 1 8 1 . 9 3 3\na. ref. 7\nb. ref. 8c. present work\nIII. RESULTS AND DISCUSSIONS\nA. Electronic structure and local magnetic moments\nAs shown in Figure 2, the band structures show strong localized and flat-band character\naround the Fermi level ( E\nF), indicating weak interactions because of the loosely connected crystal\nstructure (Figure 1(a)). Unusual for iridates, an insulating gap has opened up even withoutCoulomb interaction corrections for the AFM state (Figure 2 (a)). Upon inclusion of Coulombinteractions, the band gap remarkably increases, although the essential characteristics of the band\ndispersion are unaffected (Figure 2(b)). Furthermore, as presented in Figures 2 (c) and (d), exceptfor lifting the degeneracy of the d\nxz,yzbands [8], the main features of the band structures remain\nunchanged with or without SOC. An insulating energy gap was obtained even assumingferromagnetic (FM) ordering [8], so that the insulating nature in Na\n4IrO 4does not depend on the\nmagnetic ordering state, Coulomb parameters and SOC, rather being essentially determined by thecrystal field splitting of the unusual IrO\n4square-planar units.\nFigure 2 Band structure of Na 4IrO 4calculated within (a) GGA, (b) GGA + U, (c) GGA + SOC,\nand (d) GGA + SOC + U,w h e r e U= 2 eV. Since spin up and spin down states are degenerate in\nthe AFM state, only spin up subbands are reported in (a) and (b).\nThe detailed electronic structure can be further inspected by the projected density of states\n(pDOS). As shown in Figure 3, due to the inter-atomic interactions between the central Ir and fourligand oxygen ions, the Ir\n4+5dstates form bonding (ranging from -6 to -4 eV) and antibonding\n(from -3 to 1 eV) molecular orbitals, with the Ir antibonding states locate around EFand distinctly\nsplit off. This situation is well consistent with the typical energy level splitting of dorbitals undera square-planar crystal field [ 16] ,w h i c hi ss os t r o n gt h a tI r4+(5d5) adopts an intermediate-spin\ns t a t e( s e eF i g u r e1( b ) ) .O2 pbands are mainly located in a lower energy range, although the\npDOS shows a strong inter-atomic hybridization between Ir 5 dand O 2 ps t a t e s .A ni n s u l a t i n gg a p\nopens up between different spin channels of spin-up (spin-down) dxyand spin-down (spin-up)\ndxz,yzorbitals due to the large exchange splitting. The Ir ݀z2orbitals are the lowest-lying occupied\nstates for both spin channels. The double occupation of the ݀z2orbital (rather than the degenerate\ndxz, yz orbitals, cfr Figure 1(b) and Figure 4(d)), seemingly at variance with the expectation from\ncrystal field theory for the D 4hpoint symmetry (see the conventional energy level sequence\nschematically shown in Figure 1(b), bottom middle) [10], also occurred in another infinite-layer3doxide, SrFeO\n2, with perfect square-planar coordination [10, 15, 17]. The origin of the\n݀z2-double occupation arises from the reduction o f Coulomb repulsion interactions, due to the\nmissing oxygen ions in the direction perpendicular to the IrO 4square-plane (see Figure 1) [ 18]. In\naddition, according to the D 4hpoint group symmetry, Ir 5 ݀z2and 6 sorbital have the same a1g\nsymmetry, therefore resulting in their intra-atomic hybridization (see pDOS in Figure 3 (c)) and, in\nturn, to a large reduction of the exchange splitting for the ݀z2orbitals and, finally, to their\ndouble-occupation [ 19].\nFigure 3 Projected density of states (pDOS): (a) Ir 5d,( b )O2 pand (c) Ir intra-atomic 5 ݀z2and\n6sstates calculated within GGA; (d), (e) and (f) show Ir 5dstates calculated within GGA + U,\nGGA + SOC and GGA + SOC + U. Due to the structural symmetry, the Ir 5dyzand dzxstates\noverlap.\nSOC often significantly influences the 5 dband dispersion and plays an essential role in the\ninsulating ground state for many iridates, with the formation of novel half-filled Jeff=1/2 spin-orbit\ninsulating states [5, 6, 20,21]. The Ir4+ions in these iridates all show low-spin 5 d5(t2g5,eg0)\nelectronic configurations. However, as demonstrated in Figure 2, the difference between the bandstructures with and without SOC is small in Na\n4IrO 4. In addition, different coordination\nenvironments (square-planar vs octahedral) and related crystal fields result in distinct energy levelsplitting and orbital occupation patterns. As presented in Figure 3, just considering the\nantibonding states, the d-electron configuration in Na\n4IrO 4organizes as (z2)1<( x z ,y z )2<( x y )1<\n(x2−y2)0f o rt h es p i n - u ps t a t e s ,a n di ns e q u e n c e( z2)1<( x z ,y z )0<( x y )0<( x2−y2)0for the\nspin-down states (schematically shown in Figure 1(b)). Due to the strong crystal field splitting, the\nlowest݀z2states and the highest ݀x2−y2states are located far from other three dxy,dyz,a n d dzx\nstates (generally defined as t2gorbitals in octahedral or tetragonal crystal field). In this sense, the\nelectronic configurations of intermediate-spin state Ir4+ions in Na 4IrO 4can be viewed as reduced\nto a d3(t2g3,S= 3/2) system and the orbital degree of freedom can be thought as being quenched\n(Leff= 0) for a half filled t2gband. According to SOC Hamiltonian ˆ ˆ ˆ\nSOHS L,t h i sc o u l d\njustify why SOC does not play a dominant role in the electronic structure [ 22,23,24,25].\nTABLE II Calculated spin moment (M S) and orbital moment (M L)o fN a 4IrO 4(values in Bohrmagnetons, positive/negative signs indicate the moment directions).\nMS ML\nIr O Ir O\nGGA ±1.585 ±0.248 - -GGA + U ±1.744 ±0.24 - -\nGGA + SOC ±1.424 ±0.229 ∓0.071 ±0.017\nGGA + SOC + U ±1.592 ±0.224 ∓0.045 ±0.019\nAt variance with other 5 d5iridates [5, 26], as shown in Table II, the orbital moment of Ir4+\nions is much smaller than its spin moment in Na 4IrO 4, indicating that the orbital degree of freedom\nis indeed quenched and SOC effect is small. In addition, the orbital moments are antiparallel to thespin moments for Ir\n4+ions, whereas the orbital moments are parallel to the spin moments for O2-\nions. These results follow Hund’s third rule, according to which the orbital moment and spinmoment should be antiparallel (parallel) for a less (more) than half-filled system. Although thecalculated spin moment of Ir\n4+ions is smaller than 3 μBf o ran o m i n a l S= 3/2 intermediate-spin\n5d5-electron system, the spin moment contributions from O atoms are notably large, revealing the\nstrong inter-atomic hybridizations of Ir 5 dand O 2 pstates, consistent with the pDOS (see Figure\n3). A reduced value of spin moment is very common in iridates because of the strong inter-atomichybridizations between Ir 5 dand O 2 pstates [5, 25, 27,28,29,30,31,32]. However, the spin\nmoments are often smaller than 0.5\nBfor Ir4+ions in other octahedral-coordinated iridates [5, 25,\n27, 28, 29, 30, 31], whereas the hybridization-driven reduction is far smaller in Na 4IrO 4,r e s u l t i n g\nin large local magnetic moments. At the same time, the orbital moments are often as large as twiceof the spin moment in other iridates, where the strong SOC and the large octahedral crystal-fieldsplitting produce an effective J\neff=1/2 state for the Ir4+ion [5]. The Coulomb interactions are often\none order of magnitude smaller in iridates with respect to 3 d-based oxides, and the 5 d\ntransition-metal oxides are expected to be more itinerant because of the larger spatial extent of 5 d\norbitals [31]. However, the effective electronic correlations increase upon decreasing connectivityof IrO\n6octahedra in iridates [ 33]. Therefore, due to the peculiar cry stal structure and square-planar\ncrystal-field splitting, at variance with the expectation from the itinerant of 5 diridates, Na 4IrO 4isthe only iridate showing an intermediate-spin state with large local spin moments, as demonstrated\nby the localized flat-band structure and isolated energy levels of Na 4IrO 4(Figure 2).\nB. MCA and preferred spin orientations\nTable III Calculated MCA energy per M atom (meV) for Na 4MO 4(M = Ru, Rh, Os, and Ir). Total\nenergy values for the spin quantization axis (SAXIS) in the abplane (local [100] and [110]\ndirection) are given with respect to the energy for the SAXIS out of plane (local [001] direction),taken as reference. The SIA energy for Na\n4IrO 4with one Ir atom and three non-magnetic Si-ions\nare given in parentheses.\nSAXIS Ru Rh Os Ir\nGGA + SOC[001] 0 0 0 0\n[100] -4.87 1.80 -21.09 14.88(14.03)[110] -4.86 1.78 -21.43 12.97(12.09)\nGGA + SOC + U[001] 0 0 0 0\n[100] -4.34 1.65 -13.17 15.99(15.74)[110] -4.36 1.63 -13.86 14.69 (14.44)\nIn this paragraph we focus on the MCA in Na 4IrO 4and, for clearer insights, compare it with\nMCA in other hypothetical 4 dand 5 dcompounds with square-planar crystal field. Using the initial\ncrystal structure of Na 4IrO 4,w er e p l a c et h eI ri o n sb yR u4+(4d4), Rh4+(4d5)a n dO s4+(5d4)i o n s ,\nrespectively. For these hypothetical Na 4MO 4compounds (M = Ru, Rh, and Os), when all the\nindependent atomic internal coordinates and Bravais lattice are allowed to fully relax (includingpossible relaxation to different space group, coordination, etc), the lattice symmetry of Na\n4IrO 4\nand the related square-planar coordination are kept as ground state (in contrast to what happens for3d-based Na\n4CoO 4, where the oxygen cage around the 3 d-metal turns to tetrahedral) [8]. Our\ncalculations predict these Na 4MO 4compounds to show optimized lattice parameters very similar\nto Na 4IrO 4. As reported in Table III, the total energy within GGA + SOC (with or without U)\nstrongly depends upon the relative orientation of the spin quantization axis, leading to a sizeableMCA. While this is consistent with the strongly anisotropic coordination in IrO\n4“isolated”plaquettes, it would be interesting to experimentally investigate this aspect.\nIn particular (see Table III), for Na 4IrO 4and Na 4RhO 4with d5electronic configurations, the\nconfiguration with the spin moments parallel to the caxis (out of plane) is more stable than that\nwith the spin moments in the abplane. In contrast, for Na 4OsO 4and Na 4RuO 4with d4electronic\nconfigurations, the states with the spin moments in the abplane are energetically favored. In other\nwords, the d5compounds show an easy-axis anisotropy, whereas the d4compounds show an\neasy-plane anisotropy.\nThe MCA and preferred spin orientations can be analyzed via perturbation theory [ 34,35,\n36,37], where SOC is included to couple spin and orbital angular momentum ( ˆSand ˆL),\nresulting in the SOC Hamiltonian, ˆ ˆ ˆ\nSOHS L,λbeing the SOC constant. Employing two\nindependent coordinate systems (x, y, z) and (x’, y’, z’) for the orbital ˆLand spin ˆS,\nrespectively, the SOC Hamiltonian ˆ ˆ ˆ\nSOHS Lis rewritten as0' ˆ ˆˆ\nSO SO SOHHH ,w h e r et h e\n“spin-conserving” term\n0\n'\n'11ˆ ˆˆ ˆ ˆ (c o s s i n s i n )22\nˆˆˆ ˆ ( cos sin cos sin sin )ii\nSO z z\nzz x yHS L L e L e\nSL L L \n   \n \n(1)\nand the “spin-non-conserving” term\n'2 2\n'\n22\n'\n''1ˆ ˆˆ ˆ ˆ ( cos sin cos )22 2\n1ˆ ˆ ˆˆ ( sin cos sin )22 2\n1ˆˆ ˆ ˆˆ ( )( sin cos cos cos sin )2ii\nSO z\nii\nz\nzx yHS LL e L e\nSL L e L e\nSS L L L\n \n\n   \n \n\n\n \n \n  \n(2)\nwhereθandϕdefine the magnetization direction (z’) with respect to the (x, y, z) coordinate\nsystem [ 38]. The energy correction by SOC is given by\n2\n,ˆ\nsoc\nsoc\neg gegH e\nEEE\n(3)\nwhere gand eare the ground (occupied) and excited (unoccupied) states,gE and\neEare the corresponding unperturbed energies [33-36, 39,40].Figure 4 The pDOS for M dstates for Na 4MO 4compounds (M = Os4+,R u4+and Rh4+): (a) Os 5d,\n(b) Ru 4dand (c) Rh 4dstates calculated within GGA. Panel (d) shows the schematic energy level\nsplitting by the square-planar crystal field and the orbital occupations for the d5and d4\nconfiguration, where the SOC between unperturbed occupied and unoccupied dstates is explicitly\nhighlighted.\nAs shown in the pDOS of Figure 3 and Figure 4, electronic structure calculations indicate\nthat the crystal field splitting is the same for these isostructural 4 dand 5 dcompounds and typical\nof square-planar splitting [10, 15, 16, 17, 18]. The crystal field splitting is so strong that Ru4+(4d4),\nRh4+(4d5), Os4+(5d4)a n dI r4+(5d5) ions all adopt intermediate-spin states with double occupation\nof the݀z2orbitals. Due to the large spin exchange splitting and crystal field splitting, even\nwithout Hubbard Ucorrections, the insulating gaps open up between different (same) spin\nchannels for Na 4IrO 4and Na 4RhO 4(Na 4OsO 4and Na 4RuO 4) with d5(d4) electronic configurations.\nAs schematically shown in Figure 4 (d), with the same d5electronic configurations, Na 4IrO 4\nand Na 4RhO 4display the same energy level splitting and orbital occupations. The smallest energygap between the occupied and unoccupied levels occurs between the dxy(spin-up) and the dxz, yz\n(spin-down) levels. These levels differ in their magnetic orbital quantum number m b y1[ 3 4 ,\n35]. Because the occupied and unoccupied dstates couple within opposite-spin channels, the SOC\nHamiltonian will be governed by spin-non-conserving term'ˆ\nSOH (equation (2)), and the\nperturbation matrix element ˆ\nsocgH e will be proportional to cos[41]. As such, the\nSOC-induced interactions are maximized when the spin magnetization direction is parallel to the\norbital z-axis ( i.e.,θ=0 ° )i nN a 4IrO 4and Na 4RhO 4.\nOn the other hand, the situation is different when considering Na 4OsO 4and Na 4RuO 4with d4\nelectronic configurations. As presented in Figure 4, the d-electron configurations in Na 4OsO 4and\nNa4RuO 4align as (z2)1<( x z ,y z )2<( x y )0<( x2−y2)0for the spin-up states, and in an order of (z2)1\n<( x z ,y z )0<( x y )0<( x2−y2)0for the spin-down states. According to the schematic energy diagram\nin Figure 4 (d), the smallest energy gap between the occupied and unoccupied levels occurs nowin the same spin-up (or spin-down) channel for d\nxz,yzanddxyorbitals, differing in their magnetic\norbital quantum number m by 1 [34, 35]. SOC interactions couple occupied and unoccupied\ndstates within the same spin channel, so the SOC Hamiltonian will be governed by\nspin-conserving term ˆ\nSOH (equation (1)), and the perturbation matrix element ˆ\nsocgH e will\nbe proportional to sin[40]. In this case, the SOC-induced interactions are maximized when\nthe spin magnetization direction is perpendicular to the orbital z-axis ( i.e.,θ= 90°). Therefore,\nNa4OsO 4and Na 4RuO 4show easy-plane anisotropy, in contrast with the easy-axis anisotropy in\nNa4IrO 4and Na 4RhO 4.\nAccording to previous works [38, 39], we can further evaluate the perturbation matrix\nelement ˆ\nsocgH e and hence the energy correction by SOC. Noting that\nˆ 0soc xy H xy  , for the case of Na 4IrO 4and Na 4RhO 4with d5electronic configurations,\nthe second-order energy shift is given by\n22 2\n14 13411 111sin22 424socE           (4)\nwhere1,3 and4 are the energy gaps for the occupied and unoccupied levelsbetween the dxy(spin-up) with dxz, yz(spin-down), dxz, yz(spin-up) with dxz, yz(spin-down) and dxz, yz\n(spin-up) with dxy(spin-down) orbitals, respectively.\nFor the case of Na 4OsO 4and Na 4RuO 4with d4electronic configurations, the second-order\nenergy shift is given by\n2\n22\n11 3 4111sin24 2 4socE     (5)\nwhere1,3 and4 are the energy gaps for the occupied and unoccupied levels\nbetween the dxz, yz (spin-up) with dxy(spin-up), dxz, yz (spin-up) with dxz, yz (spin-down) and dxz, yz\n(spin-up) with dxy(spin-down) orbitals, respectively.\nAs shown in equations (4) and (5), the azimuthal ϕdependence vanished in the perturbation\ntheory up to second order for the energy shift. According to the energy level arrangements and therelated energy gaps from the pDOS of Figure 3 and Figure 4, the angle dependent parts of\nequations (4) and (5) show a\n2sindependence with positive/negative values for the Na 4MO 4\ncompounds with d5(d4) electronic configurations, indicating that the magnetization easy axis is\nout of (resides in) the abplane. We recall that the simple dependence of the total energy on the\nmagnetization angle θdeduced from second order perturbation theory [38, 42], can be expressed\nas:\n2\n01 () s i nEE K   (6)\nTo carefully evaluate the dependence of total energy on θ, we performed a series of\ncalculations by rotating the magnetization angle θ. As shown in Figure 5, the calculated results fit\nwell with what expected from equation (6) for all the Na 4MO 4(M = Ir4+,O s4+,R u4+and Rh4+)\ncompounds. The MCA energy (MAE) curves display opposite trend for the d5(Na 4IrO 4and\nNa4RhO 4)a n d d4(Na 4OsO 4and Na 4RuO 4) compounds, consistently with the opposite sign of\nequations (4) and (5) for the angle dependent parts of the energy corrections by SOC. The MAEbehavior (Figure 5 (a)) shows a minimum for Na\n4IrO 4and Na 4RhO 4at magnetization direction\nalong the crystallographic caxis, corresponding to the easy-axis anisotropy ( i.e.,t h eθ=0 °s p i n\norientation) of d5materials. In contrast, the MAE behavior (Figure 5 (b)) displays a minimum for\nNa4OsO 4and Na 4RuO 4for magnetization perpendicular to the caxis, in line with the easy-plane\nanisotropy ( i.e.,t h eθ= 90° spin orientation) of d4materials.Figure 5 Dependence of the total energy on the magnetization angle θfor the Na 4MO 4(M = Ir4+,\nOs4+,R u4+and Rh4+) compounds and fitted with a function, like Asin2(the line). The\ncalculations were performed within GGA including SOC (with or without U).\nC. Single-ion anisotropy and spin exchange interactions\nFrom a general point of view, SOC can lead to inter-site Dzyaloshinskii-Moriya (DM)\n(antisymmetric) interaction, to anisotropic exchange and to single-ion (or single-site) anisotropy(SIA). According to the crystal symmetry, the DM interaction should not appear, due to thepresence of inversion symmetry in Na\n4IrO 4; as for anisotropic exchange, we expect it to be a small\nrelativistic correction to the isotropic exchange. On the other hand, we focus on the SIA, which ismainly determined by the metal center and its first coordination crystal field [ 43]. As a further\nconfirmation of the magnitude of the SIA of a specified Ir ion, we replaced the three neighboringIr\n4+ions with nonmagnetic Si4+ions in a supercell doubled along the crystallographic caxis. In\nthis way, all other intersite NN or NNN magnetic exchange interactions vanish and the onlycontribution left is the SIA of the Ir\n4+ion. After checking that the crystal field splitting is\nunchanged with respect to the original configuration in Na 4IrO 4, our results (see Table III) show,\nas expected, a comparable magnitude of MCA and SIA energies.\nBased on DFT electronic structure calculations for various spin-ordered magnetic insulating\nstates, the spin exchange parameters can be obtained by mapping the relative energies of themagnetic ordered states onto Heisenberg or Ising Hamiltonian [ 44,45,46]. Generally, the spin\nHamiltonian can be described by the classical Heisenberg model:\n,\n,1\n2ij i j\nijHJ S S \n(7)where Sirepresents a spin operator at site iof the compound and the negative/positive value of J\ndenote AFM/FM interactions, respectively.\nThe spin exchange parameters J1,J2,a n d J3in Na 4IrO 4are illustrated in Figure 1. Based on\nthe optimized lattice parameters for the AFM unit cell as shown in Table I within GGA, weartificially construct five special magnetic ordering states ( i.e.,F M ,A F M 1 ,A F M 2 ,A F M 3a n d\nAFM4). The hypothetical FM state corresponds to a parallel alignment of all magnetic moments,whereas the other four AFM states are symmetry broken arrangements in the\n22 2 supercell.\nFigure 6 Schematic representations of the four hypothetical AFM spin ordering arrangements in\nthe 22 2 supercell (only IrO 4square-plane are shown for clarity): (a) AFM1 (all\nmagnetic moments are antiparallel to each other both in the abplane and along caxis), (b) AFM2\n(all magnetic moments are parallel to each other both in the abplane and along caxis, but the\nNNN magnetic moments are antiparallel aligned), (c) AFM3 (all magnetic moments areantiparallel to each other along caxis, but parallel aligned in the abplane ) and (d) AFM4 (all\nmagnetic moments are parallel to each other along caxis, but antiparallel aligned in the abplane).\nThe big (yellow) and small (gray) spheres denote the Ir and O atoms; the up (down) arrowsrepresent the magnetic moment orientations. The FM planes for the AFM2 magnetic ground stateare highlighted in (b), in order to better illustrate the stacking of FM planes antiferromagnetically\ncoupled along the c-axis.\nAs shown by the DFT calculation results, the Ir4+ions are in intermediate-spin states with\nformal S=3 / 2i nN a 4IrO 4(see Figure 4(d)). In terms of exchange parameters, the spin exchange\ninteraction energies (per f.u.) of the five magnetic ordering states are written as\n1239(42 )4FMEJ J J  \n11 39(2 )4AFMEJ J  \n21 2 39(42 )4AFMEJ J J  \n31 39(2 )4AFMEJ J  \n41 39(2 )4AFMEJ J \n(8)\nThus, by mapping the energy differences of these states in terms of the spin-exchange\nparameters with the corresponding energy differences from DFT calculations, we obtain\n14 12()9AFM AFM JE E \n221()18AFM FM JE E\n33 11()9AFM AFM JE E \n(9)\nTable IV Energy difference relative to the reference AFM2 state (meV/f. u.) and calculated spin\nexchange parameters (meV).\nAFM1 AFM2 AFM3 AFM4 FM J1 J2 J3 J1/J2\nGGA 9.21 0 9.05 11.33 23.74 -0.47 -1.32 0.02 0.36\nGGA + U 4.21 0 4.09 6.16 12.61 -0.43 -0.70 0.01 0.62\nGGA + SOC 8.48 0 8.27 10.50 21.93 -0.45 -1.22 0.02 0.37\nGGA + SOC + U 3.93 0 3.79 5.89 11.98 -0.43 -0.67 0.02 0.65Using the calculated energy values of the five magnetic ordering states, we obtain the\nspin-exchange parameters summarized in Table IV. SOC has a small impact on the spin exchangeparameters, whereas the Coulomb interactions show remarkable influence, because the exchangecoupling parameters Jare inversely proportional to the Hubbard U[17]. The AFM2 state is the\nmost stable, its total energy being lower than the other four magnetic states. The energies of theAFM1 state are comparable to the AFM3 state, reflecting very weak spin coupling interactions J\n3\nin the abplane. The negligible J3is consistent with the loosely connected structure and the large\nin-plane distances (about 7.2 Å) between the Ir4+ions along aorbaxis. The other two spin\nexchange interactions J1and J2are AFM, and the NN interaction J1is smaller than the NNN\ninteraction J2, showing an inverse trend with respect to the distances for the NN (about 4.7 Å\nalong caxis) and NNN (about 5.6 Å along the diagonal of the unit cell) Ir4+ions. However, this is\nreasonable, when considering the unusual crystal structure of Na 4IrO 4, where a given Ir site has\ntwo NN and eight NNN coordination Ir4+ions. It should also be noted that both the NN and NNN\ninteractions are AFM, so a geometrical magnetic frustration might arise in Na 4IrO 4, as the system\ncannot simultaneously satisfy all the NN and NNN AFM spin exchange interactions. However thelarge SIA favors the collinear alignment of t he magnetic moments. Indeed, according to the\nexperimental results, the frustration index f=|θ|/T\nNis close to 3, and the calculated ratio of J1/J2\nare far from 1 in all the cases (see Table IV), so a spin frustration does not occur, as confirmed by\nthe AFM ordering obtained from magnetic susceptibility measurements [8].\nUsing the UppASD (Uppsala Atomistic Spin Dynamics) package [ 47], we perform MC\nsimulations to capture the dynamical properties of the spin systems at finite temperatures for a 16× 16 × 16 supercell based on the classical spin Hamiltonian [17]:\n2\n,\n,1\n2ij i j i z\nij iHJ S S K S  \n(10)\nwhere the spin exchange parameters Ji,jwithin GGA + SOC + Uare summarized in Table IV,\nwhile the SIA energy2\nizKS is given in Table III. To obtain the transition temperature TN,w e\nevaluate the order parameter (i.e. staggered magnetization related to the AFM2 magnetic\nconfiguration) and the specific heat at a given temperature T. As shown in Figure 7, without\nconsidering the SIA, the order parameter and the specific heat give similar results. The criticaltemperature TNis 28 K, evaluated from the peak position of the specific heat or from the values\nwhere the order parameter becomes negligible. The critical temperature as well the height of thespecific heat peak increase upon including SIA in the MC simulations. The value of T\nNincreases\nto 57 K, similar to the case in another infinite-layer oxide SrFeO 2with square-planar coordination,\nwhere the critical temperature also increases when adding the magnetic anisotropy energy [17].The magnetism is always collinear in Na\n4IrO 4, considering the SIA, the spin moments being along\nthecaxis perpendicular to the IrO 4s q u a r e - p l a n e .I nb o t hc a s e s ,N a 4IrO 4relaxes to the same AFM2\nmagnetic ground state with FM abplanes, antiferromagnetically coupled out-of-plane,\ncorresponding to an antiparallel alignment of the spin magnetic moment of two Ir atoms in thecrystallographic unit cell, revealed by first-principles calculations to be the lowest-energymagnetic state.\nFigure 7 Order parameter (solid line and left axis) an d specific heat (dashed line and right axis)\nof Na 4IrO 4calculated as a function of temperature on the basis of the classical spin Hamiltonian\nwithout SIA (red curves) and with SIA (green curves).\nIV. CONCLUSIONS\nIn summary, the novel square-planar coordination of IrO 4plaquettes plays a crucial role in\nthe electronic structure and magnetic properties of Na 4IrO 4, as shown by our comprehensive DFT\ncalculations joint with MC simulations. The unusual square-planar crystal field and the stronghybridization effects give rise to an intermediate-spin state and to an insulating electronic structure,robust against different magnetic ordering, Coulomb parameters and even relativistic interactions.SOC produces a large MCA with an easy axis along the caxis perpendicular to the IrO\n4square-plane. When spin exchange interactions are evaluated by total energy calculations and\nmapping analysis, quite weak AFM interactions are obtained, consistent with the picture of ratherisolated IrO\n4units. Moreover, MC simulations predict a q uite low Néel temperature, consistent\nwith experiments, and a collinear long-range AFM magnetic ground state. We hope our theoreticalsimulations will stimulate experimental works aimed at detailed magnetic propertiesmeasurements and characterizations, to further understand the magnetic ground state and exploitthe large anisotropy of the uncommon square-planar coordinated Na\n4IrO 4.\nACKNOWLEDGMENTS\nWe thank P. Barone, J. Hellsvik and K. Liu for useful discussions. This work was supported by the\nCARIPLO Foundation through the MAGISTER Project Rif. 2013-0726 and by IsC43\n“C-MONAMI” Grant at Cineca Supercomputing Center. X. M. was sponsored by the China\nScholarship Council (No. 201508420180), National Natural Science Foundation of China (No.\n31600592), Scientific and Outstanding Young Science and Technology Innovation Team Program\nof Hubei Provincial Colleges and Universities (No. T201514).\nReference\n1D. Pesin and L. Balents, Nat. Phys. 6, 376 (2010).\n2W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Annu. Rev. Condens. Matter Phys. 5,\n57 (2014).\n3J. G. Rau, E. K. H. 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Wang1\n1Ames Laboratory, U.S. Department of Energy, Ames, Iowa 50011, USA\n2Department of Physics and Astronomy, University of Nebraska, Lincoln, Nebraska 68588, USA\n31137 W Emerald Ave, Mesa, Arizona 85210, USA\n4Department of Materials Science & Engineering, Iowa State University, Ames, Iowa 50011, USA\n(Dated: October 27, 2021)\nMicromagnetic simulations of alnico show substantial deviations from Stoner-Wohlfarth behavior\ndue to the unique size and spatial distribution of the rod-like Fe-Co phase formed during spinodal\ndecomposition in an external magnetic \feld. The maximum coercivity is limited by single-rod\ne\u000bects, especially deviations from ellipsoidal shape, and by interactions between the rods. Both\nthe exchange interaction between connected rods and magnetostatic interaction between rods are\nconsidered, and the results of our calculations show good agreement with recent experiments. Unlike\nsystems dominated by magnetocrystalline anisotropy, coercivity in alnico is highly dependent on size,\nshape, and geometric distribution of the Fe-Co phase, all factors that can be tuned with appropriate\nchemistry and thermal-magnetic annealing.\nThe anisotropy of most permanent magnets is of mag-\nnetocrystalline origin, meaning that hysteresis and coer-\ncivity rely on the atomic-scale interplay between spin-\norbit coupling and crystal-\feld interaction [1{3]. Alnico\nmagnets|a family of nanostructured alloys consisting\nprimarily of Fe, Al, Ni, and Co|are an exception, be-\ncause their magnetic anisotropy and hysteresis originate\nalmost entirely from magnetostatic dipole-dipole interac-\ntions [4{7]. These materials have attracted renewed at-\ntention in the context of magnetic materials that are free\nof rare-earth elements and do not contain other expensive\nelements, such as Pt [7{14]. The magnetic anisotropy of\nalnicos re\rects their peculiar nanostructure, where high-\nmagnetization rods with an approximate composition of\nFeCo (\u000b1-phase) are embedded in an essentially nonmag-\nnetic Al-Ni-rich matrix ( \u000b2-phase) [4{7, 14{18].\nThere are several grades of alnico magnets, character-\nized by di\u000berent chemical compositions and microstruc-\ntures. We focus on a high quality grade, namely, alnico 8.\nFigure 1 shows a high-angle-annular-dark-\feld (HAADF)\nscanning transmission electron microscopy (STEM) im-\nage of an alnico 8 sample along the longitudinal direc-\ntion. The\u000b1rods in the sample of Fig. 1 have diameters\nof\u001825{45 nm and lengths of \u0018100{600 nm and are uni-\nformly distributed in the \u000b2matrix. Some of the \u000b1\nrods have pointy ends and/or touch each other. The\ndetailed microstructures strongly depend on alloy com-\nposition and heat treatment conditions. Details of alnico\nalloy fabrication and microstructure characterization are\nreported elsewhere [7, 19].\nAlnico magnets have high Curie temperatures and\nmagnetizations, but their modest coercivity limits the\nperformance of this otherwise very good permanent-\nmagnet material. Surprisingly, the understanding of al-\nnico coercivity in terms of the dipolar anisotropy has re-\nmained very poor quantitatively and even qualitatively.\n\u0003Corresponding author: liqinke@ameslab.gov\n50nm\tα1\tα2\tFIG. 1. HAADF STEM image showing a side view of \u000b1rods\ndistributed in the \u000b2matrix in an alnico 8 alloy.\nThe main reason is the multiscale character of the calcu-\nlations, which involves local features having sizes of less\nthan 5 nm, but ranging to interactions on scales compa-\nrable to or exceeding the wire length of about 1 µm. Only\nrecently, computer power has become su\u000ecient to treat\ninteractions between the rods. In this letter, we present\nmicromagnetic simulations [20] to quantitatively explain\nthe coercivity of alnico magnets. In particular, we show\nand analyze how the coercivity depends on structural\nfeatures, namely, the shape of the rod ends, the spatial\narrangements of Fe-Co rods in the magnet, and the cross-\ning and/or branching of rods.\nThe conventional explanation of alnico anisotropy is\nshape anisotropy similar to that of small, elongated\nStoner-Wohlfarth particles [21, 22]. In this approxima-\ntion, an aligned magnetic rod or \\elongated \fne particle\"\nis subject to a mean-\feld-like interaction with neighbor-\ning rods, and the corresponding coercivity is given by the\nsemi-empirical formula [21]\nHc= (1\u0000p)(N?\u0000Nk)Ms: (1)\nHere,N?\u00191=2 andNk\u00190 are the demagnetization fac-arXiv:1707.04180v1  [cond-mat.mtrl-sci]  13 Jul 20172\ntors perpendicular and parallel to each rod's long axis,\nMsis the magnetization of the rods, and pis their pack-\ning fraction in the nonmagnetic matrix. However, the\ncoherent-rotation or Stoner-Wohlfarth model has several\nlimitations.\nFirst, it is limited to rods of very small diameters, less\nthan 2Rcoh\u001920 nm [23{25], whereas typical alnico rods\nhave diameters of the order of 2 R= 40{50 nm (Fig. 1).\nIn such relatively thick rods, the magnetization reversal\nstarts by magnetization curling, for which the nucleation\n\feld is described by [6, 23{27]\nHc=2K1\n\u00160Ms\u0000NkMs+c(Nk)A\n\u00160MsR2: (2)\nHere,K1is the magnetocrystalline anisotropy constant,\nAis the exchange sti\u000bness [28], and the values of c\nare 8.666 for spheres ( Nk= 1=3) and 6.678 for needles\n(Nk= 0). In alnicos, the K1term is small compared to\nthe magnetostatic terms, contributing only about 10 %\ntoHc[29], and usually neglected [13, 21]. In Eq. (1),\nthe di\u000berence N?\u0000Nk= 1\u00003Nkis positive for Nk<1=3\n(shape anisotropy), but the corresponding curling term\n\u0000NkMsin Eq. (2) is always negative. The last term\nin Eq. (2) partially compensates the coercivity loss due\nto curling but depends on the radius R. Since alnico\nrods are thick enough that reversal starts via curling,\nit is simplistic to interpret alnico anisotropy as shape\nanisotropy [25, 30]. In reality, \u0000NkMsis negative but\nsmall enough that the exchange term in Eq. (2) ensures\na positive coercivity unless the rod diameter 2 Ris very\nlarge. In practice, the diameter|determined by the spin-\nodal decomposition process|is su\u000eciently small to con-\ntribute some coercivity but not small enough to reach the\nStoner-Wohlfarth limit of Eq. (1).\nSecond, Eqs. (1{2) are only valid for ellipsoids. For\nreal structures other than ellipsoidal shapes, N?andNk\nare no longer well-de\fned, and the non-ellipsoidal shapes\nhave smaller coercivity than the ellipsoidal shape [24, 25,\n31]. This complication, commonly referred to as Brown's\nparadox, can be resolved only by explicit consideration\nof the magnet's real structure (bulk microstructure).\nThird, interactions between rods are simplistically\ntreated in Eq. (1) and formally ignored in Eq. (2). To be\nprecise, Eq. (1) replaces the complicated magnetostatic\ninteraction by a p-dependent mean \feld. The coercivity\nis the largest for distantly spaced rods ( p\u00190) but van-\nishes completely in the limit of continuous soft-magnetic\nthin \flms ( p= 1). The energy product, which suc-\ncinctly expresses the performance of a permanent mag-\nnet, reaches its maximum at p= 2=3 in this approxima-\ntion [13, 32], agreeing fairly well with experiment. The\nsame magnetostatic interaction e\u000bect could be included\nin Eq. (2) by introducing e\u000bective demagnetizing factors\n[33], but this does not alleviate the basic shortcomings\nof the approximation. In fact, the coercivity strongly de-\npends on the spatial arrangement of the magnetic rods,\nas we will see below.\nFourth, it is known experimentally that the rods or\\wires\" of the magnetic phase undergo crossing and\nbranching [6, 7, 14]. These features are likely to strongly\na\u000bect coercivity, and their treatment requires demanding\nnumerical calculations. In the theoretical literature, this\ne\u000bect has not been addressed so far.\nTo simulate the coercivity, we model the alnicos as\naligned or \\anisotropic\" magnets consisting of parallel\nmagnetic rods, thereby establishing a unique k(parallel)\naxis. This structural model reproduces the key feature\nof alnico microstructure, namely, magnetic rods in an es-\nsentially nonmagnetic matrix (Fig. 1). Some alnicos have\nrod orientations that di\u000ber from the global magnetization\naxis, i.e., the rods are misaligned [6, 7, 14]. This case is\nphysically very di\u000berent [34] from the presently aligned\nrods and goes beyond the scope of the present paper.\nOur calculations use the recently developed MuMax3\nmicromagnetics code [35]. We employed either clusters\nof particles or periodic boundary conditions (PBCs) to\ninvestigate the interactions between rods. In our calcula-\ntions, we used a 1 nm grid and 1 Oe \feld steps to produce\nthe hysteresis loops. At each \feld step, we computed\nequilibrium magnetic states by directly minimizing en-\nergy using the steepest descent method [35, 36] as imple-\nmented in MuMax3 . To simulate the (Fe-Co)-rich mag-\nnetic phase, we have assumed a saturation magnetization\nof\u00160Ms= 2:1 T and exchange sti\u000bness of A= 11 pJ=m.\nTo explore the coercivity's sensitivity to these inputs, we\nalso varied MsandAand found that Hconly changes a\nsmall amount if the ratio A=M sremains constant. For\nexample, doubling both MsandA(for an ellipsoid with\nD= 32 nm and c=a= 8) yields a coercivity di\u000berence of\nless than 10 %. Finite-temperature dynamic e\u000bects have\nnot been considered explicitly, because they are known\nto yield well-understood logarithmically small magnetic-\nviscosity corrections [3, 25].\nFigure 2 visualizes the initial stage of magnetization\nreversal, which is of the curling type, and shows the cor-\nresponding hysteresis loops. In the cuboids, the curling\nstarts at the ends of the rod, in the middle of the short\nedges, propagates along the middle of the long faces, and\neventually advances to the long edges and center of the\nrod. The curling mode in the ellipsoids is essentially delo-\ncalized throughout the rod, in agreement with exact ana-\nlytic calculations [27]. Note the nonrectangular (curved)\nshape of the cuboid hysteresis loops.\nIt is straightforward to extract the coercivities from\nthe calculated hysteresis loops. Figure 3 shows the co-\nercivities of isolated rods of aspect ratio 8 as a function\nof widthDfor the square prism or diameter D= 2Rfor\nthe ellipsoid. The overall behavior, namely, that our co-\nercivity results approach the Stoner-Wohlfarth value for\nvery thin rods and nearly vanish for thick rods, is con-\nsistent with Eqs. (1) and (2). Furthermore, the square\nprism rods have smaller coercivity than the ellipsoids, in\naccordance with Eq. (1) since the demagnetizing \feld is\ninhomogeneous in square prisms.\nFigure 4 analyzes the dependence of the coercivity on\nthe aspect ratio of the rods. For ellipsoids, Hcslowly ap-3\n(a)\n(b)\nFIG. 2. Spin structures at the initial stage of curling for iso-\nlated Fe-Co (a) square prism and (b) ellipsoid surfaces and\ncross-sections. The corresponding \felds and magnetizations\nare shown on the hysteresis loops. The surface coloring visu-\nalizes the magnetization along the \feld direction.\nproaches a plateau value, because Nkcontinues to change\nas the rods get longer. In contrast, Hcin both square\nprisms and cylinders barely changes above c=a= 4. The\nbehavior of cylinders capped by ellipsoidal tips is inter-\nmediate between cylinders and ellipsoids, and the details\ndepend on the aspect ratio of the tip, where hemispheri-\ncal tips have little impact on the coercivity, but elongated\ntips improve it. This indicates that the geometry of the\ntip ends is more important than the aspect ratio and that\nellipsoidal tips are (far) better than \rat tips.\nThe green rectangle in Fig. 3 shows the range of\nrod diameters and coercivities typically encountered in\nlaboratory-scale and industrial practice [6, 7]. Com-\npared to the Stoner-Wohlfarth predictions (dashed line),\nthe single-rod approximation of Figs. 1{3 reproduces the\ncorrect order of magnitude for coercivity. However, the\nquantitative agreement is, by no means, perfect. More\nimportantly, interactions between rods are likely to mod-\nify the coercivity in a qualitative way. On a mean-\feld\nlevel, the magnetostatic interaction corrections are ap-\nproximated by Eq. (1), but speci\fc alnico interaction\nmechanisms have not yet been considered in the liter-\nature. Two classes of interactions need to be addressed:\n0102030405060700123456789\nDD\nD(nm)Hc(kOe)\n  \nSquare Prism\nSpheroidS-W model (Prism)\nExperimentsFIG. 3. Coercivity of isolated magnetic rods of aspect ratio 8\nas a function of rod diameter. The Stoner-Wohlfarth limit for\ncuboidal rod is denoted by the dashed (blue) line. The thick-\nness of the blue and red solid lines indicates the coercivity\nvariation when the angle between the external \feld and rods\nvaries from 1 °to 5 °. Typically observed alnico Fe-Co rod sizes\nand coercivities are denoted by the shaded (green) region.\n024681012141600.511.522.5\nAspect RatioHc(kOe)\n  \nEllipsoid\nCylinder\nSquare Prism\nCylinder + Ellipsoid tips\nFIG. 4. Coercivity versus aspect ratio for di\u000berent rod ge-\nometries.D= 32 nm in all cases. The ellipsoid-capped cylin-\nders are constructed by capping a 256-nm-long cylinder with\nhemiellipsoids of various aspect ratio.\ngeometry-determined magnetostatic interactions and ex-\nchange interactions between bridging or branched rods.\nTo explore magnetostatic interaction e\u000bects, we have\ncompared two arrangements of ellipsoidal rods, namely\nsimple tetragonal (ST) rods and body-centered tetrago-\nnal (BCT) rods (Fig. 5). The ST array is constructed by\nrepeating rods along the x,y, andzdirections. The BCT\ngeometry is a staggered array, where nearest-neighbor\nparticles are shifted along the kdirection so that the\ncenter of the rods is between the tips of its eight neigh-\nboring rods. Periodic boundary conditions (PBCs) im-\nplemented using a so-called macrogeometry approach [37]\nwere applied to simulate the assembly. Figure 5 compares\nthe two arrangements for a constant packing fraction of4\n2426283032343601234\nisolated ST BCT\nD(nm)Hc(kOe)\nFIG. 5. Coercivity versus rod size for isolated ellipsoids\nand their assemblies (calculated using PBCs) with simple-\ntetragonal (ST) and body-centered tetragonal (BCT) pat-\nterns.\np\u00190:5, an aspect ratio of 8, and a vertical end-to-end\nspacing of 1{2 nm. We see that the coercivity of the\nBCT array is about twice as high as the coercivity of\nthe ST array. The reason for the di\u000berence is that tips\nwith\"magnetization act as poles and create a strong #\n\feld in the lateral neighborhood, adding to the reverse\nexternal magnetic \feld. The ST array has four laterally\ncoordinated nearest neighbors, whereas in the BCT ar-\nray, the nearest lateral neighbors are more distant. Fur-\nthermore, top and bottom tips form magnetic poles of\nopposite sign, which further reduces the lateral interac-\ntion e\u000bect for BCT array. For the coercivity of square\nprisms calculated using PBCs, the di\u000berence between ST\nand BCT arrays is smaller.\nWe have also calculated the coercivity of an isolated\ncluster of 64 ellipsoidal rods in a BCT arrangement, and\nthe coercivities are very similar to those obtained by us-\ning PBCs. The 1 \u0000pdependence of Hcon packing frac-\ntion, estimated using mean \feld as in Eq. (1), underesti-\nmatesHcin this case.\nStrong exchange interactions are established through\nbridges and branching between rods. We have consid-\nered two parallel cuboid nanorods that are connected\nthrough di\u000berent types of branching, namely, H-shaped,\nU-shaped, and O-shaped geometries. Figure 6 shows the\ncoercivities as a function of the surface-to-surface dis-\ntancedbetween rods. H-shape branching, where the\nrods are connected in the middle, is much less detrimen-\ntal to coercivity than U- or O-shaped branching, where\nthe rods are connected at the ends. This is because the\nmagnetization reversal starts at the ends of the rods and\nis made easier by U- and O-branches in the vicinity. In\nthe H-shape geometry, the reversal also starts at the rod\nends, but the tips are still fairly isolated and the branches\nin the middle have a minor e\u000bect.\nInterestingly, some of the above features are also\nencountered in \fne-particle and nanowire magnetism\n04812162024283201234\nd∥ H U O\nd(nm)Hc(kOe)FIG. 6. Coercivity of free and connected rods as a function\nof the spacing dbetween rods. The individual rods have a\nwidth of 16 nm and an aspect ratio value of 8.\n[22, 24, 38{42]. For example, \fne particles also exhibit\na transition from coherent rotation to curling, and wire-\nend features a\u000bect the coercivity [24, 25, 43]. Fine par-\nticles and embedded wires often have diameters small\nenough to approach Stoner-Wohlfarth Hcvalues but are\nnevertheless di\u000ecult to use as permanent magnets. The\nreason is that with their relatively small packing frac-\ntionpthe energy product of a permanent magnet, which\ndescribes its performance [3], is quadratic in the magne-\ntization and, therefore, quadratic in p. In contrast, the\nnanostructure of alnico results from spinodal decomposi-\ntion via speci\fc heat treatments and alnico packing frac-\ntions approach the \\ideal value\" of p= 2=3, depending on\nalloy composition. This yields relatively high magnetiza-\ntion levels but also strengthens the interactions between\nrods that can reduce coercivity.\nIn conclusion, we have used micromagnetic simulations\nto analyze the coercivity of alnico permanent magnets.\nWe \fnd strong deviations from the Stoner-Wohlfarth\nmodel caused by curling modes that are modi\fed by the\nreal structure of alnico alloys. Both the absolute coer-\ncivities and the coercivity trends are in good agreement\nwith available experimental data. The shape, size, spac-\ning, volume fraction, arrangements, and branching types\nof the magnetic Fe-Co rods in the nonmagnetic Ni-Al ma-\ntrix all a\u000bect the coercivity, but aside from the packing\nfraction, the most important features are the shape of the\nrod ends or tips and interactions between them. These\nare all factors which are controlled by the chemistry and\nthermal-magnetic annealing of these alloys [7, 14, 44].\nWe predict that sharp ellipsoidal tips and a staggered\narrangement of the rods should promote substantial co-\nercivity improvements, but this morphology may also be\nthe most di\u000ecult one to realize experimentally. Further\nresearch is necessary to see, for example, whether \feld\nannealing can be used to realize a staggered con\fgura-\ntion and if optimized draw annealing can lead to rod tip\n\\sharpening\".5\nAcknowledgment The authors are thankful to D. J.\nSellmyer, W.-Y. Zhang, H.-W. Kwon, and K.-M. Ho for\ndiscussing several aspects of the present paper. This work\nwas supported by the U.S. Department of Energy (DOE)EERE-VT-EDT program under the DREaM project at\nthe Ames Laboratory, which is operated for the U.S.\nDOE by Iowa State University under Contract No. DE-\nAC02-07CH11358.\n[1] J. F. Herbst, Rev. Mod. Phys. 63, 819 (1991).\n[2] K. 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Lograsso3,4 \n1Lawrence Livermore National Laboratory, Livermore, CA, 94551, USA  \n2Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA  \n3Ames Laboratory, Ames, IA, 50011, USA  \n4Department of Materials Science and Engineering, Iowa State University,  \nAmes , IA, 50011, USA  \n \nAbstract  \nSmCo 5 permanent magnets exhibit  enormous uniaxial magnetocrystalline anisotropy  \nenergy and have a high Curie temperature. However , a low energy product  presents a significant \ndrawback in the performance of SmCo 5 permanent magnets. In  order to increase the energy \nproduct in SmCo 5, we propose substituting fixed  amount of cobalt with iron in a new magnet, \nSmFe 3CoNi, where  inclusion of nickel metal  makes this magnet  thermodynamicall y stable. We \nfurther discuss some  basic theoretical  magnetic properties of the SmCo 5 compound. \n \n1. Introduction \nThree basic material parameters determine the intrinsic properties of hard magnetic \nmaterials: (i) spontaneous (saturation) magnetization, M s, (ii) Curie temperature, T c, and (iii) \nmagnetocrystalline anisotropy energy (MAE ), K1 [1]. These three parameters all need to be large \nfor the permanent magnet to be technologically suitable : Tc ~ ≥ 550 K, Ms ~ ≥ 1 MA/m, K 1 ~ ≥ 4 \nMJ/m3. The desired values of these properties can be achieved by combining transition -metal \n(TM) with rare -earth -metal (RE) atoms in various intermetallic compounds , where the RE atoms \ninduce a large magnetic anisotropy while the TM atoms provide a large magnetization  and high \nCurie temperature [1, 2 ]. \nSmCo 5 (in the hex agonal CaCu 5-type structure) magnets exhibit enormous uniaxial MA E \nof K1 ~ 17.2 MJ/m3, substantially higher than that of  Nd2Fe14B (Neomax) magnet s (K1 ~ 4.9 \nMJ/m3), and have  almost twice the  Curie temperature ( Tc ~ 1020 K)  compared to  Neomax  (Tc ~ \n588 K)  [3, 4]. However , Nd2Fe14B currently dominates the world market for permanent magnets \n(~ 62  %) [5, 6] , since it possesses  the highest  energy performa nce measured by  a record energy product (BH) max of 512 kJ/m3, more than twice as high as the energy  product of SmCo 5 magn ets, \n(BH) max of 231 kJ/m3 [3, 4]. Although SmCo 5 magnets  are more suitable for high temperature \napplications than Neomax , due to their  relatively low energy performance SmCo 5 magnets \noccupy  only ~ 3% of the world market  [5, 6] . \nFrom a  cost stand point , it would be beneficial  to substitute Co atoms with Fe , because Fe \nin the Earth’s crust is ~ 2000 times more abundant than Co and consequently much cheaper. In \naddition, Fe is a ferromagnetic metal with a very large  magnetization a t room temperature (1.76 \nMA/m [3]) . However , the SmFe 5 compound is thermodynamically unstable  and does not appear \nin the equilibrium Sm- Fe phase diagram, although the alloy compound Sm(Co 1–xFex)5 with the \nCaCu 5–type structure has been  synthesized  by the melt- spinning method for x  = 0.0 – 0.3 [7].  \nFurthermore,  the Curie temperature for Sm(Co 1-xFex)5 alloy s was found to increase from ~ 1020 \nK to ~ 1080 K when  increasing x from 0.0 to 0.2 [8], in contra st to Sm 2(Co 1–xFex)17 alloy s which \nexhibit a monotonic decrease in  Curie temperature with increasing Fe content [9 ].  \nThe fundamenta l purpose of the present study  is to explore the  effect of  adding Ni to the \nSmCo 5 magnet in order to stabilize Sm(Co 1–xFex)5 alloys containing  a sufficient amount of iron \nto boost  the energy product  of the Sm(Co -Fe-Ni) 5 magnet. We also investigate and report  some \ninteres ting aspects of the magnetic properties of the  SmCo 5 compound that were incorrectly \ndescribed in some previous publications  [10-13]. We employ  ab initio  calculation s using  four \ncomplementary techniques:  (i) the scalar -relativistic projector -augmented wave (PAW) method \nimplemented in the Vienna ab initio  simulation package (VASP), (ii) the fully relativistic exact \nmuffin -tin orbital method (FREMTO), (iii) the full- potential linear muffin -tin orbital method \n(FPLMTO), and (iv) the full-potential linearized augmented plane wave (FPLAPW) method . \nBoth FPLMTO a nd FPLAPW methods account for all relativistic effects. The usage of these \ndifferent methods ensures that our resul ts are robust and independent of  technical \nimplementations , while taking advantage of the strengths of each method . Pertinent details of the \ncomputational methods are described in Section 2. Results of the density -functional calculations \nof the ground state properties of Sm(Co- Fe-Ni) 5 alloys are presented in Section 3 . We present \nresults of calculation s of the magnetic properties  of the SmCo 5 compound in Section 4. Lastly, \nconcluding remarks are presented in Section 5.  \n \n2. Computational Details  Within the Vienna Ab I nitio Simulation Package (VASP) , the electron -ion interaction is \ndescribed with the projector augmented wave  (PAW) method [14 ] as implemented by Kresse, \nHafner, Furthmüller , and Joubert [15- 18]. The PAW potentials treat 5s2, 5p6, 4f5, 5d1, 6s2 (16 \nelectrons) and  4s2\n, 3d7 (9 electrons) as the valence for Sm and Co , respectively.  For the electron \nexchange and correlation energy functional, the generalized gradient approxi mation (GGA)  of \nPerdew, Burke, and Ernzerhof (PBE)  is employed [19]. A planewave energy cutoff of 350 eV is \nused in calculations. The Brillouin zone is uniformly sampled with a 10x10x11 Monkhorst -Pack \n[20] grid  (k-point spacing of 0.15 Å-1) together with a Gaussian smearing of 0.2 eV  and \nsymmetry turned off . Relaxations of the atomic positions are performed until all forces are below \n0.01 eV/Å. Energies are converged to 10-6 eV. In some cases, non- collinear calcul ations are \nperformed to evaluate spin -orbit coupling effects ( SOC ), which are accounted accord ing to the  \nscheme proposed in [21, 22]  and non- collinear local moments, with orbital moments computed \nby projection on the PAW spheres. The magnitudes of the local moments are calculated by \nprojection of the magnetization density onto the PAW spheres, for consistency  with the orbital \nmoment calculations . DFT+U on- site potentials are  included for Sm 4 f-states using the approach \nof Dudarev et al . [23] with U – J  = 4.45 eV  [11] . \nThe calculations we refer to as EMTO are performed using the Green’s  function \ntechnique based on the improved screened Korringa -Kohn- Rostoker method, where the one -\nelectron potential is represented by optimized overlapping muffin- tin (OOMT)  potential spheres \n[24, 25]. Inside the potential spheres the potential is spherically symmetric, while it is constant \nbetween the spheres. The radius of the potential spheres, the spherical potential inside these \nspheres, and the constant value in the int erstitial region are determined by minimizing (i) the \ndeviation between the exact and overlapping potentials and (ii) the errors caused by the overlap between the spheres. Within the EMTO formalism, the one -electron states are calculated exactly \nfor the OO MT potentials.  The output s of the EMTO calculation  include the self -consistent \nGreen’s function of the system and the complete, non- spherically symmetric charge density. \nFrom this , the total energy is calculated using the f ull charge- density technique [26] . We treat as \nvalence the 6s , 5p, 5d, and 4f  states for Sm  and 4s  and 3d states for Co and Fe . The \ncorresponding Kohn -Sham orbitals are expanded in terms of spdf  exact muffin -tin orbitals,  i.e. \nwe adopt an orbital momentum cutoff  l\nmax = 3. The EMTO orbitals, in turn, consist of the spdf  \npartial waves (solutions of the radial Schrödinger equation for the spherical OOMT potential \nwells) and the spdf  screened spherical waves (solutions of the Helmholtz equation for the OOMT muffin -tin zero pote ntial). The completeness of the muffin -tin basis was discussed in detail in \nRef. [2 4]. The generalized gradient approxi mation (GGA -PBE) [19] is  used for the electron \nexchange and correlation energy.  Integration over the Brillouin zone i s performed using the \nspecial k -point  technique [27] with 784 k -points in the irreducible wedge of the zone (IBZ). The \nmoments of the density of states, needed for the kinetic energy and valence charge density, are \ncalculated by integrating the Green’s function over a complex  energy contour with 2.0 Ry \ndiameter using a Gaussian integration technique with 30 points on a semi -circle enclosing the \noccupied states. In the case of the implementation of the FR -EMTO formalism, SOC is included \nthrough the four -component Dir ac equation  [28] .  \nIn order to treat compositional disorder , the EMTO method is combined with the \ncoherent potential approxi mation (CPA) [29, 30 ]. The ground- state properties of the  \nSm(Co 1–x–yFexNiy)5 alloys are obtained from EMTO -CPA calculations, including  account of the \nCoulomb screening  potential and energy [31- 33]. The equilibrium atomic density of the alloy is \nobtained from a Murnaghan fit to the total energy versus atomic volume  curve [34 ]. \nThe most accurate and fully relativistic calculations are perf ormed using a full -potential \napproach  with no geometrical approximations , where the relativistic effects, including SOC , are \naccounted through the conventional perturbative scheme [ 35] that accurately solves  the Dirac \nequation for both the light and heavy lanthanides  [36, 37]. For this purpose , we use a version of \nthe FPLMTO [38] in which the “full potential” refers to the use of non -spherical contributions to \nthe electron charge density and potential. This is accomplished by expanding the charge density \nand potential in cubic harmonics inside non- overlapping muffin- tin spheres and in a Fourier \nseries in the interstitial region. The spin moments are calculated within both these regions while the orbital moment only in the muffin- tin sphere. For Sm metal, we use two energy tails \nassociated with each basis orbital, and these pairs are different  for the semi -core  states  (5s and \n5p) and valence states ( 6s, 6p, 5d, and 4f ). With this ‘double basis’ approach, we use a total of \nsix energy tail parameters and a total of 12 basis functions per atom. Spherical harmonic expansions are carried out up to l\nmax= 6 for the basis, potential, and charge density. Spin- orbit \ncoupling (SOC) and orbital polarization (OP) are applied only to the d and f  states. The OP has \nbeen implemented in the FPLMTO as a parameter -free scheme where an energy term \nproportional to the square of the orbital moment is added to the total energy functional to account for intra -atomic interactions.  Because of the way it is constructed , the orbital polariza tion often \nbehaves as a magnific ation of the SOC , resulting in larger and improved orbital mom ents [37 ]. For the electron exchange and correlation energy functional, the generalized gradient \napprox imation (GGA -PBE) is used [19].  \nWe have also performed  relativistic full -potential linearized augmented plane wave (FP -\nLAPW) method [39]  calculations within the framework of generalized gradient approximation  \n(GGA+PBE)  [19] , onsite electron correlation  (Hubbard parameter) , and spin orbit coupling \n(GGA+U+SOC) . Calculations are performed using WIEN2K package [40]. According to Ref s. \n[10-12], we use d values of U  = 5.2 eV and J  = 0.75 eV to obtain U – J  = 4.45 eV. The k -space \nintegrations have been performed at least with 13×13×15 Brillouin zone mesh that was sufficient \nfor the convergence of total energies  and magnetic moments. According to Refs. [11- 12], f or the \nSmCo 5 compound the sphere radii for Sm and Co were set as  2.115 a.u. and 2.015 a.u., \nrespectively,  with force minimization of 3%. T he plane -wave cutoff parameters were chosen as \nRK max = 9.0 and G max=14 [11- 12]. \n \n2. Ground state properties of Sm(Co -Fe-Ni) 5 alloys  \nThe SmCo 5 compound crystallizes in the hexagonal CaCu 5-type structure with three \nnonequivalent atomic sites: Sm 1-(1a), Co 1-(2c), Co 2-(3g) (see Figure  1) with 6 atoms per formula \nunit and also per computational cell. \nTable 1 shows the equilibrium formula unit ( f.u.) volume, bulk modulus, and the  pressure \nderivative of the bulk modulus for the SmCo 5 compound computed using each of the ab initio  \nmethodolog ies.  The results of the PAW calculations are shown both without ( ‘VASP ’) and with \n(‘VASP+SOC ’) inclusion of SOC. S imilarly,  the results of the EMTO  calculations are shown \nboth without ( ‘SREMTO ’, ‘SR’ means scalar -relativistic) and with ( ‘FREMTO ’) inclusion of \nSOC. The experimental value of the equilibrium f.u.  volume is taken from Ref. [41]. One can see \nthat the equ ilibrium volumes listed in Table 1 agree quite well with the experimental data and \nthat relativistic SOC has only a minor influence.  Account for Coulomb correlations in the f -shell \n(FPLAPW) also has only a small influence on the equilibrium volume. \nFigure 2 shows the calculated heat of formation of the SmTM 5 comp ounds  (TM = Fe, Co, \nNi) as a function of the valence 3 d-electron  band occupation. The heat of formation is calculated \nwith respect to  the ground- state structures of the pure elements, i.e ., α-structure  of Sm, \nhexagonal close packed (hcp) Co , body centered cubic (bcc) Fe , and face centered cubic (fcc) Ni. \nThe heat of formation of the SmTM 5 comp ounds  drop from about  +12 mRy/atom for TM = Fe to \nabout  –1 mRy/atom for TM = Co and about –19 mRy/atom for TM =  Ni. This  tendency could be explain ed by a shift in the Fermi level due to the gradual filling of the 3 d-band from 7, to 8, and \nto 9 electrons for Fe, Co, and Ni, respectively. Both EMTO and FPLMTO methods show almost \nidentical results.  \nFigure 3 show s the results of calculation of the heat of formation of SmCo 5 compounds  \ndoped with Fe. The Coherent Potential Approximation ( CPA ) implemented within  the ab initio  \nEMTO method allows a gradual substitut ion of  Co atoms by Fe atoms on the Cu -type 3g and 2c  \nsites of the CaCu 5–type structure.  We see that EMTO -CPA calculations predict a  very small \nregion of stability  (x ≤ 0.05) for Sm(Co 1–xFex)5 alloys. FPLMTO calculations for the SmCo 5 and \nSmFe 5 pseudo- binary end points give similar  results to those given by the EMTO method; a  \nsimple straight line  interpolation  between these points gives a  region of stability  of x ≤ 0.10.  \nFigure 4  shows results  of similar calculations for  the pseudo- binary Sm(Ni 1–xFex)5 alloys. \nDue to the significant negative value of the heat of formation of the SmNi 5 compound (~ –19 \nmRy/atom) , EMTO -CPA calculations  show that approximately half of the Ni atoms in SmNi 5 \ncompound can be substituted by Fe atoms. FPLMTO  calculations for the SmNi 5 and SmFe 5 end \npoints show an even larger exten t of stable substitution of Ni by Fe. These EMTO -CPA \ncalculations are in excellent agreement  with recent SEM/EDS measurements [ 42], which find \nthat the maximum extension of SmNi 5–xFex alloys (in the CaCu 5-type structure) is about 50 at. %  \nFe, resulting in  formation of the SmFe 3Ni2 compound.  \nPrevious neutron diffraction studies  of Th(Co 1–xFex)5 alloys (also based on the CaCu 5-\ntype structure)  [43] show that the larger Fe atoms prefer to occupy the 3g- type sites, whereas the \nsmaller Co atoms prefer to occupy the 2c -type sites. Fi gure 5  show s the heat of formation  within \nthe CPA formalism of pseudo- binary SmFe 3(Ni 1–xCox)2 alloys where Fe atoms occupy all 3g-\ntype sites and the occupation  of the 2c -type sites gradually changes from pure Ni ( the SmFe 3Ni2 \ncompound) to pure Co ( the SmFe 3Co2 compound) . Present  calculations show that SmFe 3(Ni 1–\nxCox)2 alloys could remain stable until approximately half of Ni atoms are  substitute d by Co \natoms.  In the next section , we will discuss why it is important to r each maximum solubility (~ 50 \nat. %) of Fe atoms in  SmNi 5-xFex alloys (in other words to create SmFe 3Ni2 compound with Fe \natoms occup ying the 3g-type sites ) before starting substitution of Ni atoms by Co . We will also \ndiscuss  the possibility  to achieve high  axial MA E for energetically stable SmFe 3(Ni 1-xCox)2 \nalloys  using more abundant and cheap Fe and Ni than Co, and simultaneously reaching a higher \nmagnetic energy product  than that of  the initial SmCo 5 prototype.  \n 4. M agnetic properties of the SmCo 5 compound.  \nTable 2 presents previously reported results  [10- 12] of the site -projected spin, m(s), and \norbital, m(o), moments of the SmCo 5 compound calculated by FPLAPW . The calculated total \nmoment m(tot) = 9.90 µB/f.u. significantly exceeds reported experimental values of 7.80 µB/f.u. \n[44] and 8.9 µB/f.u. [45]. Also, the  calculated MAE, K 1 = 40.79 MJ/m3, substantially exceeds the \nexperimental value K 1 = 17.2 MJ/m3 [3]. We repeated these calculations using both FPLAPW \nand VASP -PAW  methods within the LDA+U formalism, using identical U – J  = 4.45 eV  as in \nRefs. [10- 12]. The results for both (001)  and (100) orientations are presented in Tables 3 and 4 \n(FPLAPW) and Tables 5 and 6 (VASP).  As in the case of  the original  FPLAPW calculations [10 -\n12], we assume parallel spin alignment of Sm and Co atoms. Our c alculated MAE are K 1 = 35.96 \nMJ/m3 for FPLAPW and K 1 = 43.43 MJ/m3 for VASP -PAW,  both significantly higher tha n the \nexperimental value K 1 = 17.2 MJ/m3 [3], consistent with the prior reported results . \nAlthough both FPLAPW  and VASP calculations qualitatively predict the correct  axial \nanisotropy of the SmCo 5 compound, we think that these MAE results are “artificial” because of \nthe incorrect parallel alignment between the spins of Sm and Co atoms. It is well established that \nspins of RE and TM atoms always align antiparallel in RE -TM compounds  [46, 47]. In order to \nconfirm this fact, we perform ed calculations of the total energy of the SmCo 5 compound as a \nfunction of the atomic volume for both parallel and antiparallel Sm and Co spin alignments. \nResults of these calculations, performed by both FPLMTO and FPLAPW methods ( with U = J  = \n0), are shown in Figures 6 and 7, respectively . The  results show that the  parallel spin alignment, \nadopted in Refs. [10- 12], represents a metastable  state, which l ies about 7 mRy/atom above the \nground state with antiparallel spin orientation. FPLAPW LDA +U calculations , performed with U \n– J = 4.45 eV , result in a similar conclusion (see Figure 8) :  within this level of theory, the \nmetastable  parallel spins state l ies ~ 4 mRy/atom above the  antiparallel  spins ground state.  \nTables  7 – 12 present the results of the site -projected spin, m(s), and orbital, m(o), moments \nof the SmCo 5 compound calculated by the different methods assuming antiparallel Sm and Co \nspin alignment. In the case of VASP+SOC, FREMTO, and FPLMTO calculations ( U = J = 0 ), as \nshown in Tables  7, 8, and 9, r espectively, the total moment is 4.18 µB/f.u., 4.60 µB/f.u., and 4.04 \nµB/f.u., respectively, which are all  less than half of the reported experimental values of 7.80 \nµB/f.u. [44] or 8.9 µB/f.u. [45]. Account ing for Coulomb correlations in the f -shell within the \nFPLAPW method ( U – J  = 4.45 eV ) produces a higher value of the total magnetic moment of  5.56 µB/f.u. (Table 11) , closer but still somewhat below the experimental values . Significant \nincrease of the total magnetic moment to  m(tot) = 6.09 µB/f.u. is achieved by incorporating orbital \npolarization (OP ) [48, 49] in the FPLMTO ( U = J = 0) method ( see Table 10 ).  \nFPLAPW ( U – J = 4.45 eV) calculations (Table 11)  predict the MAE to be  negative ( in-\nplane orientation) with  K1 ≈ –0.73 mRy/f.u. ≈ –18.75 MJ/m3. To understand the se failure s to \naccurately describe the total moment of the SmCo 5 compound (see Table 7 – Table 11) and its \nMAE  (in-plane orientation ), even when accounting for the energetically stable antiparallel  Sm \nand Co spin orientation, we need to examine several  well -established  experimental facts. \nPolar ized nuclear magnetic resonance (NMR) studies showed that the large anisotropy of the \nSmCo 5 compound originates from a  large orbital contribution from Co 1(2c) sites that are located \nin the same plane as Sm 1(1a) atoms [ 50-52]. As a result , Co 1(2c) atoms have a large positive \nanisotropy contribution while  Co 2(3g) atoms have a small negative anisotropy contribution. This \nexperimental observation has been  confirmed by several calculations for the  related  YCo 5 \ncompound [53, 54]. However, the calculations presented  above  in Table s 7 – 11 predict the  \nopposite tendency:   the orbital moment of Co 2(3g) atoms is equal  to (Table 7) or larger  than \n(Tables  8 –11) the orbital moment of Co 1(2c) atoms, which a prior i cannot pr oduce an axial \n(positive) MAE for the  SmCo 5 compound.  \nIn a previous work [55], the authors claim  that neither LDA nor LDA+U are sufficient for \ndescribing the 4 f states of  the SmCo 5 compound and, in order to describe t he crystal field (CF) \neffects of  the localized 4 f shell , a more sophisticated LDA +DMFT (dynamical mean- field \ntheory) method is necessary. A ccording to these  calculations , reproduced in Table 12, the orbital \nmoments of Co 1(2c) are larger than the orbital moments of Co 2(3g) and also the calculated total \nmoment, m(tot) = 8.02 µB/f.u., is significantly larger  than that obtained above, even  with opposite \nSm and Co spin orientations  (see Tables  7 – 11) , although still a bit lower than the experimental \nvalue of  m(tot) = 8.9 µB/f.u. [45]. We can explain the remaining  discrepancy  by the fact that, as \nseen in  Table 12, the absolute value of the spin moment of Sm atoms is still larger than  the \nabsolute value of the orbital moment of Sm atoms. As was mentioned in [46, 47], for the light \nrare earths  including Sm, ( J = L – S), the total moments  of RE (Sm) and TM (Co) atoms show \nferromagnetic behavior ( c.f., Figure 4  in Ref. [47]). Thus, we expect that f urther improvement in  \nthe description of the CF effects of  the localized 4 f electrons of Sm should result in an increase \nof the absolute value of the orbital moment of Sm , so that  (i) the value of the total moment of the SmCo 5 compound will be increased  and (ii) the ferromagneti c behavior of the total moment s of \nSm and Co atoms  will be correctly described in accor d with expe rimental observation.  \n \n5.Conclusions  \nIn this work, we suggest  a means to realize a perm anent magnet exhibiting both high  \nMAE and Curie temperature comparable to the SmCo 5 compound but with a significantly higher \nenergy produc t. The basic idea is to start from  the SmNi 5 compound (in the CaCu 5-type \nstructure) and dissolve a maximum amount of iron metal (~ 50 at. %) to form  the stable \nSmFe 3Ni2 compound in the same structure modification, where iron atoms predominantly \noccupy 3g sites (in the ideal situation iron atoms should occupy all 3g positions ). Subsequent \ngradual alloying  of the SmFe 3Ni2 compound with Co, while keeping the amount of Sm and Fe \nconstant, should allow to reach the maximum solubility of cobalt metal, which will substitute Ni \natoms  on the 2c  sites. Our calculations show that approximately half of the Ni atoms can be \nreplaced by Co atoms.   \nWe believe t hat even more accurate calculations  in the future  will show further a  \npredominant influence of the TM (2 c) sites on  the MAE of Sm(TM )5 magnets. Indeed , the \napplication of  LDA+DMFT approach to the calculation of MAE of YCo 5 magnets  by Zhu et al.  \n[56] showed that axial (positive) MAE could be obtained  only if orbital moments  on Co 1(2c) \natoms are always larger than the orbital moment s of  Co 2(3g) atoms.  The data from Ref. [55] in \nTable 12 demonstrated a good chance for correct calculation of MAE of SmCo 5 magnet by  \napplying LDA+DMFT . \n \nAcknowledgements  \n A. L. thanks A. Ruban, O. Peil, and L. Vitos for technical support.  This research is \nsupported by the Critical Materials Institute, an Energy Innovation Hub funded by the US \nDepartment of Energy, Office of Energy Efficiency and Renewable Energy, Advanced \nManufacturing Office.  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The heat of formation of the SmTM 5 compound (TM = Fe, Co, Ni) as a functions of \nthe valence 3 d-electron band occupation. \n \nSm1(1a) \nCo1(2c) Co1(2c) \nCo2(3g)  \nFigure 3. The heat of formation of pseudo- binary Sm(Co 1-xFex)5 alloys.  \n \n \nFigure 4. The heat of formation of  pseudo- binary Sm(Co 1-xNix)5 alloys.  \n \n \n \nFigure 5. The heat of formation of pseudo- binary SmFe 3(Ni 1-xCox)2 alloys . \n \n \nFigure 6. The total energy of the SmCo 5 compound as a function of the atomic volume. \nFPLMTO  calculations . \n \nFigure 7. The total energy of the SmCo 5 compound as a function of the atomic volume. \nFPLAPW  calculations. . \n \n \nFigure 8. The total energy of the SmCo 5 compound as a function of the atomic volume. \nFPLAPW  calculations . \n \n \nTables . \n \nTable 1.  Equilibrium formula unit volume, bulk modulus, its pressure derivative of the SmCo 5 \ncompound as functions of the ab initio  methodology.  \nProperty   VASP  VASP+SOC  SREMTO  FREMTO  FPLMTO  FPLAPW  \nU=J=0 eV  FPLAPW  \nU=5.20 eV  \nJ=0.75 eV  Expt.  \n[41] \nUnit cell  \nvolume \n(Å3) 84.20  84.25  85.86  85.89  84.41  85.04  85.58  85.74  \nBulk \nmodulus  \n(GPa)  134.5  134.1  133.0 130.4  123  141.1  143.1   \nBulk \nmodulus \npressure \nderivative  4.71 4.72 3.71 3.41 3.00  4.84 4.77  \n \nTable 2. Site -projected spin, m(s), and orbital, m(o), moments for the SmCo 5 compound: FPLAPW \ncalculations [10 -12]. U – J  = 4.45 eV . m(tot) = 9.90 µB/f.u. Notice that the parallel spin alignment \nof Sm and Co atoms repres ents the metastable state. \nComponent:  Sm 1(1a) Co1(2c) Co2(3g) \nm(s) (µB) + 4.46  + 1.53  + 1.56 \nm(o) (µB) - 2.80 + 0.10  + 0.10  \n \n    Table 3. Site -projected spin, m(s), and orbital, m(o), moments for the SmCo 5 compound: FPLAPW \ncalculations for (001)  orientation. U – J  = 4.45 eV . m(tot) = 10.57 µB/f.u. Notice that the parallel \nspin alignment of Sm and Co atoms represents  the metastable state .  \nComponent:  Sm 1(1a) Co1(2c) Co2(3g) Interstitial  \nm(s) (µB) + 4.880  + 1.53 4 + 1.555  - 0.404  \nm(o) (µB) - 2.277  + 0.146  + 0.127  N/A \n \nTable 4. Site -projected spin, m(s), and orbital, m(o), moments for the SmCo 5 compound: FPLAPW  \ncalculations for (100)  orientation .  U – J  = 4.45 eV . m(tot) = 10.56 µB/f.u. Notice that the parallel \nspin alignment of Sm and Co atoms represents  the metastable state . \nComponent:  Sm 1(1a) Co1(2c) Co2(3g) Interstitial  \nm(s) (µB) + 4.886  + 1.541  + 1.546  - 0.404  \nm(o) (µB) - 2.163  + 0.103  + 0.104  N/A \n \nTable 5. Site -projected spin, m(s), and orbital, m(o), moments for the SmCo 5 compound: \nVASP+SOC calculations for (001)  orientation . U – J = 4.45 eV . m(tot) = 12.41 µB/f.u. Notice that \nthe parallel spi n alignment of Sm and Co atoms represents  the metastable state.  \nComponent:  Sm 1(1a) Co1(2c) Co2(3g) Interstitial  \nm(s) (µB) + 5.055  + 1.509  + 1.495  - 0.147  \nm(o) (µB) - 3.154  + 0.145  + 0.121  N/A \n Table 6. Site -projected spin, m(s), and orbital, m(o), moments for the SmCo 5 compound: \nVASP+SOC calculations for (100)  orientation . U – J = 4.45 eV . m(tot) = 12.31 µB/f.u. Notice that \nthe parallel spi n alignment of Sm and Co atoms represents  the metastable state.  \nComponent:  Sm 1(1a) Co1(2c) Co2(3g) Interstitial  \nm(s) (µB) + 5.000  + 1.502  + 1.45  - 0.150  \nm(o) (µB) - 2.895  + 0.10 2 + 0.128  N/A \n \nTable 7. Site -projected spin, m(s), and orbital, m(o), moments for the SmCo 5 compound: \nVASP+SOC calculations. U  = J = 0. m(tot) = 4.18 µB/f.u. \nComponent:  Sm 1(1a) Co1(2c) Co2(3g) Interstitial  \nm(s) (µB) - 5.97 + 1.53  + 1.54  + 0.47  \nm(o) (µB) + 1.50  + 0.10  + 0.10  N/A \n \nTable 8. Site -projected spin, m(s), and orbital, m(o), magnetic moments for the SmCo 5 compound: \nFREMTO calculations. U  = J = 0. m(tot) = 4.60 µB/f.u. \nComponent:  Sm 1(1a) Co1(2c) Co2(3g) \nm(s) (µB) - 5.74 + 1.58  + 1.60  \nm(o) (µB) + 1.88  + 0.11  + 0.12  \n \n Table 9. Site -projected spin, m(s), and orbital, m(o), magnetic moments for the SmCo 5 compound: \nFPLMTO calculations  [55] . U = J = 0. m(tot) = 4.04 µB/f.u.  \nComponent:  Sm 1(1a) Co1(2c) Co2(3g) Interstitial  \nm(s) (µB) - 5.48 + 1.58  + 1.55  - 0.52 \nm(o) (µB) + 1.81  + 0.06  + 0.10  N/A \n \nTable 10. Site -projected spin, m(s), and orbital, m(o), magnetic moments for the SmCo 5 \ncompound: FPLMTO+OP calculations. U  = J = 0. m(tot) = 6.09 µB/f.u. \nComponent:  Sm 1(1a) Co1(2c) Co2(3g) Interstitial  \nm(s) (µB) - 4.80 + 1.53  + 1.56  - 0.84 \nm(o) (µB) + 3.30  + 0.12  + 0.15  N/A \n \nTable 11. Site -projected spin, m(s), and orbital, m(o), magnetic moments for the SmCo 5 \ncompound: FPLAPW calculations. U  - J = 4.45 eV. m(tot) = 5.56 µB/f.u. \nComponent:  Sm 1(1a) Co1(2c) Co2(3g) Interstitial  \nm(s) (µB) - 5.339  + 1.586  + 1.612  - 0.721  \nm(o) (µB) + 3.101  + 0.073  + 0.123  N/A \n \n  Table 12. Site -projected spin, m(s), and orbital, m(o), magnetic moments for the SmCo 5 \ncompound: LDA+DMFT calculations [55]. U  – J = 7.03 eV. m(tot) = 8.02 µB/f.u. \nComponent:  Sm 1(1a) Co1(2c) Co2(3g) Interstitial  \nm(s) (µB) - 3.47 + 1.54  + 1.52  - 0.39 \nm(o) (µB) + 3.26 + 0.2 2 + 0.18  N/A \n \n \n "    },    {        "title": "1708.01683v2.Tunable_dimensional_crossover_and_magnetocrystalline_anisotropy_in_Fe__2_P_based_alloys.pdf",        "content": "arXiv:1708.01683v2  [cond-mat.mtrl-sci]  22 Sep 2017Tunable dimensional crossover and magnetocrystalline ani sotropy in Fe 2P-based alloys\nI. A. Zhuravlev,1V . P. Antropov,2A. Vishina,3M. van Schilfgaarde,3and K. D. Belashchenko1\n1Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,\nUniversity of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA\n2Ames Laboratory, Ames, Iowa, 50011, USA\n3Kings College London, London WC2R 2LS, UK\n(Dated: September 18, 2018)\nElectronic structure calculations are used to examine the m agnetic properties of Fe 2P-based alloys and the\nmechanisms through which the Curie temperature and magneto crystalline anisotropy can be optimized for spe-\ncific applications. It is found that at elevated temperature s the magnetic interaction in pure Fe 2P develops a\npronounced two-dimensional character due to the suppressi on of the magnetization in one of the sublattices, but\nthe interlayer coupling is very sensitive to band filling and structural distortions. This feature suggests a natural\nexplanation of the observed sharp enhancement of the Curie t emperature by alloying with multiple elements,\nsuch as Co, Ni, Si, and B. The magnetocrystalline anisotropy is also tunable by electron doping, reaching a\nmaximum near the electron count of pure Fe 2P. These findings enable the optimization of the alloy conten t,\nsuggesting co-alloying of Fe 2P with Co (or Ni) and Si as a strategy for maximizing the magnet ocrystalline\nanisotropy at and above room temperature.\nTransition-metal pnictide alloys based on Fe 2P have at-\ntracted considerable attention due to the unusual sensitiv ity\nof their magnetic properties to temperature, pressure, ext er-\nnal magnetic field, and alloying,1–5as well as their possi-\nble magnetocaloric applications,1and they have been exten-\nsively studied theoretically.6–9Of particular interest is the\nmagnetocrystalline anisotropy (MCA) of Fe 2P, which, at 2.3\nMJ/m3, is record-high at zero temperature for systems without\nheavy elements.10,11Although ferromagnetism in Fe 2P van-\nishes through a first-order phase transition at TC=216 K,\nthis temperature can be greatly increased by alloying with S i,\nNi, Co, and other easily available elements.1,2,12–14The origin\nof such unusual TCsensitivity to alloying is not understood.\nIn combination with large MCA and appreciable coercivity\nobserved15in Fe 2P alloyed with Co, this feature makes Fe 2P-\nbased alloys interesting for permanent-magnet applicatio ns.\nHere we use first-principles calculations to study the mag-\nnetic properties of Fe 2P-based alloys. We propose that TCis\neasily tunable thanks to the two-dimensional (2D) characte r of\nthe exchange interaction developing at elevated temperatu re.\nWe also find that MCA is controlled largely by band filling\nand is maximized close to the electron count corresponding t o\npure Fe 2P. Based on these findings, we argue that co-alloying\nwith Co (or Ni) and Si is the optimal strategy to maximize the\nMCA at and above room temperature.\nWe use the Green function-based formulation of the tight-\nbinding linear muffin-tin orbital method in the atomic sphere\napproximation.16Substitutional disorder was treated using\nour implementation of the coherent potential approxima-\ntion (CPA)17with spin-orbit coupling (SOC) included and\nthe MCA energy Kcalculated following Refs. 18–20. The\natomic sphere radii were carefully chosen to reproduce the\nfull-potential band structure. Exchange and correlation w ere\ntreated within the generalized gradient approximation.21A\nuniform mesh of 12 ×12×20 points provided su fficient accu-\nracy for the Brillouin zone integration.\nAtx<0.2 the (Fe 1−xCox)2P alloy has a hexagonal struc-\nture with lattice constants that are almost independent of x.2\nTherefore, we used the experimental values for pure Fe 2P(a=5.8675 Å, c=3.4581 Å)2at all concentrations in these\nalloys, and the internal coordinates from Ref. 22.\nThere are two inequivalent Fe sites in Fe 2P: the tetrahedral,\nweakly magnetic Fe Iand the pyramidal, strongly magnetic\nFeII.4,8Co and Ni have a strong tendency to occupy the Fe I\nsite.2,23The results we report here assume 100% site prefer-\nence, but there is no qualitative di fference with equal substitu-\ntion on both types of sites.\nUsing the linear-response formalism,25we find, in agree-\nment with earlier results,7that the exchange parameters in the\nferromagnetic state do not change much with 10-15% substi-\ntution of Co, Ni, or Si, while the experimental TCincreases\nsharply. For example, TCnearly doubles at 10% Co substitu-\ntion, while the paramagnetic Curie temperature θPincreases\nonly by 14%.26The dominant exchange parameters are strong\nferromagnetic Fe II-Fe IIand a weaker but comparable Fe I-Fe II.\nMean-field theory predicts TCof order 700 K that depends\nweakly on concentration, consistent with the behavior of θP.\nThus, the dramatic influence of various alloying elements on\nTCis quite puzzling. It was suggested8that the stabilization of\nthe Fe Ilocal moments at elevated temperature is responsible\nfor the TCincrease in Si-doped Fe 2P, but it is unclear how this\nmechanism would apply to Co and Ni doping which have an\nopposite effect on band filling.\nWe propose the following scenario, which is consistent with\nfirst-principles calculations and experimental evidence. We\nobserve that the magnetic structure of Fe 2P consists of alter-\nnating layers of Fe Iand Fe IIsites. The Fe Isites are weakly\nmagnetic, and their exchange coupling to Fe IIsites is con-\nsiderably weaker than the in-plane Fe II-Fe IIcoupling, while\nthe Fe I-Fe Iexchange is negligibly small. Therefore, we ex-\npect that the Fe Isublattice magnetization declines much faster\nthan Fe IIas the temperature approaches TC. Indeed, a neutron\ndiffraction measurement found a very small Fe Imagnetization\njust below the first-order TCin 7% Ni-substituted Fe 2P.27As\na result, the Fe I-mediated coupling between the Fe IIlayers is\nstrongly suppressed near TC. On the other hand, as we will\nnow show, direct interlayer Fe II-Fe IIcoupling is weak in pure\nFe2P and strongly sensitive to band filling. Overall, this be-2\nhavior indicates a crossover from three-dimensional to qua si-\ntwo-dimensional magnetism as a function of both temperatur e\nand composition of the alloy.\nTo calculate the direct coupling between the Fe IIlayers, we\nmodel the paramagnetic state using the disordered local mo-\nment approach and calculate the Fe II-Fe IIexchange parame-\ntersJi jusing the linear response formalism.28The local mo-\nments on the Fe Isite vanish in this approach. We focus on\nthe total effective exchange J0=/summationtext\njJi jand the interlayer ex-\nchange Jz=/summationtext′\njJi j, where the prime restricts the summation\nto sites jin the Fe IIlayer that is adjacent to the one contain-\ning site i. Fig. 1shows J0andJzcalculated as a function of\nthe Fermi energy EFin the rigid-band model;29the upper axis\nshows the band filling ∆N, which is the electron count per\nformula unit, referenced from pure Fe 2P. It is seen that J0in-\ncreases smoothly by about 30% at ∆N∼1, which corresponds\nto a 50% substitution of all Fe atoms by Co. In a real alloy, J0\nis not expected to increase as much, because alloying with Co\nreduces the magnetic moments. On the other hand, although\nJzshows a similar trend, it is small at ∆N≈0 and changes\nsign at a small hole doping (see inset in Fig. 1).\nThis result suggests that the magnetic structure e ffectively\nbecomes quasi-2D at elevated temperatures. In this scenari o,\nTCcan be strongly suppressed from its mean-field value,30\nwhile nothing special happens to θP. Moreover, Fig. 1shows\nthat the degree of two-dimensionality is sensitive to elect ron\ndoping, and we expect the e ffect of alloying or another per-\nturbation on TCshould correlate with its e ffect on Jz. While\nthe electron (hole) doping makes the exchange interaction l ess\n(more) two-dimensional, specific alloying elements can als o\naffectJzin ways that are unrelated to electron count, such as\nthrough the structural distortions induced by the size e ffect.\n\u0001\u0001\n\u0001\u0001\u0001\u0001\n-\u0001\u0002\u0003-\u0001\u0002\u0004\u0001\u0001\u0002\u0004\u0001\u0002\u0003\u0001\u0002\u0005\u0001\u0003\u0006\u0007\b\u0004\u0001-\u0004-\u0001\u0002\t\u0001\u0001\u0002\t\u0004\n\u0002-\u0002\u0002(\n\u000b)\u0001(\u0001\u0002\u0003)△\u0003\n-\u0001\u0002\u0001\u0003\u0001\u0001\u0002\u0001\u0003\u0001\u0001\u0002\u0003\u0001\u0002\u0004\nFIG. 1. Dependence of exchange interaction on Fermi level po sition\nin paramagnetic Fe 2P alloys between pyramidal Fe sites. Black: total\nexchange interaction ( J0); blue: interlayer exchange between two\nnearest layers of pyramidal Fe IIsites ( Jz). Inset: enlargement of Jzat\nsmall doping. Top axis: band filling ∆N.\nGoing beyond the rigid-band model, Fig. 2shows J0and\ntheJz/J0ratio computed using CPA in paramagnetic Fe 2P al-\nloyed with Co, Ni, or Si, taking substitutional25and spin28\ndisorder on the same footing. If the lattice constants are ke pt\nfixed (solid lines), the CPA results are similar to the predic -tions of the rigid-band model. However, it turns out the latt ice\ndistortion induced by Si overwhelms its e ffect on the elec-\ntron count, and overall Si increases Jz(blue square). On the\nother hand, the structural distortion induced by alloying w ith\nCo or Ni is negligible. Thus, Co, Ni, and Si allincrease Jz,\ndespite their opposite e ffects on the band filling. All these al-\nloying elements also sharply increase TC, which supports the\nidea that the degree of magnetic two-dimensionality correl ates\nwith TCand is otherwise hard to explain, given the opposite\n(and small) effects of Co and Si on J0.\n■ ■\n●●●●▲▲▲\n◆◆◆(\u0001)\n-\u0001\u0002\u0003\u0004-\u0001\u0002\u0003-\u0001\u0002\u0001\u0004\u0001\u0001\u0002\u0001\u0004\u0001\u0002\u0003\u0001\u0002\u0003\u0004\u0001\u0002\u0005\u0001\u0001\u0002\u0001\u0006\u0001\u0002\u0001\u0007\nΔ\u0001\u0001\u0001/\u0001\u0001■ ■\n●●●●▲▲▲\n◆◆◆\n(\u0002\u0003\u0001-\u0002\u0004\u0005\u0002)\u0003\u0006\n◆(\u0002\u0003\u0001-\u0002\u0007\b\u0002)\u0003\u0006\n●\u0002\u0003\u0003(\u0006\u0001-\u0002\t\b\u0002)(\n)\n\b\u0002\u0004\t\t\u0002\u0004\u0007\u0001\u0001(\u0001\u0002\u0003)\nFIG. 2. Dependence of the exchange coupling on band filling ∆N\nin paramagnetic Fe 2P-based alloys. (a) Total exchange interaction\nJ0. (b) Ratio of interlayer exchange between two nearest layer s of\npyramidal Fe sites and J0. Symbols connected by lines: at exper-\nimental lattice parameters for Fe 2P. Blue squares: at experimental\nlattice parameters13for Fe 2P0.9Si0.1.\nWe now turn to magnetocrystalline anisotropy. Fig. 3shows\nthe dependence of Kin Fe 2P on the electron doping in the\nrigid-band model, as well as the CPA results for various al-\nloys, all plotted as a function of the electron count ∆N. First,\nwe focus on three-component alloys with Co, Ni, Si, or B.\nAs in the case of the exchange coupling, the rigid-band ap-\nproximation agrees quite well with CPA calculations for the se\nalloys. This agreement suggests that the MCA energy in Fe 2P\nis not dominated by spin-orbit “hot spots,” which would be\nstrongly suppressed by disorder.20Further, we see that the be-\nhavior of Kis primarily controlled by band filling. For exam-\nple,Kbehaves very similar in (Fe 1−xCox)2P and (Fe 1−xNix)2P\nalloys when plotted against ∆N, which means that the e ffect\nof Ni is equivalent to twice as much Co. The e ffect of Si or B\nalso fits closely with the band filling trend.\nThe small offset of the maximum in Fig. 3is likely an arti-\nfact of the atomic sphere approximation, which slightly ove r-\nestimates the exchange splitting. While there is no intrins ic\nreason for the maximum to occur exactly at ∆N=0, we\nfound that a full-potential rigid-band calculation reprod uces\nthe trend seen in Fig. 3, but the maximum occurs almost ex-\nactly at∆N=0. The decline of the MCA energy with the\nincreasing concentration of Ni or Co agrees with the experi-\nmental data.10,11,26\nTo identify the mechanisms of MCA and its dependence\non band filling, we first consider the site and spin decompo-3\n■\n■\n■\n■\n■\n■■■■■\n■\n■■■◆\n◆\n◆\n◆\n◆\n◆\n◆\n▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲\n▲\n▼▼▼▼▼▼▼▼▼▼▼\n▼\n▼\n▼\n▼\n▼■\u0001\u0002\u0001(\u0003\u0002-\u0003\u0004\u0005\u0003)\n◆\u0001\u0002\u0001(\u0003\u0002-\u0003\u0006\u0003)▲(\u0001\u0002\u0002-\u0003\u0007\b\u0003)\u0001(\u0003\u0004\u0005\u0006\u0004\u0005\u0004\u0005\u0002)\n○(\u0001\u0002\u0002-\u0003\t\u0005\u0003)\u0001(\u0003\u0004\u0005\u0006\u0004\u0005\u0004\u0005\u0002)\n●(\u0001\u0002\u0002-\u0003\u0007\b\u0003)\u0001\u0003\n□(\u0001\u0002\u0002-\u0003\t\u0005\u0003)\u0001\u0003\n\n\u0005\u000b\u0005\f\r\u000e\u000f\u0010\f\n\u0007\b*\n-\u0001\u0002\u0003-\u0001\u0002\u0001\u0004\u0001\u0001\u0002\u0001\u0004\u0001\u0002\u0003\u0001\u0002\u0003\u0004\u0001\u0002\u0005\u0001\u0002\u0005\u0004\u0001\u0002\u0006\u0001\u0002\u0004\u0001\u0002\u0007\n\u0001 \u0000 \u0002\n\u0003 \u0004 \u0005\n\u0006 \u0007 \b\nΔ\u0001\u0001(\u0001\u0002\u0003)\nFIG. 3. Dependence of Kon band filling∆Nin different alloys\n(see legend) calculated in CPA. Inverted brown triangles (l abeled\nCo∗): (Fe 1−xCox)2(P0.9Si0.1) with experimental lattice parameters13\nfor Fe 2(P0.9Si0.1). Diamonds: rigid-band approximation for Fe 2P.\nsition of the anisotropy of the SOC energy.19,31We have ver-\nified that, as expected, this quantity closely follows the co n-\ncentration dependence of Kin these alloys. The dominant\nterm in the spin decomposition comes from the mixing of the\nminority-spin states of Fe IIby the ˆSzˆLzterm in SOC, while the\nmajority-spin and the spin-o ff-diagonal term are fairly small\nin the interesting range of concentrations. The contributi on\nof Fe Ibehaves similarly but is a few times smaller. Given\nthat the diagonal minority-spin contribution dominates, t he\nMCA is approximately proportional to the orbital moment\nanisotropy,31,32which may be easier to measure.32\nThe orbital-resolved density of states for a Fe IIatom in pure\nFe2P (Fig. 4) shows two closely-spaced peaks of the xyand\nx2−y2character (or m=±2 in the Ylmbasis) separated by the\nFermi level. Mixing of these states by the ˆLzoperator leads to\nthe MCA maximum seen in Fig. 3.\n3z\n2-1\nyz,zx\nxy,x\n\t-y\n\n-\u0001-\u0002-\u0003-\u0004\u0005\u0004\u0003-\u0003-\u0004\u0005\u0004\u0003\n\u0001-\u0001\u0001(\u0006\u0007)\u0001\u0002\u0003(\u0004\u0005-\u0001)\nFIG. 4. Orbital-resolved partial density of 3 dstates for a Fe IIatom.\nWe now examine the contributions to Kcoming from differ-\nent regions in the Brillouin zone.19Fig.5shows the difference\nof the minority-spin single-particle energies for magneti zation\nalong the xandzaxes in pure Fe 2P, resolved by k. We see thatthe MCA accumulates over a fairly large part of the Brillouin\nzone, with the most important contribution coming from the\nregion with kz≈k1/3=|ΓA|/3 (bright red area in Fig. 5).\nNote that an earlier analysis6focused only on high-symmetry\ndirections in the Brillouin zone, leading to an erroneous co n-\nclusion that the dominant contributions to Kcome from band\nsplittings along the KM and ΓA lines. This observation under-\nlines the need to examine the contributions coming from the\nentire Brillouin zone.\nFIG. 5. Brillouin zone map of the k-resolved minority-spin contri-\nbution to Kfor pure Fe 2P. Color intensity indicates the magnitude of\npositive (red) or negative (blue) contributions. The Γ′and M′points\nare at kz=|ΓA|/3 on theΓA and ML lines, respectively.\nThe origin of the dominant positive contribution from the\nvicinity of the kz≈k1/3plane can be understood by examining\nthe partial minority-spin spectral function for the transi tion-\nmetal site. Fig. 6shows this spectral function for pure Fe 2P,\nresolving 3 dorbital contributions by color. We see two inter-\nsecting bands with the Fermi level cutting through them. The\nSOC strongly splits these bands for M/bardblz[panel (b)], while\nthe splitting is much weaker for M/bardblx(not shown). Although\nthere is no exact spin-orbital selection rule, the states ne ar the\nFermi level are predominantly of the xyandx2−y2character\n(see Fig. 4). These orbitals are strongly mixed by the ˆLz(but\nnotˆLx) operator, which gives a positive contribution to K.\nFIG. 6. Partial minority-spin spectral function in the kz=|ΓA|/3\nplane for the transition-metal site in Fe 2P. (a) Without SOC. (b) With\nSOC for M/bardblz. The intensities of the red, blue, and green color\nchannels are proportional to the sum of m=±2 (xyandx2−y2), sum\nofm=±1 (xzandyz), and m=0 (z2) character, respectively.4\nThe dependence of Kin (Fe 0.85Co0.15)2P on the c/aratio is\nsimilar to the results of Ref. 6for pure Fe 2P: a 5% increase in\nc/aresults in a 30% increase in MCA, while the magnetiza-\ntion is slightly decreased. With increasing volume the MCA\nincreases slightly, at a rate of about 2% per 1% volume in-\ncrease, while the magnetization is nearly constant.\nAt room temperature the anisotropy field and coercivity in\n(Fe 1−xCox)2P are maximized at certain Co concentrations.15\nThis is because there is a tradeo ffbetween low TCat low con-\ncentrations and low MCA, even at T=0, at high concen-\ntrations. For permanent-magnet applications, it is of inte rest\nto maximize the MCA at the operating temperature. First,\nwe considered the prospects of increasing the MCA of the\n(Fe 0.85Co0.15)2P alloy by substituting a small amount of Fe\nby an additional alloying element, such as one with a stronge r\nSOC. We found that Mn, Tc, and Re increase Kwhen equally\nsubstituted on both Fe sublattices, with Mn having the large st\neffect and Re the smallest. For example, a 5% substitution of\nRe increases Kby about 15%, and that of Mn by as much as\n70%. However, if Mn fully segregates to the pyramidal site\n(which it strongly prefers14), the MCA is only enhanced by\n20%. The effect of Mn is then essentially that of the reduced\nband filling, which could be achieved by simply reducing the\namount of Co. Equal substitution of both Fe sites by Ru and\nOs has almost no effect on K, while Rh and Ir reduce it. Thus,\ndouble substitution of Fe by Co and another transition-meta l\nelement is not promising.\nOn the other hand, given that Kis sensitive to band filling\nand is maximized close to the electron count of pure Fe 2P,\nthe following strategy suggests itself: co-alloy Fe 2P with Co\nor Ni and Si (on Fe and P sublattices, respectively) to main-\ntain the optimal band filling while increasing TCwell above\nthe operating temperature. For example, as shown in Fig.\n3, the MCA in (Fe 1−xXx)2P0.9Si0.1alloys, where X is either\nCo or Ni, has the same dependence on band filling as the\nthree-component alloys. Although the lattice distortion c orre-\nsponding to Fe 2P0.9Si0.1reduces the maximal MCA, it is still\nachieved at the same band filling.\nWe have examined the Bloch spectral functions in these al-\nloys and found that, in all cases, all the bands seen in Fig.\n6are easily identifiable and their broadening is fairly small ,\nwhile the band filling determines the location of the Fermi\nlevel. The band broadening does, however, reduce the MCA in\n(Fe 1−xXx)2P0.9Si0.1compared to (Fe 1−xXx)2P at the same band\nfilling. Therefore, for alloys with the optimal band filling, co-\nalloying with Co (or Ni) and Si still comes with a tradeo ff\nbetween the increasing TCand decreasing the MCA at T=0.\nAs mentioned above, the shift of the maximum in Kfrom\n∆N=0 is likely due to a slightly overestimated exchange\nsplitting. Thus, we predict that MCA is maximized close to\nthe 1:2 doping ratio for Co and Si, or 1:4 for Ni and Si.\nCPA calculations show that the ground-state magnetization\nin Fe 2P alloys with Co, Ni, and Si, in the relevant range of\nconcentrations, is approximately a linear function of the b and\nfilling: M≈M0+A∆N, where A≈−0.85µB/f.u. In par-\nticular, at T=0 the “optimal” co-doped alloys should have\napproximately the same magnetization as pure Fe 2P.\nConsider the series of (Fe 1−xCox)2P1−¯ySi¯yalloys where ¯ y≈2xmaximizes KatT=0 for the given x. It is of practical\ninterest to find xthat maximizes K(T,x) in this alloy at the\ngiven T. To find this maximum, we calculate K(0,x) in CPA\nand approximate the temperature dependence as follows.\nIt is often possible to calculate K(T) using the disordered\nlocal moment method,20,33but we do not have a quantitative\nmodel to represent the statistical distribution of the magn etic\nmoments and their orientations in the present system with\na weakly magnetic sublattice and a dimensional crossover.\nTherefore, we turn to the experimental data on K(T), which\nwas measured at several concentrations in Ni-doped Fe 2P.10,11\nTheK(T/TC) curves at 0, 10, and 20% Ni are monotonic and\nessentially identical when scaled by K(0). Therefore, we as-\nsume that K(T,x)/K(0,x)=f(T/TC) is the same function of\nT/TCat any x, and we take it from experiment.10,11\nThe missing piece is the dependence of TConx. Consider\nTC(x,y) in the alloy where xandyare unrestricted concen-\ntrations of Co and Si. Since we are considering the series\nof alloys at optimal band filling, we expect that the increase\nofTC(x,¯y) with xis primarily due to the increase in Jzdue\nto the structural distortion introduced by Si. In Fe 2(P1−ySiy),\nthe hole doping reduces the e ffect of this structural distor-\ntion. Therefore, we can use TC(0,¯y) from the experimental\ndata for Fe 2(P1−ySiy) as the lower bound for TC(x,¯y) in the\nlower-bound estimate for K(T,x)=f(T/TC)K(0,x).\nThe resulting lower-bound estimates for K(T,x) are shown\nin Fig. 7for three different temperatures. At room temper-\nature, MCA reaches a maximum of about 0.26 meV /f.u. at\nx≈0.09. At higher temperatures this maximum shifts to\nlarger concentrations, while the maximum MCA declines.\nThus, correcting for the 2% shift of the K(∆N) curve, we pre-\ndict that the optimal alloy for permanent-magnet applicati ons\nin the 300-400 K operating range has 7-10% Co (or 3.5-5%\nNi) and a compensating amount of 14-20% Si. This target\ncomposition should be the starting point for experimental v er-\nification. More generally, understanding of the mechanisms\nby which alloying a ffects the Curie temperature and MCA\nprovides a path for optimizing the composition of Fe 2P-based\nalloys for specific applications.\nFIG. 7. Concentration dependence of Kat 273K (black), 323K (blue)\nand 400K (red) in (Fe 1−xCox)2P1−¯ySi¯yalloys with optimal band filling\nreached at ¯ y≈2x. See text for details.\nThe work at UNL was supported by the National Sci-5\nence Foundation through Grants No. DMR-1308751 and\nDMR-1609776 and performed utilizing the Holland Comput-\ning Center of the University of Nebraska. Work at Ames Lab\nwas supported by the Critical Materials Institute, an Energ yInnovation Hub funded by the US DOE. Ames Laboratory is\noperated for the US DOE by Iowa State University under Con-\ntract No. DE-AC02-07CH11358. MvS was supported by the\nEPSRC CCP9 Flagship Project No. EP /M011631/1.\n1O. Beckman and L. 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Chem. 35, 1963\n(1973)."    },    {        "title": "1708.08549v2.Electric_field_induced_changes_of_magnetic_moments_and_magnetocrystalline_anisotropy_in_ultrathin_cobalt_films.pdf",        "content": "1 \n Electric -field-induced changes of magnetic moments and \nmagnetocrystalline anisotropy in ultrathin cobalt films \n \nTakeshi Kawabe1*, Kohei Yoshikawa1*, Masahito Tsujikawa2,3‡, Takuya Tsukahara1, \nKohei Nawaoka1, Yoshinori Kotani4, Kentaro Toyoki4, Minori Goto1,5, Motohiro \nSuzuki4, Tetsuya Nakamura4, Masafumi Shirai2,3, Yoshishige Suzuki1,5, and Shinji \nMiwa1,5† \n1Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka \n560-8531, Japan  \n2Research Institute of Electrical Communication, Tohoku University, Sendai, Miyagi \n980-8577, Japan  \n3Center for Spintronics Research Network (CSRN), Tohoku University, Sendai, Miyagi \n980-8577, Japan  \n4Japan Synchrotron Radiation Research Inst itute (JASRI), Sayo, Hyogo 679 -5198, \nJapan  \n5Center for Spintronics Research Network  (CSRN) , Osaka University, Toyonaka, Osaka  \n560-8531, Japan  \n*These authors contributed equally to this work  \n†miwa@mp.es.osaka -u.ac.jp  \n‡t-masa@riec.tohoku.ac.jp  \n \nIn this study, t he m icroscopic origin s of the voltage -controlled magnetic anisotropy \n(VCMA) in 3d-ferromagnetic metals  are revealed. Using in-situ X-ray fluorescence \nspectroscopy  that provides a high quantum e fficiency , electric -field-induc ed changes in 2 \n orbital magnetic moment and magnetic dipole Tz terms in ultrathin Co films  are \ndemonstrated . An orbital magnetic moment difference of 0.013 μB. was generated in the \npresence of electric fields of ±0.2 V/nm. The VCMA  of Co was properly e stimated by \nthe induced change in orbital magnetic moment , accord ing to the perturbation theory \nmodel . The induc ed change in magnetic dipole Tz term only slightly contribut ed to the \nVCMA  in 3d-ferromagnetic metals . \nPACS: 85.75. -d, 75.30.Gw, 78.70.Dm, 78.20.Ls  \n \n \n \nVarious magne toelectric effects in ultrathin ferromagnetic  metals have been reported , \nsuch as the electric -field-induced modifications of magnet ocrystalline  anisotropy  energy \n(MAE)  [1–3], Curie temperature [ 4,5], exchange bias  [6] and antisymmetric exchange \ninteraction [ 7]. In particular , the electric -field-induced modification of MAE , which is \ncalled voltage -controlled magnetic anisotropy (VCMA), in ferromagnetic 3d-metals has \nbeen intensively studied  because  of its great potential for enabling the construction o f \nultralow -power -consumption electric devices  [3]. As has been reported, one mechanism \nto ex plain the MAE of low-dimensional ferromagnetic 3d-metals  is the magnetization \ndirection dependence of the orbital angular momentum  of their 3d-state. This anisotropy \nin the orbital magnetic moment influences the MAE through spin-orbit interactions  [8]. \nThis mechanism is often regarded as the Bruno mechanism. Hence, the VCMA can be \nexplained by the change of the orbital magnetic moment by selective electron /hole \ndoping into electron orbital s of atoms at the interface. Several  theoretical studies have \nreport ed that the hybridization and/or modulation of the 3d-orbital s is important for the \nVCMA  [9–13]. However, an electric -field-induc ed change of the orbital magnetic 3 \n moment has not been experimentally confirmed . \nIt has been experimentally confirmed that an electrochemical reaction, namely, O2− \nmigration , induces a magnetoelectric effect  in FeCoO x/MgO [ 14], FePt /ion-gel [15], \nCo/Gd2O3 [16,17], and Fe/BaTiO 3 [18] system s. Because such an electrochemical \nreaction requires a thermal activation process and has a limited operatin g speed  in the \nsub-millisecond range  [17], it is hardly the microscopic origin of the aforementioned \nVCMA  in 3d-ferromagnetic metals with high -speed (< 0.5 ns) operation [3,19–21]. \nIn addition to this, recent research has shown the significance of the magnetic dipole \nTz term in the 5d-state to the VCMA in Pt with proximity -induced spin polarization  [22]. \nIn a Fe/Pt/MgO system, the Tz term induction , correlat ing with electric quadrupole \ninduction , induces a VCMA  through the second -order perturbation term including the \nspin-flip process  [23]. From the first -principles study, this quadrupole mechanism has a \ncomparable contribution to the VCMA  in the Fe/Pt/MgO system as the Bruno \nmecha nism. To the best of our knowledge, h owever, there has been n o report \nexperimentally demonstrat ing the relative significance between the Bruno and the \nquadrupole mechanism s in 3 d-ferromagnetic metals because of a lack of direct \nobservation s of the electric -field-induced change of orbital magnetic moment  in \nferromagnetic metals . \nHere , we report direct evidence of the electric -field-induced change of orbital \nmagnetic moment of  ultrathin Co films in terms of in-situ X-ray magnetic circular \ndichroism (X MCD ) spectroscopy  with ultrahigh efficiency . Utilizing an epitaxial  \nFe/Co/MgO multilayer system, we obtain an induced orbital magnetic moment \nanisotropy change of 0.0 13μB in the presence of electric field s of ±0.2 V/nm.  Moreover, \nthe VCMA  in the system  is estimated well by the induced change of the orbital 4 \n magnetic moment anisotropy in Co in accordance with the Bruno’s perturbation theory \nmodel [ 8]. \nThe f ollowing three sets of samples were prepared  for the stud ies. The f irst set of \nsample s were tunnel junction s to characterize the electric -field-induced change of \nXMCD  using the  partial fluorescence yield (PFY) method  and a silicon drift detector \n(SDD)  with a large solid angle . The d evice structure is illustrated in Fig. 1. A multilayer  \nconsisting of fcc-MgO(001) substrate /fcc-MgO (001)  buffer (5  nm)/bcc-V(001)  \n(30 nm)/bcc-Fe(001)  (0.4 nm)/Co (0.14  nm)/fcc-MgO (001)  barrier (2  nm) was prepared  \nusing  electron beam deposition in an ultrahigh vacuu m. The MgO(001) buffer was \nemployed to block the diffusion of carbon from the surface of the MgO substrate into \nthe V/Fe/Co multilayer  [24]. The MgO substrate was annealed at 800  °C for 10 min, \nand the V layer was post -annealed at 500  °C for 30 min. We confirmed formation s of \nepitaxial and flat interfaces of each layer  using the results of reflection high -energy \nelectron diffraction  [25]. The 0.14 -nm Co, correspond ing to a monatomic Co  layer , was \ngrown coherently onto the bcc -Fe(001) surface while maintaining its two -dimensional \nsquare  lattice structure. Then , the Co layer was covered by the fcc -MgO(001) epitaxial \nlayer. Following the removal of the sample from the ultrahigh vacuum, a SiO 2 \n(5 nm)/Cr (2 nm)/Au (5  nm) layer was deposited. Then , the multilayer was patterned \ninto a  160-μm diameter tunnel junction  [26]. The s econd set of sample s were continuous \nfilms used to characterize the X-ray absorption spectra (XAS) and its XMCD ; the total \nelectron yield (TEY) method was employed to calibrate the XMCD  obtained by the \nPFY method . A similar (001) -oriented epitaxial multilayer s that consists of MgO \nsubstrate/MgO buffer  (5 nm)/V  (30 nm)/ Fe (0.4 nm)/Co (0 nm, 0.14 nm)/MgO  (2 nm), \nwithou t a top SiO 2/Cr/Au layer , was prepared . The t hird set of sample s were magnetic 5 \n tunnel junction s used to characterize the MAE  and VCMA . A similar (001) -oriented \nepitaxial multilayer that consists of MgO substrate/MgO buffer  (5 nm)/V  (30 nm)/ Fe \n(0.3 nm, 0.4 nm, 0.5  nm)/Co (0 nm, 0.14 nm)/MgO barrier (1.4  nm)/Fe (10 nm)/Au \n(5 nm) was prepared and patterned into tunnel junction s that have a 2×5 μm2 junction . \nThe XAS /XMCD measurements were conducted at the soft X -ray beamline, BL25SU \nof SPring -8. Details of t he measurement setup and condition s are  reported elsewhere  \n[26–28]. The degree of circular polarization was previously estimated to be 96% [28] \nand was used as the correcting factor in the sum rule analysis. The XAS with right and \nleft helicities, μ+ and μ−, respectively, were recorded. The s elf-absorption effects in the \nfluorescent XAS/XMCD were corrected by refer ring to those recorded by the TEY \nmethod, which are less affected by the self -absorption effect  [29]. In the estimations of \nthe magnetic moment s using the sum rule , the accuracy errors of the physical \nparameters are approximately 10%. However, the errors that emerge due to the relative \nchanges , depending on the electric field and/or magnetization direction , i.e., precision \nerrors, are much smaller than the accuracy errors and were used to determine the error \nbars for this study . All the XAS/XMCD measurements were conducted at room \ntemperature.  \nThe inset of Fig. 2(a) depic ts the magnetization as a function of the external magnetic \nfield H, which was obtained using the XMCD defined as (μ+ − μ−) at the Co L3 edge \n(778.4 eV) with the PFY method.  The m agnetic field direction was θ = 20°. From th e \ninset of Fig. 2(a) , it can be seen that the magnetization of the Fe /Co layer is easily \nsaturated in the magnetic field direction  because the in-plane shape anisotropy energy \nand perpendicular MAE almost cancel each other  out in the Fe /Co layer . Figure s 2(a) \nand 2(b) present the polarization -averaged XAS defined  as (μ+ + μ−)/2 and the XMCD 6 \n spectra  around the L3- and L2-edges of the Co using the PFY method , respectively . \nMagnetic field s of ±1.9 T were applied at θ = 20° to saturate the magnetization of the \nFe/Co layer . The i nstrumental asymmetries of the nonmagnetic origin w ere removed by \nmeasuring the spectra with an opposite magneti c field direction.  \nThe m agnetic properties of the V/Fe (0.4 nm)/Co (0.14  nm)/MgO  (2 nm) and V/Fe  \n(0.4 nm)/MgO  (2 nm) are summarized in Table 1 , where  mL, mS, mT, and μB are the \norbital magnetic moment, spin magnetic moment, magnetic dipole moment , and Bohr \nmagneton, respectively. Each magnetic moment was characterized by the XAS/XMCD \nresults obtained using the TEY method with  sum rule analysis . For the sum  rule analysis  \n[30–32], 2.29 and 3.39 were employed as the number of holes in the 3 d-orbitals of Co \nand Fe, respectively [ 32]. The m agnetic moments of both Co and Fe in the \nV/Fe/Co/MgO sample were larger than th e reported values for pure Co and Fe [32], \nwhile those  of Fe in the V /Fe/MgO sample were comparable  to the values of pure Fe . \nSuch large magnetic moment s were reported in a Fe –Co alloy [ 33] and Co monatomic \nlayer [ 34–36]. The interfacial MAE is defined as the MAE with no bulk -induced effects  \n[2], and was characterized  by the resonant field of the ferromagnetic resonance  in the \nmagnetic tunnel junctions [ 25]. A positive MAE is defined as prefer ring perpendicular \nmagnetization. The VCMA  (fJ/Vm) was characterized by the electric -field-induced shift \nin the resonant field  and is defined as the MAE (mJ/m2) per unit electric field (V/m) in \nthe 2 -nm MgO.  \nIt was reported that the (0001 )-oriented ultrathin hcp-Co film exhibits orbital \nmagnetic moment anisotropy, which  explains the perpendicular MAE [ 35]. From Table \n1, the orbital magnetic moment of the 0.14-nm Co film in our experiment is relatively \nlarge, but its anisotropy is negligible. In co ntrast to the Co, the Fe exhibits an orbital 7 \n magnetic moment anisotropy of about 20% in both the V/Fe/Co/MgO and V /Fe/MgO \nsamples.  Hence,  the interfac ial MAE of 0.4 mJ/m2 in V/Fe/Co/MgO  and 0.5 mJ/m2 in \nV/Fe/MgO  may be attributed to the MAE of the Fe. The VCMA of  Fe/Co/MgO  \n(−82 fJ/Vm) is more than twice that of Fe/MgO  (−31 fJ/Vm ). Because the Co insertion \nat the Fe–MgO interface significantly increase s the VCMA , the Co should be \nresponsible for the VCMA  in the system . \nThe changes in the Co-L3 (778.4  eV) and -L2 (793.8  eV) peak heights of the XMCD \nsignals induced by the electric field were characterize d and are displayed in Figs. 3 (a) \nand 3(b), respectively, which were obtained using the PFY method. A magnetic field of \n±1.9 T with θ = 20 ° was applied during the measurement  to sa turate the magnetization  \nof Fe /Co layer in the magnetic field direction . In our system, an external voltage of \n±3 V corresponds to an electric field of ±0. 2 V/nm according to the capacitance model  \n[2]. The positive (negative) voltage induces electron s (hole s) at the Co–MgO interface. \nThe s olid and dashed line s indicate forward and backward voltage sweep s, respectively . \nAfter the measurements, w e can confirm that there is no significant change in the \nXAS/XMCD before and after the measurement s, which implies  that sample degradation \nby the electric field application is negligible.  The same measurement with a magnetic \nfield of ±1.9 T with θ = 70 ° was also conducted.  In contrast to the case of Fe /MgO  \n[26,37], where  the induced changes of XMCD at the Fe-L2 and -L3 absorption edges \nwere negligible, it  should be noted that significant changes were observed at the Co-L2 \nand -L3 absorption edge s in Fe/Co/MgO . \nThe electric -field-induced changes in the magnetic moments of Co were determined \nusing sum  rule analysis [ 30–32] and are presented in Fig s. 3(c) and 3(d). We assumed \nthat the XMCD integrals a t the L2- and L3-edges of Co, calculated by the TEY method , 8 \n were proportional to their peak int ensities  displayed in Figs. 3(a) and 3(b). In the \nsum-rule analysis equations (Eqs. (1) −(3) in Ref. 22), hole number of d-orbitals is \nproportional to the XAS integrals at the L3- and L2- edges. However, the magnetic \nmoments are not dependent of the changes of the XAS integrals. Therefore, although \nXAS and hole number can be modulated by the electric -field, the influence of the \ninduced changes in XAS on th e changes in magnetic moments could be neglected . As \nshown in Fig. 3 (c), the mL of Co with a voltage of −3 V is larger than that corresponding \nto 3 V . Moreover, the induced change of mL with θ = 20° is larger than that with θ = 70°. \nOur experiment demonstrates that an orbital magnetic moment anisotropy change  of \n(0.013±0.00 8)μB between magnetization angles of θ = 20° and 70 ° was generated in the \npresence of electric fields of ±0.2 V/nm.  The electric -field-induced change in the \neffective spin magnetic moment mS−7mT is shown in Fig. 3(d). Similar to mL, \nmS−7mT is enhanced at negative voltage application. Moreover, the \nelectric -field-induced change of the magnetic moment is anisotropic. In contrast to mT, \nit is known that mS is insensitive to the magnetization direction . Hence , the anisotropic \npart of the induced change of the magnetic moment is attributed to mT, that is, the \nmagnetic dipole Tz term [ 23,31,36]. A similar trend in the induced change of mS−7mT \nwas reported in Pt with proximity -induced spin polarization [ 22]. Note that the \nenhanced mS−7mT at a negative bias voltage, which induce s hole s at the Co–MgO \ninterface , cannot be explained by the electric -field-induced change in mS because of  an \nelectrochemical reaction following oxygen migration . In the case of oxygen migrat ion, \nhole accumulation decrease s the magnetic moments  as reported in Co/GdO x [16]. \nIn Figs. 3(c) and (d), the electric -field-induced changes of the magnetic moments may \nbe nonlinear ; however, such nonlinear components are within the error bars. During the 9 \n XMCD measurement, the sample current (power) density in the tunnel junction was \napproximately  0.5 mA/cm2 (1.5 mW/cm2). Therefore , the  change of the sample \ntemperature would be negligible, i.e., much lower  than 1K [ 38]. Therefore, a sample \ntemperature change is not likely to cause such nonlinear behavior. Moreover, we have \nconfirmed that VCMA in Fe/Co/MgO is linear as reported in our previous studies [ 39], \nwhere a linear electric -field-induced change in orbital magnetic moment is expected. \nTherefore, in the following discussion, we employed the values at ±3 V , in order to \nminimize error s in the analysis.  \nTo analyze the MAE, the following equation  is employed to address the second -order \nperturbation of the spin -orbit interaction  [23]: \n T\nex2\nBL, L,\nB'\n221\n4'mEm m E   \n \n.                 (1) \nThe perpendicular MAE (ΔE > 0 for perpendicular easy axis ) is defined as the MAE of \nthe in -plane magne tized film subtracted from that of  the perpendicularly magne tized \nfilm. Here,  \n 90\nL,0\nL, L, s s s m m m  and \n 90\nT0\nT T m m m  express the changes in the \norbital magnetic moment and magnetic dipole moment between the perpendicularly ( θ = \n0°) and in -plane ( θ = 90°) magne tized films , respectively. Moreover , ΔmL,↓(↑) represents \nthe contribution from the minority  (majority) spin -band. The measured orbital magnetic \nmoment  \n\nLm  is equal  to \n \n L, L,m m , and  λ’ is the effective spin-orbit interaction \ncoefficient  in the 3d-bands . In contrast to the case of Pt with proximity -induced spin  \npolarization [22], we first assume that we can neglect the second term corresponding to \nthe spin-flip perturbation process  between the exchange -split Eex majority and minority \nspin-bands , insofar  as Eex is large and λ’ is small in the Co. In the case of Co, w e can 10 \n also assume that we can neglect the term  related to mL↑ [36], because the majority \nspin-band is almost occupied . Thus, the perpendicular MAE is proportional to the \nmeasured changes in the orbital magnetic moment; this relation is known as the Bruno \nmodel [8], ΔE ≈ (λ’ ΔmL)/4μB. \nThe λ’ for the Bruno model is not identical to the spin-orbit interaction coefficient  of \nan atom , because λ’ depends on the band structure. The λ’ of the Co for the Bruno \nmodel was reported to be 3.3 meV in  the Au/Co(3  monolayer, ML)/Au multilayer [ 36]. \nIf only the interfacial 2ML -Co is responsible f or the observed orbital magnetic moment \nanisotropy,  the intrinsic λ’ for the Co would be 5 .0 meV . In our study, using the Bruno \nmodel, if λ’ = 5.0 meV, we obtained that the induced change in perpendicular MAE is \n(0.039 ± 0.023) mJ/m2 when the electric field is switched from +0.2 V/nm to −0.2 \nV/nm; the experimentally obtained ΔmL = (0.017 ± 0.010)μB was used.  We employed a \nsimple assumption, \n  2 90\nL2 0\nL L sin cos   m m m . Table 1 (−82 fJ/ Vm) shows that the \nexperimentally obtained induced change in perpendicular MAE of the Fe /Co/MgO  \nstructure at ±0.2 V /nm is 0.03 mJ/m2, which is in good agreement with the induced \nchange in perpendicular MAE obtained using t he Bruno model. Hence, our experiment \nprovides  direct evidence for the application of the Bruno model to the VCMA  in the \n3d-ferromagnetic metals . \nFrom the discussion above, the change of the orbital magnetic moment anisotropy in \nthe Co seems to explain the VCMA . However, the impact of the change of the magnetic \ndipole Tz term mT shown in Fig.  3(d), on the VCMA  remains to be seen. Figure 4 (a) \npresents a cross -section al view of our model for the first -principles study. The \nFe/Co/MgO multilayer was modeled by a periodic slab  supercell with  3ML -Cu, 4ML -V, \n3ML -Fe, 2 ML-CoFe, 5ML -MgO , and a 26-Å-thick vacuum layer. To reproduce the 11 \n magnetic properties in Table 1, we did not employ the Fe/Co/MgO multilayer w ith an \nideal Co–MgO interface , but instead used that with intermixed CoFe at the MgO \ninterface. To simplify the model, we assume that the Co coverage and Fe cover age in \nthe atoms at  the MgO interf ace are both 0.5. Details of the computation method are \nreported elsewhere [ 22]. For the in -plane lattice constant of the atoms , the value of \n0.286  nm, which is identical to that of bulk Fe , was employed.  Because of the strong \nscreening effect of metals, the MAE change  in the system is dominated by atoms at the \nMgO interface. Figure 4 (a) also depicts the induced change of charge density. The \ninduced change of charge density at an electric field of +0.2 V/nm in the MgO is \nsubtracted from that at −0.2 V/nm. The blue and r ed regions indicate the hole \naccumulations and depletions, respectively.  From Fig. 4 (a), note that the induced change \nof the charge density in the metals is dominant in the interfacial atoms with the MgO . \nTable 2 lists the magnetic properties obtained from the first -principles study. The values \nof the magnetic moments are those of the atoms at the MgO interface. The \nfirst-principles study qualitatively reproduces the experimental results listed in Table 1. \nTo discuss the effects of the electric -field-induced mL and mT on the VCMA , the \ninduced change of perpendicular MAE from the second -order perturbation to the \nspin-orbit interaction is calculated directly from the first -principles study  [40]. We \nemployed the following equation as the perpendicular MAE  [41]: \n\n\nuo so sux z\nssE EsoL su soL su\nE\n, , ',2 2\n2\n'//,, //,', ,, ,',\n\n.           (2) \nFor Co and Fe, 69.5 and 54.4 meV, which is the spin -orbit coupling of the atoms, are \nemployed for  λ in Eq. 2 , respectively . The values of the electric -field-induced changes \n(δ) to the perpendicular MAE in the Co and Fe atoms at the  MgO  interface , arising from 12 \n the spin -conserved term ( s's = ↑↑ or s's = ↓↓) and spin -flip term ( s's = ↓↑ or s's = ↑↓), are \nderived and shown in Fig. 4(b). The MAE at +0.2 V/nm is subtracted from that at −0.2 \nV/nm. As discussed in previous studies  [22,23], the perpendicular MAE from the \nspin-conserved terms (ΔE↑↑ and ΔE↓↓) in Eq. 2 corresponds to the first terms of Eq. 1: \nΔmL,↓ −ΔmL,↑. Further, the perpendicular MAE from the spin -flip terms (ΔE↓↑ and ΔE↑↓) \nin Eq. 2 corresponds to the second term of Eq. 2: ΔmT. First, f rom Fig. 4 (b), the \nelectric -field-induced change of the perpendicular MAE  in Co is approximately three \ntimes  larger than that of Fe. Second, the change in the electric -field-induced \nperpendicular MAE from the spin-flip terms ( δE↓↑ + δE↑↓) is negligible and that from \nthe spin-conserved terms ( δE↑↑ + δE↓↓) dominates the MAE  change . From the se results , \nnote that the induced perpendicular MAE change from the spin-conserved terms of Co \ndominates the MAE change  in the system. In other words, rather than the \nelectric -field-induced change of the magnetic dipole Tz term δmT, the induced change of \nthe orbital magnetic moment anisotropy  δmL is responsible for the VCMA . \nIn this study, the m agnetic moments of Co we re experimentally characterized using \nin-situ XMCD at ±0.2 V/nm external electric field s. With the electric -field-induced \nchanges in magnetic moments in the 3d-state of Co, an i nduced change in orbital \nmagnetic moment of 0.0 13μB was confirmed . The VCMA  in the Co was suitably \nestimated by the induced change of the orbital magnetic moment anisotropy , according \nto the Bruno’s perturbation theory model. The induced change of the magnetic dipole Tz \nterm of  Co was confi rmed ; however, the first -principles study indicates that the VCMA  \nin the system is determined by the induced change of the o rbital magnetic moment \nanisotropy rather than that of the magnetic dipole Tz term.  This study provides new \ninsight into electric -field control of condensed matter.  13 \n  \nWe thank E. Tamura of Osaka University, T. Nozaki of AIST, and Y . Shiota of Kyoto \nUniversity for their discussion s. This work received support from the ImPACT Program \nof the Council for Science, Technology and Innovation (Cabinet Office, Government of \nJapan), and from JSPS KAKENHI  (Grant Nos. JP26103002 and JP15H05420). 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The magnetic moments are  \ncharacterized by the XAS/XMCD results obtained using the TEY method with sum rule \nanalysis , and they are the averaged values in the total thickness. In the estimations of the \nmagnetic moments using the sum rule, the accuracy errors of the physical parameters \nare approximately 10%. However, the errors that emerge due to the relative changes , i.e. \nprecision errors,  are much smaller than the accuracy errors and were used to determine \nthe error bars. The interfac ial MAE  is defined as the MA E with  no bulk-induced effects. \nThe VCMA  is defined as the change in MAE per unit electric field i n the 2 -nm MgO.  \n Fe (0.4 nm)/Co  (0.14 nm)/MgO  Fe (0.4 nm)/MgO  \n Fe Co Fe \nmL/μB θ = 0° 0.12  ± 0.005  0.28  ± 0.005  0.10  ± 0.005  \nθ = 70° 0.10  ± 0.005  0.28  ± 0.005  0.08  ± 0.005  \n(mS−7mT)/μB θ = 0° 2.63  ± 0.01  2.28  ± 0.01  2.09  ± 0.01  \nθ = 70° 2.23  ± 0.01  2.19 ± 0.01  2.05  ± 0.01  \nInterfacial MAE  0.4 ± 0.05 mJ/m2 0.5 ± 0.05 mJ/m2 \nVCMA  −82 ± 8 fJ/Vm  −31 ± 3 fJ/Vm  \n \nTABLE II. Calculated magnetic properties of Fe/Co/MgO and Fe /MgO systems. The \nvalues of the magnetic moments are those of the atoms at the MgO interface. The \nperpendicular MAE is defined as the difference in MAEs between the perpendicularly \nand in -plane magnetized states. The VCMA  is defined as the change in MAE per unit \nelectric field in MgO.  \n Fe/Co/MgO  Fe/MgO  \n Fe Co Fe \nmL/μB θ = 0° 0.112  0.144  0.127  \nθ = 90° 0.098  0.129  0.102  \n(mS−7mT)/μB θ = 0° 2.814  1.869  2.976  \nθ = 90° 2.637  1.670  2.782  \nPerpendicular MAE  0.20 mJ/m2 0.98 mJ/m2 \nVCMA  −256 fJ/Vm  −90 fJ/Vm  \n  19 \n  \nFIG. 1 (Color online) Schematic of the sample structure and measurement configuration. \nAn external  electric field was applied to the ferromagnetic ultrathin Co film with  a \ntwo-dimensional square lattice via a dielectric consisting of MgO and SiO 2. \nXAS/ XMCD measurements were performed using the PFY method with a n SDD . \n  \n20 \n  \nFIG. 2 (a) Polarization -averaged X-ray absorption  (μ+ + μ−)/2 and (b) its XMCD (μ+ − \nμ−) spectra obtained using the PFY method a round the Co-absorption edge. An e xternal \nmagnetic field of ±1.9 T was applied to saturate the magnetization of  the Fe/Co layer in \nthe magnetic field direction. The i nset shows the magnetization as a function of the \nexternal magnetic field H measured by XMCD at the Co-L3 edge  (778.4 eV) . The \nmeasurements were  conducted in a magnetic field with θ = 20°.  \n21 \n  \nFIG. 3 (Color online) External -voltage -induced changes in XMCD at the (a) Co-L3 \n(778.4 eV) and (b) Co -L2 (793.8 eV) edges  obtained using the PFY method . \nExternal -voltage -induced changes in the (c) orbital magnetic moment  mL and (d) \neffective spin magnetic moment mS−7mT of Co . An external voltage of ±3 V \ncorresponds to an external electric field of ±0.2 V/nm in the 2 -nm MgO  dielectric . An \nexternal magnetic field of ±1.9 T was applied to saturate the magnetization of Fe/Co \nlayer in the magnetic field direction . A negative bias voltage, where holes accumulate at \nthe Co-MgO interface, increases both the mL and mS−7mT. \n  \n22 \n  \nFIG. 4 (Color online) (a) Computational model with the induced change of charge \ndensity. The induced change of the charge density at an electric field of +0.2 V/nm in \nthe MgO is subtracted from that at −0.2 V/nm. The blue and red areas represent the hole \naccumulation and deplet ion, respectively. (b) Electric -field-induced changes to the \nperpendicular MAE of Co  and Fe atoms at the MgO interface  calculated with Eq. 2. The \nMAE at +0.2 V/nm in the MgO is subtracted from that at −0.2 V/nm. The \nspin-conserved -term-induced values of  the MAE change (δE↑↑ + δE↓↓) in Co provide the \ndominant contribution to the VCMA . \n"    },    {        "title": "1708.09179v1.Micrometer_thick_soft_magnetic_films_with_magnetic_moments_restricted_strictly_in_plane_by_negative_magnetocrystalline_anisotropy.pdf",        "content": "Micrometer thick soft magnetic films with magnetic moments restricted \nstrictly in\n \nplane \nby\n \nnegative magnetocrystalline anisotropy\n \nTianyong Ma, Juanying Jiao, Zhiwei Li*, Liang Qiao, Tao Wang\n*\n, Fashen Li\n \nKey Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, \nLanzhou 730000, China\n \nCorresponding author: \nzwei\nl\ni@lzu.edu.cn\n \nand \nwtao@lzu.edu.cn\n \n \n \nAbstract\n \nStripe domains or any other \ntype\n \ndomain structures with part \nof their magnetic moments deviating from the film \nplane, which usually occur above a certain film thickness, are known problems that limit their potential applications for\n \nsoft magnetic thin films (\nSM\nTFs). \nIn this work, \nwe report the growth of micrometer t\nhick c\n-\naxis oriented hcp\n-\nCo\n84\nIr\n16\n \nSM\nTFs with their magnetic moments restricted strictly in plane by negative magnetocrystalline anisotropy.\n \nExtensive \ncharacterization\ns\n \nhave been performed on these films, which show that they exhibit very good soft magnetic properties \neven for our micrometer thick films. Moreover, the anisotropy properties\n \nand high\n-\nfrequency properties were th\no\nroughly \ninvestigated and our results show ve\nry promising properties of these \nSM\nTFs for future \napp\nlications.\n \n \nKeywords: \nnegative magnetocrystalline anisotropy\n, \nmagnetic moments\n, \nperpendicular anisotropy\n, \nhigh\n-\nfrequency \nproperties\n \n \n \n1.\n \nIntroduction\n \nIn recent years, \nstudy of s\noft magnetic thin films \n(SMTFs) \nhas\n \nbecome one of the hot topic\ns\n \nin the\n \nfield of\n \nmagnetism because of their potential applications in high\n-\nfrequency fields, such as miniature inductors, \nmicro\n-\ntransformers and noise suppressors\n \n[1\n-\n3]\n. In high\n-\nfrequency application\ns\n, there are two \nbasic \ndemands\n \nfor \nthe \nSMTFs, which include small switching field and high permeability before the cut\n-\noff frequency. In order to reach these \nrequirements, the magnetic material\ns\n \nmust have both\n \nhigh saturation magnetization \ns\nM\n \nand \nlow co\nercivity \nc\nH\n.\n \nE\nxtensive studies ha\nve\n \nbeen devoted to\n \nFe\n-\n \nand Co\n-\nbased\n \namorphous and nanocrystalline SMTFs\n \nwith negligible \nmagnetocrystalline anisotropy [4,5]\n \nand \nthese\n \nSMTFs\n \nexhibit\n \nexcellent static and high\n-\nfrequency magnetic properties.\n \nHowever, \ni\nn practical uses of \nmagnetic \ndevice\ns, \nthe\n \nSMTFs\n \nmust have \nsufficient thickness (\nusually \nabove micrometer) to \nobtain enough magnetic flux signal\n \n[6]\n. \nF\nor the \nwell\n-\nstudi\ned \nFe\n-\n \nand Co\n-\nbased\n \na\nmorphous and nanocrystalline SMTFs, a \nconsiderable\n \nperpendicular anisotropy \nwill \nalways occur\n \ndue to\n \ndefects and/or \ninternal stress \ncoming from\n \nthe growth\n procedure\n \nof the\nse\n \nSMTF\ns [7\n-\n9]. T\nh\nis\n \nperpendicular anisotropy\n \nwill\n \nresult\n \nin \nthe \nappearance\n \nof a stri\npe\n-\ntype domain \nstructure \nwhen\n \nthe thickness of th\nese\n \nSMTFs\n \nexceeds a few \nhundred nanometers. \nSuch a stripe domain structure will \nlead to\n \ntwo unfavorable consequences\n \nthat hinders practical applications \n[10,11]\n:\n \n(1) \na significant increase \nin\n \ncoercivity\n,\n \nwhich \nlead\ns\n \nto a\n \ndeterioration of the soft magnetic properties; (2) the magnetic moments exhibit a large vertical \ncomponent rather than strictly lie \nwith\nin the \nfilm \nplane. \nIn practical uses, f\nor\n \nexample, the inductance \nand \nthe \nquality \nfactor of \nthe miniat\nure \ninductors will be significantly reduced\n \nand the loss will be extr\ne\nmely large when SMTFs\n \nwith \nstripe domain \nstructure \nwere\n \nused. \nTherefore, preparation of thick enough SMTFs with its \nmagnetic moments strictly \nlying \nparallel to\n \nthe\n \nfilm\n \nplane \nis essentia\nl for them to have\n \nexcellent \nsoft magnetic\n \nproperties and thus is \ncrucial for \npractical application\ns\n \nof \nthese \nSMTFs\n.\n \n \nIn our previous work, we have \nstudi\ned \nCoIr\n-\nbased \nSMTFs\n \nwith a strong negative magnetocrystalline anisotropy \ngrain\nu\nK\n, which \nshow\ns that the c\n-\nplane\n \nof \nthe \nCoIr crystal\n \nis an easy magnetization plane and the c\n-\naxis is the \ncorresponding hard axis. In other words, \nrotation of \nthe \nmagnetic moments from the \nc\n-\nplane\n \nto \nthe c\n-\naxis\n \ndirection must \novercome\n \nan\n \neffect\nive magnetic\n \nfield\n \ns\ngrain\nu\ngrain\nu\nM\nK\nH\n2\n\n \nof the order of\n \n~\n8 kOe\n \n[12\n,13\n] in addition to the \ndemagnetization field 4\nπ\nM\ns\n. \nTherefore, we anticipate that the critical thickness for stripe type domain structure to appear \nshall be greatly enhanced i\nf \nthe CoIr \nSMTFs\n \nwere grow\nn\n \nwith\n \ntheir c\n-\nplane\n \nparallel to the film\n \nplane\n.\n \nIn this work, to \nv\ne\nrify our idea, we report the growth of oriented hcp\n-\nCo\n84\nIr\n16\n \nSMTFs\n \nup to micrometer thick. Our results show that these \nfilms with film thickness ranging from 100 nm to 1.12 \nμ\nm all have excellent soft magnetic properties with small \ncoercivity and\n, more importantly,\n \nwith the\n \nmagnetic moments \nof these films lie\n \nparallel to \nthe film plane.\n \nThe intrinsic \nmagnetocrystalline anisotropy\n \nand \ntotal effective out\n-\nof\n-\nplane anisotropy have been investigated, and our high\n-\nfrequency \npermeability measurements show very promising properties of these \nSMTFs\n \nfor future practical \napp\nlications\n.\n \n2.\n \nExperiment\n \nAll our samples were prepared by DC \nperpendicular\n \nmagnetron sputtering method with a layered structure of\n \nsubstrate/Ti/Au/Co\n84\nIr\n16\n \n(see inset of Fig. 1 (a)). Si wafer with (100) surface orientation was used as substrate, and the \nseed layer of \nTi(8 nm) / Au(25 nm) was deposited first in order to induce the c\n-\naxis orientation of the Co\n84\nIr\n16\n \nsoft \nmagnetic layer. A variety of oriented Co\n84\nIr\n16\n \nfilms with different layer thickness\nes \nranging from 100 nm to 1120 nm \nwere deposited using the same seed layer. \nI\nn this work, for clarity, the samples were denoted as CoIr100, CoIr300, \nCoIr500, CoIr950, and CoIr1120 corresponding to samples with Co\n84\nIr\n16\n \nlayer thicknesses of 100, 300, 500, 950,\n \nand \n1120 nanometer. The seed layer was deposited using Ar as working gas with a pressure of 0.25 Pa\n.\n \nThe Co\n84\nIr\n16\n \nsoft \nmagnetic layer was grown with an Ar pressure of 0.3 Pa\n \nby using Co target with five Ir chips symmetrically placed on \nthe erosion race\n-\ntr\nack. To induce an in\n-\nplane uniaxial anisotropy, the substrate was inclined by 5\no\n \nwith a wedge during \nthe sputtering process. The crystalline structure \nand \ncross section\n \nmorphology \nwere investigated by X\n-\nray diffraction \ntechnique (XRD) and scanning electron\n \nmicroscope (SEM). The out\n-\nof\n-\nplane magneti\nzation component\n \nin zero\n-\nfield was \ndetected through magnetic force microscopy (MFM). The static magnetic properties were measured with a vibrating sample magnetometer (VSM) and the dynamic magnetic properties were\n \nmeasured with an electron spin resonance \nspectrometer (ESR). The microwave permeability measurements were performed by a vector network analyzer (VNA) \nusing the shorted microstrip method. \n \n \n3.\n \nResults and discussion\n \nThe XRD patterns of the samples with indic\nated Co\n84\nIr\n16\n \nlayer thicknesses measured with \n\n\n2\n\nscan \nare\n \nshown \nin Fig. 1 (a). Only two diffraction peaks corresponding to Au (111) plane and Co\n84\nIr\n16\n \n(002) plane can be seen for all the \nsamples. The schematic layer structure of the samp\nles is shown in the inset of Fig. 1 (a). Similar to our previous work \n[\n13\n], the 8 nm Ti layer provides an amorphous surface that assists the oriented growth of the Au layer with (111) plane \nparallel to the thin film which was expected to induce the oriente\nd growth of Co\n84\nIr\n16\n \nsoft magnetic layer with its c\n-\naxis \nnormal to the film surface. This was proved working very well by the only diffraction peak of the Co\n84\nIr\n16\n \n(002) plane \nfor all our samples. To further verify the goodness of the c\n-\naxis alignment of the Co\n84\nIr\n16\n \nlayer, rocking scans on the (002) \npeak were made for the CoIr100 and CoIr1120 samples as shown in Fig.1 (\nb\n). The full width at half maximum (FWHM) \ndete\nrmined from \nLorentz\n \nfits to the experime\nn\ntal data amount to 1.9\no\n \nand 1.7\no\n \nfor the CoIr100 and CoIr1120 samples, \nrespectively.\n \nT\nh\ne small values of FWHM\n \nreveal \nthe good alignment of\n \nthe Co\n84\nIr\n16\n \nlayer [1\n4\n]. Interestingly, the even \nsmaller value for the CoIr1\n120 sample dem\non\nstrates that the misalignment which might come from defects or lattice \nmismatch can be further reduced for thicker samples. In Fig. 2 we present \na typical \ncross\n-\nsection image from SEM \nmeasurements for the 950 nm sample. The good alignment o\nf the c\n-\naxis of the Co\n84\nIr\n16\n \nlayer can be directly seen from \nthe columnar\n-\nlike crystalline \nstructure\n.\n \nThe in\n-\nplane magnetic hysteresis loops of the oriented hcp\n-\nCo\n84\nIr\n16\n \nSMTFs\n \nare shown in Fig. 3 (a)\n-\n(e). These loops \nwere measured with the applied magnetic\n \nfield parallel or perpendicular to the easy axis\n \nwith\nin \nthe \nfilm plane\n. For all \nsamples, the hysteresis loops are totally different when measured in the two directions\n, which suggest\ns that\n \nall films \npossess an in\n-\nplane uniaxial magnetic anisotropy\n. The \nuniaxial anisotropy induced by\n \nthe\n \noblique deposition technique \ncan be\n \nexplained \nwith \nthe framework of \nthe \nso\n-\ncalled self\n-\nshadowing model\n \n[15]. According to this \nmodel\n, the region \nbehind a growing crystallite is in the \n“\nshadow\n”\n \nof the crystallite and \nthus \nis prevented from \ncontinuous\n \nreceiving metal \nvapor, so the crystallites \nwill \nform\n \na\n \ntwo\n-\ndimensional array of chains \nwhich leads to \nthe uniaxial anisotropy\n \nof the \nSMTFs\n. \nThe almost rectangular loops are observed along the easy axis and\n \nthe \nremanence ratios \nare all above 0.92\n.\n \nThe \nlarge\n \nremanence ratio \nsuggests\n \nthat nearly all \nthe magnetic moments lie parallel to the film plane even for our \nmicrometer thick sample. This is in sharp contrast to the\n \nFe\n-\n \nand Co\n-\nbased amorphous and nanocrystalline SMTFs\n \nin \nwhich \nstrip domain structure will appear when the film thickness exceeds only a few hundred nanometers\n \n[7\n-\n9]. \nAs \nshown in Fig.3 (f), the saturation magnetization 4\nπ\nM\ns \nof all the films are close to 1\n4\n \nKOe which agrees well with \nreported values in literature [1\n6\n].\n \nThe coercivity \nc\nH\n \ndetermined from the easy axis loop decrease gradually from 35 Oe \nto 16 Oe with increasing film thickness. This decreasing tendency, which commonly exists in many magnetic thin films, can be well explained by the random anisotropy model for polycrystalline\n \nferromagnetic films [17]. When the grain size \nis larger than the natural exchange length, the coercivity is inversely\n \nproportion\nal\n \nto the grain size and can be described \nas \n1\n\n\nD\nH\nc\n \nwith nearly constant \n4π\nMs and nearly invariable\n \nin\n-\nplane un\niaxial anisotropy field\n \ninduced by the same \ncondition. \n \nThe zero\n-\nfield MFM images tha\nt\n \nwere\n \na\ncquired\n \nin the as\n-\ndeposited state\n \nat the film surface of sample CoIr950 and \nCoIr1120 are shown in Fig. 4. No obvious magnetic contrast can be seen in these \nz\nero\n-\nfi\neld MFM images, indicating the \nabsence of stripe type domains and/or any other type of domain structures that involve with partial magnetic moments \ndeviating from the others that lie parallel to the film plane. \nThis\n \nmatches well with the analysis from the \nhysteresis loops \nand is another evidence that most of the magnetic mom\ne\nnts are restricted within the film plane.\n \nThis is exactly the \nsituation that we anticipated for the o\nriented hcp\n-\nCo\n84\nIr\n16\n \nSMTFs\n \nwith a strong intrinsic negative magnetocrystalline \nanisotropy of the order of ~ \n-\n10\n6\n \nJ/m\n3\n.\n \nTogether with the demagnetization field, this will provide an effective out\n-\nof\n-\nplane \nanisotropy of the order of ~ 20 kOe. On the other hand, the perpendicular anis\notropy that leads to the stripe type domain \nstructure is much smaller, ~ 10\n4\n_\n10\n5\n \nJ/m\n3\n. For example, \nthe reported perpendicular anisotropy is about 9.5 kJ/m\n3\n \nfor\n \nNiFe \nfilms\n \n[\n18\n], and 11.3\n \nkJ/m\n3\n \nfor\n \nFeCoNbB amorphous film\n \n[\n19\n]\n, where stripe domain structure \nappears above several \nhundred nanometers\n. \nTherefore, the magnetic moments in our \nSMTFs\n \nare completely \ncompel\nl\ned\n \nto lie within the film \nplane by the strong effective out\n-\nof\n-\nplane anisotropy field.\n \n \nTo quantify this effect and to get\n \nthe\n \nvalue of the\n \nintrinsic \nmagnetocrystalline anisotropy \ngrain\nu\nK\n \nand the \ntotal \neffective out\n-\nof\n-\nplane anisotropy field \n\nH\n \nfor our samples, we performed ESR measurements on all the films with a \n9 \nGHz\n \nmicrowave magnetic field applied in\n-\npl\nane but perpendicular to the easy axis direction. The DC magnetic field is \napplied parallel to the film plane and forming an angle of \nH\n\n \nwith the easy axis. The measured resonance fields \nr\nH\n \nare shown in Fig. 5 (a)\n-\n(e\n) as a function of \nH\n\ntogether with theoretical fits. It is well known that \nr\nH\n \nis smallest when \nthe applied magnetic field is parallel to the easy axis and the largest when perpendicular. From published works [\n13\n,\n20\n],\n \nthe relationship between the resonance field and the angle \nH\n\n \ncan be expressed as\n:\n \n\n\n\n\n\n\n\n\n\n\nH\nr\nu\nH\nr\nu\nH\nH\nH\nH\nH\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n0\n0\n0\n0\n2\n2\ncos\n2\ncos\ncos\ncos\n,      \n \n(\n1\n)\n \nwhere \n\n \nis the angular frequency, \n\n \nis the gyromagnetic ratio, \n0\n\n \nis the equilibrium positions of the magnetization. \nThe solid lines in Fig. 5 (a)\n-\n(e) are numerical fits to the experimental data using equation (\n1\n). The apparent twofold \nsymmetry indicates a well defined in\n-\nplane uniaxial anisotropy i\nn agreement with \nour\n \nVSM measurements [21]. The \nobtained total out\n-\nof\n-\nplane anisotropy field \n\nH\n \nare shown in Fig. 5 (f) as a function of film thickness. As can be seen, \n\nH\ndecrease gradually from 27 kOe to 23 kOe with\n \nincreasing film thickness.\n \nTo get a clear understanding of this trend, \nwe express t\nh\ne total out\n-\nof\n-\nplane anisotropy field \nas \np\ngrain\ns\nH\nH\nM\nH\n\n\n\nu\n4\n\n\n, where \ngrain\ns\nH\nM\nu\n4\n\n\n \ncan be treated as constant. Then the decrease of \n\nH\n \nis coming from \np\nH\n, the so called perpendicular anisotropy field in the \nfavor of stripe domain structure. The corresponding perpendicular anisotropy constant can be determined as \n2\ns\np\np\nM\nH\nK\n\n \nwhich we beli\ne\nve is originating fr\nom the columnar structure and any defects or stress of t\nhe \nas\n-\ngrown\n \nfilm [22, 23]. Following the above definition, \ngrain\ns\nH\nM\nu\n4\n\n\n \nis determined to be 27.5 kOe and the intrinsic \nmagnetocrystalline anisotropy \ngrain\nu\nK\n \nwas calculat\ned to \nbe \n-\n753 kJ/m\n3\n \nwhich agrees well with reported values [\n12\n,13\n]. \nThen the thickness\n \ndependence of the perpendicular anisotropy \np\nK\n \nwas calculated and shown in Fig. 5 (f). Although \np\nK\n \nis much larger than that for the\n \nNiFe \nfilm and the \nFeCoNbB amorphous film\n \nwhere \np\nK\n \nis in the order of 10 kJ/m\n3\n, \nthey are still much smaller than \ngrain\nu\nK\n. Then, the net energy density gain \nP\ngrain\nu\nK\nK\n\n \nis still smaller when the \nmagnetic moments lie parallel to the film plane even for the micrometer thick film. In other words, one can expect, the \nmagnetic moments will be restricted within the film plane by the neg\na\ntive magnetocrystalline anisotropy provi\nded \ngrain\nu\nP\nK\nK\n\n.\n \nTo v\ne\nrify the potential applicability, we measured the microwave permeability spectra of our SMTFs by VNA. T\nhe \nfrequency dependence of the permeability can be described by the following \nfor\nmula deduced from the Landau\n–\nLifshitz\n–\nG\nilbert (LLG) equation [\n24,25\n]: \n \n)\n(\nM\n4\n1\n2\n1\n2\n2\n1\n1\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\ni\ni\ns\n.\n   \n(\n3\n)\n \nin which \n)\nH\n(H\n)\n(\nu\n1\n\n\n\n\n\n\n\n\nu\ngrain\nu\ns\nH\nH\nM\n4\nγ\nω\n, \nu\nH\n\n\n\n2\n, where \n\n \nis the angular frequency, \n\n \nis \nthe damping constant\n \nand \nu\nH\nis \nthe \nin\n-\nplane uniaxial anisotropy field\n. Considering the values of \ns\nM\nand\n\nH\nhave been \nobtained\n \nalready\n, we fit\nted\n \nthe experiment\nal\n \ncurves by \nequation (\n3\n)\n \nusing \nu\nH\nand \n\n \nas \nfree \nfitting parameter\ns as \nshown in Fig. 6 (a)\n-\n(e)\n. \nThe extracted values for \nu\nH\n \nare \n115, 115, 120, 114, 110 \nOe \nfor\n \nour\n \nfilms\n \nin the order of \nincreasing film thickness\n. Obviously, the\nse\n \nvalues are approximately a constant \nwhich is consistent \nwith the fact that \nthey\n \nwere\n \ninduced by the same\n \ngrowth \ncondition\n, namely the same incline angle ~ 5\no\n \nduring the sputtering process [\n15\n]\n. \nOne \ncan see that all these films exhibit a similar behavior with a nearly constant real part before the gradually decr\neasing \nresonance frequency with increasing film thickness. The initial permeability matches well with the \ncalculated\n \nvalu\nes \nby \nequation\n \nu\ns\ni\nH\nM\n\n\n4\n1\n\n\n, which is approximately 1\n2\n0 using the measured saturation magnetization \ns\nM\n\n4\n \nand \nthe extracted in\n-\nplane uniaxial anisotropy field \nu\nH\n. \nThe natural resonance frequency \nexp\nr\nf\ncorresponding to the \nmaximum value in the imaginary part, as shown in Fig.\n \n6 (f)\n, \nfollows the same trend of the calculated v\nalues by \nthe \nextended Kittel\n’\ns\n \nequation\n \n\n\n\nH\nH\nf\nu\nr\n2\n\n \nwhich can be understood by the decrease of \n\nH\n \nwith increasing film thickness. Th\ne weak \ndiscrepancy\n \nof t\nhe experimental \nand \ncalculated data may result from the revised\n \nKittel\n’\ns\n \nequation\n \nderived from the LLG equation by neglecting the damping effect [\n26\n]. \n \nThe damping constant \n\n \nderived from the fitting, which increase with film thickness, were plotted in Fig. 6 (f). It \nis well known that \n\n, \nwhich is an important parameter for high frequency applications [\n27\n]\n,\n \nconsists of two parts, \nnamely, the intrinsic and extrinsic contributions. The intrinsic part is largely depends on fundamental properties such a\ns \nMs and spin\n-\norbit coupling effect [\n28\n], and the extrinsic part is generally explained by the defect\n-\ninduced t\nwo\n-\nmagnon \nmodel and/or the local resonance model [\n29\n,\n30\n], both of which is associated with magnetic inhomogeneities within the \nmaterial (e.g.\n \ndue to anisotropy dispersion and surface or interface roughness).\n \nHere, the intrinsic part may not influence \nthe d\namping constant since \ns\nM\n \nis almost a constant. Therefore, the extrinsic part must play an important role in \nincreasing the damping constant. Apart from this, it was reported that eddy currents also affect the permeability spectru\nm \nand cause a broadening of the resonance peak for \nthicker films [3\n1\n,3\n2\n]. From the above considerations, we beli\ne\nve that \nboth magnetic inhomogeneities and eddy currents are involved in the increase of the damping constant, especially for the \nmicrometer thick films. \n \n4.\n \nConclusions\n \nIn summary,\n \nc\n-\naxis oriented \nhcp\n-\nCo\n84\nIr\n16\n \nmagnetic films with strong negative magnetocrystalline anisotropy were \nfabricated\n \nin the range of\n \n100 nm to 1.12 \n\nm.\n \nAll these films were shown to have high saturation magnetization and high \nrectangular ratio with low coercivit\ny\n \nin the easy di\nrection. More importantly, all these films have a high degree of c\n-\naxis \norientation even up to micrometer thick. 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Wang,and Jongill Hong 1999\n \nMagnetic and Microstru\nctural Characterization of FeTaN High \nSaturation Materials for Recording Heads\n \nIEEE Trans. Magn. \n35\n \n2\n \n[\n23\n] \nR.A. Morris, Y. Inaba, J.W. Harrell\n \nand \nG.B. Thompson\n \n2010\n \nInfluence of underlayers on the c\n-\naxis \ndistribution in Co\n80\nPt\n20\n \nthin films Thin Solid Films. \n518\n \n4970\n \n[\n24\n] \nN.N. Phuoc, F. Xu, C.K. Ong\n \n2009\n \nTuning magnetization dynamic properties of Fe\n–\nSiO2 multilayers by \noblique deposition\n \nJ. Appl. Phys. \n105\n \n113926\n \n[\n25\n] \nS. Ge, S. Yao, M. Yamaguchi, X. Yang, H. Zuo, T. Ishii, D. Zhou\n, and F.Li\n \n2007\n \nMicrostructure and magnetism \nof FeCo\n–\nS\niO2 nano\n-\ngranular films f\nor high frequency application\n \nJ. Phys. D: Appl. Phys.\n \n40\n \n3660\n \n[\n26\n] \nShujuan Yuan, Baojuan Kang, Liming Yu, Shixun Cao, Xinluo Zhao\n \n2009\n \nIncreased ferromagnetic resonance \nlinewidt\nh and exchange anisotropy in NiFe/FeMn bilayers\n \nJ. Appl. Phys. \n105\n \n063902\n \n[\n27\n] \nN. Fujita, N. Inaba, F. Kirino, S. Igarashi, K. Koike, H. Kato\n \n2008 Damping constant of Co/Pt multilayer \nthin\n-\nfilm media\n \nJ. Magn. Magn. Mater\n. \n320\n \n301\n9\n \n[\n28\n] \nV. Kambersky\n \n1970\n \nOn\n \nthe Landau\n-\nLifshitz relaxation in ferromagnetic metals\n \nCan. J. Phys. B\n.\n \n48\n \n2906\n \n[\n29\n]\n \nR. Arias and D. L. Mills\n \n1999\n \nExtrinsic contributions to the ferromagnetic resonance response of ultrathin \nfilms\n \nPhys. Rev. B\n.\n \n60\n \n7395\n \n[\n30\n]\n \nC. Chappert, K. L. Dang, P. Beauvillain, H. Hurdequint, and D. Renard\n \n1986\n \nFerromagnetic resonance studies \nof very thin cobalt films on a gold substrate Phys. Rev. B\n.\n \n34\n \n3192\n \n[\n31\n]\n \nJ. Ben Youssef\n, N. Vukadinovic, D. Billet and M. Labrune \n2004\n \nThickness\n-\ndep\nendent magnetic excitations in \nPermalloy films with nonuniform magnetization Phys. Rev. B\n.\n \n69\n \n174402\n \n[\n32\n] \nN. Fujita, N. Inaba, F. Kirino, S. Igarashi, K. Koike\n \nand\n \nH. Kato\n \n2008\n \nDamping constant of Co/Pt multilayer \nthin\n-\nfilm media\n \nJ. Magn. Magn. Mater\n. \n320\n \n3019\n \n \n  \nFig.\n \n1\n \n(\na\n) XRD patterns of the oriented hcp\n-\nCo\n84\nIr\n16\n \nfilms with indicated layer thicknesses. The inset shows the \nsch\ne\nmatic layer structure of the sample. (\nb\n) XRD rocking curves at the (002) peak of the 100 nm and 1120 nm films\n.\n \n \n \nFig.\n \n2\n \nTypical SEM cross\n-\nsection image of the oriented hcp\n-\nCo\n84\nIr\n16\n \nfilm of 950 nm thick.\n \n \n     \nFig.\n \n3\n \n(a\n-\ne) In\n-\nplane magnetic hysteresis loops of the oriented hcp\n-\nCo\n84\nIr\n16\n \nfilms with the applied field parallel or \nperpendicular to the easy axis. (f) The thickne\nss dependence of Hc determ\ni\nn\ne\nd from the easy axis loops and \n4\nπ\nMs for \nthe oriented hcp\n-\nCo\n84\nIr\n16\n \nfilms.\n \n \n \n \nFig.\n \n4\n \nZero\n-\nfield MFM images of the oriented hcp\n-\nCo\n84\nIr\n16\n \nfilms with 950 nm and 1120 nm thicknesses. The image size \nare both 20 \nμ\nm \n×\n \n20 \nμ\nm.\n \n \n \n \n \n   \nFig. 5\n \n(a)\n-\n(e) Angle dependence of the resonance field\n \n(Hr)\n \nfor the oriented \nhcp\n-\nCo\n84\nIr\n16\n \nfilms. The dots are \nexperimental data and the curves\n \nare the fitted results. (f) The thickness dependence of\n \n \nH\nθ\n \n \na\nnd \nK\np\n.\n \nT\nhe \nred\n \nline \nis\n \nthe\n \nfit\nting\n \ncurve.\n \n \n \n \nFig. \n6\n \n(a)\n-\n(e) \nPermeability spectra of the oriented \nhcp\n-\nCo\n84\nIr\n16\n \nfilms with indicated film thicknesses and the fitted \ndamping constant \nα. \n(f) Thickness dependencies of the resonance frequency and the damping constant for the oriented \nCo\n84\nIr\n16\n \nfilms. f\nr \nis\n \nthe \ncalculated resonance frequency and f\nr\nexp\n \nis that determined from the measured imaginary part of \nthe permeability.\n \n \n \n \n \n \n \n"    },    {        "title": "1709.09965v1.Magnetization_and_Anisotropy_of_Cobalt_Ferrite_Thin_Films.pdf",        "content": " 1 Magnetization and Anisotropy of Cobalt Ferrite Thin Films  \n \nF. Eskandari1,2, S.B. Porter1, M. Venkatesan1, P. Kameli1,2, K. Rode1 and J.M.D. Coey1 \n \n1School of Physics and CRANN, Trinity College, Dublin 2. Ireland.  \n2Department of Physics, Isfahan University  of Technology, Isfahan 84156 –83111, Iran.  \n \n \nAbstract:  \n \nThe magnetization of thin films of cobalt ferrite frequently falls far below the bulk value of \n455 kAm-1, which corresponds to an inverse cation distribution in the spinel structure with a \nsignifican t orbital moment of about 0.6 µ B that is associated with the octahedrally -coordinated \nCo2+ ions. The orbital moment is responsible for the magnetostriction and magnetocrystalline \nanisotropy, and its sensitivity to imposed strain. We have systematically inv estigated the \nstructure and magnetism of films produced by pulsed -laser deposition on different substrates  \n(TiO 2, MgO, MgAl 2O4, SrTiO 3, LSAT, LaAlO 3) and as a function of temperature (500 -700C) \nand oxygen pressure (10-4 – 10 Pa). Magnetization at room -temperature ranges from 60 to 440 \nkAm-1, and uniaxial substrate -induced anisotropy ranges from +220 kJm-3 for films on \ndeposited on MgO  (100)  to -2100 kJm-3 for films deposited on MgAl 2O4 (100) , where the room -\ntemperature anisotropy field reaches 14 T.  No rearrangement of high -spin Fe3+ and Co2+ cations \non tetrahedral and octahedral sites can reduce the magnetization below the bulk value, but a \nswitch from  Fe3+ and Co2+ to Fe2+ and low -spin Co3+ on octahedral sites will reduce the low -\ntemperature magnetization to 120 kAm-1, and a consequent reduction of Curie temperature can \nbring the room -temperature value to near zero. Possible reasons for the appearance of low -spin \ncobalt in the thin films are discussed.  \n \nKeywords; Cobalt ferrite, thin films, pulsed -laser deposition, low -spin Co3+, strain engineering \nof magnetization.  \n  2 1 Introduction  \nThe spinel ferrites are an important family of insulating ferrimagnetic oxides, widely \nused as soft high -frequenc y magnetic materials. Their general formula is MFe 2O4 where the \niron is ferric Fe3+ and M is a divalent transition metal cation, such as Mg2+, Mn2+, Fe2+, Co2+, \nNi2+ or ½(Li++Fe3+) [1]. All the ferrites are ferrimagnetic insulating with a high Curie \ntemperature Tc, and. with the exception of Fe 3O4 all are insulating. None except CoFe 2O4 \nexhibits much anisotropy.  \nThe cubic spinel structure, space group Fd 3̅m illustrated in Fig. 1, is formed of a cubic \nclose -packed array of ox ygen anions slightly displaced from their ideal positions, with the \ncations occupying tetrahedral 8 a sites [A -sites] , which have cubic  4̅3m point symmetry and \noctahedral 16 d sites {B -sites} , which have trigonal 3̅m point symmetry.  There are 56 atoms in \nthe unit cell. The ‘normal’ cation distribution has the divalent cations on A -sites, but the \n‘inverse’ distribution is more common, where one of the two ferric ions occupies the A -sites, \nand the  other ferric ion and the  M2+ cations are on B -sites. The ferri te of interest to us here is \nCoFe 2O4 (CFO), which usually has a near -ideal inverse cation distribution in the bulk  \n[Fe3+]{Co2+Fe3+}O4,        (1)              \nwith Fe3+ cations occupying the 8 a sites. The cation distribution can be modified by heat \ntreatme nt [2], and quenching increa ses the occupancy of A -sites by Co2+. The accepted value \nof the lattice parameter is 839.2 pm, although the value varies slightly with the sample \nstoichiometry and preparation method.  \nFe3+ (3d5; t2g3eg2) has S = 5/2 and a spin moment of 5 µ B. High -spin Co2+ (3d7 t2g5eg2) \nhas S = 3/2 and a spin moment of 3 µ B, but in B -sites the cobalt can also ha ve a significant \nunquenched orbital moment of ~ 0. 6 µB [3-5], which is responsible for the strong cubic \nanisotropy K1c ≈ 290 kJm-3 with <100> easy directions. Although the moment of an isolated  3 Co2+ ion aligns along a local <111> trigonal axis [3], the resultant bulk anisotropy lies along \n<100> [6]. The net moment is about 3.6 µ B per formula at room temperature (RT), and the \nmagn etization of bulk samples is 455 kAm-1 or 86 Am2kg-1 (86 emu/g) , based on the X -ray \ndensity of 5290 kgm-3. The ferrimagnetic Néel temperature of CoFe 2O4 is 790 K, so the ground \nstate, T = 0 values are slightly higher.  \nAnother consequence of the unquenched orbital moment on the Co2+ is an \nexceptionally -large magnetostriction. Originally measured by Bozorth in 1955 [7], the value \nof (3/2)100 for a Co 0.8Fe2.2O4 crystal was found to be -885 ppm, corresponding to  a \ntetragonality ( a|| - a)/a of almost 1 %. The tetragonality has been measured in unsaturated bulk \nmaterial by synchrotron X -ray diffraction [8]. A similar value of magnetostricton ( -845 ppm) \nwas recorded recently for a crystal of nominal composition CoFe 2O4 [9], but, the \nmagnetostriction of a Co -rich crystal, Co 1.1Fe1.9O4, was much less, ( -375 ppm) [7], and the \nvalues can be quite variable.  A result of the magnetostriction is that an i mposed strain ε along \n<100> leads to a uniaxial anisotropy [10]   \nKu ≈ (3/2)100εE        (2) \nwhere E is Young’s modulus (C 11), which is 257 GPa [11]. A 1% biaxial compression therefore \nleads to an easy -plane anisotropy Ku ≈ -2 MJm-3. In an alternative formulation for biaxially \nstrained cubic films is [12]  \nKu = (3/2)100 (C11- C12) (a|| - a)/a     (3) \nwhere C 12 = 106 GPa [11]  \nThe magnetic anisotropy of thin films of CoFe 2O4 is exceptionally -sensitive to substrate -\ninduced stra in [10, 13 -18].  4 Thin films of CoFe 2O4 with good chemical and mechanical stability have attracted \nsome attention as potential perpendicular recording media [19], as tunnel -barrier spin filters \n[20-22] due to the spin -dependent bandgap [23], as magneto -optic media [1] and as \nmagnetostrictive films [24]. \nThere have been many reports of preparation of CoFe 2O4 in thin film form using pulsed -\nlaser deposition (PLD) [10, 13 -15, 19, 24 -31], and the literatur e also includes reports of films \nproduced by rf sputtering [16, 17, 32 -34], MBE [18, 35 -37], CVD [38, 39]   and ALD [40]. \nAuthors use a variety of substrates, deposition temperatures, oxygen pressures, laser fluence, \nsample thickness and thermal treatments, and find a wide range of magnetization, anisotropy \nand hysteresis. A summary of some o f the earlier PLD work is provided in Table 1. What is \nremarkable is that the magnetization found at room -temperature is usually much less than the \nbulk value, and only comes close to it in a few instances. This is puzzling, because we cannot \nlower the mag netization by rearranging the Co2+ and Fe3+ cations on A - and B -sites in a \ncollinear ferrimagnetic structure.  \nIn this work, we have systematically investigated the effects of thin film growth \nconditions on structural and magnetic properties of CFO thin fil ms deposited by PLD on \nvarious substrates, principally with the aim of explaining the anomalously small values of \nmagnetization that are usually found in CFO films, but often ignored by plotting the y -axis of \nthe magnetization curves in reduced units. We d iscuss our results in terms of the presence of \nlow-spin Co3+ ions on B -sites, such as are found in Co 3O4 [41].  \n   5 Table 1. Reports of magnetization of CoFe 2O4 films prepared by pulsed laser deposition.  \n  \nSubstrate  Substrate \nTemperature, \nTs  (oC) Thickness \nt (nm)  Oxygen \nPres sure \n P (Pa) Magnetization  \nM (kAm-1)* Reference  \nMgO (100)  600 \n800 400 4 335 \n426 [25] \nSTO//CoCr 2O4 \n 600/1000p  140-430 0.1 380 [10] \nMAO  500/1000p  65-900 0.1 ? [26] \n \nMgO (100)  \nSTO (100)  700 \n550 80 10-5 180 – 220 \n370 \n [13] \n \nAl2O3 (0001)  \nSiO 2 800 \n550 40-200 \n33 0.2 ? \n260 [19] \n \n \nSTO (100)  500-700 70 0.002 -0.01 ? [27] \n \nMAO(100)  175-690 200-220 1.3  \n(15% \nozone)  450 ( 5K)  [28] \n \n \nMgO(100)  \nSTO(100)  450 200 1.3 300 \n140/4 80 [29] \n \n \nSi(100)/SiO 2 250-600 \n250 135 2.9 \n0.7-7.0 \n 130-270 \n130-220 [24, 30]  \n \nMgO(100)  \nSTO(100)  \nBTO(100)  \nLAO(100)  ? 13-100 7 260 \n390 \n210 \n280 \n [14] \n \n \nMgO(100)  \n 400 50-400 2 100-185 [15] \n \n \nPt(111)  550-750 247-290 9 180-220 [31] \n \n* 1 kAm-1 is equivalent to 1 emu/cc or 1.26 G  p post annealed  \n   6 2. Experimental Methods  \nThe films were deposited by PLD onto various single -crystal substrates  from a sintered \nCoFe 2O4 target prepared by sol -gel synthesis. A KrF excimer laser (248 nm wavelength with \n25 ns pulse width, Lambda Physics) was used to ablate the ceramic target with a laser fluence \nof about 2  J cm-2 and a pulse repetition rate of 10 Hz.  The distance between the substrate and \ntarget was fixed at 6 cm. Before deposition, the chamber was evacuated to 2×10-4 Pa and the \nsubstrate was heated to 900 C for 1 hour, followed by cooling to the required deposition \ntemperature Ts. After deposition, the films were cooled to room temperature at a rate of about \n5C min-1 at constant oxygen pressure.  They were normally deposited at 600  C, but some \nfilms wer e deposited at higher or lower temperature (500 -700C). The oxygen pressure was \nvaried in the range 2 × 10-4 – 10 Pa. In addition, we have deposited films on many other \nsubstrates – MgO (100), MgO (110), MgO (111), SrTiO 3 (STO) (001), MgAl 2O4 (MAO) (001),  \nLaAlO 3 (LAO) (001), (La,Sr)(Al,Ta)O 3 (LSAT) (100) and TiO 2 (001).  The objective was to \ndetermine optimum conditions, which would yield films of the highest quality, magnetization \nand anisotropy.  Reflection  high-energy electron diffraction (RHEED) was obse rved during and \nafter film growth.  2 X-ray diffraction (XRD), -scans and reciprocal space mapping \n(RSM) analysis were carried out to check the crystallinity, orientation and strain of the thin \nfilms using a Bruker D8 X -ray diffractometer (Cu K α1; λ = 154.05 pm). Film thicknesses were  \ndetermined by small -angle X -ray reflectivity (XRR) and confirmed in some cases by \nellipsometry. Morphology of the films was examined using contact -mode atomic force \nmicroscopy (AFM), and composition was checked by energy dispersive X -ray analysis (EDX). \nMost magnetic measurements were made using a 5  T SQUID magnetometer (MPMS 5 XL, \nQuantum Design) on films mounted in clear plastic straws with magnetic field applied parallel \nor perpendicular to the film plane. Measurements on selected films were made using a  7 vibrating -sample magnetometer in fields of up to 14  T (PPMS, Quantum Design) when the \nanisotropy field was very large.  \n \n3. Experimental Results  \n        We have examined 60 films of cobalt ferrite. Since there are so many experimental \nvariables, we make  progress by changing them one at a time — substrate temperature, oxygen \npressure, laser fluence, substrate material. Film thickness was generally in the range 30 – 50 \nnm; we did not investigate ultra -thin films, but we included one 15 nm film on MgO(100) under \npreferred conditions (600oC, vacuum).  Firstly, in order to find the optimum substrate \ntemperature ( Ts) for CFO thin film growth, two sets of samples were prepared  on MgO(100) \nsubstra tes, one under vacuum (2×10-4 Pa) and the other in an oxygen pressure of 2 Pa . In each \ncase the ma gnetization was greatest when Ts ≈ 600oC. The lattice parameter of MgO, which \nhas an ideal cubic close -packed oxygen lattice, is 421 pm or slightly more than half that of \nCFO (839 pm). The X -ray diffraction patterns of CFO thin films deposited under  vacuu m on \nMgO (100) substrates at different Ts of 500 - 700 C are shown in Fig. 2a,b. T he films are all \nhighly -oriented along the [00l] direction, as evidenced by the single CFO  (008) peaks visible \nin Fig 2. Our -scans showed four -fold in -plane symmetry, indi cating that the film growth on \nMgO is quasi -epitaxial. The CFO films on MgO are under in -plane tensile stress, so  the out -\nof-plane lattice parameter ( a = 837 pm) is smaller than the cubic bulk value . The tetragonality \nis ≈ -0.6 %. In Fig. 2b, it can be se en that the CFO  peak width decreases on increasing Ts from \n500 to 650 C.  \n        The oxygen pressure inside the chamber during film deposition is another important factor \ncontrolling the film structure and magnetization. Therefore, we deposited the CFO fi lms at \ndifferent oxygen pressures, ranging from vacuum (2×10-4 Pa) to 10 Pa at a fixed substrate  8 temperature of 600 C. All these films grow epitaxially. Fig. 1c shows that the position of the \n(008) peak shifts to lower angles with decreasing oxygen partial  pressure. Thus, the out -of-\nplane lattice parameter a and the tetragonality decrease with increasing oxygen partial \npressure.  \n      The strain state of the films has been investigated in more detail by reciprocal space \nmapping (RSM) around the MgO (113) and CFO (226) peaks for samples grown at 600 C in \nvacuum . The MgO substrates are often twinned, but different in -plane and out -of-plane lattice \nparameters are seen for CFO, as well as nonuniformity of the out -of-plane parameter across \nthe film thickness fo r all except the deposition in vacuum. The RSMs of Fig. 3 correspond to \nCFO films which are fully strained in the plane of the film with a|| = 842 pm. The out -of-plane \nlattice parameter a is uniform across the film thickness (56 nm) for CFO deposited at 6 00C \nin vacuum  (Fig. 3a) and it is equal to 837 pm.   There is an increasingly broad distribution of a \nwith increasing oxygen pressure. The double MgO spots seen for films deposited in 2 or 10 Pa \nof oxygen are simply due to substrate twins, but the vertica l streaking of the CFO (226) \nreflection indicates a heterogeneous distribution of a across the thickness. In 2 Pa, the \nvariation is from 832 to 838 pm, whereas for the 10 Pa film of similar thickness, the variation \nis from 832 to 842 pm. The vertical stra in is progressively relaxed in oxygen, but not in vacuum. \nFigure 3 also shows RHEED patterns of the same three films, which establish good crystal \nquality . RMS r oughness measured by AFM was < 1 nm. The RHEED indicates that the disorder \nof the surface incre ases with increasing oxygen pressure. This is suggested by the \ndisappearance of the faint Kikuchi lines and broadening of the diffraction pattern into streaks \ncorresponding to a decrease in long -range order at the surface in the plane of the film as the \npressure is increased to 2 Pa, and the collapse of the pattern into spots at 10 Pa, possibly  9 indicating the existence of ordered islands, or the result of transmission through small \ncrystalline structures on the surface.  \n The composition of selected films ha s been checked by EDX analysis. The ceramic \ntarget was stoichiometric, with an Fe:Co ratio of 1.98(2). Analysis at multiple points on four \nfilms deposited on MgO showed that their composition was uniform, but with a slightly \ndifferent Fe:Co ratio of 1.80(2 ) indicating that the films  are Co -rich.  \n            Figure 4 show s room -temperature magnetization (M -H loops) of CFO thin films \ndeposited on MgO (001) in vacuum at substrate temperatures of 500 and 600 C. The saturation \nmagnetization in both cases is sig nificantly less than the value of 86 Am2kg-1 (455 kAm-1) \nmeasured for the bulk ceramic target used for the PLD. Plots of Ms vs Ts for films deposited in \nvacuum and in a 2 Pa pressure of oxygen are shown in Fig. 5.  The moments in vacuum are \nlarger, but in  both cases, the maximum falls at Ts = 600C, which is why we limit further studies \nto films produced at this temperature. We find that there is some effect of laser fluence. \nIncreasing the fluence from 2.1 to 2.7 Jcm-2 leads to an increase of Ms from 397 kAm-1 to 453 \nkAm-1, which is the bulk valu e. The magnetization of these films on MgO lies perpendicular \nto the film plane, with coercivity of up to 400 kAm-1 for the high -moment films. Coercivity in \nthe low -moment films is greater. The magnetization curve of the 453 kAm-1 sample, which we \ncan rega rd as a benchmark, is included in Figure 8. There is no hysteresis when the field is \napplied in -plane, but four -fold anisotropy is associated with the tetragonal symmetry of the \nfilm.  The intrinsic perpendicular anisotropy Ku can be estimated from the eq uation  \n Ku  =  K -Ks        (4) \nwhere the parameter Kis deduced from the area between the perpendicular and parallel \nmagnetization curves, and the shape anisotropy Ks is -½μ0Ms2  The anisotropy energy \nassociated with each term is of the form E = Ksin2θ, where θ is the angle between the direction  10 of uniform magnetization and the film normal.  From the data in Fig. 4, Ms =  397 kAm-1 so Ks \n= - 99 kJm-3. The measured value of Kis 127 kJm-3, hence it follows that Ku = 226 kJm-3. \nSubstantially larger values of Ku have been reported by Niizeki for samples with very similar \nmagnetization curves using torque measurements [42] or simple estimates of the anisotropy \nfield µ 0Ha = 2Ku/Ms [16]. Our estimates of Ku from magnetization curves that exhibit hysteresis \ncan underestimate Ku by up to 35% because of effects of magnetic viscosity in hysteretic  easy \ndirection.  \n Figure 6 shows magnetization curves for samples deposited at 600 C in two diffe rent \noxygen pressures. The oxygen decreases the magnetization, as summarized in Fig. 7. The \nmagnetization of the sample grown in an oxygen pressure of 10 Pa is only 60 kAm-1. This is \npartly because the Curie temperature of the sample is rather low. The oxy gen pressure \ndependence is opposite to that found in NiFe 2O4 and YIG [43, 44] . From the temperature -\ndependence of Ms, Tc is estimated to be about 450 K. The magnetization of these low -moment \nfilms on MgO is perpendicular to the plane, and the coercivity becomes very large at low \ntemperature, exceeding 5 T at 100 K. The 15nm film prepared at 600 C in vacuum is different \nto all the other, thicker films, insofar as the moment lies in -plane. Its magnetization is 389 \nkAm-1. \n Next, we turn to results for the other substrates. Films deposited at 600 C in vacuum \non the other cuts of MgO, (111) or (110), show lower magnetization than on (100), 280 kAm-\n1 and 254 kAm-1, respectively, with less pronounced anisotropy. The results for ot her \nsubstrates, shown in Figure 8, are more interesting. Films on TiO 2 (100)  and LAO  (100)  \nexhibited magnetization of 444 kAm-1 and 412 kAm-1 respectively, close to the bulk value, but \nthese substrates have a large lattice mismatch of ±10% so there is no e pitaxy. The films are  11 polycrystalline. Magnetization lies in -plane and the anisotropy is due to shape, with negligible \nKu contribution from substrate -induced lattice strain.  \n        The films on spinel, MgAl 2O4 substrates, are subject to 3.7% lattice mism atch, leading to \nstrong perpendicular expansion. The RSM illustrated in Fig. 9 shows that the in -plane lattice \nparameter does not follow the template provided by the spinel lattice, but there is nonetheless \na broad, correlated distribution of lattice param eters centred at a|| = 831 pm and a = 850 pm, \ncorresponding to an in -plane compression of  1%, and a similar perpendicular expansion. The \ntetragonality is 2%. Magnetization curves plotted in Fig. 8 show a magnetization of CFO on \nMAO of 315 kAm-1, with a coercivity of 380 kAm-1 and an enorm ous in -plane anisotropy \nestimated from the area between the curves as Ku = -1.12 MJm-3, far exceeding the shape \nanisotropy Ks = - 62 kJm-3.  The anisotropy field μ 0Ha is 14 T, and the anisotropy energy, \nestimated simply as -½μ0HaMs is -2.2 MJm-3. Both Ms and Ku are reduced in magnitude in a 2 \nPa oxygen atmosphere, to 262 kAm-1 and -0.58 MJm-3, respectively. Films deposited on LSAT \nbehave similarly, with strong negative Ku = -0.65 MJm-3 for films made in vacuum.  \n       STO is different. Here the film is ori ented, but the magnetization  is only 40% of the bulk \nvalue. The perpendicular lattice parameter expands a little, but the net anisotropy is \nperpendicular to the plane, as for MgO, but very weak. The effect of substrate strain is 20 – 40 \ntimes less than for  MAO or LSAT, and of opposite sign. The anisotropy and magnetization of \nthe films deposited on different substrates is summarized in Table 2.  \n3 Discussion.  \n3.1 Anisotropy.  \nWe confirm the idea, originally proposed for films on MgO and MAO [10, 36]  and \nconfirmed by electronic structure and crystal field calculations [23, 45 -49]that the magnetic \nanisotropy in cobalt ferrite films is largely governed by substrate -induced strain. All our films  12 show 001 oriented growth, except those deposited on TiO 2 (001) or LaAlO 3 (100) substrates, \nwhich have in -plane lattice parameters that are completely mismatched to that of CoFe 2O4, \nbeing about 10% bigger or smaller. In these two cases, there is effec tively no substrate strain, \nand shape anisotropy dominates (Table 2). The magnetization of these unconstrained films is \nclose to that of bulk CFO. They exhibit easy -plane magnetization with Ks ~ 100 kJm-3 and \nisotropic coecivity, Hc ≈ 120 kAm-1.  \nTable 2. Anisotropy of cobalt ferrite films deposited at 600 C in vacuum on different 100 \noriented substrates.  \nSubstrate  a Parameter \n(pm)  Growth  Ms \n(kAm-1) Ks \n(kJm-3) Ku \n(kJm-3) Anisotropy  \nTiO 2 919 Unconstrained   413  -107    ~ 0 Shape  \nin-plane  \nMgO  842 Quasi -epitaxial   397    -99    226 Strain  \nperpendicular  \nMgAl 2O4 808 Oriented, \nstrained   317    -63 -1120  Strain  \nin-plane  \nSrTiO 3 782 Oriented   183    -21   ~ 30 Strain, \nweakly  \nperpendicular  \nLSAT  775 Oriented, \nstrained   270    -46   -650 Strain  \nin-plane  \nLaAlO 3 758 Unconstrained   444  -123     ~ 0 Shape  \nIn-plane  \nThe blue line separates the parameters that are greater than or less than that of CFO, 839 pm.  \nNext come the MAO and LSAT substrates where there is still a large lattice mismatc h, \nbut there is now clear 001 axis orientation of CFO, with the wide spread of correlated a|| and \na parameters, shown in Fig. 9. The CFO is significantly strained, and according to Eq.3 the \ntetragonality of 2 .2% should lead to uniaxial hard -axis anisotropy of -2.9 MJm-3 on account of \nthe cobalt magnetostriction, However, the magnitude of 100 in CFO is often much less than \nthe frequently -cited 590 ppm [7], and the resulting Ku will be correspondingly reduced. For  13 example, a value s = 225 ppm reported for polycrystalline CFO [50] corresponds to 100 =  \n256 ppm . The tetragonali ty of CFO on LSAT is 1.9 %. \nThe origin of 100 is the high -spin Co2+ on B-sites, so we can expect that there are \nsignificant variations in the amount of this ion in different films, which provides  a link between \nanisotropy and magnetization. High -spin Co2+ on A -sites does not contribute to the anisotropy \nbecause there is no orbital moment in cubic 4̅3m point symmetry. In DFT calculations, cation \ndistributions were fo und to be sensitive to epitaxial strain [51]. Furthermore, a study of the \ninfluence of substrate strain on Ku in CoCr 2O4, a spinel with the normal cation distribution, \nreveals that its sign is opposite to that for CoFe 2O4 [52].   \nMagnetoelastic strain -related anisotropy of CFO on MAO has been discussed \npreviously [34], including  a study of 5 nm MBE films which were found to grown epitaxially \non MAO [36], but the extremely large easy -plane anis otropy we have observed on spinel is \nunprecedented; the anisotropy field is more than double anything reported previously [36, 5 3]. \nThat for CFO on LSAT is also very large. The magnetization of CFO on both substrates is \nabout 2/3 of the bulk value.  \nSTO is a different story. Here the perpendicular 004 and 008 X -ray reflections are \nbroad, but the perpendicular lattice parameter of 841.8 pm indicates a dilation of just 0.3%. \nFurthermore the magnetization is much reduced, to  40% of the bulk value. The shape \nanisotropy is therefore weak, Ks = -21 kJm-3, and it appears from the magnetization curves in \nFig 8 that the uniaxial anisotropy Ku is of similar magnitude but opposite sign, leading to a \nfeeble perpendicular anisotropy Kfor the film. The anisotropy mechanism based on the \nmagnetostriction of B -site Co2+ does not apply in this case. We think the small moment signals \na cation distribution that is quite different to that of bulk CFO.   14 The films grown on MgO at 600oC in vacuum are of good epitaxial character according \nto the RSM of Fig 3a, and the RHEED pattern of Fig. 3d which shows traces of Kikuchi fringes. \nRMS surface roughness for the 56 nm thick film is  0.7 nm. The films have 85 % or more of the \nbulk magnetization, and the tetragonality of -0.6% corresponds to a positive Ku of 800 kJm-3 \naccording to eq. 3, significantly more  than our measured value of Ku of 226 kJm-3 (Table 2). \nThe discrepancy may arise because the value of (3/2)100 in our films is reduced to about 250 \nppm, or because  our method of evaluating Ku from the magnetization curves underestimates \nthe value.  Torque measurements generally  give greater  values  [16, 34 ]. Figure 1 2 summarize s \nthe relation between tetragonality  or perpendicular strain  and anisotropy Ku in the films.  \nWe have not performed a systematic study of anisotropy vs film thickness for CFO \nfilms on MgO, but we found that a 15 nm film was easy -plane, whereas films with 3 0 ≤ t ≤ 70 \nnm are all easy -axis. A low -moment 5 nm MBE film was also found to be easy -plane [36]. \nEasy -axis anisotropy persists up to 300 nm, but the thickest PLD films become easy -plane \nagain as the strain is eventually relaxed, and the structure  become s cubic [54]. We should \ntherefore include another term i/t in the anisotropy expression (4) to take account of the \ninterface an isotropy; K= Ku-Ks + i/t. From the first spin reorientation thickness, we \nestimate that i is easy -plane and approximately - 6 mJm-2.  Figure 10 sketches the evolution \nof the anisotropy of CFO films on MgO, marking the dominant anisotropy term in e ach of three \nthickness regimes.  \n3.2 Magnetization.  \nA remarkable feature of the cobalt ferrite films produced by PLD and other methods is \nthe magnetization, which frequently falls short of the bulk value and is remarkably sensitive to \nsample preparation con ditions. A well -known consequence of thermal treatment of bulk ferrites \nis to induce deviations from the ideal inverse cation distribution [Fe3+]{Fe3+Co2+}O4 by moving  15 some Co2+ ions over to A -sites, and an equal number of Fe3+ ions onto B -sites. The formu la is \nthen   \n[Fe3+1-x Cox]{Fe3+1+x Co2+1-x }O4      (5) \nand the moment is thereby increased from 3.5µ B per formula unit to 3.5(1 + x)µ B, assuming an \nA-site Co moment of 3µ B.  In any case, no permutation of these cations can reduce  the net \nmagnetization of a collinear ferrimagnetic spin structure. The slight excess of cobalt detected \nin the EDX analysis of the films, which was independent of the preparation method, does \nreduce the magnetization a bit. The inverse formula is then [Fe3+]{Fe3+1-y Co2+1+y }O4, where \nan Fe/Co ratio of 1.8 corresponds to y = 0.07. The magnetization falls to 80% of the bulk value, \nbut permuting the cations only increases the value.  \nHow then can we get the low values of room -temperature magnetizati on seen in Table \n2 and Figs 5,7 . The average of Ms for 35 different CFO films was 240 kAm-1, just 53% of the \nbulk value. There are several possibilities to consider, namely low magnetic ordering \ntemperatures, noncollinear magnetic structures including the effects of antiphase boundaries, \nlow-spin cobalt and oxygen stoichiometry  \nMagnetic ordering temperature . The great majority of our magnetization measurements were \nmade at room -temperature. The reason for not measuring systematically at low temperature is \nthe Curie law paramagnetism of iron  impurities in the MgO substrates used for more than half \nthe depositions. The effect is small and linear above 100 K, but increasingly important and \nnonlinear below. Since Tc of CFO is 790 K, thermal effects at RT/ Tc  ≈ 0.37 are expected to \nreduce the magnetization by about 5% [55]. We checked the Curie temperature of the 600oC \nsample with the smallest magnetization, prepared in 10 Pa oxygen pressure (Fig 7), which was \nestimated to be roughly 460 K, from magnetization measured at temperatures up to 380 K.  For \nall other samples with reduced moment, such as those produced in an oxygen pressure of 2 Pa,  16 Tc is higher and finite temperature effect s have little impact on the  magnetization at  room -\ntemperature.  \nNoncollinear magnetic structures. An explanation commonly advanced for the low moments \nof spinel ferrite films deposited on MgO is that antiphase boundaries [17, 36, 37, 5 6] are \nincorporated into CFO during film growth. Originally proposed to explain the low \nmagnetization and slow approach to saturation of magnetite [57, 58], an antiphase boundary \nforms where two crystallites that have nucleated independently on MgO  (100) grow together. \nThere is therefore a plane of antiferromagnetic B – O – B interacti ons at the interface which \nare hard to overcome when the crystallites themselves are of nanoscale dimensions. These \neffects may well be present in our films grown at low temperatures, but the observation of near -\nbulk values of magnetization for samples gro wn at 600 C in vacuum (Fig 8a) and the high \ncrystalline quality of these films (Fig 3a) suggest that antiphase bondaries  may be important \nonly in films grown  at low Ts. These antiphase boundaries are not expected in films on MAO \nwhich has the same spinel s tructure and crystal symmetry as CFO, although  there is another \ntype associated with misfit dislocations that has been identified in Fe 3O4 films on MAO [59]. \n       Another possible reason for a reduced moment in an inverse spinel would be a noncollinear \nor canted spin structure of the B -site cations, which form the majority sublattice. The spins in \nzinc ferrite, for example, freeze in a random arra ngement below about 15 K, but ZnFe 2O4 is a \nnormal spinel, with nonmagnetic Zn2+ cations on the A -sites. The presence of A-site Fe3+ in \ninverse spinels ensures that the antiferromagnetic 135o A – O – B superexchange interactions \nare strong, and determine th e collinear Néel state [60].  \nThe next and most pl ausible explanation for the reduced moment relates to the spin \nstate of the cobalt. In the bulk, B -sites are populated by Fe3+ and high -spin Co2+, which is \nstabilized by the trigonal crystal field at the distorted octahedral sites, which have 3̅m symmetry   17 on account of the deviation of the oxygen 32 e site special position  parameter  u from 3/8. One \npossibility is that substrate -induced strain switches the Co2+ on B -sites to a low -spin state, 3 d7 \nt2g6eg1 with S = ½ and a spin moment of 1µ B. The Curie tempera ture may be similar, but the \nmagnetization will be much lower, ~ 130 kAm-1 or 24 Am2kg-1. Some admixture with the \nnormal spinel configuration [Co2+]{Fe3+2}O4 is possible, but as usual it will increase the \nmoment. Cation size, crystal -field stabilization en ergy and temperature are all factors that \ndetermine the spin state, but Co3+ is often low -spin on octahedral sites, where it adopts a stable \n3d6 t2g6 configuration with no moment, S = 0 [61]. The normal cobalt spinel Co 3O4 has \nnonmagnetic low -spin Co3+ on B-sites [41, 6 2]. Another possibility is therefore the replace ment \nof Fe3+ and high -spin Co2+ on B -sites by Fe2+ and low -spin Co3+. This also has the effect of \nreducing the net moment per formula to 1 µ B but it makes A -site Fe3+ the majority sublattice. \nFigure 11 illustrates the spin states of Co ions on the two site s in a one -electron energy level \npicture. The splitting of the t2g triplet  on octahedral B-sites corresponds to compression  of the \noctahedra  along the local <111>  axis [3] . Table 3 compares the ionic radii and the spin moments \nin tetrahedral and octahedral sites. It can be seen there that the mean B -site radius of 70 pm in \nthe inverse structure is reduced to 65 pm or 66.5 pm in with the first or second of these cobalt \nlow-spin hypotheses, marked in blue or green on the table, respectively. Any substrate which \nexerts imperfectly -relaxed compressive strain, those below the blue line in Table 2, may be \nexpected to favour some low -spin cobalt, which can account for the reduced moments observed \non MAO, STO and LSAT , and on MgO prepared in oxygen . \nThe correlation of the low moment observed in CFO films with l ow-spin cobalt means \nthat strain -induced anisotropy Ku is correspondingly reduced. Low -spin cobalt can provide an \nexplanation for the anomalous behaviour of the films on STO. They have a low moment of 185 \nkAm-1 and a greatly reduced anisotropy with a posit ive sign (see Table 2). Their low moment  18 suggests conversion of much of the cobalt to a low -spin state, which will reduce the strain - \ninduced anisotropy, and may even change its sign [16, 45, 48] . Furthermore, depositing films   \n \nTable 3.  Ionic radii (in pm) and spin states of Fe and Co cations on in tetrahedral or octahedral \noxygen sites  \n Fe2+ Fe3+ Co2+(Hs) Co3+(Hs) Co2+ (Ls) Co3+ (Ls) \n \nTetrahedral \nIonic r adius  \n \nspin  \n58    \n \n \n2  \n49/49/49     \n \n \n5/2  \n56   \n \n \n3/2  \n47    \n \n \n2  \n49           \n \n \n3/2  \n45  \n \n \n1 \n \n \nOctahedral \nIonic radius  \n \nspin  \n78 \n \n \n2  \n65/65  \n \n \n5/2  \n75 \n \n \n3/2  \n61    \n \n \n2  \n65 \n \n \n1/2  \n55 \n \n \n0 \n \n \non substrates like STO or MgO with an undeformed cubic oxygen sublattice may modify \noxygen special position parameter u, hence the trigonal distortion of the oxygen around the B -\nsites, thereby reducing the orbital moment and the magnetostriction 100 that controls Ku. \n Finally, we consider the effect of oxygen pressure, which strongly reduces the moments \nof films on MgO (Fig 7) and MAO. The spinel lattice is known to accommodate excess oxygen \nby means of B -site cation vacancies, oxidizing divalent  cations there to the trivalent state. The \ntextbook example is γFe2O3, which is really the cation -deficient spinel  \n[Fe3+]{Fe3+5/3 ☐ 1/3}O4,        (6) \nwhere ☐ is a vacant B -site. A similar effect in CFO leads to a formula [Fe3+]{Fe3+7/9 Co3+8/9 \n☐ 1/3}O4. If the Co3+ is low -spin, the ferrimagnetic net moment is 1.1 µ B and the A -sites form  19 the majority sublattice. Furthermore, nonmagnetic low -spin Co3+ means that 11/18 of the B -\nsites are nonmagnetic, which will reduce the magnetic ordering temperature to 350  K or less. \nWe can therefore expect oxygen pressure to reduce both Ms and Tc, although admittedly E q 6 \nrepresents an extreme case.  \n The plot in Fig. 12 shows Ku vs tetragonality.  The films p repared in vacuum with Ms \n> 300 kAm-1 show roughly a linear rela tion. These films have mostly high spin Co2+ on B-sites, \nwhich provides the strain -sensitive anisotropy. However, the two films with a low moment Ms \n< 200 kAm-1 (MgO at 2 Pa, STO) are different, we think because of the important fraction of \nlow-spin Co3+ in these materials.  \n4 Conclusions.  \n Our systematic broad -brush examination and analysis of cobalt ferrite thin films has \nrevealed a great variety of magnetic properties, often quite different to those of the bulk. The \ninfluence of the substrate is evidentl y critical, as are the deposition temperature and oxygen \npressure. Strain engineering is especially rich in CFO, because it works two ways. First is to \ninduce tetragonality in the films, which deforms the strong {100} cubic anisotropy K1c \nassociated high -spin Co2+ in octahedral sites to give a uniaxial anisotropy Ku, which may be \neasy-axis or easy plane, depending on the substrate. Second, a triaxial compressive strain can \nconvert cobalt from a high -spin to a low -spin state, thereby modifying the magnetization, Ms, \nshape anisotropy Ks and magnetocrystalline anisotrop y K1c.  There is therefore a new \nopportunity to modify anisotropy with strain. The example of CFO on STO shows the cobalt \nferrite teetering on an edge, where a slight strain -induced anisotropy change could have a big \neffect. This suggests suggests a  possib ility of using  piezoelectric substrates to switch, based on \ncobalt spin -state transitions and the related anisotropy   20   The next step should be to look carefully for the effects postulated using methods that \nexhibit atomic -scale sensitivity. These include e lectron microscopy with atomic scale \nresolution to evaluate the substrate -induced strain profiles and extended lattice defects, as well \nas spectroscopic methods such as XMCD to determine the cobalt spin and orbital moments,. \nCare will be needed to distingu ish near -surface ions which can be subject to a reduced crystal \nfield from those deeper in the film where the full crystal field will induce a low spin state. \nCo3O4 thin films will be useful reference samples here. Ferromagnetic resonance will be useful \nto evaluate the anisotropy fields and magnetizations independently.  \n         Cobalt ferrite illustrates the man y ways in which a thin film of an oxide containing Co2+ \ncan differ from bulk material, and it offers new opportunities for strain engineering of ma gnetic \nlayers in oxide electronics.  \nAcknowledgements. 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Shvets, Physical Review B 72, 134404 (2005).  \n[59] M. Luysberg, R. G. S. Sofin, S. K. Arora, and I. V. Shvets, Physical Review B 80, 024111 (2009).  \n[60] C. M. Srivastava, G. Srinivasan, and N. G. Nanadikar, Physical Review B 19, 499  (1979).  \n[61] R. G. Burns, Cambridge University Press, 1993.  \n[62] V. A. M. Brabers and A. D. D. Broemme, Journal of Magnetism and Magnetic Materials 104, \n405 (1992).  \n \n \n \n \n \n \n \n \n  23  \n \n \n \n \n \n \n \n \n \nFig. 1.  The spinel structure , illustrating the tetrahedral A -sites (black) and octahedral B -sites \n(red)  \n \n \n \n \n \n \n \n \n \nOxygenc) b)\na)a)a)\na)\nOxygen\nOctahedral site\nTetrahedral site 24  \n \n \n \n \n \n \n \n \n \nFig. 2.  (a), (b) XRD patterns of CFO films on MgO(100) prepared at different substrate \ntemperatures Td in vacuum, and (c) at different pressures at  Td = 600 oC . \n \n \n \n \n \n \n \n20 30 40 50 60 70 80 90 100MgO(004)\n650 oC700 oC\n600 oC\n550 oCIntensity (a.u.)\n2 (degree)MgO500 oC\nMgO(002)\nCFO(008)\n90 92 94 96 98100 102 104 106 108 110MgO(004)\n650 oC700 oC\n600 oC\n550 oCIntensity (a.u.)\n2 (degree)MgO500 oC\nCFO(008)\n90 92 94 96 98100 102 104 106 108 110MgO (004)\nVacuum\n0.009  bar\n0.2 bar\n2 bar\n20 barIntensity (a.u.)\n2(degree)MgO100 bar\nCFO (008)(a) (b) (c)  25  \n \n \n \n \nFig. 3.  Reciprocal space maps about the (113) reflection showing in -plane and out of plane lattice \nparameters for CFO films deposited on MgO (001) at 600 C in (a) vacuum (a) 2 Pa c) 10 Pa and \nthe respe ctive RHEED patterns d), e), f).  \n \n \n \n \n \n \n \n \n 26  \n \n \n \n \n \n \n \n \n \nFig.4. Magnetization of CFO films deposited on MgO (001) in vacuum at 500 and 600 oC.  \n \n \n \n \n \n \n \n \n \n \n-5 -4 -3 -2 -1 0 1 2 3 4 5-400-300-200-1000100200300400\n600o CM (kAm-1)\n0H (T) in-plane\n out-of-plane500o C 27  \n \n \n \n \n \n \n \n \n \nFig.5. Magnetization as a function of substrat e temperature for CFO films deposited on MgO \n(001) in vacuum (solid circles), and in 2 Pa solid squares. Laser fluence was 2.1 Jcm-2, except for \nopen circle (2.7 J Jcm-2). \n \n \n \n \n \n \n \n \n \n500 550 600 650 7000100200300400500\n2 Pa(kAm-1)Vacuum\ns (oC) 28  \n \n \n \n \n \n \n \nFig.6. Magnetization of CFO films deposited on MgO (001) at 600 oC in 2 Pa  and vacuum. The \nlaser fluence was 1.8 Jcm-2.  \n \n \n \n \n \n \n \n \n \n \n \n-5 -4 -3 -2 -1 0 1 2 3 4 5-400-300-200-1000100200300400\nVacuumM (kAm-1)\n0H (T) in-plane\n out-of-planeP(O2) = 2 Pa 29  \n \n \n \n \n \n \n \n \n \nFig. 7. Room temperature magnetization for CFO films deposited on MgO (001) at 600 oC and \ndifferent oxygen pressures. Laser fluence was 2.1 Jcm-2. \n \n \n \n \n \n \n \n \n1E-5 1E-4 1E-3 0.01 0.1 1 100100200300400500Ms (kAm-1) \nP(O2) (Pa) 30  \n \n \n \n \n \n \n \n \n \nFig. 8. Room temperature magnetization of CFO films deposited on six different (100) substrates \nat 600C in vacuum. The graphs are plotted on the same vertical scale and expanded  horizontal \nscales to emphasize the influence of the different substrates.  \n \n \n \n \n \n \n \n \n \n-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14-500-400-300-200-1000100200300400500M (kAm-1)\n0H (T) in-plane\n out-of-planeCFO/MAO\n-5 -4 -3 -2 -1 0 1 2 3 4 5\n0H (T) in-plane\n out-of-planeCFO/LAO\n-5 -4 -3 -2 -1 0 1 2 3 4 5\n0H (T) in-plane\n out-of-planeCFO/STO\n-5 -4 -3 -2 -1 0 1 2 3 4 5-500-400-300-200-1000100200300400500M (kAm-1)\n0H (T) in-plane\n out-of-planeCFO/MgO-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14\n-5 -4 -3 -2 -1 0 1 2 3 4 5\n0H (T) in-plane\n out-of-planeCFO/TiO2\n0H (T) in-plane\n out-of-planeCFO/LSAT 31  \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 9 The reciprocal space map of CFO on MgAl 2O4 (100) grown under optimised \nconditions. MgAl 2O4 imposes a strong in -plane compression on the CoFe 2O4 film, which causes a \ndramatic elongation of the out -of-plane lattice parameter to conserve the volume  of the unit cell.  \n \n \n \n 32  \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 10. Magnetization direction for CFO films grown on MgO  (100)  as a function of film \nthickness. In the thinnest films, interface anisotropy is dominant, but in the thickest ones, strain \nrelaxation eliminates t he tetragonal magnetocrystalline anisotropy, and shape anisotropy then \npredominates.  \n \n \n \n \n \n \n \n \n \n \n1 10 100 1000      In-plane\nShape anisotropy Ks           In-plane\nInterface anisotropy i     Perpendicular\nTetragonal magneto-\ncrystalline anisotropy Ku \nt (nm) 33  \n \n \n \n \n \n \n \n \n \n \n \nFigure 1 1. One electron energy diagrams for Co2+ and Co3+ on tetrahedral or octahedral sites in \nhigh and low spin states.  \n \n \n \n \n \n \n \n \n 34  \n \n \n \n \n \n \n \n \n \n \n      Figure 12. Plot of uniaxial anisotropy vs tetragonality for CFO films on different substrates.  \n \n \n \n-1 0 1 2-1000-5000500\nEasy plane\nMAO LSAT LAO STO \nTiO2 MgO 2 Pa Ku (kJm-3)\nTetragonality (%)MgO Easy axis"    },    {        "title": "1710.01532v2.Ferromagnetic_order_in_dipolar_systems_with_anisotropy__application_to_magnetic_nanoparticle_supracrystals.pdf",        "content": "arXiv:1710.01532v2  [cond-mat.mes-hall]  10 Oct 2018CondensedMatterPhysics,2017,Vol.20,No3,33703:1–10\nDOI:10.5488/CMP.20.33703\nhttp://www.icmp.lviv.ua/journal\nFerromagneticorderindipolarsystemswith\nanisotropy:applicationtomagneticnanoparticle\nsupracrystals∗\nV.Russier†,E.Ngo\nICMPE,UMR7182CNRSandUPE2-8rueHenriDunant94320Thiais ,France\nReceivedJune14,2017,infinalformJuly19,2017,corrected October9,2018\nSingledomainmagneticnanoparticles(MNP)interactingth roughdipolarinteractions(DDI)inadditiontothe\nmagnetocrystallineenergymaypresentalowtemperaturefe rromagnetic(SFM)orspinglass(SSG)phaseac-\ncordingtotheunderlyingstructureandthedegreeofordero ftheassembly.Westudy,fromMonteCarlosim-\nulationsintheframeworkoftheeffectiveone-spinormacro spinmodels,thecaseofamonodisperseassembly\nofsingledomainMNPfixedonthesitesofaperfectlatticewit hfccsymmetryandrandomlydistributedeasy\naxes.Welimitourselvestothecaseofalowanisotropy,name lytheonsetofthedisappearanceofthedipolar\nlong-rangeferromagnetic(FM)phaseobtainedintheabsenc eofanisotropyduetothedisorderintroducedby\nthelatter.\nKeywords:magneticnanoparticles,MonteCarlosimulation,ferromag neticorder\nPACS:75.50.Tt,64.60.De\n1.Introduction\nThe physics of nanoscale magnetic materials is still a very a ctive field of research both in view\nof the potential applications, especially in nanomedicine [1], and from the fundamental point of view\n[2, 3]. The fundamental aspects are all the more important th at magnetic nanoparticles (MNP) can be\nsynthesized in a wide range of size and shapes and their assem blies obtained with different structures:\ncolloidal suspensions, or ferrofluids embedded in non-magn etic materials where one can tune the in-\nterparticle interactions through the concentration, or as powders. Moreover, when the size dispersion\nis sufficiently narrow, MNP can self-organize in long-range o rdered 3D supracrystals, namely crystals\nof nanoparticles [4–6], conversely to the discontinuous me tal insulator multilayers (DMIM) where the\nunderlying structure is disordered [7–9]. The interplay be tween mutual interactions and the degree of\norder of the underlying structure is a key factor in understa nding the extrinsic properties of single domain\nMNP assemblies [3]. Indeed, due to frustration effects, acco rding to whether the MNP are arranged or\nnot with a long-range order at high concentration, a ferroma gnetic or a spin glass state is expected at a\nlow temperature [3, 7] (respectively super-ferromagnetic , SFM and super-spin glass, SSG according to\nthe nanoscale magnetism nomenclature). Hence, one of the pu rposes of the modelling is to understand\nunder which conditions the SFM or the SSG state can be reached .\nThe modelling of the magnetic properties of both nanostruct ured materials and materials including\nnanoscale particles is a multiscale problem since, on the on e hand, the local magnetic structure within\nMNP at the atomic site scale presents nontrivial features an d, on the other hand, the interactions between\nMNP play an important role. An important simplification in th e modelling occurs for particles of the\nradius under a critical size rsddepending on their chemical composition, typically of a few dozens on\n∗Paper dedicated to the memory of Dr. J.-P. Badiali.\n†E-mail: russier@icmpe.cnrs.fr\nThisworkislicensedundera CreativeCommonsAttribution4.0InternationalLicense .Furtherdistribution\nofthisworkmustmaintainattributiontotheauthor(s)andt hepublishedarticle’stitle,journalcitation,andDOI.33703-1V.Russier,E.Ngo\nnanometers, where they reach a single domain regime avoidin g both multidomains and vortex structures\n(typical values for rsdare 15 nm for Fe, 35 nm for Co, 30 nm for γ-Fe2O3[2]). In this case, the properties\nof the MNP ensemble can be described through an effective one- spin model (EOS) where each MNP is\ncharacterized by its moment and magnetocrystalline anisot ropy energy. However, both the moment and\ncharacteristics of the anisotropy term must be considered a s effective quantities taking into account some\nof the atomic scale features. Hence, the effective saturatio n magnetization Msessentially differs from\nits bulk value due to crystallinity and/or surface defects, the so-called dead layer, and is expected to be\nsmaller than the latter. As concerns the characteristics of magnetocrystalline anisotropy energy (MAE),\nthe situation is more complex since even its symmetry, eithe r uniaxial or cubic, can differ from that of the\nbulk material. It is worth mentioning that the non-collinea rities of the surface spins can be represented\nby an additional cubic term in the MAE in the framework of the E OS [10].\nThe EOS type of approach appears then as a simplifying level o f description but, nevertheless, a nec-\nessary step to model the interacting MNP assemblies, especi ally when dipolar interaction are included. In\nthe framework of the EOS models, the theoretical descriptio n of MNP assemblies make use of the standard\nmethods of statistical physics since one deals with a set of p articles interacting through a pairwise potential\nin a one-body potential including both the MAE and the Zeeman term due to the applied external mag-\nnetic field. For MNP coated with a non-magnetic layer, the int erparticle interaction includes an isotropic\nshort-range term and the dipole-dipole interaction (DDI), since the exchange term coming from the direct\ncontact between MNP (referred to as superexchange) vanishe s. Hence, one is faced with the models\napparented to the dipolar hard or soft spheres, according to the short ranged potential (DHS and DSS,\nrespectively) widely developed in a more general framework of dipolar fluids including molecular polar\nfluids to which one adds the MAE contribution in the form of a on e-body potential. An important distinc-\ntion is to be made concerning the structure of the system unde r study according to whether it is a liquid-like\nsuspension, a ferrofluid, or a frozen system which can be an en semble of MNP in a frozen non-magnetic\nembedding medium or a powder sample. In this latter case, the thermally activated degrees of freedom\nare the MNP moments and averages over frozen structure reali zations that are done in a second step.\nIn highly concentrated systems, the nature of the state indu ced at low temperature by the collective\nbehavior resulting from the DDI strongly depends on the unde rlying structure and dimensionality due\nto the anisotropic character of the DDI. For a well ordered sy stem such as a fully occupied perfect\nlattice free of MAE or with a MAE characterized by aligned eas y axes, a long-range ferromagnetic (face\ncentered cubic, fcc, body centered cubic, bcc or body center ed tetragonal, bct) or anti-ferromagnetic\n(simple cubic, sc) ordered state is reached [11–16]. Notice that the solid dipolar ferromagnetic ground\nstate presents a bct structure [17], a result in agreement wi th the solid state phase diagram of the dipolar\nhard sphere system [18]. Conversely disordered systems pre sent a spin-glass like (SSG) [7, 19, 20] as\nhas been experimentally evidenced [21–25]. It is worth ment ioning that not only the perfect lattices of\nDHS or DSS [15, 26] orient spontaneously at low temperature o r high pressure. This is also the case for\nthe dense liquid state [12, 27–30] where a ferroelectric (or equivalently a FM) nematic state has been\nreported in the bulk or in slab geometry.\nOn a perfect lattice, the structural disorder may be introdu ced either through the MAE contribution\nwith a random distribution of easy axes or through dilution. In the limiting case of an infinite magnitude\nof the MAE, the DHS on a lattice reduces to the dipolar Ising mo del, where the exchange coupling\nconstants Jijfollow the dipolar energy form for moments aligned on the eas y axes. In the perfect aligned\ndipole (PAD) case, the ordered/disordered transition take s a SG character when reducing the lattice\noccupation rate [16], while in the random axes dipoles (RAD) , the transition is a SG one whatever the\nlattice occupation [31, 32].\nWell ordered MNP supracrystals with a fcc structure [4, 5] ar e presumably good candidates for\nexperimental realization of a dipolar induced super-ferro magnetic (SFM) phase. The SFM phase was\nalready looked at through the behavior of the hysteresis in t erms of temperature on supracrytals made\nof iron oxide nanoparticles [6]. A ferrimagnetic phase was e videnced involving the ferromagnetically\nordered MNP core moments and the spin canting at the MNP surfa ce. In this work we investigate from\nMonte Carlo simulations the effect of the magnitude of the MAE on the dipolar SFM state reached by\nan ensemble of DHS on a fcc lattice when the MAE easy axes are ra ndomly distributed. We also show\nthat the partial alignment of the easy axes leads to the recov ery of the SFM state. We only focus on the\nSPM-SFM transition.\n33703-2Ferromagneticorderindipolarsystemswithanisotropy\n2.ModelandMonteCarlosimulations\nWe model an assembly of MNP uncoupled exchange, interacting through dipolar interactions (DDI),\ncharacterized by their magnetocrystalline anisotropy (MA E) and self-organized on long-range ordered\nsupra crystals of fcc symmetry. Although this simple model i s not at all specific, we have in mind more\nprecisely the case of Co or γ-Fe2O3MNP. The MAE is assumed to be of uniaxial symmetry. This is\njustified from a crystallographic point of view in the case of hcp-Co, and from the general experimental\nfinding forγ-Fe2O3where the shape or the surface effect leads to an effective unia xial anisotropy. The\neasy axes distribution is random unless otherwise mentione d. In the framework of an effective one-spin\nmodel, we consider a monodisperse ensemble of up to N=2048 dipolar hard spheres located on the\nsites of a fcc lattice with the volume fraction φ. The total energy in a reduced form, after introducing a\nreference temperature T0andβ0=1/(kBT0)reads\nβ0E=ǫd/summationdisplay.1\ni<jˆmiˆmj−3(ˆmiˆrij)(ˆmjˆrij)\n(rij/d)3−ǫu/summationdisplay.1\ni(ˆniˆmi)2, (2.1)\nwhere ˆ niand ˆmiare the easy axes and the unit vectors in the direction of the m oments, respectively. The\ncoupling constants are related to the saturation magnetiza tion Ms, the uniaxial anisotropy constant Ku\nand the MNP volume v(d)[the MNP moment is µ=v(d)Ms] by\nǫd=β0µ0\n4π(π/6)M2\nsv(d), ǫ u=β0Kuv(d), (2.2)\nǫdis a usual dipolar constant. We also introduce more relevant coupling parameters through a reference\ndistance, dref=aφ−1/3, and a reduced temperature\nλd=ǫd/parenleftbiggd\ndref/parenrightbigg3\n, λ u=ǫu/λd, T∗=T/(T0λd), (2.3)\nwhereφis the volume fraction and a convenient choice for the propor tionality constant is a=(φ(0)\nM)1/3,φ(0)\nM\nbeing the maximum value of φfor a given structure here taken as the fcc lattice [ φ(0)\nM=φ(fcc)\nM=√\n2π/6].\nWe perform Monte Carlo simulations in the framework of the pa rallel tempering scheme [33] with a\ndistribution of temperatures bracketing the superparamag netic (SPM)-ordered phase transition. The set\nof temperatures {Tn}is chosen either from a geometric distribution, Tn+1/Tn=const or according to the\nconstant entropy increase scheme [34] which leads to a more h omogeneous rate of configuration exchange\nin the tempering scheme. Periodic boundary conditionsand E wald summations are used with the so-called\nconducting external conditions, according to which the sys tem is embedded in a medium characterized\nby an infinite permeability, which eliminates the shape depe ndent depolarizing effects [13, 35]. We\nemphasize that this is a necessary condition to get a spontan eous magnetization. At this point, we refer a\nreader to the analysis and comments in [27, 36].\nWe calculate, on the one hand, the mean value of the energy, th e spontaneous magnetization, per\nparticle\n∝an}bracketle{tm∝an}bracketri}ht=/angbracketleftBig/barex/barex/barex1\nN/summationdisplay.1\niˆmi/barex/barex/barex/angbracketrightBig\n, (2.4)\nand the nematic order parameter λ[13], revealing the occurrence of an orientational order. T he mean\nvalue of the magnetization, defined in equation (2.4) is the o rder parameter of the SFM phase. When\nan anti-ferromagnetic phase is expected, as is the case for t he simple cubic (sc) lattice, the staggered\nmagnetization should be considered in place of ∝an}bracketle{tm∝an}bracketri}htwith\n∝an}bracketle{tm∝an}bracketri}htst=/angbracketleftBigg/braceleftBigg/summationdisplay.1\nα=1,3/bracketleftBigg\n1\nN/summationdisplay.1\niˆmiα(−1)piβ+piγ/bracketrightBigg2/bracerightBigg1/2/angbracketrightBigg\n, (2.5)\nwhere piαdenotes the layer in the direction αto which the lattice site ipertains.\n33703-3V.Russier,E.Ngo\nOn the other hand, we calculate the heat capacity Cvand the susceptibility from the polarization\nfluctuation\nχm=Nβ∝an}bracketle{tδm2∝an}bracketri}ht=Nβ/parenleftbig∝an}bracketle{tm2∝an}bracketri}ht − ∝an}bracketle{t m∝an}bracketri}ht2/parenrightbig, (2.6)\nboth showing a characteristic peak at the transition temper ature. Finally, the location of transition temper-\nature is determined from finite size scaling analysis throug h the crossing point of the Binder cumulant [37]\nBm=1\n2/parenleftbigg\n5−3∝an}bracketle{tm4∝an}bracketri}ht\n∝an}bracketle{tm2∝an}bracketri}ht2/parenrightbigg\n, (2.7)\nand the location of the Cvand∝an}bracketle{tδm2∝an}bracketri}htversus temperature peaks. Here, as in [38], Bmcorresponds to\nthe normalization such that its limiting values in the (anti )-ferromagnetic and paramagnetic phases are\nBm=1.0 and Bm=0.0, respectively.\nSince we consider a random distribution of easy axes, by incr easingλuwe gradually increase the\namount of disorder. Starting from λu=0 toλu≫λd, the model goes from the perfect lattice of dipoles\nto the random axes dipoles (RAD) model where the moments are a ssigned in the direction to the easy\naxes ˆ mi=siˆniand behave as Ising spins si=±1. The former presents a well ordered ferromagnetic\nphase at low temperature [13], while the latter presents a sp in glass state (SSG) [31]. According to this\npicture, we expect the system to present a SPM →SFM and a SPM →SSG transition for small and large\nvalues ofλu, respectively.\n3.Results\nIn what follows, for the sake of simplicity, we shall denote t he SPM and SFM as paramagnetic (PM)\nand ferromagnetic (FM) phases, respectively. Our purpose i s twofold: first, the determination of the\nPM-ordered phase transition temperature in the weak λuregime, and, secondly, the characterization of\nthe ordered phase. In this work we do not focus on the SSG phase which is known to occur at large values\nofλu[31].\nIn the simulations, we use the parameters of cobalt MNP of ca 8 nm in diameter and the volume\nfraction corresponding to the experimental supracrystals elaborated in [4] in order to fix the dipolar\ncoupling constant. Doing this, we get λd≃2.15. On the other hand, the MAE coupling constant is\nseen as a variable parameter. It is clear that the conclusion s hold for systems characterized by other\nvalues of the dipolar coupling, since we can adjust the value of the reference temperature T0. Taking into\nconsideration the fact that the system free of anisotropy pr esents a FM phase [13–15], we expect the\nordered phase to keep its FM character at small values of λuor at least to present the characteristics of a\nquasi-long-range order (QLRO) magnetic phase as it is the ca se for the random anisotropy model (RAM)\nin the weak anisotropy regime [38].\nFirst of all, in the absence of anisotropy, we determine the P M/FM transition temperature in the\nframework of the finite size scaling analysis, namely from th e crossing point in T∗of the Binder cumulant\ngiven by equation (2.7), corresponding to different system s izes, see figure 1. We locate the PM/FM\ntransition at T∗\nc=0.618±0.005. This result, leading to ρµ2/(kBTc)=2.283 whereρis the number\ndensity, introduced by relating (d/dnn)3of equation (2.3) to ρ, is in close agreement with the early finding\nof Bouchaud and Zerah [14]. Moreover, as shown in figure 1, we fi nd that T∗\ncis very well bracketed by the\nCvandχmpeaks, namely T∗\np(Cv)<T∗\nc<T∗\np(χm)with[T∗\np(χm)−T∗\np(Cv)] ≃0.02. A precise determination\nofBmat the PM-ordered transition, namely its non-trivial fixed p oint value, B∗\nm, is beyond the scope of\nthe present work. It is worth mentioning that this is a difficul t task, which is not unambiguous since it\ndepends on non-essential parameters of the transition [26, 39–41]. Nevertheless, as expected we get a\nvalue, B∗\nm=0.76±0.02 atλu=0, compatible with that of either the Heisenberg model on the lattice\n(0.7916 [42]) or the dipolar hard sphere model ( ≃0.772 to 0.812 [26]).\nThen, we compare the behavior of the simple cubic lattice sc t o the behavior of the face centered cubic\nfcc lattice. It is well known that the sc lattice is antiferro magnetic [11, 43], and thus one should compare\nthe staggered spontaneous polarization, given by equation (2.5), of the sc to the usual one, equation (2.4),\n33703-4Ferromagneticorderindipolarsystemswithanisotropy\n 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0.5 0.53 0.56 0.59 0.62 0.65Bm\nT*λu = 0.0N =  256\n     864\n    2048\n 0 2 4 6 8 10 12\n 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.5 1 1.5 2 2.5\n χmCv/NkB\nT*N =  256\n     864\n    2048\nFigure1. (Color online) Left: the Binder cumulant for the moment, equ ation (2.7) forλu=0. Right:\nsusceptibility,χm, and heat capacity Cv, right forλu=0.\nof the fcc. The result for both the polarization and the nemat ic order parameter λ, displayed in figure 2\nfor simulation boxes of similar size shows that the ordering behaviors of the two lattices present very\nsimilar characteristics, at least in the absence of anisotr opy. This differs from the situation of the Ising\nPAD model corresponding to λu→∞ with aligned easy axes in the zdirection where the PM-AFM and\nPM-FM transitions are found at T∗\nc≃2.5 and 4.0 for the sc and fcc lattices, respectively [43].\nThen, we have performed simulations for the fcc lattice and λuin the range 0/lessorequalslantλu/lessorequalslant1.28 which\nis still in the low anisotropy regime. We still determine the PM-ordered transition temperature from the\nfinite size behavior of the Binder cumulant and the location o f the Cvandχmpeaks. An important result\nis a very weak dependence of the transition temperature with λu; indeed, we get T∗\nc=0.62±0.007 and\n0.610±0.007 forλu=0.465 and 1.28, respectively. The nature of the ordered phase c an be deduced from\nthe low temperature behavior of the magnetization ∝an}bracketle{tm∝an}bracketri}htand the shape of the Cv(T∗)curve. We expect the\nmagnetization in the FM phase, on the one hand, to continuous ly increase with a decrease of T∗down to\nvanishing temperatures and, on the other hand, to be size ind ependent below a threshold temperature. In\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0  0.2  0.4  0.6  0.8  1<m> , λ\nT*\nFigure2. (Color online) Comparison of the polarization (fcc) ∝an}bracketle{tm∝an}bracketri}htor staggered polarization (sc) ∝an}bracketle{tm∝an}bracketri}htst,\nopen symbols, and nematic order parameter λ, solid symbols, on the fcc and sc lattices. Triangles: fcc\nlattice with N=1372, squares: sc lattice with N=1000,λu=0.\n33703-5V.Russier,E.Ngo\n 0.1 0.3 0.5 0.7 0.9\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9< m >\nT* 0.85 0.95\n 0.01 0.04 0.07\nFigure3. Polarization of the fcc lattice in terms of T∗forλu=0 and N=256, squares; 864, circles and\n2916, triangles. Inset: low temperature behavior of the nem atic order parameter λ, bottom and ∝an}bracketle{tm∝an}bracketri}ht, top.\nparticular, the continuous increase of ∝an}bracketle{tm∝an}bracketri}htdown to T∗≃0 is indicative of the absence of freezing of the\nsystem at any finite temperature. These two criteria are clea rly reached in the absence of anisotropy, as\ncan be seen in figure 3. On the other hand, a lambda-like shape o fCvtogether with the size dependence\nof the Cvpeak, as displayed in figure 1 for λu=0 is indicative of the onset of QLRO state from the PM\none. Indeed, the Cvcurve shape of typical SG strongly differs from a lambda-like one and, moreover,\ndoes not follow the characteristic finite size scaling of the PM/FM (or PM/AFM) transition [16, 32].\nForλu=0.465, we get a behavior for both the magnetization ∝an}bracketle{tm∝an}bracketri}htand the nematic order parameter λ\nvery close to that observed in the absence of anisotropy, λu=0 and the above two criteria are satisfied.\nMoreover, the dependence of ∝an}bracketle{tm∝an}bracketri}htwith respect to the number of particles, Nis negligible, at least for\nN/lessorequalslant2048, indicating that the mean spontaneous polarization at low temperature remains finite on an\ninfinite system. Therefore, we can conclude that the ordered phase is FM in this case. This is obviously no\nmore the case when λu=1.28, as shown in figure 4 where we compare the corresponding spo ntaneous\npolarization in terms of T∗for different system sizes. Increasing λu, up toλu=1.28, the behavior of ∝an}bracketle{tm∝an}bracketri}ht\nstrongly differs and the first point is the appearance of a plat eau in the ∝an}bracketle{tm∝an}bracketri}ht(T∗)curve at low temperature.\n 0.1 0.3 0.5 0.7\n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.3 0.5 0.7 0.9\n< m >\nT*λu = 0.465\nλu = 1.28N = 256\n 500\n 864\n 2048\nFigure4. Comparison of the fcc lattice polarization in terms of T∗forλu=0.465 and 1.28 and different\nvalues of Nas indicated in the figure.\n33703-6Ferromagneticorderindipolarsystemswithanisotropy\n 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0.1 0.2 0.3 0.4 0.5 0.6 0.7Bm\nT*256\n500\n864\n2048\n 0.8 1 1.2 1.4 1.6 1.8 2 2.2\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Cv/NkB\nT*256\n864\n1372\n2048\nFigure5. (Color online) Left: Binder cumulant for λu=1.28 and different values of Nas indicated in\nthe figure. Right: heat capacity for λu=1.28 and different values of Nas indicated in the figure.\nWe also get a different behavior for the Binder cumulant, disp layed in figure 5. Indeed, if there is still\na crossing point at the onset of the PM-ordered phase, (which is, however, less pronounced), the low\ntemperature first Bmno more reaches the limiting value Bm=1, and its value decreases with an increase\ninNwhich results from the onset of a decrease of Bmwith T∗beyond some value of N. Such a behavior\nresembles the one obtained in [44] where it was analyzed as th e occurrence of a reentrant SG phase. It is\nworth mentioning that the onset of the PM-ordered transitio n still presents a FM character, as indicated\nby the shape of the Cvcurve displayed in figure 5 but it is presumably a QLRO one inst ead of a FM\nphase as indicated first by the size dependence of Cv, strongly weakened when compared to the PM-FM\ntransition case. To confirm this point, we show in figure 6 the b ehavior of the ∝an}bracketle{tm∝an}bracketri}htin terms of Nat a low\ntemperature. From the fits of ∝an}bracketle{tm∝an}bracketri}ht(N)at fixed T∗in the form ≃b.N−band the values of b(λu)obtained\n(b=0.01379±0.0009, 0.04936±0.006 and 0.09904±0.01 forλu=0.5, 1.0 and 1.28, respectively),\nsee figure 6, we conclude that the spontaneous magnetization at a low temperature is likely to vanish in\n 0.6 0.9\n 100  1000< m >\nNλu = 1.28; T* = 0.14\nT* = 0.11\nλu = 1.0; T* = 0.094\nλu = 0.5; T* = 0.094\nFigure6. Log log plot of the magnetization at low temperature in terms ofN. The lines are fits of the\nform∝an}bracketle{tm∝an}bracketri}ht=aN−bwith b=0.01379±0.0009 and 0.04936±0.006 at T∗=0.094 forλu=0.5 and\n1.0, respectively, and b=0.09355±0.0.006 and 0.09904±0.01 forλu=1.28 and T∗=0.14 and 0.11,\nrespectively.\n33703-7V.Russier,E.Ngo\n 0.1 0.3 0.5 0.7 0.9\n 0.2  0.6  1  1.4  1.8< m > , λ\nT*λu = 1.86; θm = π/6\nFigure7. (Color online) Spontaneous polarization (squares) and nem atic order parameter (circles) for a\nsystem with a textured distribution of easy axes centered on thez-axis with a solid angle limited by a\ngaussian with variance Θm=π/6;λu=1.86.\nthe limit N→ ∞ forλu/greaterorequalslant0.5. From the low temperature moments configurations obtained with 2048\nparticles andλu=1.28, we cannot conclude on the onset of the domain formation at least in this range\nof sizes; on the other hand, the ∝an}bracketle{tˆµi.ˆµj∝an}bracketri}htcorrelation function still presents a FM character with no s ign\ninversion but a noticeably reduced range when compared to th eλu=0 case.\nFinally, we show that the FM ordered phase is recovered when w e consider a preferentially oriented\neasy axes distribution, with probability\np(Θ) ∝sin(Θ)e−Θ2/2σ2\naround the z-axis where one recovers a usual random distribution for σ≫1. In figure 7 we show an\nexample withσ=π/6 in the caseλu=1.86 a value for which the long-range FM order is lost when the\neasy axes is randomly distributed.\nTo conclude, we have shown that an assembly of MNP located on a perfectly ordered fcc lattice\npresents a FM phase at low temperature only in a reduced range of MAE amplitude, λu/lessorequalslant0.5 for a\nrandom distribution of easy axes. However, there is an onset of a QLRO phase, at a temperature T∗\ncvery\nclose to the PM-FM transition of the system free of anisotrop y. This QLRO phase is characterized by a\nspontaneous magnetization vanishing in the limit of an infin ite system. The FM phase can be recovered\nby aligning, at least partially, the easy axes.\nAcknowledgements\nWe acknowledge important discussions with Dr. J. Richardi, Dr. I. Lisiecki, Dr. T. Ngo, Dr. C. Salze-\nmann, Dr. S. Nakamae, Dr. C. Raepsaet, Dr. J.-J. Weis and Pr. J .J. Alonso.\nV. Russier, who has been one of the Dr. J.-P. Badiali’s PhD stu dents, acknowledges the huge importance\nDr. J.-P. Badiali had on his carrier. The collaboration whic h lead to about 15 co-authored publications\naimed at a theoretical description of the electrochemical d ouble layer which brought together statistical\nphysics physicists and electrochemists. It was during this activity that J.P. Badiali urged V. Russier to the\nso-called dipolar hard sphere fluid as a model for molecular s olvents. The work presented here dealing\nwith another application of this model is in some sense the co ntinuation of this initial training and\ncollaboration.\nWork performed under grant ANR-CE08-007 from the ANR French Agency. This work was granted\nan access to the HPC resources of CINES under the allocation 2 016-A0020906180 made by GENCI.\n33703-8Ferromagneticorderindipolarsystemswithanisotropy\nReferences\n1. Pankhurst Q., Thanh N., Jones S.K., Dobson J., J. Phys. D: A ppl. Phys., 2009, 42, 224001,\ndoi:10.1088/0022-3727/42/22/224001.\n2. 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B, 2005, 72, 214402, doi:10.1103/PhysRevB.72.214402.\n33703-9V.Russier,E.Ngo\nФеромагнiтнийпорядокудипольнихсистемахз\nанiзотропiєю:застосуваннядомагнiтнихнаночастинкових\nсупракристалiв\nВ.Рус’є,Е.Нго\nIнститутхiмiїтаматерiалiвприунiверситетi“Париж-Схiд ”(УПС),дослiдницькийцентр7182\nНацiональногоцентрунауковихдослiдженьтаУПС,вул.Анрi Дюнана,2-8,94320Тьє,Францiя\nОднодоменнiмагнiтнiнаночастинки(МНЧ)iздипольноюмiжч астинковоювзаємодiєюможутьокрiмма-\nгнiтокристалiчноїфазизнаходитисяунизькотемпературнi йферомагнiтнiй(ФМ)абоспiново-склянiйфа-\nзахзалежновiдструктуритаступенювпорядкуваннясистеми .МетодоммоделюванняМонте-Карлов\nрамкаходноспiновоїчимакроспiновоїмоделейавторамидос лiджуєтьсямонодисперснасистемаодно-\nдоменнихМНЧ,розташованихнавузлахiдеальноїґраткизсим етрiєютипуfcciздовiльнорозподiленими\nосями.Розглядаєтьсявипадокнизькоїанiзотропiї,зокрем апочаткузникненнядипольноїдалекосяжної\nФМфази.\nКлючовiслова:магнiтнiнаночастинки,моделюванняМонте -Карло,феромагнiтневпорядкування\n33703-10"    },    {        "title": "1712.02139v2.First_principles_investigation_of_magnetocrystalline_anisotropy_oscillations_in_Co___2__FeAl_Ta_heterostructures.pdf",        "content": "First-principles investigation of magnetocrystalline anisotropy oscillations in\nCo2FeAl/Ta heterostructures\nJunfeng Qiao,1, 2Shouzhong Peng,1, 2Youguang Zhang,1, 2Hongxin Yang,3and Weisheng Zhao1, 2,\u0003\n1Fert Beijing Institute, BDBC, Beihang University, Beijing 100191, China\n2School of Electronic and Information Engineering, Beihang University, Beijing 100191, China\n3Key Laboratory of Magnetic Materials and Devices,\nNingbo Institute of Materials Technology and Engineering,\nChinese Academy of Sciences, Ningbo 315201, China\n(Dated: July 11, 2018)\nWe report \frst-principles investigations of magnetocrystalline anisotropy energy (MCAE) oscilla-\ntions as a function of capping layer thickness in Heusler alloy Co 2FeAl/Ta heterostructures. Substan-\ntial oscillation is observed in FeAl-interface structure. According to k-space and band-decomposed\ncharge density analyses, this oscillation is mainly attributed to the Fermi-energy-vicinal quantum\nwell states (QWS) which are con\fned between Co 2FeAl/Ta interface and Ta/vacuum surface. The\nsmaller oscillation magnitude in the Co-interface structure can be explained by the smooth potential\ntransition at the interface. These \fndings clarify that MCAE in Co 2FeAl/Ta is not a local property\nof the interface and that the quantum well e\u000bect plays a dominant role in MCAE oscillations of the\nheterostructures. This work presents the possibility of tuning MCAE by QWS in capping layers, and\npaves the way for arti\fcially controlling magnetic anisotropy energy in magnetic tunnel junctions.\nI. INTRODUCTION\nWith the increasing demand for high-speed and low-\npower-consumption storage devices, intensive researches\nhave been made on spin-transfer-torque magnetic ran-\ndom access memory (STT-MRAM). The core structure\nof MRAM is magnetic tunnel junction (MTJ),1which\nis composed of an insulating barrier sandwiched by two\nferromagnetic (FM) electrodes. The relative orienta-\ntion of two FM electrodes represents two states and\ncan be utilized to store one bit information. To realize\nhigh storage density, the manufacturing process is scaling\ndown to nanometer regime. However, the increasing pro-\ncess variations in the fabrication pose serious challenges\nto fundamental physics,2especially magnetocrystalline\nanisotropy energy (MCAE), which is critical for the ther-\nmal stability of the relative magnetization orientation\nof two FM electrodes. Previous work reported that to\nachieve a retention time of 10 years, an interfacial perpen-\ndicular magnetic anisotropy (PMA) of 4 :7 mJ=m2is re-\nquired for device sizes scaling down to 10 nm.3However,\nat present the most widely used FM electrode, CoFeB,\ncan commonly reach an interfacial PMA of 1 :3 mJ=m2\nwhen interfaced with MgO tunneling barrier.1,4{6At the\nsame time tunneling magnetoresistance (TMR) can reach\na value of 120 % in CoFeB/MgO/CoFeB MTJ,1which\nneeds to be improved as well.\nTo further promote the development of STT-MRAM,\nother FM materials are under investigation. Heusler\nalloys are a big family of ternary intermetallic com-\npounds with nearly 1500 members.7According to their\nchemical composition, Heusler alloys can be separated\ninto two classes, full Heusler with chemical composition\n\u0003weisheng.zhao@buaa.edu.cnX2YZ (L21structure) and half Heusler XYZ ( C1bstruc-\nture), in which X and Y are transition metals, and Z\nis main group element.8By virtue of the broad choices\nof elements and stoichiometry, many Heusler compounds\nexhibit interesting properties, such as half-metallicity,9\nvarious Hall e\u000bect,10{12thermoelectric e\u000bect,13topo-\nlogical e\u000bect12and superconductivity,14etc. Among\nHeusler alloys, Co 2FeAl (CFA) has attracted lots of at-\ntention due to its high spin polarization and low mag-\nnetic damping constant.15,16TMR ratio can reach up\nto 700 % at 10 K and 330 % at room temperature (RT)\nin Co 2FeAl/MgO/CoFe MTJ.17Magnetic damping con-\nstant,\u000b, can reach as low as 0 :001,18which is bene\fcial\nfor reducing STT switching current. Another merit of\nCFA is its \fne lattice matching with MgO. As a result,\nepitaxial growth of CFA(001)[110] kMgO(001)[100] can\nbe achieved in experiment.19All these advantages make\nCFA a promising candidate for MTJ electrode material.\nRegarding magnetic anisotropy energy (MAE) of CFA,\nexperimental and theoretical results con\frmed that the\nCo2FeAl/MgO interface can reach around 1 mJ =m2.20{22\nHowever, as discussed above, MAE needs to be optimized\nfurther.Besides, it is crucial to \fnd out e\u000bective ways to\narti\fcially control MAE.\nRecently, experimental and theoretical results showed\nthat heavy metals (HM) can induce large variations of\nphysical properties including MAE when interfaced with\nFM materials.23{30In practical MTJs, a bu\u000ber layer at\nthe bottom and a capping layer on the top are necessary\nto improve and protect the FM/MgO/FM core structure.\nConsequently, the choice of capping layer provides us a\nunique way to control MAE of the whole structure. On\nthe other hand, when the thickness of these multilayers\nreaches down to atomic scale, quantum mechanical (QM)\ne\u000bects start to dominate. One of the most well-known\nQM e\u000bect is quantum well (QW), in which the wave func-\ntions of the quantum particle are con\fned by potentialarXiv:1712.02139v2  [cond-mat.mtrl-sci]  9 Feb 20182\nbarriers and the energy levels are quantized. In spintron-\nics, the milestone e\u000bect, giant magnetoresistance (GMR),\nand its closely related phenomenon, interlayer exchange\ncoupling (IEC), are deeply related to QW. These e\u000bects\nhave been successfully explained by quantum interfer-\nences due to re\rections at the spacer boundaries.31In\nterms of the in\ruence of quantum well states (QWS) on\nMAE, early theoretical works, using tight-binding for-\nmalism and a perturbation treatment to spin orbit cou-\npling (SOC), reported the oscillation of MAE with re-\nspect to Pd layer thickness in Co/Pd system.32While\nthere also exists other work which supports interfacial-\nMAE in Pd/Co/Pd(111) structure.33Other than HM\nPd, MAE oscillations with respect to both Co and Cu\nwere found in Co/Cu system.34Since the IEC e\u000bects are\nprominent in these structures,35,36the formation of QWS\nare well con\frmed. Indeed, 10 years later, MAE oscilla-\ntions were observed in Cu(001)/Co, Ag(001)/Fe37and\nFe/Cu, Co/Cu structures,38and the origin of these os-\ncillations were attributed to QWS. Also, QWS induced\noscillatory IEC was found in Co/MgO/Co PMA MTJ.39\nRecent \frst-principles studies have correlated QWS with\nMCAE in Ag/Fe and IEC in Fe/Ag/Fe structures.40,41\nThese works indicate that the in\ruence of QWS on mag-\nnetic properties, speci\fcally MAE, may become salient\nin some structures.\nIn this paper, we report ab-initio calculations of\nMCAE in CFA/Ta structures and observe MCAE oscilla-\ntions associated with the Ta layer thickness. These oscil-\nlations are further proved as induced by both majority-\nspin and minority-spin QWS con\fned in Ta layers. The\norigin of the signi\fcant MCAE oscillation is attributed\nto the repeated traversing of QWS across Fermi energy\nand the large SOC of Ta. In all, QWS formed in the\ncapping layer provide us a unique method to tune MAE\nin the MTJ structure.\nII. METHODS\nCalculations were performed using Vienna ab ini-\ntiosimulation package (VASP) based on projector-\naugmented wave (PAW) method and a plane wave ba-\nsis set.42The exchange and correlation terms were\ndescribed using generalized gradient approximation\n(GGA) in the scheme of Perdew-Burke-Ernzerhof (PBE)\nparameterization.43We used a kinetic energy cuto\u000b of\n520 eV and a Gamma centered Monkhorst-Pack k-point\nmesh of 25\u000225\u00021. The convergence of MCAE relative\ntok-point has been checked carefully, the variation of\nMCAE is about 0 :05 meV when changing k-point mesh\nfrom 20\u000220\u00021 to 25\u000225\u00021, which is at least a mag-\nnitude smaller than the oscillation amplitude of MCAE.\nThe energy convergence criteria of all the calculations\nwere set as 1 :0\u000210\u00007eV, and all the structures were re-\nlaxed until the force acting on each atom was less than\n0:01 eV=\u0017A. All the structures have at least 15 \u0017A vacuum\nspace to eliminate interactions between periodic images.Bulk CFA has a cubic L21crystal structure. Af-\nter fully relaxing bulk structure in volume and shape,\nthe lattice constant is found to be abulk = 5:70\u0017A,\nperfectly matches the experimental value 5 :73\u0017A.44For\nCFA/Ta heterostructure, an in-plane lattice constant of\na=abulk=p\n2 = 4:03\u0017A is adopted for the unit cell, which\nis rotated by 45 degrees from the conventional cell of bulk\nCFA. For all the CFA/Ta structures, 9 monolayers (ML)\nof CFA are used, and 1 to 12 ML of Ta layers are put\non top of CFA, as shown in Fig. 1. We use CFA/Ta[ n]\nto label structures of di\u000berent Ta ML, where nis the\nnumber of Ta ML, ranging from 1 to 12. As for the in-\nterface between CFA and Ta, there exist two kinds of\ncon\fgurations and both of them have been investigated.\nFeAl-CFA/Ta is used as the label when FeAl layer of CFA\ndirectly contact with Ta, while Co-CFA/Ta is used when\nCo layer of CFA contact with Ta.\nFIG. 1. Crystal structure of (a)FeAl-CFA/Ta[9], (b)Co-\nCFA/Ta[9], (c)FeAl-CFA/Ta[5] and (d)Co-CFA/Ta[5]. Only\n4 structures are shown as illustrations. In other structures,\nonly the numbers of Ta ML are varied, ranging from 1 to\n12. The dashed green rectangle box highlights the area of the\ninterfaces.\nTo calculate MCAE, two-step procedures were\nadopted. Firstly, charge density was acquired self-\nconsistently without taking into account SOC. Secondly,\nreading the self-consistent charge density, two non-self-\nconsistent calculations were performed including SOC,\nwith magnetization pointing towards the [100] direction\nand the [001] direction, respectively. Finally, MCAE was\ncalculated by MCAE =E[100]\u0000E[001], positive MCAE\nstands for PMA while negative MCAE for in-plane mag-\nnetic anisotropy.\nTo get a deeper understanding of the origin of oscil-3\nlation, MCAE is decomposed into k-space. According\nto force theorem,45{47the main contribution of MCAE\noriginates from the di\u000berence of eigenvalues between two\nmagnetization directions. Indeed, we found that the ion\nEwald summation energy, Hartree energy, exchange cor-\nrelation energy and external potential energy are exactly\nthe same between two magnetization directions, the dif-\nference of total energy only comes from the di\u000berence of\neigenvalue summation, this testi\fes the feasibility of k-\nspace decomposition of MCAE. Speci\fcally, this can be\nexpressed as\nMCAE (k) =X\nin[100]\ni;k\u000f[100]\ni;k\u0000X\ni0n[001]\ni0;k\u000f[001]\ni0;k(1)\nwhere kis thek-point index, i;i0are the band indexes\nof magnetization direction along [100] and [001], respec-\ntively.ni;kis occupation number of this band, \u000fi;kis the\nenergy of band iatk-point k.\nIII. RESULTS AND DISCUSSION\nA. MCAE oscillation\nUnlike the FM/oxide structure where the MCAE can\nbe accounted as local hybridization of the interfacial Fe-\n3dorbital and the interfacial O-2 porbital,6the MCAE of\nCFA/Ta structure varies strongly when the Ta thickness\nchanges. In this circumstance, MCAE cannot be treated\nas a local property of the interface. We observe a strong\noscillation of MCAE in FeAl-CFA/Ta structure relative\nto the thickness of capping layer Ta, as shown in Fig. 2.\nThe oscillation period is approximately 4 ML, and the\noscillation amplitude decreases as the number of Ta ML\nincreases. This is due to that the con\fnement e\u000bect of\nQW will become less prominent when the width of QW\nincreases and the bulk states of Ta will account for a\nlarger proportion in all the electron states. Interestingly,\nthe oscillation is smaller in the Co-CFA/Ta structure, the\nreason for this phenomenon will be discussed later.\nTo comprehend the origin of the oscillations, we man-\nually tweak the strength of SOC in the structures. Since\nMCAE only comes from SOC, switching o\u000b the SOC of\nTa will totally screen out the contribution of Ta to MCAE\nof the whole system. For the FeAl-CFA/Ta structure, by\nsuppressing the SOC of Ta while still keeping the SOC of\nCFA, oscillation of the MCAE relative to Ta layer thick-\nness disappears [see cyan lines in Fig. 2]. For the Co-\nCFA/Ta structure, a smaller oscillation exists and the\ntweaking of the SOC of Ta has little in\ruence on the\nMCAE [see red lines in Fig. 2]. These strongly indicate\nthat the electron states in Ta play the determinant role\nin the MCAE oscillations of CFA/Ta structures.\nA further analysis can be carried out by de\fning the\nMCAE di\u000berence\nMCAEdiff(n) =MCAE (n)\u0000MCAETa-off(n) (2)\nFIG. 2. MCAE oscillation with respect to Ta ML number.\nCyan color for FeAl-interface structure, red for Co-interface\nstructure. Circle mark for MCAE calculation with SOC of\nTa switched o\u000b while SOC of CFA still included, square mark\nfor normal MCAE calculation. Lines in the inset are de\fned\nby Eq. (2). The numbers in the inset mark the oscillation\nmagnitude of the \frst period and the second period.\nwherenis the number of Ta ML, MCAETa-off(n)\nis the result calculated with SOC of Ta switched o\u000b.\nTheMCAEdiff(n) will only contain MCAE contribu-\ntion originated from Ta layers, as plotted in the inset of\nFig. 2. Note three major di\u000berences can be discerned.\nFirstly, the oscillation magnitude of Co-CFA/Ta vanishes\nmuch faster than that of FeAl-CFA/Ta. Secondly, we\ncan de\fne an oscillation period of 4 ML in FeAl-CFA/Ta\nbut it is harder to clearly de\fne an oscillation period for\nCo-CFA/Ta. Thirdly, in Co-CFA/Ta, the mean value of\nMCAEdiff(n) is essentially zero while the mean value\nofMCAEdiff(n) of FeAl-CFA/Ta largely deviates from\nzero. The oscillation of physical properties relative to\n\flm thickness is a hallmark of QWS, and these three re-\nmarkable di\u000berences suggest that for FeAl-CFA/Ta, the\nelectron states in Ta layers may form QWS and explain\nthe MCAE oscillation. While for Co-CFA/Ta, since the\nMCAE oscillation is less prominent, there is less prob-\nability to correlate MCAE oscillation with QWS in Ta\nlayers. The subsequent paragraph will concentrate on\nthe analysis of MCAE with special electron states and\nevidence of the existence of QWS in FeAl-CFA/Ta struc-\nture. The same analytic procedures are also applied to\nCo-CFA/Ta in the Supplemental Material.\nB. Critical k-points and band structure\nEmploying k-space resolved method, we dissect MCAE\ninto two-dimensional Brillouin zone (2D-BZ) for FeAl-\nCFA/Ta[n]. Comparing k-resolved graphs for structures\nwith di\u000berent Ta layer thickness, two critical k-points,\nwhich have large contributions to MCAE, can be se-4\nlected out, i.e. k-points at [ kx;ky] = [\u00000:48;\u00000:2] and\n[kx;ky] = [0:48;\u00000:2], as shown in Fig. 3. (with number\nofk-points set as 25 \u000225\u00021,kxandkyranging from\n\u00000:48 to 0:48). The SOC breaks the symmetry of 2D-\nBZ, contributions to total MCAE slightly di\u000ber from each\nother between these two k-points. In fact, when consid-\nering the symmetry of the crystal structure, these two k-\npoints are identical and locate at the center of Mpoint\nandXpoint of the 2D-BZ, and they will be called as\ncriticalk-point in the following text.\nFIG. 3. k-resolved graphs of MCAE in 2D-BZ for (a)\nFeAl-CFA/Ta[3], (b) FeAl-CFA/Ta[6], (c) FeAl-CFA/Ta[9]\nand (d) FeAl-CFA/Ta[12]. Critical k-points at [ kx; ky] =\n[\u00000:48;\u00000:2] and [ kx; ky] = [0 :48;\u00000:2] are labeled by red cir-\ncles, which have the dominant contributions to total MCAE.\nAccording to second order perturbation theory, the\nperturbation of SOC to one-electron energies can be writ-\nten as\n\u000e\u000fi=X\ni06=ijhi0jHSOCjiij2\n\u000fi\u0000\u000fi0(3)\nEaxis\ncorr=X\nini\u000e\u000fi=1\n2X\niX\ni06=ini\u0000ni0\n\u000fi\u0000\u000fi0jhi0jHaxis\nSOCjiij2;\naxis = [100];[001]\n(4)\nwherei;i0is the quantum number of one-electron\nstates,\u000fiis the one-electron energy and HSOC is the\nHamiltonian of SOC, axis is the magnetization direction,\nandEaxis\ncorris the energy correction to unperturbed state\ncaused by SOC.\nThis expression indicates that only electron states near\nFermi energy have maximal impact on MCAE. To extract\nout more information about states contributing most to\nthe MCAE oscillation, we systematically examine spin-\nresolved band structures along di\u000berent directions at this\nk-point. As an example, we draw the band structurealong lineky= 2:57kx+ 1:03 , with band index ranging\nfrom 209 to 214, as shown in Fig. 4(b) for spin-up elec-\ntrons, and band 187 to 193 for spin-down electrons, as\nshown in Fig. 4(d). We \fnd that the spin-up band with\nindex 212 and spin-down band with index 190 traverse\nFermi energy along most of the directions in 2D-BZ. As a\nreminder, this particular number has no physical mean-\ning but only labels the order of Kohn-Sham eigenvalues\nin the calculation result. Note due to exchange split-\nting, the energy of spin-down band are higher than the\ncorresponding spin-up band which has identical band in-\ndex with the spin-up band, so the Fermi-energy-vicinal\nbands are di\u000berent for spin-up and spin-down electrons\nand should be considered separately.\nFor the Co-CFA/Ta, we \fnd that di\u000berent from FeAl-\nCFA/Ta, the rapid variations in 2D-BZ make the ascrip-\ntion of MCAE oscillation to a speci\fc electron state a bit\nmore di\u000ecult. We can still select out critical k-points but\nthe magnitude of the peak does not have a sharp contrast\nwith otherk-points, as shown in Fig. S1 of Supplemental\nMaterial.\nFIG. 4. (a) 3D spin-up band structure of FeAl-CFA/Ta[9]\nin a rectangle region around the critical k-point [ kx; ky] =\n[\u00000:48;\u00000:2], which is marked as a blue dot in the horizontal\nplane. Color represents the relative magnitude of energy in\nthe same band, larger value in red, smaller value in blue. (b)\nSpin-up band structure along line: ky= 2:57kx+ 1:03 as an\nexample, i.e., the red arrowed line in (a). (c) 3D spin-down\nband structure of FeAl-CFA/Ta[9] in the rectangle region. (d)\nSpin-down band structure along line: ky= 2:57kx+1:03. The\nnumbers in legend of (b) and (d) are the band indexes. These\nbands are vicinal to Fermi energy and have large contributions\nto total MCAE.5\nC. Characterization of QWS\nTo explore the nature of these speci\fc electron states,\nthe band-decomposed charge densities of these Fermi-\nenergy-vicinal states are plotted, and we conclude that\nthese are the quantum well states con\fned between the\nFeAl-CFA/Ta interface and the Ta/vacuum surface, as\nshown in Fig. 5 for spin-up electrons and Fig. 6 for spin-\ndown electrons. Note spin-up electrons of band 209 to\n212 are Fermi-energy-vicinal, while for spin-down Fermi-\nenergy-vicinal states, their band indexes are from 188 to\n190. For the Fermi-energy-vicinal states of both major-\nity spin and minority spin, i.e. spin-up and spin-down\nelectrons, all of them are mostly con\fned in Ta layers, as\nindicated by the orange lines in Fig. 5 and green lines\nin Fig. 6. With increasing band index, more wave crests\nare formed, which are the characteristic feature of QWS.\nThe energies of QWS depend on the width of the well,\nnamely, the thickness of the Ta layer. By increasing or\ndecreasing Ta layer thickness, the QWS will fall or rise\nthrough Fermi energy, consequently lead to the oscilla-\ntion of the total MCAE, as suggested by Eq. (4).\nFIG. 5. Charge densities of energy bands at index (a) 209,\n(b) 210 and (c) 212. Green color for spin-down electron, or-\nange for spin-up electron and blue for total charge density.\nThe horizontal axes of these \fgures correspond to zaxis of the\ncrystal structure. Note the energies correspond to the spin-up\nelectrons of these bands are vicinal to Fermi energy, while en-\nergies correspond to the spin-down electrons are higher than\nFermi energy due to exchange splitting. The spin-up electrons\nare mostly con\fned in Ta layers.\nFor the Co-CFA/Ta structure, the band-decomposed\ncharge densities of Fermi-energy-vicinal states do not\nperfectly resemble that of an idealized one-dimensional\nquantum well with in\fnite potential barriers, the charac-\nter of QWS is less apparent than FeAl-CFA/Ta. Band-\nFIG. 6. Charge densities of energy bands at index (a) 188,\n(b) 189 and (c) 190. Note the energies correspond to the spin-\ndown electrons of these bands are vicinal to Fermi energy. The\nspin-down electrons of these bands are con\fned in Ta layers,\nwhile spin-up electrons couple to the states in CFA to some\nextent.\ndecomposed charge densities of Co-CFA/Ta[9] are plot-\nted in Fig. S3 and Fig. S4 of Supplemental Material.\nBefore concluding the charge density analysis, we\nwould like to clarify that both Fig. 5 and Fig. 6 are\nband-decomposed, while the total charge density should\nbe the sum of all the occupied bands. So the crests in the\nband-decomposed charge densities do not imply a large\nantiferromagnetic coupling between Ta layers and CFA.\nThe magnetic moments of all the atoms are plotted in\nFig. S5 for FeAl-CFA/Ta[9] and S6 for Co-CFA/Ta[9]\nfor the interested reader.\nD. Interface potential\nThe phenomenon that only the FeAl-CFA/Ta struc-\nture has strong MCAE oscillation can be understood by\nthe interface potential di\u000berence. The potential drop at\nthe interface determines the magnitude of con\fnement\nof electrons. Potential of FeAl layer di\u000bers substantially\nfrom Co layer. As shown in Fig. 7, a larger mismatch of\nthe potential between Ta and FeAl-CFA is found while\nthe mismatch between Ta and Co-CFA is much smoother.\nSince the constructions of initial structures and the pro-\ncesses of atomic relaxations inevitably lead to a small dis-\nplacement along zaxis between two structures of di\u000berent\ninterfaces, and the potential di\u000berence strongly relies on\nthe origins of coordinates in two structures, we choose\npart of Ta layers as sampling and minimize the square\nerror so as to accurately align these two structures, i.e.6\nthe potential di\u000berence is calculated as\nVdiff(z) =VCo(z)\u0000VFeAl(z+\u000e) (5)\nwhere\n\u000e= arg min\n\u000ffZ\n(VTa\nCo(z)\u0000VTa\nFeAl(z+\u000f))2dzg(6)\nwherezis the coordinate along zaxis,\u000eis the displace-\nment of FeAl-interface structure that accurately aligns\ntwo structures, VCoandVFeAl are the potentials of Co-\nCFA/Ta[9] and FeAl-CFA/Ta[9], respectively. VTa\nCoand\nVTa\nFeAl are the potentials of Ta layers of Co-CFA/Ta[9]\nand FeAl-CFA/Ta[9], respectively. Vdiff(z) is the poten-\ntial di\u000berence plotted in Fig. 7(c).\nFIG. 7. Potentials along crystallographic zdirection\nof (a) Co-CFA/Ta[9] and (b) FeAl-CFA/Ta[9]. (c) Poten-\ntial di\u000berence between Co-CFA/Ta[9] structure and FeAl-\nCFA/Ta[9] structure, where the upper horizontal axis is for\nCo-CFA/Ta[9] and the lower horizontal axis is for FeAl-\nCFA/Ta[9]. The black arrows in (a) and (b) mark the location\nof the interfaces. The dashed orange and green lines repre-\nsentzcoordinates of each layer in Co-CFA/Ta[9] and FeAl-\nCFA/Ta[9], respectively. A potential di\u000berence of 4 :842 eV\ncan be found between two structures at the interfaces.By carefully aligning Ta layers of FeAl-interface and\nCo-interface structures to the same position, little dif-\nference is found in Ta part between both structures.\nBut at the interface, the potential is 4 :8 eV higher in\nFeAl-CFA/Ta than Co-CFA/Ta [see Fig. 7(c)]. This\nlarger mismatch ultimately makes con\fnement e\u000bect\nmore prominent in the FeAl-CFA/Ta structure, thus ex-\nplaining the larger magnitude of MCAE oscillation. The\nsmoother transition of potential in Co-CFA/Ta does not\nstrongly con\fne electrons into Ta layers, consequently\nMCAE oscillation relative to Ta layer thickness vanishes\nfaster in the Co-interface structure.\nIV. CONCLUSIONS\nBy means of \frst-principles calculations, we observed\na signi\fcant oscillation of MCAE as a function of heavy\nmetal layer thickness in the CFA/Ta structure with FeAl\ninterface. The origin of this oscillation can be attributed\nto electron states con\fnement in Ta layers. Through k-\nspace analysis, states having the largest contribution to\nMCAE can be traced down into special k-points located\nat the center of Xpoint andMpoint in 2D-BZ. More-\nover, it was unveiled that the Fermi-energy-vicinal states\ncontribute the most to total MCAE. The wave crests\nand troughs appearing in these bands indicate that these\nare the quantum well states con\fned in Ta layers. The\nsmaller oscillation magnitude in Co-interface structure\ncan be explained by the smoother potential transition at\nthe Co/Ta interface, which imposes less con\fnement on\nelectrons in Ta layers. This work clari\fes that due to the\nQWS in capping layer, MCAE in CFA/Ta cannot be ac-\ncounted as a local property of the interface. 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Mater. 159, 337 (1996)."    },    {        "title": "1712.09504v1.Effective_reduction_of_the_coercivity_for_Co___72__Pt___28___thin_film_by_exchange_coupled_Co___81__Ir___19___soft_layer_with_negative_magnetocrystalline_anisotropy.pdf",        "content": "arXiv:1712.09504v1  [physics.app-ph]  27 Dec 2017Effective reduction of the coercivity for Co 72Pt28thin film by exchange coupled\nCo81Ir19soft layer with negative magnetocrystalline anisotropy\nJ. Y. Jiao,1T. Y. Ma,1Z. W. Li,1,∗L. Qiao,1Y. Wang,1T. Wang,1,†and F. S. Li1\n1Institute of Applied Magnetics, Key Lab for Magnetism and Ma gnetic Materials of the Ministry of Education,\nLanzhou University, Lanzhou 730000, P.R. China\n(Dated: October 13, 2018)\nWe report on the investigation of coercivity changes of the C o72Pt28/Co81Ir19exchange-coupled\ncomposite (ECC) media with negative soft-layer (SL) magnet ocrystalline anisotropy (MA) . Our\nresults show that the hard-layer (HL) of our sample exhibits a columnar type microstructure with\nwell isolated grains and the SL with hcp-structure grows on t op of the HL with the same texture.\nTherefore, strong coupling of the two layers have been reali zed as evidenced by the magnetic char-\nacterization. Importantly, we observe a more effective redu ction of the coercivity of the ECC media\nby using SLs with negative MA when compared to the use of SLs wi th positive or negligible MA.\nThe experimental results are corroborated by theoretical c alculations.\nPACS numbers: 75.70.-i, 75.50.Ss\nI. INTRODUCTION\nIn order to solve the so called ”magentic record-\ning trilemma” problem, different approaches such\nas exchange-coupled composite (ECC) media1–4and\nheat/microwave/light assisted magnetic recording5–7,\nhave been used to reduce the coercivity of the record-\ning media with very high uniaxial anisotropy constant.\nAmong them, the ECC media, in the form of mag-\nnetically hard/soft multi-layer structure, have attracted\nmuch attention due to their promising potential in re-\nsolving the above mentioned trilemma challenge1–4. Ex-\ntensive theoretical work have been performed to investi-\ngate the effects of soft-layer (SL) thickness8,9, anisotropy\nconstant10–12, hard-to-soft layer coupling strength3,13–15,\nand the magnetization reversal process14,16,17. Experi-\nmentally, many work have been realized with FePt18–23,\nCoPt24–28due to their high uniaxial magnetocrystalline\nanisotropy(MA) which isessentialtomaintainhigh ther-\nmal stability in high density magnetic recoding. Consid-\nerablereductionofthecoercivitywereachievedbychang-\ning the SL composition21,24,25which alters its anisotropy\nconstant, and by changing the SL and/or interlayer\nthickness18–20,26–28whichoptimizesthehard-to-softlayer\ncoupling strength, thus improving the writability of the\nmedia while keeping high thermal stability.\nHowever, all these experimental work were done with\npositive or negligible SL MA, Ks, in accordance with\nthe theoretical work of Suess et al.11. From the work\nof Suess et al., optimal reduction of the switching field\ncan be realized with small but positive SL MA and the\nswitching field will increase by increasing the anisotropy\nconstant or by decreasing it towards the negative direc-\ntion (see Fig. 2 in Ref.11). On the other hand, using\nthe model proposed by Victora et al.29, Wang et al.12\nhave shown that negative SL anisotropy actually has a\nbeneficial contribution in decreasing the switching field.\nTo resolve the different theoretical predictions between\nSuess et al. and Victora et al. and to further optimizethe magnetic properties of the ECC media, it is very in-\nteresting to experimentally investigatehowdoes negative\nSL MA affect the coercivity of the ECC system.\nTherefore, we report detailed studies on the reduction\nof its coercivity of the Co 72Pt28/Co100−xIrxECC sys-\ntem. By changing the Ir content of the Co 100−xIrxSL,\nits MA can be tuned30,31. Here, positive MA means that\nthe easy axis of the magnetic material is along its crys-\ntallographic c-axis and negative MA, on the other hand,\nmeans that the easy axis is within its ab-plane. Thus, by\noriented-growth of the hcp-Co 81Ir19(negative MA with\nx= 19) layer with its crystallographic c-axis parallel to\nthe normal of the film surface, its magnetic moments can\nbe restricted strictly in-plane by its negative MA, thus\nprovidingaverystrongnegativetotalin-planeanisotropy\nfor our ECC media32,33. A series of samples with differ-\nent SL thickness and composition were investigated and\nwe will show that negative SL MA is more efficient in\nreducing the coercivity of the ECC media, thus our re-\nsults agree with the theoretical prediction of Victora et\nal.12,29and open a new playground for further improve-\nments of the magnetic properties of the ECC media for\nhigh density magnetic recording.\nII. EXPERIMENTS\nCo72Pt28/Co81Ir19ECC media system ( Ks=−6×\n105J/m330) were prepared by DC perpendicular mag-\nnetronsputteringmethodwithalayeredstructureofsub-\nstrate/Ta(5nm)/Pt(10nm)/Ru(15nm)/Co 72Pt28(20nm)\n/Co81Ir19(tsnm) with ts=0∼15nm (see inset of Fig. 1).\nFor simplicity, our sample will be denoted as HL/SL19\nwhere HL/SL stands for hard-layer/soft-layer and 19\nrepresents the Ir content. Si(100)-orientation wafer\nwith surface oxidation were used as substrate and\nthe Ta(5nm)/Pt(10nm) layer was grown to promote\nRu(002) texture which induces the columnar growth of\nthe Co 72Pt28layer with well isolated grains34,35. For2\nthe growth of the HL and SL, Ar pressures of 4Pa and\n0.3Pa were used, respectively. The base pressure before\ndeposition was lower than 2 ×10−5Pa. For comparison\nreasons, ECC media systems with different SL composi-\ntions were also prepared with a similar fashion. To be\nexact, SL=Fe, Co and Co 100−xIrxwith x=23, 29 and 33\nsystems were prepared and these samples will be denoted\nas HL/(Fe,Co) and HL/SL(23,29,33), respectively. Note,\nthat all Co 100−xIrxwith x=19, 23, 29 and 33 samples\nhave the same crystal structure30,31.\nChemicalcomposition ofourthin films havebeen char-\nacterized by energy dispersive spectrometer. Grain mor-\nphology of the sample was measured using a transmis-\nsion electron microscope(TEM). The crystalstructure of\nour sample was characterized by x-ray diffraction (XRD)\nwith Cu Kα1radiation. Characterization of the mag-\nnetic properties were done with a vibrating sample mag-\nnetometer (VSM). Dynamic magnetic properties were\nmeasured with an electron spin resonance spectrome-\nter (ESR) to determine the intrinsic magnetocrystalline\nanisotropy constant and the detailed procedure can be\nfound inourpreviouslypublished works32,33. Thesemea-\nsurements were all done at room temperature and our\nsamples were stored under vacuum when the measure-\nments are done.\nIII. RESULTS AND DISCUSSION\nIn Fig.1we present the typical TEM image of the cross\nsection of sample HL/SL19 with ts=5nm. Columnar\ngrowth of the Co 72Pt28layer with well isolated grains\ncan be seen from the image and the well isolated na-\nture of these grains are also reflected by the tilted mag-\nnetic hysteresis loop of the HL sample as shown later\nin Fig.3(a)35. The XRD pattern of our sample are\nshown in Fig. 2. All samples exhibit three main peaks\nat 39.6o, 42.1o, and 43.1owhich correspond to Pt (111),\nRu (002), and Co 72Pt28/Co81Ir19(002) peaks30,34, re-\nspectively. The minor peak at 47.7ocomes from the Si\nsubstrate. A closer check on the third peak at 43.1o,\none can find that the peak is actually composed of two\npeaks corresponding to Co 72Pt28and Co 81Ir19layers as\ndenoted by the vertical green dash-doted lines in Fig. 2.\nHowever, trying to fit this peak with two sub-peaks was\nnot successful due to their closeness and thus we fitted\nthis peak with only one peak. The determined peak full\nwidth at half maximum (FWHM) was plotted as a func-\ntion of the SL thickness, as shown in the inset of Fig. 2.\nOne can see that FWHM increases with the SL thickness\nbeing consistent with the gradual increase in intensity\nof the SL sub-peak, inline with the gradual increase of\nthe volume fraction of the SL. Importantly, the only ob-\nservable (002) peak for the Co 72Pt28and Co 81Ir19layers\nindicate the oriented-growth of the these layers. Since\nCo72Pt28(Co81Ir19)layerexhibitpositive(negative)MA,\nthe easy magnetic direction of the Co 72Pt28(Co81Ir19)\nlayer is parallel (perpendicular) to the normal of the film\nFIG. 1: (color online) TEM image of the cross section of the\nHL/SL19 sample with the SL thickness of 5nm. Inset shows\nthe schematic layer structure.\n/s51/s56 /s52/s48 /s52/s50 /s52/s52 /s52/s54 /s52/s56 /s53/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48\n/s48 /s51 /s54 /s57 /s49/s50 /s49/s53/s48/s46/s55/s48/s48/s46/s55/s53/s48/s46/s56/s48/s48/s46/s56/s53\n/s72/s76/s47/s83/s76/s49/s57/s32/s49/s53/s32/s110/s109\n/s72/s76/s47/s83/s76/s49/s57/s32/s49/s48/s32/s110/s109/s67/s111\n/s56/s49/s73/s114\n/s49/s57/s40/s48/s48/s50/s41/s82/s117/s40/s48/s48/s50/s41\n/s67/s111\n/s55/s50/s80/s116\n/s50/s56/s40/s48/s48/s50/s41\n/s72/s76/s47/s83/s76/s49/s57/s32/s52/s32/s110/s109\n/s72/s76/s47/s83/s76/s49/s57/s32/s50/s32/s110/s109\n/s72/s76/s47/s83/s76/s49/s57/s32/s48/s32/s110/s109/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s50 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s72/s76/s47/s83/s76/s49/s57/s32/s54/s32/s110/s109/s80/s116/s40/s49/s49/s49/s41\n/s70/s87/s72/s77/s32/s40/s100/s101/s103/s114/s101/s101/s41\n/s83/s76/s49/s57/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\nFIG. 2: (color online) XRDpatterns oftheHL/SL19samples\nwith indicated soft layer thicknesses. The solid lines repr esent\nprofile fitting as described in the text. Inset shows the soft\nlayer thickness dependence of the HL/SL19 (002) peak width.\nsurface30,32,34.\nNormalized magnetic hysteresis loops of our sample\nwere shown in Fig. 3(a)-(d). In Fig. 3(a) we present the\nmeasurements for Co 72Pt28and Co 81Ir19separate lay-\ners grown on the same seed-layer. The measurements\nwere done with the applied magnetic field along the\neasy magnetic direction of the HL (perpendicular to the\nfilm plane) or SL (parallel to the film plane) sample.\nFor the HL sample, the easy-axis coercivity and satu-\nration magnetization were determined to be 0.51T and\n1.1×106A/m, respectively. The MA of the HL was cal-\nculated to be 1.25 ×106J/m3by the area difference be-\ntweenthehard-axisandeasy-axishysteresisloops. These\nmagnetic parameters are close to reported values of sim-\nilar composition34,36. The tilt of the hysteresis loop sug-\ngests a weak inter-granular coupling of the HL, which is\nfavorable for the application in perpendicular magnetic3\n/s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s40/s100/s41 /s40/s99/s41/s40/s98/s41\n/s32/s72/s76/s32/s111/s117/s116/s45/s112/s108/s97/s110/s101/s77/s47/s77\n/s83\n/s48/s72/s32/s40/s84/s41/s40/s97/s41\n/s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s48/s32/s110/s109\n/s32/s50/s32/s110/s109\n/s32/s52/s32/s110/s109\n/s32/s54/s32/s110/s109\n/s32/s49/s48/s32/s110/s109\n/s32/s49/s53/s32/s110/s109/s77/s47/s77\n/s83\n/s48/s72/s32/s40/s84/s41/s83/s97/s109/s112/s108/s101/s32/s72/s76/s47/s83/s76/s49/s57\n/s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s72/s76/s32/s50/s48/s32/s110/s109\n/s32/s72/s76/s47/s67/s111/s40/s53/s32/s110/s109/s41\n/s32/s72/s76/s47/s70/s101/s40/s53/s32/s110/s109/s41\n/s32/s72/s76/s47/s83/s76/s49/s57/s40/s53/s32/s110/s109/s41/s77/s47/s77\n/s83\n/s48/s72/s32/s40/s84/s41/s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s83/s97/s109/s112/s108/s101/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115\n/s116/s32/s61/s32/s53/s32/s110/s109\n/s32/s72/s76/s47/s83/s76/s49/s57\n/s32/s72/s76/s47/s83/s76/s50/s51\n/s32/s72/s76/s47/s83/s76/s50/s57\n/s32/s72/s76/s47/s83/s76/s51/s51/s77/s47/s77\n/s83\n/s48/s72/s32/s40/s84/s41/s45/s48/s46/s48/s53 /s48/s46/s48/s48 /s48/s46/s48/s53\n/s32/s83/s76/s49/s57/s32/s105/s110/s45/s112/s108/s97/s110/s101/s48/s72/s32/s40/s84/s41\nFIG. 3: (color online) (a) Magnetic hysteresis loops of the\nseparate HL and SL samples grown on the same seed-layer\nwith applied magnetic field along the easy magnetic directio n\nof the sample indicating good hard/soft magnetic propertie s.\nNote the very different µ0Hscale. (b) Magnetic hysteresis\nloops of sample HL/SL19 with different SL thicknesses. (c)\nComparison of the magnetic hysteresis loops between sample s\nwithnoSLandCo(5nm), Fe(5nm)andSL19(5nm)astheSL.\n(d) Magnetic hysteresis loops of samples HL/SL(19-33) with\nvariousSLanisotropyconstant. Notethatthehysteresislo ops\nin (c)-(d) have been measured with the external field applied\nperpendicular to the film plane.\nrecording35. For the SL sample, an easy-axis coercivity\nof 0.004T and saturation magnetization of 1.2 ×106A/m\nwere obtained which are close to the values of our previ-\nous report30. These results prove that, under the current\nfilm growth conditions, HL and SL films with good struc-\ntural and magnetic properties can be obtained.\nIn Fig.3(b), we present the measurements for our\nECC sample HL/SL19 with different SL thicknesses,\nts=0∼15nm. Clearly, the coercivity of the ECC system\nweregreatlyreduced. Noobviousstepwereobserveddur-\ning the magnetization reversal process, indicating strong\ncoupling between the hard and soft layers being con-\nsistent with the absence of any interlayers between the\ntwo layers, which shows the uniform switching of the SL\nwith respect to the HL. However, as the SL thickness ts\nincreases, the perpendicular magnetic anisotropy of the\nECC media is significantly degraded as the rectangular\nloops tend to tilt more and more with increasing ts. This\ncorresponds to a reduction of the squareness ratio of the\nhysteresis loops, which indicates the domination of the\nnegative anisotropy of the SL, resulting in a reduction of\nthe remanence of hysteresis loops25.\nFor comparison, we also prepared HL/Co(5nm) and\nHL/Fe(5nm) ECC samples with small but positive SL\nMA and their magnetic hysteresis loops are shown to-\ngether with that of HL/SL19(5nm) in Fig. 3(c). Obvi-\nously, the SL19 layer with negative MA is more efficient\nin reducing the coercivity of the ECC system. Addition-\nally, to study how the reduction ofthe coercivitydependson the strength of the anisotropy constant of the SL for\nthe ECC media, we prepared different samples with dif-\nferent Ir content in the SL, thus different SL MA30,31, as\nshown in Fig. 3(d). One can see that the strength of the\nanisotropy constant does not have significant impacts on\nthe media coercivity as the SL thickness does.\nTo see the coercivity reduction more clearly, we sum-\nmarized our results in Table Iand Fig. 4together with\npublished data on the CoPt system with different SLs.\nObviously, as shown in Fig. 4(a), Co 81Ir19SL with neg-\native MA is more effective in reducing the ECC media\ncoercivity for any defined SL thickness. In Fig. 4(b), we\npresent the reduction of the media coercivity as a func-\ntion of the SL MA constant. The data have been nor-\nmalized to the value of the most right side data point\natKs=-3.5×105J/m3for better comparison with calcu-\nlations. The enhanced reduction with decreasing Ksis\ninconsistent with the predictions by Suess et al.11; how-\never being in agreement with the results of Victora et\nal.12,29.\nTABLE I: Published data on the coercivity µ0HCreduction\nfor the Co 100−xPtxsystem using different soft layers. t sde-\nnotes the hard/soft-layer thicknessand *indicates the cur rent\nwork.\nHard/soft-layer t s(nm)µ0HC(T) ref.\nCoPt-SiO 2/Co-SiO 220/0-20 0.7-0.3526\nCoCrPt/CoTb 20/0-11 0.4-0.337\nCo71Pt29-TiO2/Co-TiO 215/0-15 0.47-0.227\nCo71Pt29-TiO2/Co93Pt7-TiO215/0-15 0.47-0.327\nCo71Pt29-TiO2/Co83Pt17-TiO215/0-15 0.47-0.4227\nCo74Pt22Ni4/Ni73O2712/0-20 0.75-0.328\nCo72Pt28/Co81Ir19 20/0-15 0.52-0.052 *\nTo understand this results more clearly, we would like\nto revisit the theoretical model used by Suess et al.\nand Wang et al.. In the work of Suess et al.11, they\ncalculated the switching field with a simple expression\nHs=max(Hp,Hn) with\nHn=2Ks\nµ0M1+2Aπ2\n4t2sµ0M1\nHp=1\n4×2(Kh−Ks)\nµ0M2, (1)\nwhereµ0Hnandµ0Hprepresent the nucleation field in\nthe soft layer and pinning field at the interface, respec-\ntively.KsandKhdenote the anisotropy constants of\nthe soft and hard layers. Ais the intragranular exchange\nconstant and M1/M2is the saturation magnetization of\nthe soft/hard layer. tsis the soft layer thickness. On\nthe other hand, Victora et al.12,29used a more complete\nmodel by minimizing the total energy of the ECC sys-\ntem. In this model, the total magnetic energy of the4\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s45/s54/s46/s48 /s45/s53/s46/s53 /s45/s53/s46/s48 /s45/s52/s46/s53 /s45/s52/s46/s48 /s45/s51/s46/s53/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52\n/s40/s98/s41/s72\n/s99/s40/s116/s41/s47/s72\n/s99/s40/s116/s61/s48/s32/s110/s109/s41\n/s116\n/s115/s32/s40/s110/s109/s41/s40/s97/s41/s125/s67/s117/s114/s114/s101/s110/s116/s32/s119/s111/s114/s107/s67/s111\n/s55/s49/s80/s116\n/s50/s57/s45/s84/s105 /s79\n/s50/s47/s67/s111\n/s56/s51/s80/s116\n/s49/s55/s45/s84/s105 /s79\n/s50\n/s67/s111/s67/s114/s80/s116/s47/s67/s111/s84/s98\n/s67/s111\n/s55/s49/s80/s116\n/s50/s57/s45/s84/s105 /s79\n/s50/s47/s67/s111\n/s57/s51/s80/s116\n/s55/s45/s84/s105 /s79\n/s50\n/s67/s111/s80/s116/s45/s83/s105 /s79\n/s50/s47/s67/s111/s45/s83/s105 /s79\n/s50\n/s67/s111\n/s55/s52/s80/s116\n/s50/s50/s78/s105 \n/s52/s47/s78/s105 \n/s55/s51/s79\n/s50/s55\n/s32/s67/s111\n/s55/s49/s80/s116\n/s50/s57/s45/s84/s105 /s79\n/s50/s47/s67/s111/s45/s84/s105 /s79\n/s50\n/s32/s67/s111\n/s55/s50/s80/s116\n/s50/s56/s47/s67/s111\n/s56/s49/s73 /s114\n/s49/s57/s32/s67/s111\n/s55/s50/s80/s116\n/s50/s56/s47/s70/s101/s32/s67/s111\n/s55/s50/s80\n/s116/s50/s56/s47/s67/s111/s72\n/s99/s40/s75\n/s115/s41/s47/s72\n/s99/s40/s45/s51/s46/s53/s120/s49/s48/s53\n/s32/s74/s47/s109/s51\n/s41\n/s75\n/s115/s32/s40/s120/s49/s48/s53\n/s32/s74/s47/s109/s51\n/s41/s32/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116/s115\n/s32/s83/s117/s101/s115/s115/s39/s115/s32/s116/s104/s101/s111/s114/s121\n/s32/s86/s105/s99/s116/s111/s114/s97/s39/s115/s32/s116/s104/s101/s111/s114/s121\nFIG. 4: (color online) (a) Reduction of the coercivity of the\nECC media for various HL/SL systems as a function of the\nSL thickness. For comparison reasons, the data were normal-\nized to the initial value with zero SL thickness. Data with HL\ncomposition of Co 71Pt29-TiO2was taken from Ref.27, CoPt-\nSiO2was taken from Ref.26, CoCrPt was taken from Ref.37,\nand Co 74Pt22Ni4was taken from Ref.28. (b) Influence of the\nstrength of the SL MA on the coercivity of the ECC media\nwith a SL thickness of 5nm. Theoretical calculations were\ndone using equation 1 (Suess’s theory) and equation 2 (Vic-\ntora’s theory) for comparison as described in the main text.\nECC system can be expressed by\nE=HM1cosθ1V1+HM2cosθ2V2+K1sin2θ1V1\n+K2sin2θ2V2−Jecos(θ1−θ2),(2)\nwhereµ0H,Mi,Ki, andVidenote the external field, sat-\nuration magnetization, anisotropy constant and volume\nof the particle, respectively. Sub-index 1, 2 indicate the\nsoft, hard region, respectively. Jeis the exchange con-\nstant and θiindicates the angle between the magnetic\nmoments and the external filed. In addition, demagne-\ntizing energy was included properly in this model which\nwas omitted by Suess et al. in their calculation11.\nIn the calculation work of Wang et al., Victora’s model\nwas used and the negative SL anisotropy was assumed\nto come from the demagnetizing field which is limited\nby the largest demagnetizing factor ( N= 1) by set-\nting the MA to zero12. Here, we further push the total/s48 /s49 /s50 /s51 /s52/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48\n/s40/s98/s41/s72\n/s115/s119/s32/s40/s84/s41\n/s72\n/s101/s32/s40/s84/s41\n/s32 /s72\n/s107/s49/s32/s61/s32/s48/s46/s48/s49/s32/s84\n/s32 /s72\n/s107/s49/s32/s61/s32/s45/s48/s46/s57/s52/s32/s84\n/s32 /s72\n/s107/s49/s32/s61/s32/s45/s50/s46/s52/s48/s56/s32/s84/s40/s97/s41\n/s48 /s49 /s50 /s51 /s52/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48/s71/s97/s105/s110/s32\n/s72\n/s101/s32/s40/s84/s41\nFIG. 5: (color online) (a) Switching field H swas a func-\ntion of the interlayer exchange field H ewith various soft layer\nanisotropy H k1. (b) Dependence of the gain factor ξon He\nwith various H k1.\nSL anisotropy towards the negative direction by using\nCo1−xIrxwith negative MA (total in-plane anisotropy\nfield can be expressed as Hk1=−Ms+ 2Ks/µ0Ms).\nAs shown in Fig. 5, we recalculated the switching field\nµ0Hswand Gain factor ξas a function of the interlayer\nexchange field µ0Heusing the same procedure as done\nby Wang et al12with the total in-plane anisotropy field\nµ0Hk1up to a negative value of −2.408T. The Gain fac-\ntor, which is a reflection of the thermal stability, is de-\nfined as ξ= 2∆E/(HswMaV), where ∆ E,MaandV\nare the total anisotropy energy, average saturation mag-\nnetization, and total volume, respectively. From Fig. 5\n(a)-(b), one can see that in the strong coupling regime\n(biggerµ0He), the more negative of the SL MA the more\nefficient in reducing the switching field and the bigger of\nthe Gain factor can be obtained. According to these re-\nsults, the sameconclusioncan be drawnasin reference12,\nthat is by using a SL with negative Kscan further reduce\nthe switching field of the ECC system and can further in-\ncrease the Gain factor ξeven up to very large negative\nvalues of the MA ( Ks=-6×105J/m3in our case).\nIn order to reproduce the same trend of our experi-5\n/s45/s50 /s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s116\n/s115/s32/s40/s110/s109/s41/s72\n/s99/s40/s116/s41/s47/s72\n/s99/s40/s116/s61/s48/s32/s110/s109/s41\n/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s49/s48/s48/s48/s84/s104/s101/s114/s109/s97/s108/s32/s115/s116/s97/s98/s105/s108/s105/s116/s121/s32/s102/s97/s99/s116/s111/s114/s32/s40/s75\n/s117/s86/s47/s107\n/s66/s84/s41\nFIG. 6: (color online) Thickness dependence of the normal-\nized coercivity (left axis) and thermal stability factor (r ight\naxis) of our sample HL/SL19 .\nmental results, we calculated the SL MA dependence of\nthe coercivity reduction using the above two theories as\nshown in Fig. 4(b). Clearly, in opposite to the results of\nSuessetal., Victora’stheorypredictsthesametrendwith\nour experiments. For the calculation, A= 10−11J/m3\nand exchange field of µ0He= 4T were used. The SL\nthickness was set to be 5nm which is the same as our\nexperiments, the anisotropy constant and the saturation\nmagnetization of the HL and SL were taken from the\nmeasured values in this work. First, calculation of the\nswitching field were done separately using equation ( 1)11\nand equation ( 2)12. Then, they were normalizedtogether\nwith experimental data at the most right point ( Ks=-\n3.5×105J/m3) for comparison of the trends with chang-\ningMA.Asshownbyequation( 2), Victora’smodelstarts\nfrom a more general way and the switching field can be\ncalculated by minimizing the total energy against θ1and\nθ212,29. Themagnetostaticfieldcomingfromothergrains\nof the media is considered as external field. Most impor-\ntantly, it points out that the demagnetizing energy is\nnot a negligible factor for the ECC media since it has\nan obvious contribution to the anisotropy of the SL. We\nfurther notice that Suess’s model predicts an decrease of\nthethermalstabilityoftherecordingmediawithnegative\nmagnetocrystalline anisotropy of the SL11. On the other\nhand, in our calculation using Victora’s model, we can\nget enhanced thermal stability for stronger inter-layer\ncoupling regimes (larger Gain factor in our calculation\nas discussed above.).\nTo check the thermal stability of our ECC media, we\nmeasured the remnant coercivity µ0Hras a function of\nwaiting time to study the magnetic decay of our sample.\nFirst, a negative saturation field of 2T was applied, then\nthe field was increased to positive fields around the coer-\ncive field for some waiting time t, after that the field was\nset to zeroandmeasurethe magnetic moment ofthe film.\nThe remnant coercivity µ0Hr(t) can be obtained by lin-\near fitting of the measured M(H) curve for each waitingtimet. The thermal stability factor KuV/kBTcan then\nbe evaluated by analyzing the time dependent µ0Hr(t)\nusing Sharrock’s equation38,\nHr(t) =Hr(0){1−[kBT\nKuVln(f0t\nln2)]1/n},(3)\nwheref0is called the attempt frequency ( ∼1010Hz for\nmagnetic systems38), n=1.5 was fixed39.KuVis the en-\nergy barrier at zero applied field, and kBTis the thermal\nenergy required to flip the magnetization. The obtained\nKuV/kBTare shown in Fig. 6together with the coerciv-\nityreductionofourHL/SL19sample. Ascanbeseen,dif-\nferent with the monotonic decrease behavior of the coer-\ncivity, the thermal stability factor first increase with the\nSL thickness up to 6nm and then decrease with further\nincreaseoftheSLthickness. SimilartotheFePt-C/FePt4\nsystem, the initial increase can be attributed to the in-\ncreaseoftheswitchingvolumeduetothestrongexchange\ncoupling between the HL and SL, and the decrease above\n6nm is mainly due to the demagnetizing effect. Typical\nKuV/kBTfor the current perpendicular recording me-\ndia using CoCrPt system is around 8040, which is much\nsmallerthanourvalueof226atzeroSLthickness. There-\nfore, our ECC media show very good thermal stability\nand writability than the current CoCrPt perpendicular\nmedia. Finally, we would like to emphasize the advan-\ntage of Victora’s model in the study of ECC media sys-\ntems and the rather regretful fact that no experimental\nwork using SLs with negative MA have been employed\nto reduce the coercivity of the ECC media. We hope\nthat our work will promote further experimental work in\nthis direction and promote further improvements of the\nwritability for high density magnetic recording.\nIV. SUMMARY\nIn conclusion, we have prepared Co 72Pt28/Co81Ir19\nECC media with negative SL MA. TEM image shows\nthe columnar growth of the Co 72Pt28HL and XRD mea-\nsurements shows the oriented-growth of the SL on top\nof the HL with hcp-structure. Good hard/soft magnetic\nproperties of the hard/soft layers have been proved by\nour VSM measurements. We observe a single step mag-\nnetization reversal of the ECC media, suggesting strong\ncoupling of the hard and soft magnetic layers. More in-\nterestingly,weobserveanenhancedreductionofthecoer-\ncivity of the ECC media by using SLs with negative MA\nwhen compared to the use of SLs with small but positive\nMA. These results have been qualitatively reproduced by\ntheoretical calculations. Therefore, our results provide a\nnew direction for us to further improve the writability of\nthe ECC media for future high density magnetic record-\ning.6\nV. ACKNOWLEDGEMENT\nThis work was supported by the National Natural\nScience Foundations of China (Nos. 11574122 and\n11204115) and the Fundamental Research Funds for the\nCentralUniversities(Nos. lzujbky-2017-k20andlzujbky-2017-31).\nReferences\n∗Electronic address: zweili@lzu.edu.cn\n†Electronic address: wtao@lzu.edu.cn\n1D. Suess, T. 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Their large saturation magnetization, strong magne-\ntocrystalline anisotropy, and high Curie temperature originate from the interaction between the T-3d electrons and R-4f\nelectrons. This article discusses the magnetism of rare-earth magnet compounds. The basic theory and first-principles\ncalculation approaches for quantitative description of the magnetic properties are presented, together with applications\nto typical compounds such as Nd 2Fe14B, Sm 2Fe17N3, and the recently synthesized NdFe 12N.\n1. Introduction\nModern permanent magnets are the consequences of the\nfine combination of various magnetic and nonmagnetic ma-\nterials, as well as micro-, macro-, and metallographic struc-\ntures.1)Thus, quantum theory tells only part of the story of\nrare-earth magnets. Nevertheless, since magnetism is one of\nthe most prominent manifestations of the quantal nature of\nelectrons,2)quantum theory must be a key player in studying\npermanent magnet materials. In this review, we will concen-\ntrate mostly on the electronic and magnetic properties of sin-\ngle crystals of rare-earth magnet materials, discussing some\nselected topics that may be essential in terms of developing\npermanent magnets.\nRare earths have particular importance in modern perma-\nnent magnets. The reason why rare-earth elements are so im-\nportant in magnets is that one of the necessary conditions for\na ferromagnet to be a permanent magnet material is magnetic\nanisotropy. The magnetic anisotropy originates from either\ncrystalline or shape anisotropy; the latter can never be strong\nenough for modern magnets. The former is the result of spin-\norbit coupling (SOC) which eventually sticks spins to a crys-\ntal structure. The strength of single-electron SOC of 4f elec-\ntrons of rare earths is \u00180:5 eV . For Fe-3d electrons, it is one\norder of magnitude smaller. Although rare-earth magnets con-\ntain only a small amount of rare earths, e.g., less than 1 /7 of\nthe whole in the case of Nd 2Fe14B magnets, adding them en-\nhances the magnetic anisotropy at the working temperature by\n\u001850% (naturally much more at low temperatures), which is\nalready a huge increase from the technological point of view.\nUnfortunately, as is widely recognized, there is no estab-\nlished way to treat the electronic and magnetic properties of\n4f electron systems from first principles. This makes the theo-\nretical treatment of rare-earth magnets rather di \u000ecult. At best,\nwhat we can do now is to compromise and to add some kinds\nof ad hoc treatments, under several assumptions, each of them\nbeing not based on an approximation of the same level, on\ntop of the standard first-principles theory. In the subsequent\nsections, we will review the recent development of quantum\nmechanical approaches to the problem of rare-earth magnets,\n\u0003t-miyake@aist.go.jpwhich mostly follow such types of incomplete approaches.\nIn Sect. 2, we briefly review typical rare-earth magnets\nand the basic idea of their magnetism. The framework of\nfirst-principles approaches that are used to describe rare-earth\nmagnets is explained in Sect. 3. Some examples of the ap-\nproaches are also given. In particular, the magnetic anisotropy\nof Sm 2Fe17Nxis discussed in detail. The finite-temperature\nproperties of rare-earth magnets are discussed in Sect. 4 to-\ngether with some methodological aspects. Several di \u000berent\nbut complementary methods are explained with some recent\nresults. Section 5 deals with NdFe 12N and related compounds.\nNdFe 12N has been synthesized recently, and turned out to\nhave excellent intrinsic magnetic properties surpassing those\nof Nd 2Fe14B. Section 6 summarizes the review.\n2. Rare-Earth Magnet Compounds\nRare-earth magnet compounds3, 4)are mainly composed of\ntransition-metal ( T) and rare-earth ( R) elements. The major-\nity component is 3d transition metals, which are essential\nfor a large saturation magnetization and high Curie temper-\nature, while Relements are responsible for strong magne-\ntocrystalline anisotropy. Figure 1 shows the electronic states\nschematically. As the electron configuration of R-4f electrons\nfollows Hund’s rule, the orbital magnetic moment appears\nin the presence of SOC. The resultant electron distribution\nslightly deviates from a spherical shape. The nonspherical\ncomponent is subjected to a crystal electric field produced by\nother electrons and ions, which determines the direction of the\n4f orbital moment. Once the direction is fixed, the direction of\nthe spin moment is also fixed by the LS coupling. The 4f elec-\ntrons are coupled to 5d electrons by the intraatomic exchange\ninteraction, consequently, their spin moments are aligned par-\nallel to each other. Since the 5d orbitals are spatially extended,\nthey hybridize with the T-3d orbitals antiferromagnetically.\nTherefore, the T-3d spin is antiparallel to the R-4f spin.\nThe above consideration indicates that the strength of the\ncrystal field at the rare-earth site is a good measure of the mag-\nnetocrystalline anisotropy of a rare-earth magnet compound.\nIn crystal-field theory,5, 6)the magnetic anisotropy constant\nK1, defined by\nE(\u0012)'K1sin2\u0012; (1)\n1arXiv:1801.03455v1  [cond-mat.mtrl-sci]  10 Jan 2018J. Phys. Soc. Jpn.\n4f(S)3d(S)5d(S)\n4f(L)hybridizationexchangeLSlight RETM\n4f(S)3d(S)5d(S)\n4f(L)hybridizationexchangeLSheavy RETM\nFig. 1. (Color online) Interaction between 3d electrons of transition metals\nand rare-earth 4f electrons in rare-earth magnet compounds.\ncan be expressed as\nK1=\u00003J(J\u00001)\u000bJhr2iA0\n2nR; (2)\nwhere\u0012is the angle between the easy axis and the magnetiza-\ntion, Jis the total angular momentum, \u000bJis the first Stevens\nfactor,hr2iis the spatial extent of the 4f orbital, A0\n2is the\nsecond-order crystal-field parameter, and nRis the rare-earth\nconcentration. Here, Jand\u000bJare constants depending on the\nRion, while A0\n2reflects the electronic states of the compound.\nTable I shows the ground-state properties of trivalent rare-\nearth ions. The orbital angular momentum ( L) is parallel (an-\ntiparallel) to the spin angular momentum ( S) when the 4f shell\nis more (less) than half-filled as shown in Fig. 1. Hence, the\ntotal angular momentum J=L+Shas a larger magnitude\nin heavy rare-earth elements than in light rare-earth elements.\nThis is the reason why heavy rare-earth elements, e.g., Dy, en-\nhance the magnetocrystalline anisotropy of a rare-earth mag-\nnet. Meanwhile, the magnetic moment is suppressed as the\nheavy rare-earth concentration increases since the 4f magnetic\nmoment partially cancels the magnetic moment of the T-3d\nelectrons. Therefore, heavy rare-earth elements generally sup-\npress the performance of a permanent magnet, although they\nimprove coercivity through enhancement of the magnetocrys-\ntalline anisotropy. The first Stevens factor \u000bJis also shown in\nTable I. A positive (negative) \u000bJmeans that the electron dis-\ntribution of the 4f electrons is elongated (compressed) in the\ndirection of the orbital moment, and the magnitude of \u000bJgives\nthe degree of asphericity. We see that the sign of \u000bJof Nd3+\nis opposite to that of Sm3+. Assuming that the crystal-field\nparameter A0\n2is insensitive to the Rion, as a rule of thumb, a\nNd-based compound shows uniaxial (basal-plane) anisotropy\nwhen the corresponding Sm-based compound has basal-plane\n(uniaxial) anisotropy.\nThe situation is more complicated in some cases. For exam-\nple, cerium has a mixed-valence state. There are no f electrons\nin most Ce-based magnet compounds, hence, a simple argu-\nment based on Eq. (2) does not hold. A Ce compound will\npossess strong magnetocrystalline anisotropy if we can make\nCe trivalent because the magnitude of the Stevens factor is\nlarge. Samarium is also a di \u000ecult element to treat theoreti-\ncally. Because the energy splitting between the Jmultiplets\nis small, excited Jstates would a \u000bect the finite-temperatureRion fnL S J g J \u000bJ\nCe3+f13 1 /2 5 /2 6 /7\u00002=(5\u00017)\nPr3+f25 1 4 4 /5\u000022\u000113=(32\u000152\u000111)\nNd3+f36 3 /2 9 /2 8 /11\u00007=(32\u0001112)\nPm3+f46 2 4 3 /5 2\u00017=(3\u00015\u0001112)\nSm3+f55 5 /2 5 /2 2 /7 13 =(32\u00015\u00017)\nEu3+f63 3 0 0 0\nGd3+f70 7 /2 7 /2 2 0\nTb3+f83 3 6 3 /2\u00001=(32\u000111)\nDy3+f95 5 /2 15 /2 4 /3\u00002=(32\u00015\u00017)\nHo3+f106 2 8 5 /4\u00001=(2\u000132\u000152)\nEr3+f116 3 /2 15 /2 6 /5 22=(32\u000152\u00017)\nTm3+f125 1 6 7 /6 1 =(32\u000111)\nYb3+f133 1 /2 7 /2 8 /7 2 =(32\u00017)\nTable I. Number of f electrons, orbital momentum L, spin momentum S,\ntotal angular momentum J, the Land ´e g factor and the first Stevens factor \u000bJ\nof trivalent rare-earth ions.\nmagnetism of Sm systems. The hybridization e \u000bect between\n4f and other orbitals is also to be considered, which will\nbe discussed in detail in the following section by taking\nSm 2Fe17N3as an example. Contributions from transition-\nmetal sublattices to magnetocrystalline anisotropy are another\nfactor, which will be discussed in Sect. 4.\nThe rare-earth magnet compounds are classified into sev-\neral families depending on their chemical composition. The\nsimplest one is RT5having the CaCu 5structure [Fig. 2(a)]. It\nhas a hexagonal unit cell containing one formula unit. There\nare two Tsites, 2 cand 3 g. The 2 csites form a honeycomb lat-\ntice, and Ris located at the center of a hexagon. The 3 gsites\nform a kagom ´e lattice. Because of this characteristic crystal\nstructure, flat bands exist in the electronic band dispersion.7)\nThe first-generation rare-earth magnets YCo 58)and SmCo 59)\nbelong to this family.\nBy replacing nout of mrare-earth sites with a pair of\ntransition-metal sites (“dumbbell”), Rm\u0000nT5m+2nis obtained.\nThere are two structures for ( m;n)=(3,1). One is the rhombo-\nhedral Th 2Zn17structure shown in Fig. 2(b). The other is the\nhexagonal Th 2Ni17structure. In the former case, the dumb-\nbellTsites are arranged in a sequence of ABCABC along the\ncdirection, while the stacking sequence is ABABAB in the lat-\nter structure. Sm 2Co17with the Th 2Zn17structure belongs to\nthisR2T17family. Since it contains a higher Co content than\nSmCo 5, the saturation magnetization is larger. So is ( BH)max,\nalthough its magnetocrystalline anisotropy is weaker. While\nSm 2Co17is a strong magnet compound, Sm 2Fe17shows\nbasal-plane anisotropy. However, the anisotropy is changed\nby adding nitrogen. Interstitial nitrogenation induces strong\nuniaxial magnetic anisotropy as well as an increase in the\nmagnetization and Curie temperature.10, 11)\nIf half of the Rsites in RT5are substituted with dumbbell\nTpairs, RT12is obtained [( m;n)=(2,1)]. The crystal structure\nis ThMn 12-type with the body-centered tetragonal structure\n[Fig. 2(c)]. This family has been studied intensively in recent\nyears. Notably, it was reported that NdFe 12N has a larger sat-\nuration magnetization and anisotropy field than Nd 2Fe14B.12)\nWe will discuss this family in Sect. 5.\nBoth Sm 2Fe17N310, 11)and NdFe 12N contain nitrogen. In\nfact, light elements vary the magnetic properties of rare-\nearth transition-metal intermetallics. Historically, boron pro-\nvided a breakthrough before the development of nitrogenated\n2J. Phys. Soc. Jpn.\nsystems. By adding boron to Nd 2Fe17, Sagawa et al. in-\nvented a sintered neodymium magnet whose main phase was\nNd2Fe14B.13)Croat et al. independently developed a melt-\nspun neodymium magnet.14)The neodymium magnet has\nbeen the strongest magnet in the last three decades. Sagawa’s\nintention was to raise the Curie temperature of Nd 2Fe17by\ninserting boron. He thought that boron would increase the\nFe-Fe distance, which may lead to an increase in the Curie\ntemperature. Indeed, the Curie temperature was raised, but\nthe microscopic mechanism was di \u000berent from what Sagawa\nhad expected. The chemical formula was not Nd 2Fe17Bx\nbut Nd 2Fe14B, and the crystal structure was a complicated\ntetragonal one containing four formula units in a unit cell\n[Fig. 2(d)].\n3. First-Principles Calculation\nFirst-principles electronic structure calculation commonly\nmeans calculation based on the local density approximation\n(LDA) or its slight extension, the generalized gradient approx-\nimation (GGA), within the framework of density functional\ntheory (DFT).15, 16)A standard theoretical investigation of the\nproperties of condensed matter starts from a first-principles\nelectronic structure calculation as a first step. This is particu-\nlarly true for magnetic materials since ground-state magnetic\nproperties are well treated by first-principles calculations in\nmost cases. However, for rare-earth magnets, such an ap-\nproach fails in many cases. This is explained in the following.\nWhen applied to the element magnets Fe, Co, Ni, both the\nLDA and GGA produce mostly reasonable results. An excep-\ntional case is the LDA applied to Fe: the LDA predicts in-\ncorrect crystal and magnetic structures for the ground state of\nFe. The failure of the LDA for Fe, however, simply originates\nfrom the fact that the LDA has a tendency of overbinding the\natoms, predicting too small an equilibrium lattice constant\nof Fe. In this sense, it is not so fatal as might be supposed.\nThe applicability of the LDA /GGA is also similar for vari-\nous magnetic intermetallic compounds and compounds such\nas transition-metal chalcogenides, pnictides, and halides. In\nthe vicinity of the border separating the high-spin state from\nthe low-spin (or nonmagnetic) state, and also near the region\nwhere the metal-insulator transition occurs, these approaches\noften fail to predict the correct ground states.17)However, we\nmay say that the LDA /GGA calculation correctly describes\nthe overall behavior of these materials in general.\nThe above is not true for rare-earth magnets: the f-states of\nrare-earth elements cannot be properly treated in the frame-\nwork of the LDA /GGA. For example, let us consider SmCo 5,\nwhich is a prototype rare-earth permanent magnet compound.\nSm has five f-electrons in its trivalent state as is usually the\ncase in a crystal. The corresponding atomic LS multiplet of\nthe lowest energy is6H5=2. Even in a crystal, such an atomic\nconfiguration of Sm 4f electrons is well preserved since the\nhybridization of the 4f state with the f-symmetry states com-\nposed of s, p, and d states of neighboring sites is fairly small.\nIn this situation, together with the fact that the f-states must\nbe partially occupied, what would be expected is that all the\n4f states are pinned at the Fermi level. This implies that the\nenergy required for valence fluctuation to take place is quite\nsmall and the LS multiplet loses its meaning. In reality, for\nthese narrow states, the e \u000bects of the electron-electron inter-\naction are so strong that the electronic states are not any moreextended. The electronic states split into occupied and un-\noccupied states, the former being pushed down rather deep\ninside the Fermi sea and the latter being pushed up above\nthe Fermi sea. This situation can hardly be reproduced by\nthe LDA /GGA, where all the Kohn–Sham orbitals are gener-\nated with a single common e \u000bective potential. Since there are\nno schemes that improve this situation in a fundamental way,\neven state-of-the-art calculations have to solve the problem in\nan adhoc way.\nThere are several easy fixes. The first one is the “open-core”\napproach, where the f states are dealt with as open-shell core\nstates. In all electron approaches such as the full-potential\nlinearized augmented plane wave (FLAPW) method and the\nKorringa–Kohn–Rostoker (KKR) Green’s function method,\nthese core states with positive energy eigenvalues have to be\ncalculated explicitly with a rather artificial boundary condi-\ntion, e.g., a zero or zero-derivative boundary condition on ra-\ndial wave functions. Suitable care has to be taken so as not\nto include these core states within the valence f-states. In the\nKKR, this can be done by removing the resonances, which\ncorrespond to the virtual-bound f states from the atomic t-\nmatrix. Another possible way to obtain open cores is to simply\nshift the potential for the f-states downward so that the energy\neigenvalue could be negative even under a natural boundary\ncondition. In pseudopotential codes, open-core treatment is\neasily done by including the f-states as open-shell core states\nwhen constructing pseudopotentials.\nThe second approach is to apply self-interaction corrections\n(SIC) to the f-states. This scheme obviously is beyond the\nscope of density functional theory in a strict sense. Never-\ntheless, it can be a reasonable approach if the targeted states\nactually localize. For such localized states, at least the self-\nexchange energy can be calculated exactly and hence may\ngive a better description of the exchange-correlation energy.\nThe SIC causes orbital splitting as naturally expected, and\nremedy the shortcoming of the LDA /GGA.\nThe third one is the so-called LDA +U method,18)which\nis nothing but a local Hartree–Fock approximation. If the f-\nstates are known to be localized, which is the same situation\nas needed for SIC to be applicable, the local Hartree–Fock ap-\nproximation might not be a bad approximation. The method,\nhowever, is not one that takes account of the “strong corre-\nlation”: it merely introduces the e \u000bects of a strong electron\ninteraction by hand.\nAs an example, Fig. 3 shows the calculated density of states\nof Nd 2Fe14B using the GGA, the open-core, and the SIC\nscheme, respectively. All the calculations are performed using\nKKR codes, where the open-core and SIC schemes are imple-\nmented with the GGA (PBE) exchange-correlation energy.19)\nThe spin-orbit coupling is included on top of the scalar rela-\ntivistic approximation.\nThe resulting magnetic properties are compared in Table II.\nNote that the calculated spin and orbital magnetic moments\nare the values projected to ( L;ML;S;MS) states. Therefore,\nthe magnetic moment that can be obtained from the total an-\ngular momentum Jwould be slightly larger in the present\ncase. For example, Msobtained for the open-core calculation\nis 1.90 T instead of 1.87 T if the spin-orbitals be used.\nThe magnetic properties do not depend strongly on the\nschemes except for the orbital magnetic moment Morb. This\nindicates that the f states of Nd do not contribute significantly\n3J. Phys. Soc. Jpn.\nFig. 2. (Color online) Crystal structures of rare-earth magnet compounds. (a) CaCu 5structure, (b) Th 2Zn17structure, (c) ThMn 12structure, and (d)\nNd2Fe14B structure. The red balls represent rare-earth atoms. The green balls in (d) are boron. Other balls represent transition metals.\n4J. Phys. Soc. Jpn.\nFig. 3. (Color online) Density of states of Nd 2Fe14B calculated using (a)\nGGA (PBE), (b) open-core scheme, and (c) SIC. The spin-down core states\nof Nd in (b) appear slightly above the Fermi level. The Nd local f densities\nof states are indicated by blue lines. The states around \u000018 eV are the Nd\nsemicore p-states.\nScheme Mspin(\u0016B)Morb(\u0016B)Ms(T) TC(K) Etotal(Ry)\nGGA 100.9 31.3 1.63 1184 -296731.6327\nopen-core 100.8 50.7 1.87 1157 -296730.3632\nSIC 100.1 49.6 1.85 1160 -296731.2779\nTable II. Spin magnetic moment ( Mspin), orbital magnetic moment ( Morb),\nsaturation magnetization ( Ms), Curie temperature ( TC), and total energy\n(Etotal) of Nd 2Fe14B calculated by three di \u000berent treatments of Nd f-states:\ntreating them as valence states (GGA), as partially occupied core states\n(open-core), and with self-interaction correction (SIC).\nin determining the magnetic properties as a whole. However,\nas is implied by the considerable di \u000berence in Morb, the mag-\nnetic anisotropy, which is not calculated here, could be af-\nfected by the scheme of treatment. It is also pointed out that\nthe equilibrium lattice constants depend on the treatment of\nthe f states (in the above calculation the lattice constants are\nfixed to the experimental values). In general, the LDA gives\nthe smallest equilibrium volume. The volume becomes larger\nfor the GGA, SIC, and open-core treatments in this order. The\nvolume change naturally a \u000bects the magnetic properties con-\nsiderably.\nValues of Curie temperature TCare estimated by the mean\nfield approximation assuming a Heisenberg model; the ex-\nchange coupling constants Ji jare calculated using the scheme\nobtained by Oguchi et al.20)and by Liechtenstein et al.21)It\nshould be noticed that TCdepends on details of the calcula-\ntions as well as the choice of the exchange-correlation energy.\nFor example, if the LDA were used, TCwould be 1049 K in-\nstead of the GGA(PBE) value of 1184 K.\nOne of the strategies in developing permanent magnet ma-\nterials is improving their performance by forming alloys. For\nexample, a common way to improve the high temperature\nperformance of Nd 2Fe14B is to introduce some Dy that sub-\nstitutes for Nd. The calculation of the electronic structure\nof such substitutional alloys can be conveniently performed\nin the framework of the coherent potential approximation(CPA). Such types of calculations are also possible for substi-\ntutional alloys between vacancies and atoms , Vc 1\u0000xAx, where\nVc indicates vacancy. An example is Sm 2Fe17Nx(0\u0014x\u00143),\nwhere N randomly occupies one of the three vacant interstitial\nsites adjacent to Sm.\nIn the following, we review the results of recent calcula-\ntions on Sm 2Fe17Nx.22)Sm 2Fe17N3with the Th 2Zn17struc-\nture shows a much larger magnetic anisotropy and a higher\nCurie temperature than Nd 2Fe14B, although its saturation\nmagnetization is slightly smaller than that of the latter. Ex-\nperimentally, adding N to Sm 2Fe17increases the saturation\nmagnetization by 12% and the Curie temperature by 93%,\nand changes its magnetic anisotropy from in-plane to uni-\naxial, thus making it suitable for a permanent magnet mate-\nrial.10)Unfortunately, Sm 2Fe17N3decomposes at high tem-\nperatures,23)which prevents us from producing sintered mag-\nnets. For this reason, it has never replaced Nd 2Fe14B. How-\never, studying Sm 2Fe17N3will provide us with some hints that\nmight be useful when seeking new high-performance perma-\nnent magnet materials.\nThe electronic structure was calculated by using the KKR\nGreen’s function method with the LDA (MJW parametriza-\ntion) of density functional theory. The relativistic e \u000bects are\ntaken into account within the scalar relativistic approxima-\ntion. The SOC (only the spin diagonal terms) is included. The\nSIC scheme for the Sm-f states is exploited. The nonstoichio-\nmetric content of N is treated as mentioned above using the\nCPA, i.e., the 9 esite in the Th 2Zn17structure is randomly oc-\ncupied by N or a vacancy with the probability corresponding\nto the content of N. Three types of di \u000berent sets of lattice pa-\nrameters are used: structure A has the experimental parame-\nters of Sm 2Fe17N3,24)structure B has those of Sm 2Fe17,25)and\nstructure C has the same volume as structure A and the same\natomic positions as structure B. The atomic sphere approxi-\nmation (ASA) is employed and the ratio of the radii among\nthem is taken to be 1:1:0.5 for Sm:Fe:N. Note that this ratio\nsometimes a \u000bects the results considerably. In the following\ncalculation the ratio was not in particular adjusted. The max-\nimum angular momentum of the atomic scattering t-matrix\nof KKR is 3 for Sm and 2 for the others. Higher angular mo-\nmenta are taken into account as non-scattering states that con-\ntribute in determining the Fermi level.\nFigure 4 shows the density of states of Sm 2Fe17N3with\nstructure A. Sm-f spin-down states split into two parts. The\noccupied f states further split, showing a finer structure. The\ntotal number of electrons in the occupied Sm-f states is 5.76\nwhen calculated using the above mentioned ratio of the ASA\nradii, and thus the configuration is more or less Sm2+. This\nresult is rather definite and also consistent with the results ob-\ntained by LDA calculations, although the occupied f states are\nlocated at an energetically much deeper position than those\nin the LDA result. The fact that Sm is likely to be divalent\ncontradicts the usual assumption that Sm is more or less triva-\nlent. However, we have to be particularly cautious about the\nvalency for metallic systems such as Sm 2Fe17N3. Firstly, al-\nthough f states are fairly localized, they still have positive en-\nergies and extend to the interstitial region. Thus, the number\nof f electrons strongly depends on the volume assigned to the\nSm atom, while the volume itself is to some extent arbitrary.\nSecond, for such systems, we do not know how to define the\nvalency that corresponds to the concept of chemical valency\n5J. Phys. Soc. Jpn.\nin the chemistry sense. Therefore, the best we can do is to\ncompare the predicted and observed spectroscopic data that\nmay reflect the electron configuration, without asking about\nthe valency.\nFig. 4. (Color online) Density of states of Sm 2Fe17N3calculated using\nLDA with SIC. Sm local f densities of states are indicated by blue lines.\nThe states around \u000018 eV originate from the Sm semicore 5p states, those\naround -15 eV originate from the N semicore 2s states.\nFigure 5 shows the calculated magnetocrystalline anisot-\nropy constant K1of Sm 2Fe17Nx. Here, K1was evaluated from\nthe total energy of the system as a function of the direction of\nthe magnetization or, conversely, the direction of the crystal\naxes. The anisotropy energy is fitted to Eq. (1).\nThe calculation was performed using the ASA, i.e., the\ncrystal is filled with atomic spheres, the sum of whose vol-\numes is the crystal volume, centered on each atomic site.\nSince the potential inside each atomic sphere is assumed to be\nspherically symmetric, electrons do not feel any anisotropic\nelectrostatic field, namely, no crystal field e \u000bect arises. There-\nfore, in this calculation, all the magnetocrystalline anisotropy\nstems from the band structure that reflects the e \u000bects of SOC.\nThis band structure e \u000bect, which is also understood as the ef-\nfect of the hybridization with ligands, is distinguished from\nthe crystal field e \u000bect. The former is usually more important\nthan the latter for transition-metal ions but this is not neces-\nsarily the case for rare earths. In the present system, although\nit is a matter of course that the anisotropic electrostatic field\nalso could be an important source of the magnetocrystalline\nanisotropy, the band structure e \u000bect makes a significant con-\ntribution. The overall trends of the behavior of magnetocrys-\ntalline anisotropy are reasonable. In particular, the behavior\nofK1, starting from nearly zero at x=0, and increasing\nwith increasing x, is well reproduced. There is a consider-\nable discrepancy in the absolute values of K1between the\ncalculation and experiment. It is noted, however, that the cal-\nculated K1easily varies by\u000650% depending on the calcula-\ntion details. For example, if the SIC procedure proposed by\nPerdew and Zunger26)is adopted instead of that by Filippetti\nand Spaldin,27)which was used here, the value of K1becomes\nFig. 5. (Color online) Magnetocrystalline anisotropy constant K1of\nSm2Fe17Nxas functions of N concentration x. The red and green circles show\nthe results calculated for structures A and B, respectively.\n10.2 MJ /m3, which is considerably smaller than the present\nresult of 20.3 MJ /m3.\nThe mechanism by which uniaxial anisotropy occurs is\nschematically shown in Fig. 6. Without N atoms, the hy-\nbridization of Sm-f states with surrounding atoms is rather\nsmall and the f states keep a feature of narrow atomic-like\nstate irrespective of the relative angle between the magne-\ntization and crystal axes. In this situation, the rotation of\ncrystal axes has little e \u000bect on the Sm-f states and causes\nno significant magnetic anisotropy. When N atoms are in-\ntroduced, the hybridization between Sm-4f and N-2p states\noccurs. When the magnetization lies along the c-axis, the\nstrongest hybridization occurs between N-2p and Sm-4f with\nmagnetic quantum number m=\u00063 states. On the other hand,\nfor the in-plane magnetization, the strongest hybridization is\nbetween the Sm m=0 state and N-2p states. Comparing these\ntwo cases, we may say that an energy gain is expected only\nwhen the hybridization occurs between m=\u00003 and the N-2p\nstates. This is because the SOC pushes up only the m=\u00003\nstate above the Fermi level and thus causes the energy gain\ndue to the lowering of the occupied state energy levels. The\nimportance of hybridization is also proven by the fact that,\nif the open-core scheme for Sm-f states be adopted, where\nno hybridization occurs between Sm-f and N-2p states, K1\ntakes a small negative value. The mechanism of such energy\ngains is the same as the superexchange working between two\nlocal magnetic moments: the virtual process to unoccupied\nstates plays a role. The energy gain, and hence the magnetic\nanisotropy energy per N atom, due to hybridization is thus\n6J. Phys. Soc. Jpn.\ngiven by\n\u0001E\u0018\u0000jVj2\nESm(m =\u00003)\u0000EN2p; (3)\nwhere Vis the hybridization energy between the Sm-4f state\nwith m=\u00003 of energy ESm(m =\u00003)and N-2p states of energy\nEN2p. A similar e \u000bect caused by the hybridization between\nthe unoccupied N-2p states and occupied Sm-4f states also\nexists, and it actually counteracts the above mechanism, i.e.,\nless hybridization for the case of magnetization along the c-\naxis. However, this would not a \u000bect the magnetic anisotropy\nsignificantly because the unoccupied N-2p states, which are\nthe antibonding states formed between the N-2p and Fe-3d\nstates, are orthogonal to the N-2p states that hybridize with\nSm-4f states, and contribute little to this mechanism.\nFig. 6. Energy diagram of N 2p and Sm 4f states and their hybridizations\nfor magnetization (a) along c-axis and (b) perpendicular to c-axis. Since the\nSm f states for\u00002\u0014m\u00143 are fully occupied, the hybridization of these\nstates with N 2p states does not contribute to the energy gain due to hybridiza-\ntion. Thus, only the m=\u00003 state contributes to the energy gain, which is the\norigin of the magnetic anisotropy in this case.\n4. Finite-Temperature Magnetism\n4.1 Local moment disorder method\nThere are no general methods so far to treat the finite-\ntemperature magnetism of metallic systems from first prin-\nciples. However, several schemes that can potentially incor-\nporate finite-temperature magnetism into a first-principles ap-\nproach have been proposed and even applied to permanent\nmagnet materials. One of them, which represents the most re-\ncent developments, may be schemes using dynamical mean\nfield theory (DMFT) combined with first-principles calcula-\ntion.28–32)In this method the e \u000bects of electron correlation are\ntreated locally more or less (depending on the solver) exactly\nin the framework of the local model Hamiltonian. The band\nstructure is fully taken into account using the framework of\nfirst-principles calculation.\nAnother conceivable way may be to apply the spin-\nfluctuation theories33)developed for the tight-binding model(or Hubbard model) to the Kohn–Sham equations. In the\nframework of the tight-binding model, a standard scheme to\ndeal with the finite-temperature magnetism of itinerant elec-\ntron systems is based on the functional integral method. This\napproach was first applied to the ferromagnetism of narrow\nd-bands by Wang and co-workers,34, 35)and by Cyrott and\nco-workers36, 37)In this approach the Stratonovich–Hubbard\ntransformation,38, 39)which maps the problem of an interact-\ning electron system to that of a non-interacting system with\nfluctuating auxiliary fields that are to be integrated out in the\nsense of a functional integral, is used to calculate the grand\npotential.39)\nAlthough Moriya et al.33)went a little further in the frame-\nwork of the functional integral method for general discussion,\nmost other works used the static, single-site, and saddle-point\napproximations in performing the procedure implied by this\nmethod. In particular, Hubbard40–42)and Hasegawa43, 44)inde-\npendently developed the theory of ferromagnetism for Fe, Co,\nand Ni within the above approximations. Once these approx-\nimations are exploited, the procedure is reduced to the cal-\nculation of the electronic structure of random substitutional\nalloys. For this reason, such approaches are also called alloy\nanalogy .\nThe so-called local moment disorder (LMD) method (often\ncalled the disordered local moment (DLM) method) is a typ-\nical scheme using the alloy analogy. The method is viewed\nas, but with a slight nuance, being based on the functional\nintegral method. Since the method is equally applied to the\nground state, the use of the LMD method is not restricted to\nthe study of the finite-temperature properties: the method was\nused by Jo45, 46)to describe the quantum critical point of mag-\nnetic alloys in the tight-binding model and later used in the\nframework of KKR-CPA-LDA to discuss similar problems by\nAkai and co-workers.47)The LMD approach combined with\nKKR-CPA-LDA was also applied to discuss the magnetism\nabove TCof Fe and Co by Oguchi et al.20)and by Pindor\net al.48)The approach was further developed by Gyor \u000by et\nal.49)and Staunton and co-workers.50)The major di \u000berence\nbetween the calculations based on the LMD combined with\nKKR-CPA-LDA and the classical calculations in the tight-\nbinding model is that the former is based on DFT, and hence,\nall the dynamical e \u000bects are assumed to be incorporated in\nthat framework.\nIn the prototype LMD scheme, two local magnetic states,\none aligned parallel to the magnetization, the other antipar-\nallel, are considered and the system is supposed to be a ran-\ndom alloy composed of atoms of these two distinct local mag-\nnetic states. It may be said that it simulates the paramagnetic\nstate above TC. The energy di \u000berence between the ferromag-\nnetic and random alloy (LMD) states then gives an estimate of\nTC. If one assumes that the system is described by a Heisen-\nberg model, TCis given by 2 =3 of the energy di \u000berence per\nnumber of magnetic ions. Thus, calculated TC’s usually show\nreasonable correspondence with ones obtained by the scheme\nusing Ji jmentioned earlier in this section. The above alloy-\nanalogy-type scheme is suitable above TCwhere the rotational\nsymmetry in the spin space is preserved, and hence the Ising-\nlike treatment becomes exact in the single-site treatment. This\nscheme may still be feasible even below TC, where the local\nrotational symmetry in the spin space breaks, for the calcu-\nlations of some insensitive quantities such as magnetization.\n7J. Phys. Soc. Jpn.\nHowever, this certainly is not true for the magnetic anisotropy,\nwhere the vector nature of spins is essential. In such cases, the\ndirectional distribution of spin in the whole solid angle has\nto be considered; each angle corresponds to each constituent\natom of the alloy .\nOne of the attempts that are of great relevance to the study\nof permanent magnet materials is the studies by Staunton and\nco-workers51, 52)An important feature of their approach in the\npresent context is that they include the SOC in the frame-\nwork, which renders the calculation of the temperature de-\npendence of the magnetic anisotropy tractable. The informa-\ntion obtained from such calculations can be utilized for other\ncompletely di \u000berent approaches, which will be explained in\nthe following subsection.\n4.2 Spin-model analysis\nAnother approach to finite-temperature magnetism is anal-\nysis using a spin model. An e \u000bective spin Hamiltonian of a\nrare-earth magnet compound is expressed as follows:\nH =HT+HR+HRT+Hext; (4)\nHT=\u00002X\nhi;ji2TJTT\ni jSi\u0001Sj\u0000X\ni2TDT\ni\u0010\nSz\ni\u00112; (5)\nHR=X\ni2RX\nl˜\u0012Ji\nlAml\nl;ihrliiˆOml\nl:i; (6)\nHRT =2X\nhi;ji;i2R;j2TJRT\ni j(gJ\u00001)Ji\u0001Sj; (7)\nHext =\u0000\u00160X\nimi\u0001Hext: (8)\nHere,HTis the Hamiltonian in the Tsublattices. The first\nterm in Eq. (5) is the magnetic exchange coupling between\ntheith and jth sites, and the second term is the single-ion\nanisotropy.HRrepresents the single-ion anisotropy at the R\nsite, where ˜\u0012Ji\nlis the Stevens factor, Aml\nl;iis the crystal-field pa-\nrameter, and ˆOl:iis the Stevens operator equivalent. HRTis the\nexchange coupling between the Rsite and Tsite, where Jiis\nthe total magnetic momentum at the Rsite, and Sjis the spin\nmomentum at the Tsite. Finally,Hextexpresses coupling be-\ntween the external magnetic field Hextand the magnetic mo-\nment at the ith site, mi. Although this form of the Hamiltonian\nhas been known for a long time, it is only recently that a quan-\ntitative calculation has been carried out. Matsumoto et al. have\nevaluated the parameters in Eqs. (4)-(7) for NdFe 12N by first-\nprinciples calculation, and solved the derived spin Hamilto-\nnian by the classical Monte Carlo method.53)They found that\nthe anisotropy field at high temperatures is sensitive to the\nmagnetic exchange coupling between RandT, namely JRT\ni jin\nEq. (7).\nA similar simulation was carried out for Nd 2Fe14B by Toga\net al.54)They computed e \u000bective parameters from first prin-\nciples except for Am\nl. For the crystal-field parameters, exper-\nimentally deduced values for l=2,4,6 and m=055)were used.\nThe calculated magnetization obtained by Monte Carlo sim-\nulation successfully reproduced the spin reorientation tran-\nsition at\u0018140 K. The Curie temperature was calculated to\nbe 754 K, which is in reasonable agreement with the exper-\nimental value of 585 K. The magnetocrystalline anisotropyenergy can be computed using the constrained Monte Carlo\nmethod.56)In this method, the direction of the total magneti-\nzation is fixed to a given angle \u0012. The direction of the spin\nmagnetic moment at each site is changed under this con-\nstraint, and the thermal average is taken by Monte Carlo sim-\nulation. The free energy F(\u0012) is then obtained from\nF(\u0012)=Z\u0012\nd\u00120[n(\u00120)\u0002T(\u00120)]\u0001@n(\u0012)\n@\u0012; (9)\nwhere T(\u0012) is the magnetic torque and n(\u0012) is the unit vector\nin the direction of the total magnetization. Figure 7(a) shows\nthe magnetic anisotropy constants obtained by fitting to the\nfollowing equation:\nF(\u0012;T)=KA\n1(T) sin2\u0012+KA\n2(T) sin4\u0012+KA\n3(T) sin6\u0012+const:;\n(10)\nwhere Tis the temperature. We see that both KA\n1(T) and\nKA\n2(T) are in good agreement with the experiment57)ex-\ncept for T<100 K, where the quantum e \u000bect would be-\ncome significant. Figure 7(b) shows the magnetocrystalline\nanisotropy energy FAas a function of temperature. Contribu-\ntions from Nd and Fe sites are also plotted, which are evalu-\nated by hypothetically putting DT=0 and Am\nl=0, respectively.\nThe anisotropy energy becomes weaker as the temperature in-\ncreases. At low temperatures, the magnetic anisotropy orig-\ninating from Nd sites is stronger than that from Fe sites.\nThis is naturally understood because strong magnetocrys-\ntalline anisotropy originates from single-ion anisotropy at R\nsites. As the temperature is raised, however, the Nd contri-\nbution decays quickly, whereas the Fe contribution decreases\ngradually with linear dependence against temperature. The\nquick decay of the Nd contribution was explained by Sasaki\net al. in the molecular-field approximation as follows.58)The\nexchange field acting on Rfrom Tbecomes weaker with in-\ncreasing temperature because of thermal fluctuation. Then,\nthe energy splitting between the mstates in the Jmultiplet\nbecomes smaller. Excited mstates are easily occupied by ther-\nmal excitation, and the 4f electron distribution approaches a\nspherical distribution. As a consequence, the crystal-field ef-\nfect becomes ine \u000bective, which results in a decrease in the\nmagnetic anisotropy at R. This implies that the magnetic ex-\nchange field is more important than the single-ion anisotropy\nfor magnetic anisotropy at high temperatures.\n5.RFe12-Type Compounds\nRFe12-type compounds with the ThMn 12structure have\nbeen investigated actively in the past few years. This class\nof compounds was studied as possible strong magnet com-\npounds in the late 80’s when iron-rich phases were synthe-\nsized.59–61)Among them, SmFe 11Ti was developed by Ohashi\net al.62)Subsequently, it was found that interstitial nitrogena-\ntion improves magnetic properties: the magnetization is en-\nhanced and the Curie temperature rises by 100–200 K. Mag-\nnetocrystalline anisotropy is also a \u000bected significantly. Yang\nand co-workers found that NdFe 11TiN\u000eis a good magnet com-\npound having reasonably high saturation magnetization.63, 64)\nHowever, the magnetization is smaller than that of Nd 2Fe14B,\nwhich had already been developed. Hence, RFe12-type com-\npounds have not been studied extensively for two decades.\n8J. Phys. Soc. Jpn.\n01 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0Temp. [K]−5051015KAi[MJ/m3]KA1KA2KA4\n0100200300400500600700800Temp. [K]012345678FA[MJ/m3]TotalNdFe\nFig. 7. (Color online) (a) Anisotropy constants KA\ni(i=1,2,3) of Nd 2Fe14B\nobtained by constrained Monte Carlo simulations for the classical spin\nHamiltonian. Experimental values for KA\n1andKA\n2are also plotted for com-\nparison. (b) Temperature dependence of the magnetic anisotropy energy. The\nfigures are taken from ref. [54]. Copyright 2016 by the American Physical\nSociety.\nThe RFe12-type compounds contain a high Fe content.\nThis is advantageous for achieving large saturation magne-\ntization. Nevertheless, NdFe 11TiN has lower magnetization\nthan Nd 2Fe14B because of the presence of Ti, which sub-\nstitutes for one of Fe sites. Miyake and co-workers studied\nNdFe 11TiN and NdFe 12N by first-principles calculation.65, 66)\nThey found that NdFe 11TiN has substantially smaller mag-\nnetization than NdFe 12N because (1) the spin is negatively\npolarized at the Ti site, and (2) the magnetic moments at Fe\nsites in the vicinity of Ti are suppressed on average. As a re-sult, the reduction of the magnetization by Ti substitution is\nmore significant than the naive expectation from the change\nin the iron concentration. On the other hand, the A0\n2param-\neter in NdFe 12N is comparable to that of NdFe 11TiN, sug-\ngesting that NdFe 12N has reasonably large magnetocrystalline\nanisotropy [Fig.8]. In both NdFe 11TiN and NdFe 12N, inter-\nstitial nitrogenation enhances the A0\n2parameter drastically.\nThis is because a weak chemical bond is formed between\nNd and N, and the electron density increases between them\n[Fig.8(b)]. It pushes away the Nd-4f electrons in the perpen-\ndicular direction, which induces uniaxial magnetocrystalline\nanisotropy. Subsequently, Hirayama and co-workers synthe-\nsized NdFe 12N on a MgO substrate with a W underlayer, and\nreported that the compound has larger saturation magnetiza-\ntion and anisotropy field than Nd 2Fe14B.12)\nInterstitial light elements have a strong influence on the\nmagnetism of rare-earth magnet compounds. Kanamori dis-\ncussed the role of B in Nd 2Fe14B.67, 68)When B is added to\niron compounds, the B-2p state hybridizes with the Fe-3d\nstates. Since the B-2p energy level is located higher than the\nFe-3d level, the antibonding state, having strong B-2p char-\nacter, appears above the Fermi level. Then, the Fe-3d state is\npushed down by p-d hybridization. This suppresses the spin\nmagnetic moment of Fe sites neighboring B. This is called\ncobaltization . Meanwhile, the 3d orbital at cobaltized Fe sites\nhybridizes with 3d orbitals at surrounding Fe sites. Then, the\nspin magnetic moment is enhanced at the latter sites. These\nchemical e \u000bects can have a sizable e \u000bect on the total magneti-\nzation of the compound. Harashima et al. studied these e \u000bects\nin the hypothetical compound NdFe 11TiB.69)They confirmed\nthat the change in the spin magnetic moment at each Fe site\nis explained by the cobaltization mechanism. They also found\nthat the net change in the total spin magnetic moment is neg-\native, namely the chemical e \u000bect induced by B reduces the\ntotal magnetic moment. As a matter of fact, the total mag-\nnetic moment of NdFe 11TiB is larger than that of NdFe 11Ti,\nbut this is attributed to magnetovolume e \u000bect [Fig.9]. As the\nlight element Xis changed from B to C or N, the magnetic\nmoment shows a jump between X=C and X=N. A similar\nresult has been reported in related systems.70, 71)This Xde-\npendence originates from a chemical e \u000bect. The antibonding\nstate between the X-2p and Fe-3d states is downshifted as\nthe atomic number of Xincreases. Eventually, the hybridized\nstate crosses the Fermi level. It is partially occupied in the\nmajority-spin channel for X=N, leading to enhancement of\nthe magnetic moment. (The symmetry of the hybridized state\nchanges from antibonding to bonding character as the state\ncrosses the Fermi level.)\nAn important issue of RFe12-type compounds is how to sta-\nbilize the bulk phase. This is achieved by substituting part of\nthe Fe sites with another element Msuch as Ti, V , Cr, Mn, Mo,\nW, Al, or Si. However, these stabilizing elements decrease the\nmagnetic moment. The search for a stabilizing element that\ndoes not lead to significant magnetization reduction is a hot\ntopic.72)\n6. Concluding remarks\nAlthough research on rare-earth magnets has a long\nhistory, the quantitative understanding is still insu \u000ecient. A\ntheoretical framework of first-principles calculation is under\ndevelopment, mainly because of the di \u000eculty in treating\n9J. Phys. Soc. Jpn.\nFig. 8. (Color online) (a) Magnetization and second-order crystal-field parameter of NdFe 11Ti, NdFe 12, NdFe 11TiN, and NdFe 12N. (b) Di \u000berence in the\nelectron density between NdFe 11TiN and NdFe 11Ti. In the latter case, nitrogen is removed from the former by a fixing structure.65, 66)The electron density\nincreases (decreases) by >0.001 /(Bohr)3in the red (blue) region.\n10J. Phys. Soc. Jpn.\nNdFe11TiENdFe11TiX\n2424.52525.52626.52727.528\nBCNOFmagneticm o m e n t[µB/f.u.]NdFe11TiXNdFe11Ti (struc. NdFe11TiX)chemical effect\nfor NdFe11Ti withits optimized structuremagnetovolume effectNdFe11Ti\nFig. 9. (Color online) Calculated spin magnetic moment and magnetiza-\ntion (per volume) of NdFe 11Ti and NdFe 11TiXforX=B, C, N, O, and F.\nNdFe 11TiEis the result for NdFe 11Ti in which Xis removed from NdFe 11TiX\nby a fixing structure. The figure is taken from ref. [69]. Copyright 2015 by\nthe American Physical Society.\nthe 4f electrons of rare-earth elements. Accurate description\nof magnetism is an important issue yet to be resolved.\nAnother challenge is applications to grain boundaries. Real\npermanent magnets contain additive elements, impurities,\ndefects, and various subphases. Of particular interest are\nthe interfaces between the main phase and grain boundary\nphases, which are believed to play a crucial role in coercivity.\nThe recent development of supercomputers enables us to\ndirectly compute the interfaces by first-principles calculation.\nExploration of a new magnet is also a major challenge. The\nneodymium-based magnet has been the strongest permanent\nmagnet for the last thirty years. NdFe 12N has superior\nintrinsic magnetic properties; however, its thermodynamic\ninstability prevents industrial application. Exploration of a\nwide range of compounds is anticipated. The recent devel-\nopment of materials-informatics may help us discover new\nmagnet compounds e \u000eciently.\nWe acknowledge collaboration and fruitful discussions\nwith Kiyoyuki Terakura, Yosuke Harashima, Hiori Kino,\nShoji Ishibashi, Taro Fukazawa, Shotaro Doi, Munehisa\nMatsumoto, Yuta Toga, Seiji Miyashita, Akimasa Sakuma,\nMasako Ogura and Satoshi Hirosawa. 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Miyake: Jour-\nnal of Applied Physics 120(2016) 203904.\n12"    },    {        "title": "1801.04162v1.Control_of_oxidation_and_spin_state_in_a_single_molecule_junction.pdf",        "content": "Control of oxidation and spin state in a\nsingle-molecule junction\nBenjamin W. Heinrich,yChristopher Ehlert,z,{Nino Hatter,yLukas Braun,y\nChristian Lotze,yPeter Saalfrank,\u0003,zand Katharina J. Franke\u0003,y\nyFachbereich Physik, Freie Universit at Berlin, Arnimallee 14, 14195 Berlin, Germany\nzInstitute of Chemistry, Universit at Potsdam, Karl-Liebknecht-Strasse 24{25, 14476\nPotsdam, Germany\n{Department of Chemistry, Wilfrid Laurier University, 75 University Ave W, Waterloo,\nON N2L3C5, Canada\nE-mail: peter.saalfrank@uni-potsdam.de; franke@physik.fu-berlin.de\nAbstract\nThe oxidation and spin state of a metal-organic molecule determine its chemical\nreactivity and magnetic properties. Here, we demonstrate the reversible control of the\noxidation and spin state in a single Fe-porphyrin molecule in the force \feld of the tip of\na scanning tunneling microscope. Within the regimes of half-integer and integer spin\nstate, we can further track the evolution of the magnetocrystalline anisotropy. Our\nexperimental results are corroborated by density functional theory and wave function\ntheory. This combined analysis allows us to draw a complete picture of the molecular\nstates over a large range of intramolecular deformations.\nThe oxidation state of an atom describes the number of its valence electrons in a very\nsimple picture. Within this framework, redox reactions are understood as changes of the\n1arXiv:1801.04162v1  [cond-mat.mes-hall]  12 Jan 2018oxidation state by integer numbers (of charges). In metal-organic complexes, such reac-\ntions typically involve a change in the number of bonds to neighbouring atoms. They are\nparticularly important in nature, where they occur, e.g., in the oxygen metabolism by O 2\ncoordination to haemoglobin. Furthermore, they are important in the perspective of spin-\ntronics, because an oxidation reaction also involves a change of the spin state.\nSurface-anchored metalloporphyrins are model systems, where the interplay between lig-\nand geometry, oxidation state, magnetism and (spin) transport has been studied in great\ndetail down to the single molecule level.1{10Scanning tunneling microscopy and spectroscopy\n(STM/STS) has been used to show that additional ligands like H, CO or Cl can induce\nchanges in the oxidation and spin state of the core metal ion.11{14Most STM experiments\nassume that the tip is a non-invasive tool in the investigation. However, the STM tip can\nperturb the molecular properties when it is brought in su\u000eciently close proximity to the\nmolecule. As such, the coupling of a molecule to the surface can be modi\fed, which can lead\nto the appearance/disappearance of a Kondo resonance in case of magnetic molecules.15,16\nSimilar changes had previously been only detected in two-terminal break junction experi-\nments, where the stress upon opening/closing the junction is assumed to be very large.17\nRecently, it has been shown that far before contact formation, an STM tip may already a\u000bect\nintramolecular properties. In the far tunneling regime, the forces exerted by the STM tip are\nalready su\u000eciently large to induce small conformational changes within the molecule, which\na\u000bect the crystal \feld splitting of the central ion, and, hence, its magnetic anisotropy.14\nHere, we show that the tunneling regime bears even more degrees of freedom, which can\nbe reversibly controlled by the STM tip. For this purpose, we adsorb iron(III) octaethyl-\nporphyrin chloride (Fe-OEP-Cl) on a Pb(111) surface. In the absence of the tip, the d5\ncon\fguration of FeIIIgives rise to an S= 5=2 spin state in the ligand \feld. We make use\nof the possibility to controllably and reversibly manipulate the Fe{Cl bond by the force\n\feld of an STM tip. We show that the forces do not only lead to a change in the magnetic\nanisotropy as was reported previously,18but that they can also be used to induce a reversible\n2-5 0 50123\n-320 pm\n-160 pm dI/dV (arb. units)\nSample bias (mV)-90 pmc-2∆ 2∆\n2 nm\n-400 -300 -200 -100 010-3\nConductance (2e2/h)\n∆z (pm)10-2\n10-4a\nb\nΙΙΙε\nFigure 1: Fe-OEP-Cl on Pb(111). (a) Topography of a monolayer island. Two types of\nmolecules are observed: Fe-OEP (dim center) and Fe-OEP-Cl (bright center). (b) Conduc-\ntance at 10 mV vs. tip-sample distance \u0001 zmeasured with the tip place above the center of a\nFe-OEP-Cl molecule. A sudden drop in the conductance is observed at \u0001 z\u0019\u0000130 pm. (c)\ndI=dV spectra at di\u000berent tip-sample distances \u0001 zshow signatures of inelastic excitations\nat di\u000berent energies.\nredox reaction, implying the reversible change from a half-integer to an integer spin state.\nBoth states are not a\u000bected by sizable scattering from substrate electrons, i.e., their spin\nstate is not quenched by the Kondo e\u000bect. In combination with the use of superconducting\nelectrodes, which increase the excited state spin lifetime compared to metal electrodes,18\nthis renders this system suitable for potential spintronic applications.\nThe experiments are supported by theoretical simulations. By means of Density Func-\ntional Theory (DFT) and Wavefunction Theory (WFT) calculations, we determine the rel-\native energies of di\u000berent spin manifolds of Fe-OEP-Cl as a function of the Fe{Cl distance.\nOur simulations reproduce the experimentally observed changes in the magnetocrystalline\nanisotropy and indicate that at large Fe{Cl distances, the integer S= 2 state is favored.\n3Results and Discussion\nWe deposited Fe-OEP-Cl molecules on an atomically clean Pb(111) surface held at \u0019240 K.\nThis leads to self-assembled mixed islands of iron-octaethylporphyrin-chloride (Fe-OEP-Cl)\nand iron-octaethylporphyrin (Fe-OEP) as can be seen in the STM images obtained at low\ntemperature (Fig. 1a). The individual molecules can be identi\fed by the eight ethyl legs. The\nchlorinated species appears with a bright protrusion in the center, whereas the dechlorinated\nspecies appears with an apparent depression in the center. The dechlorination process is\nprobably catalytically activated by the surface and occurs upon adsorption. The magnetic\nexcitation spectra of both species have been characterized previously.14,18As mentioned\nabove, in Fe-OEP-Cl the Fe center lies in a +3 oxidation state with a spin of S= 5=2.\nDechlorination changes the oxidation state to +2 with the resulting spin state being S= 1.\nThis change of oxidation and spin state is naturally expected when the Fe center is subject\nto such a drastic modi\fcation as removing one ligand. It was also observed that the STM tip\ncan have an in\ruence on both individual species. The Fe ions are subject to crystal \felds,\nwhich split the dlevels and induce a sizable magnetocrystalline anisotropy and, hence, a\nzero \feld splitting (ZFS). By approaching the tip to the molecule, for both species the ZFS\nhas been shown to be modi\fed by the local potential of the tip.14\nHere, we explore if the tip potential can be used to change the local energy landscape\neven further such that a reversible change not only of the ZFS, but also of the spin state takes\nplace. For this, we target at the Fe-OEP-Cl molecule because it bears more \rexibility with\nits additional ligand than the dechlorinated species. We \frst explore the change of junction\nconductance upon approaching the tip on top of the Fe-OEP-Cl molecule in Figure 1b. After\na \frst exponential increase of the conductance with reduced distance (\u0001 z < 0), which is\ncharacteristic for the tunneling regime (regime I), there is a reversible drop in the conductance\nin the order of 30% of the total signal. This can be linked to a sudden change of the junction\ngeometry and/or the electronic structure of the complex. Further decreasing the distance\nincreases the conductance again, yet with a smaller and decreasing slope (regime II). At\n4-5 0 50-100-200- 300\n∆z (pm)\nSample bias (mV)-5 0 5-200-240\n∆z (pm)\nSample bias (mV)a b\n dI/dV\n (arb. units)03 -280 pm-2∆ 2∆ -2∆ 2∆\nΙΙΙ\nΙΙFigure 2: (a) Pseudo-2D color plot of dI=dV spectra of the molecule in Fig. 1b and 1c as a\nfunction of \u0001 z. (b) Pseudo-2D color plot of dI=dV spectra of a di\u000berent molecule with a\ndi\u000berent tip. \u0001 zincrement: 5 pm. The spectrum at \u0001 z=\u0000280 pm is shown on top and\nevidences a low-energy excitation close to the superconducting gap edge. For additional data\non di\u000berent molecules and with di\u000berent tips see SI.\n\u0001z\u0019\u0000330 pm, the conductance increases again with increasing slope. The di\u000berent slopes\nindicate di\u000berent structural/ electronic con\fgurations of the molecule in the junction. To\ncharacterize the magnetic properties of the molecule in these regimes, we record dI=dV\nspectra at di\u000berent \u0001 z(Fig. 1c). In all spectra, we observe an excitation gap with a pair\nof quasiparticle excitation peaks at \u0006(\u0001tip+ \u0001 sample ), with \u0001 tipand \u0001 sample being the real\npart of the order parameter of the superconducting tip and sample, respectively. In regime\nI (\u0001z=\u000090 pm), there are two additional pairs of resonances outside the gap. These\nresonances are due to inelastic spin excitations on a superconductor.18,19Replicas of the\nquasiparticle resonances appear at the threshold energies for the opening of an inelastic\ntunneling channel. The \frst inelastic peak corresponds to a transition between the M s=\n\u00061=2 to the M s=\u00063=2 state within the S= 5=2 manifold. The second excitation from the\nMs=\u00063=2 state to the M s=\u00065=2 state becomes possible at large current densities due\nto spin pumping.18In regime II, i.e., after the sudden change in the molecular conductance,\nboth inelastic excitations vanish. Instead, an excitation of lower energy is observed close to\nthe superconducting gap edge (see, e.g., the spectrum at \u0001 z=\u0000160 pm). At even smaller\ntip{molecule distance (exemplary spectrum at \u0001 z=\u0000320 pm), we observe an inelastic\nexcitation of even lower energy, which merges with the coherence peak.\n5Mol F 2015 August\ndZ=-150pm; right after SCO\n0.4b\n0 1 2\nBz (T)0.60.81.0\nEnergy (mV)a0.5 T\n1.5 T\n2.5 T\n0.60.81.0\n dI/dV (arb. units)\n-1 0 1\nSample bias (mV)Figure 3: (a) B-\feld dependent dI=dV spectra at \u0001 z=\u0000150 pm. A shift of the excitation\nstep to higher energies is observed with increasing B\feld. (b) Step energies as a function\nofB-\feld (black squares). The purple asterisk indicates the energy as extracted from the\nspectrum recorded with tip and sample in the superconducting state. The excitation shifts\nto higher energies with increasing B\feld.\nTo track the evolution of the inelastic excitations we record a series of spectra while\napproaching the STM tip in small intervals. A 2D color plot of these spectra is shown in\nFig. 2a. Within regime I, we observe a shift of both excitations to larger energies upon tip\napproach as a result of an increased ZFS. This has been qualitatively explained as a response\nto a distortion of the ligand \feld.14The sudden drop in conductance at the transition to\nregime II is also re\rected in an abrupt change of the dI=dV spectra. The two excitations of\ntheS= 5=2 manifold disappear and a new excitation appears. It shifts to lower energies\nand merges with the quasiparticle excitation peaks when approaching the STM tip to \u0001 z\u0019\n\u0000200 pm. Eventually, a faint excitation emerges from the quasiparticle excitation resonances\nfor \u0001z <\u0000220 pm and a second excitation departs from the gap edge at \u0001 z\u0019\u0000290 pm. A\nhigh-resolution 2D plot of regime II (for a di\u000berent molecule) is shown in Fig. 2b. It resolves\nthe latter two resonances more clearly (note that their appearance is shifted in \u0001 zwith\nrespect to the molecule shown in Fig. 2a).\nWhereas the origin of the two excitations in regime I is already understood, regime II was\nnot explored in earlier experiments. We note that regime II does not indicate the desorption\nof the Cl ligand, because Fe-OEP shows characteristic spin excitations at larger energies,14\nwhich are absent here (see Fig. S2 of the SI). Furthermore, the junctions are stable: when\n6b\nFe-Cl distance (pm)Energy (meV)\n210 310 285 260 2350.00.51.01.52.0\nMS0 1 2aa\n-200-100100300\n200\n0\nElectronic energy (kJ/mol)\n210\nFe-Cl distance (pm)310 285 260 235S=5/2S=3/2\nS=1\nS=2Fe(II)Fe(III)Figure 4: (a) Potential energies of di\u000berent spin manifolds as a function of Fe{Cl distance,\ndetermined by a combination of DFT optimization and WFT single-point calculations (see\ntext). Half-integer and integer Svalues refer to the neutral (FeIII) and anionic (FeII) com-\nplexes, respectively. (b) ZFS of the S= 2 manifold. The colors indicate contributions of\nthe eigenstates of the spin projection operator. We calculated the average ( \u0016Ms) as the sum\nof the absolute eigenvalues multiplied by the weights of contributing eigenstates. Note that\nat 210 pm the lowest two eigenstates are almost degenerate, while for Fe{Cl distances larger\nthan 285 pm, the two highest states are almost degenerate.\nwe retract the tip, we reversibly enter into regime I. Indeed, we can precisely and reversibly\ntune the junction conductance and, hence, the energy of the excitations. We \frst determine\nwhether the new excitations are of magnetic origin. By applying an external magnetic \feld\nperpendicular to the sample surface, superconductivity in substrate and tip are quenched.\nHere, inelastic excitations give rise to steps in the dI=dV spectra at the excitation threshold as\nexpected for a normal metal substrate (Fig. 3a). Importantly, we observe a shift of the steps\nto larger energies in response to an increasing external \feld. The extracted step energies\nagree with a Zeeman shift (Fig. 3b), which evidences their magnetic origin.\nSummarizing the experimental results, there is evidence for a sharp transition from the\nS= 5=2 state to a di\u000berent spin state when approaching the STM tip on top of Fe-OEP-Cl.\nIn principle, this could either be a spin crossover to another half-integer spin state or a\n7change to an integer spin state, which necessarily needs to be accompanied by a change in\noxidation state. In order to gain further insights, we performed DFT calculations using the\nB3LYP exchange-correlation functional20as well as WFT based simulations in the form of\nthe so-called CASSCF/NEVPT2 method21{25according to Refs.26,27Details are described in\nthe Methods section and in the SI. We assume that the tip-molecule interaction, in the sense\nof a Lennard-Jones potential, is attractive at not too short tip-molecule distances. Upon\napproaching with the STM tip, the Cl ion will thus be attracted by the tip. We model the\ne\u000bect of such a potential in an isolated Fe-porphin molecule by increasing the Fe{Cl distance.\nIn porphin, the eight ethyl groups are replaced by H atoms. We have previously checked\nthat the ethyl legs do not a\u000bect the electronic and magnetic properties of the complex (see\nFig. S6 in the SI).\nWe used DFT optimization to compute the relaxed structure while scanning and keeping\nthe Fe{Cl distance \fxed in three di\u000berent spin states of the neutral Fe(III)-complex, i.e., with\nspin quantum numbers S= 5=2;S= 3=2;S= 1=2. Based on these results, we calculated\nthe potential energies by the CASSCF/NEVPT2 method at the DFT geometries (Fig. 4a;\ncorresponding pure DFT potential curves are shown in Fig. S8 of the SI). In agreement with\nprevious experiments and calculations, we \fnd that the ground state exhibits a spin state\nofS= 5=2. Here, the Fe{Cl distance amounts to 220 pm. This con\fguration represents\nthe unperturbed molecule on the surface, i.e., when the tip is far away. While the energy\ndi\u000berence between the S= 5=2 andS= 3=2 states decreases, the latter remains at larger\nenergies throughout the whole range of reasonable Fe{Cl distances. Hence, a spin crossover\nscenario as an explanation for the sudden change in experimental spin excitations from\nregime I to regime II can be excluded1. TheS= 1=2 spin state lies even higher in energy\nand can therefore be excluded to play a role as well.\nHence, we also calculated the potential energies of the anionic Fe(II)-complex in its\n1We can also experimentally exclude such a spin crossover, because the appearance of two excitations in\nregime II cannot be captured in a S= 3=2 system.\n8S= 2 andS= 1 spin states2. TheS= 2 state is found at signi\fcantly lower energies than\ntheS= 1 state. Within increasing Fe{Cl bond length, the potential energy increases but\nnot with the same slope as in the S= 5=2 state. As a result, the energy di\u000berence between\nS= 5=2 andS= 2, i.e., the electron a\u000enity of the neutral complex increases with increasing\nFe{Cl distance. Together with a resulting image charge stabilization of the anionic complex\nand a relatively low work function of Pb of about 4 eV, this is the main driving force for an\nelectron transfer from the tip or substrate to the molecule at a certain critical, elongated\nFe{Cl distance.\nTo check whether the experimentally observed spin excitations in regime II match with\nthis scenario, we calculated the ZFS of the anionic system. The results are shown in Fig. 4b.\nOverall it is observed that the state energies are very sensitive with respect to geometrical\nvariation. The discussion of this behavior can be split in two parts: a \frst part (i) with Fe{Cl\nbond length shorter than 285 pm and a second part (ii) with Fe{Cl bond lengths longer that\n285 pm.\nIn part (i) the ground state is composed of two states, which are almost degenerated. The\neigenvalues of the spin projection operator on the zaxis have mainly Ms=\u00062 character.\nThe third and fourth state have eigenvalues of Ms=\u00061, while the \ffth state has the Ms= 0\neigenvalue. The splitting can be easily rationalized with the help of an e\u000bective Hamiltonian\nmethod28and related to so-called axial and rhombic ZFS parameters DandE, respectively,\nas explained in the SI.\nWith increasing bond length the third, fourth and \ffth state decrease in energy and\nmixing occurs between the states. Inelastic electron tunneling requires \u0001 Ms= 0;\u00061. Hence,\none can observe excitations from the Ms=\u00062 ground states to the Ms=\u00061 states. The\nenergy of this excitation is in the order of 1 meV with decreasing energy until the bond\nlength amounts to 285 pm. This behavior of one excitation with decreasing energy is found\nin experiment at the initial stages of regime II, i.e., right after the spin transition.\n2TheS= 0 state can be disregarded as it would not show any spin excitations.\n9When the Fe{Cl bond length becomes longer than 285 pm [part (ii)], the lowest two\nstates are again almost degenerate, but have changed their character. Both states are now\nmixtures of several eigenvectors of the spin projection operator. The \frst one is dominated\nbyMs= 0, the second by Ms=\u00061. The third state is mainly of Ms=\u00061 character, while\nthe fourth and \ffth are dominated by Ms=\u00062. The energy of the three highest states\nincrease now monotonously with elongated Fe{Cl bond lengths. Since the nearly degenerate\nground state has Ms=\u00061 andMs=0 character, inelastic excitations are allowed to both\ntheMs=\u00061 andMs=\u00062. These two excitations are indeed found in experiment in the\nlater part of regime II. In agreement with theory, their energies increase with increased Fe{Cl\nbond length. The lower intensity of the second excitation in experiment may be ascribed to\nan underestimated energy splitting of the Ms=\u00061 andMs=0 state in the calculations. A\n\fnite splitting would require the second state to be thermally occupied in order to be able\nto observe an inelastic transition and thus be reduced in intensity.\nIn conclusion, we have established a fully reversible switching of the spin and oxida-\ntion state in a single molecule including the possibility to \fne-tune the magnetocrystalline\nanisotropy. By approaching the STM tip on a Fe-OEP-Cl molecule, we have observed dras-\ntic changes in the spin excitation spectra. The force exerted by the STM tip leads to a\ndeformation of the molecule, in particular to an elongation of the Fe{Cl bond length. Ini-\ntially, this a\u000bects the magnetocrystalline anisotropy of the S= 5=2 state. At a critical\nbond length, a sudden change of oxidation and spin state occurs, which leads to a distinctly\ndi\u000berent \fngerprint in the excitation spectra. Our combined DFT and CASSCF/NEVPT2\nsimulations allow for an unambiguous identi\fcation of the experimentally observed excita-\ntions. Their behavior under the force \feld agrees well with the experiment. Our experiments\nprovide an intriguing method for a reversible and controlled way of changing not only the\nmagnetic anisotropy, but also and more drastically, the spin state. They open new pathways\nfor controlling magnetic properties by external forces.\n10Methods\nAll experiments were performed in a SPECS JT-STM, a commercial low-temperature scan-\nning tunneling microscope (STM) working under UHV conditions at a base temperature\nof 1.2 K. We obtained clean and superconducting surfaces of the Pb(111) single crystal by\ncycles of Ne+ion sputtering and subsequent annealing to \u0019430 K. In order to circumvent\nthe Fermi-Dirac limit in energy resolution, superconducting, Pb-covered W tips were used,\nwhich allow us to improve the energy resolution down to \u001960\u0016V when employing elab-\norated RF-\fltering and grounding schemes.29All spectra are recorded in constant-height\nmode with an o\u000bset \u0001 z(as indicated in the \fgures) in the tip{sample distance relative to\nthe set point: U= 50 mV;I= 200 pA. The di\u000berential conductance dI=dV is measured\nusing standard lock-in technique with a modulation frequency f= 912 Hz and a voltage\nmodulation Umod= 35\u0016Vrms.\nWe used the ORCA (version 3.0.3) program package30to optimize geometries and to\ncalculate potential energy surface scans along the Fe{Cl distance for several spin multiplic-\nities. The B3LYP density functional20in combination with the def2-TZVP basis set31was\nutilized.\nFor the geometries along the potential energy scans, the Zero-Field parameters have\nbeen computed by using a correlated WFT methodology28,32based on the complete active\nspace self-consistent \feld method (CASSCF).21In order to take into account dynamical\ncorrelation, the NEVPT222{25method has been used on top of CASSCF.\nSubsequently, the e\u000bects of spin-orbit and spin-spin interactions were included within the\nframework of quasi-degenerate perturbation theory. 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W.; Braun, L.; Pascual, J. I.; Franke, K. J. Nature Physics 2013 ,9,\n765{768\n(19) Berggren, P.; Fransson, J. EPL (Europhysics Letters) 2014 ,108, 67009\n13(20) Becke, A. D. The Journal of Chemical Physics 1993 ,98, 5648{5652\n(21) Malmqvist, P.- \u0017A.; Roos, B. O. Chemical Physics Letters 1989 ,155, 189{194\n(22) Angeli, C.; Cimiraglia, R.; Evangelisti, S.; Leininger, T.; Malrieu, J.-P. The Journal\nof Chemical Physics 2001 ,114, 10252{10264\n(23) Angeli, C.; Cimiraglia, R.; Malrieu, J.-P. Chemical Physics Letters 2001 ,350,\n297{305\n(24) Angeli, C.; Cimiraglia, R.; Malrieu, J.-P. The Journal of Chemical Physics 2002 ,\n117, 9138{9153\n(25) Angeli, C.; Bories, B.; Cavallini, A.; Cimiraglia, R. The Journal of Chemical\nPhysics 2006 ,124, 054108\n(26) Atanasov, M.; Ganyushin, D.; Pantazis, D. A.; Sivalingam, K.; Neese, F. Inorganic\nchemistry 2011 ,50, 7460{7477\n(27) Stavretis, S. E.; Atanasov, M.; Podlesnyak, A. A.; Hunter, S. C.; Neese, F.; Xue, Z.-\nL.Inorganic Chemistry 2015 ,54, 9790{9801\n(28) Ganyushin, D.; Neese, F. The Journal of Chemical Physics 2006 ,125, 024103\n(29) Ruby, M.; Heinrich, B. W.; Pascual, J. I.; Franke, K. J. Physical Review Letters\n2015 ,114, 157001\n(30) Neese, F. WIREs Computational Molecular Science 2011 ,2, 73{78\n(31) Weigend, F.; Ahlrichs, R. Physical Chemistry Chemical Physics 2005 ,7, 3297\n(32) Atanasov, M.; Aravena, D.; Suturina, E.; Bill, E.; Maganas, D.; Neese, F. Coordi-\nnation Chemistry Reviews 2015 ,289, 177{214\n(33) Neese, F. The Journal of Chemical Physics 2005 ,122, 034107\n14Supplementary Information\nAdditional experimental data\nTip approach on di\u000berent Fe-OEP-Cl molecules with di\u000berent Pb\ntips\nIn the main manuscript, we investigate the in\ruence of the tip of the STM on the spin\nand oxidation state of Fe-OEP-Cl molecules at di\u000berent tip-sample distances. In order to\ncheck the reproducibility of the experiments, di\u000berent microscopic and macroscopic Pb tips\nas well as di\u000berent molecules on di\u000berent preparations were tested. We observe a reversible\ntransition from regime I to regime II with all Pb tips and on all Fe-OEP-Cl molecules tested,\nalbeit at slightly di\u000berent junction conductances. The stability in regime II and thus the\nextent of this regime di\u000bers for di\u000berent tips. Only those experimental series, for which we do\nnot observe any di\u000berence in the topography and dI=dV spectra under standard tunneling\nconditions ( I= 200 pA,U= 50 mV) before and after the tip approach are used for evaluation\nand presented in the manuscript and Supplementary Information.\n-5 0 50-100-200- 300\n∆z (pm)\nSample bias (mV)a\n-5 0 50-100-200- 300\n∆z (pm)\nSample bias (mV)b\nΙΙΙ\nΙΙΙ\nFigure S1: Pseudo-2D color plot distance-dependent dI=dV spectra for two di\u000berent Fe-\nOEP-Cl molecules on Pb(111). Both junctions are stable to at least \u0001 z=\u0000360 pm. Set\npoint:V= 50 mV;I= 200 pA.\nFigure S1 presents two additional experimental series acquired with di\u000berent tips and\non di\u000berent molecules. While the di\u000berent experiments show qualitative agreement, there\nare di\u000berences with respect to the relative tip-sample distance \u0001 z, for which the transition\n15between regime I and regime II occurs. In Fig. S1a, the transition occurs at \u0001 z\u0019\u0000150 pm,\nwhile it happens at \u0001 z\u0019\u0000170 pm in Fig. S1b (starting from the same tunneling set point).\nFurthermore, the evolution of the spin excitation patterns within regime II di\u000bers. For all\nmolecules, we \frst observe a low-energy excitation, which shifts towards the superconducting\ngap edge. After this minimum of excitation energy, two excitations shift away from the\nsuperconducting gap edge. The shift of these excitations di\u000bers between the molecules. In\nsome cases, a faint third excitation is observed. We ascribe the di\u000berent excitation energies\nto small di\u000berences in the adsorption sites which are present within the island1and modify\nthe crystal \feld of the Fe ion. Furthermore, di\u000berent tip geometries alter the potential acting\non the molecule during the tip approach. However, all observations agree with the S= 2\nstate and the composition of the mixed spin projection states as described in the main text\nand in Fig. 4b of the main text.\nComparison of Fe-OEP with Fe-OEP-Cl in regime II\n3006009001200\n0dI/dV (nS)\n-10 0 10\nSample bias (mV)20 -20Fe-OEP-Cl\n-110 pmFe-OEP-Cl-160 pmFe-OEP -150 pm\nregime II\nregime I\nFigure S2: dI=dV spectra in a larger energy range of Fe-OEP, as well as of Fe-OEP-Cl in\nregime I and II, respectively. Set point: V= 50 mV;I= 200 pA. The spectrum of Fe-OEP\nshows two pairs of inelastic excitations at sample bias V\u0019\u000612mV andV\u0019\u000614 mV, which\nare not observed for Fe-OEP-Cl, independent of the tip approach.\nFigure S2 compares dI=dV spectra of Fe-OEP with those of Fe-OEP-Cl in both regimes.\nThere are two pairs of excitations at \u0019\u000612 mV and\u0019\u000614 mV observed in the case of\n16Fe-OEP. These show small variations depending on the adsorption site of the molecule.\nFurthermore, they vary in energy with the tip-sample distance, but variations are restricted\nto less then 10%.1For Fe-OEP-Cl, we do not observe excitations with energies >7 meV,\nwhich holds true for all junction con\fgurations and reversible tip approaches. This is a\nfurther proof that regime II of the Fe-OEP-Cl junction is not equivalent with a Fe-OEP\njunction. The Cl ligand is still present in the junction and actively in\ruences the potential\nlandscape.\nTip approach in the normal state\n-5 0 50-100-200- 300\n∆z (pm)\nSample bias (mV)-5 0 5∆z (pm)\nSample bias (mV)a b\n-100-200- 300Mol F 2015 0 to -330 pm\n0.5 T\n0.5 T 0 Tc\n-5 0 5\nSample bias (mV)dI/dV (arb. units)\n1.01.52.0\n-130 pm-160 pm-220 pm-300 pm\nΙΙΙ\nΙΙΙ\nFigure S3: Pseudo-2D color plot of distance-dependent dI=dV spectra on Fe-OEP-Cl on\nPb(111) at 0 T (a) and 0.5 T (b), respectively. Same molecule as in Fig. 2a of the main\nmanuscript. In (b), tip and substrate are in the normal, non-superconducting state. (c)\nExamples of dI=dV spectra as presented in (b) at various distances. Set point: V= 50 mV;\nI= 200 pA.\nFigure S3 presents a comparison of the approach experiment for tip and sample being\nin the superconducting (Fig. S3a) and in the normal state (Fig. S3b and c). Similar to\nthe observations in the superconducting state, the inelastic excitations also shift in energy\nwhen approaching with the STM tip in the normal state. Because of the reduced energy\nresolution in the normal metal state, which is caused by the temperature-induced Fermi-\nDirac broadening, these shifts are less well resolved compared to the experiments in the\nsuperconducting state. Yet, there is a qualitative agreement between both experiments.\n17After the transition into regime II at \u0001 z\u0019\u0000150 pm, the excitation gap in the normal-state\ncase closes with decreasing tip-sample distance, which corresponds to the excitation peak\nmoving closer to the BCS coherence peaks in the superconducting state. Then, for \u0001 z <\n\u0000220 pm, the gap widens again in the normal state, in agreement with the superconducting\ncase, where excitations move away from the coherence peaks to higher energies.\nNote that in regime I the second pair of excitations is absent in the normal state, but\npresent in the superconducting state. This is due to the increased lifetime in the latter,2\nwhich allows for spin pumping.\nZeeman shift of inelastic excitations\n 0.5 T\n 1.0 T\n 1.5 T\n 2.0 T\n 2.5 T\n 3.0 T 0.0 T\n 0.5 T\n 1.0 T\n 1.5 T\n 2.0 T\n 2.5 T\n 3.0 T\nSample bias (mV) Sample bias (mV)-3 0 1 3 2 -2 -1dI/dV (arb. units)\n0.41.0\n0.60.8dI/dV (arb. units)\n0.41.0\n0.60.8\n-3 0 1 3 2 -2 -1a b c\nB-field0Energy\n 3/2\n 1/2 5/2\nFigure S4: (a) B-\feld dependent dI=dV spectra on Fe-OEP-Cl in regime I ( S= 5=2) at\na constant tip-sample distance. Set point: V= 50 mV;I= 200 pA; \u0001 z=\u0000100 pm. (b)\nSimulated excitation spectra using D= 0:72 meV,E= 0 meV,g= 1:8, and an e\u000bective\ntemperature Te\u000b= 1:3 K. (c) B-\feld dependent excitation diagram for a S= 5=2 spin\nsystem with in-plane anisotropy and the B \feld oriented parallel to the anisotropy axis.\nAdapted from Supporting Information of Ref.1\n.\nIn the following, we discuss the Zeeman shift of the spin eigenstates in Fe-OEP-Cl. Fig-\nure S4a shows a B-\feld-dependent measurement of the excitation spectrum in the normal\nstate in regime I at constant distance. The B-\feld is oriented in zdirection. With increas-\ning \feld strength, the excitation steps at approximately \u00061:5 mV at 0:5 T move to lower\nenergies with increasing \feld. Additionally, a dip appears at the Fermi energy EFand gains\nweight with \feld strength. The excitation spectra can be simulated using an e\u000bective spin\n18Hamiltonian model:3\n^HS=g\u0016B^SB+D^S2\nz+E(^S2\nx\u0000^S2\ny): (S1)\nThe \frst term accounts for the Zeeman splitting (with gbeing the Land\u0013 e g-factor, \u0016B\nthe Bohr magneton, B the magnetic \feld, and ^S= (^Sx;^Sy;^Sz) the spin operator). The\nsecond term describes the zero-\feld splitting in the presence of axial anisotropy with param-\neterD, while the last term also takes transverse anisotropy into account with the rhombic\nanisotropy parameter E. Using a temperature broadened Fermi-Dirac distribution, we can\nsimulate the inelastic excitation spectra of the S= 5=2 spin state with in-plane anisotropy\n(D= 0:72 meV) in B-\felds of various strength parallel to the anisotropy axis. The simulation\nis done using the code by M. Ternes.4The result shown in Fig. S4b is in excellent agreement\nwith the measurements presented in Fig. S4a. Figure S4c shows a scheme of the correspond-\ning energy levels of the spin eigenstates of the S= 5=2 system with in-plane anisotropy (B\nparallel to the anisotropy axis). Because of the selection rules for inelastic spin excitations\n(\u0001Ms=\u00061; 0), the only ground state excitation possible at zero \feld is Ms=\u00061=2!\u0006 3=2.\nAtB6= 0, a second excitation becomes possible ( Ms=\u00001=2!+1=2), which increases in\nenergy with B. It gives rise to the dip at EFin the excitation spectrum because the limited\nenergy resolution at 1.1 K does not allow to resolve the step-like features (see also Theory\nsection).\nIn regime II, we also test for B-\feld-dependent excitations (Fig. 3 of the main manuscript).\nHere, we present a series of excitation spectra at di\u000berent \feld strength at a smaller tip-\nsample separation (\u0001 z=\u0000340 pm, Fig. S5). Again, a B-\feld dependence is observed, the\nexcitation step becoming wider with increasing B-\feld. This is most likely due to the shift-\ning of two or more super-positioned steps, which cannot be resolved due to the thermal\nlimitation in the energy resolution. Hence, we prove the magnetic origin of the excitation.\n190.5 T\n-2 0 2\nSample bias (mV)dI/dV (arb. units)\n0.71.01.3 2.5 T\n1.5 T\n1.0 T2.0 T3.0 TFigure S5: B-\feld dependent spectra at \u0001 z=\u0000340 pm (same molecule as in Fig. 2a and\nFig. 3 in the main text). Set point: U= 50 mV;I= 200 pA.\nTheory\nModels\nWe considered free molecules as model compounds for our quantum chemical investigations.\nFe-OEP-Cl was used in experiments. As shown below, the ethyl groups do not have a\nsigni\fcant e\u000bect on the crystal \feld of the Fe center, and, hence, on the magnetic properties.\nTherefore, we used Fe-P-Cl, where \\P\" stands for porphin, the simplest porphyrine with\neight peripheral H atoms instead of the ethyl groups, as a model system.\nFor the Fe-P-Cl molecules, a neutral variant (with Fe in oxidation state +3) and an\nanionic variant (with Fe in oxidation state +2) were considered. The chosen coordinate\nsystem is such the Fe-Cl bond is aligned along the z-axis.\nComputational methodology\nAs mentioned also in the Methods section of the main manuscript, the ORCA (version 3.0.3)\nprogram package5was used to optimize geometries and to calculate potential energy scans\nalong the Fe-Cl distance for several spin multiplicities. These are the S= 0,S= 1, and\n20S= 2 multiplicities for the anionic system, and S= 1=2,S= 3=2 andS= 5=2 for the\nneutral molecular model. For the scans, Fe-Cl distances in a range between 2.1 and 3.2 \u0017A\n(in steps of 0.05 \u0017A) were considered.\nFirst of all, the scans were done on the DFT level of theory, utilizing the B3LYP density\nfunctional6in combination with the def2-TZVP basis set.7To speed up the calculations, the\nRIJCOSX approximation8(in combination with the def2-TZVP/JK auxiliary basis set) was\nused.\nFor the calculation of spin-orbit coupling, zero-\feld splitted (excited) states, but also for\ncorrected potential energy curves of a given spin multiplicity, single-point multi-reference\nWFT methods were used, following established protocols.3,9,10For this purpose, Complete\nActive Space Self Consistent Field (CASSCF) calculations11were performed. In all calcula-\ntions, the orbitals were optimized in a state-averaged fashion and the quasi-restricted orbitals\nof the DFT calculations were used as guess orbitals. In order to take into account e\u000bects\nof dynamical correlation, the NEVPT212{15method has been used on top of the CASSCF\nresults. The number of calculated states with respect to the given spin quantum num-\nber,N(S), is, for the most important spin multiplicities as follows: N(5=2) = 1;N(3=2) =\n24;N(2) = 5;N(1) = 45.\nThe e\u000bects of spin-spin and spin-orbit interactions included in the form of quasi-degenerated\nperturbation theory using the following Hamiltonian:\n^H=^H0+^Hsoc+^Hsp\u0000sp+^HZe: (S2)\nHere, ^H0is the usual, Born-Oppenheimer electronic Hamiltonian, ^Hsocis the spin-orbit\ncoupling operator, approximated by an e\u000bective one electron spin-orbit mean \feld operator\n(SOMF),16^Hsp\u0000spaccounts for spin-spin interaction and ^HZe=1\n2\u0016Bge\u0010\n^L+ 2^S\u0011\nBis the\nZeeman term which describes magnetic \feld interaction ( \u0016Bis the Bohr magneton, geis the\ng-factor of the free electron, ^Land ^Sthe total angular and spin moment operators and B\n21represents the magnetic \feld).17To accomplish this, the MRCI module of the ORCA program\nhas been used. Within the CASSCF/NEVPT2/MRCI calculations the RI approximation\n(with the def2-TZVP/C auxiliary basis set) has been used.18,19\nFor systems with spin S\u00151, zero-\feld splitting leads to splitting of spin states that\nare otherwise degenerate. For analyzing this e\u000bect in a simpli\fed manner, e\u000bective spin\nHamiltonians are often used. The E\u000bective Hamiltonian method9,10is used to construct the\nspin Hamiltonian for a proper orientation (see discussion above and Eq. S1). The eigenvalues\nof this Hamiltonian reproduce the splitting of a spin ground state, into a (2 S+ 1)-fold\nmultiplet. For example, neglecting the E-term, theS= 2 state (of, e.g., a Fe2+ion), splits\ninto states with Ms= 0,Ms=\u00061, andMs=\u00062, separated by Dand 3D, respectively.\nLikewise, the S= 5=2 state (of, e.g., a Fe3+ion), splits into states with Ms=\u00061=2,\nMs=\u00063=2, andMs=\u00065=2, separated by 2 Dand 4D, respectively. While this splitting\nbehavior can be that simple (see below, for S= 5=2), more complicated cases arise for E >0\nand involve a mixing of Msstates (see main text for S= 2, Fig. 4b).\nWe analyzed the character of 2S+1 states in Fig. 4b of the main text by calculating an\naveraged spin-projection eigenvalue along the z-axis:\n\u0016Ms=1\n2S+ 12S+1X\nijwi\u0001Ms;ij (S3)\nwhere,wiis the weight of the eigenstate of the spin projection operator along the z-axis with\nthe eigenvalue Ms;i.\u0016Msis calculated for all 2S+1 states.\nE\u000bect of neglecting the ethyl groups\nTo make sure that our simpli\fed Fe-P-Cl model system is able to represent the characteristics\nof the Fe-OEP-Cl system, a test calculation for the neutral compound at its equilibrium\ngeometry has been performed. We compare the resulting Fe-Cl bond length as well as the\nenergies of the magnetic states (and the anisotropy parameters). The results are shown\n22together with the geometry of the Fe-P-Cl complex in Fig. S6. We conclude the ethyl side\ngroups have a negligible e\u000bect on the electronic and magnetic properties of the system.\nThis is the reason why in all other calculations presented in this work, the simpli\fed and\ncomputationally less demanding, Fe-P-Cl model system has been used.\nFe-OEP-Cl Fe-P-Cl \u0001\nr(Fe-Cl) [ \u0017A] 2.2335 2.2250 0.0085\nD [meV] 0.151 0.153 0.002\nE/D 0.001505 0.001739 0.000234\nE(M s=\u00063=2) [meV] 0.303 0.307 0.004\nE(M s=\u00065=2) [meV] 0.908 0.920 0.012\nFigure S6: Left: Optimized structure of the Fe-P-Cl model. Right: Comparison between\ngeometrical and anisotropy parameters. The deviations, \u0001, between both models are rather\nsmall. As a consequence, the computational less demanding model (Fe-P-Cl) has been used\nin our study.\nPotential energy curves at the B3LYP level of theory\nAs already stated in the main text, geometries were optimized at selected, \fxed Fe-Cl dis-\ntances with the B3LYP method. The electronic energies of the DFT calculations as a function\nof the Fe-Cl distance are plotted in Fig. S7. Since the relative stability of low- vs.high-spin\nstates are highly dependent on the amount of added \\exact\" (Hartree-Fock-like) exchange\nin the DFT functional,20we prefer to show the NEVPT2 energies (at B3LYP geometries) in\nthe main text instead (Fig. 4a there). The main conclusions of this work are una\u000bected by\nthis choice.\nZero-\feld splittings and anisotropy parameters for S= 5=2\nAt all points along the potential energy surface scan, the zero-\feld splitting was calculated.\nThe results (state energies and anisotropy parameters) for the S= 5=2 state of the neutral\n(Fe3+) complex are shown in Fig. S8.\nFor the state energies in Fig. S8a, we observe a splitting of the six-fold degenerate ground\n232.10 2.35 2.60 2.85 3.10\nFe-Cl distance [ ˚A]−250−200−150−100−50050100150Electronic energy [kJ/mol]\nS=5/2\nS=3/2\nS=2\nS=1\nS=1/2Figure S7: Potential energy curves for various spin states of neutral Fe-P-Cl (Fe3+,S=\n1=2;3=2;5=2) and anionic Fe-P-Cl (Fe2+,S= 1;2;S= 0 is at high energy and not shown),\ncalculated with B3LYP/def2-TZVP.\nstateS= 5=2 into three Kramer doublets with Ms=\u00061=2;\u00063=2 and\u00065=2, respectively,\nas expected, when spin-orbit coupling and the spin-spin interaction are considered. At the\nequilibrium Fe-Cl distance of 2.20 \u0017A, the computed state energies, relative to the ground\nstate, are E 1= 0:3 meV and E 2= 0:91 meV. Consequently, the excitation energies are\n\u000f1= 0:3 meV (E 1-E0) and\u000f2= 0:61 meV (E 2-E1), respectively. These excitation energies\nare lower than in the experiment (approx. 1/5 of the experimental value). Since lower\ncalculated excitation energies have been observed also for other iron(III) porphyrin halides\nby Stavretis et al.9who used a very similar method as we do, this seems to be a systematic\ne\u000bect.\nFurther, we see that with increasing Fe-Cl bond length, the excitation energies become\nlarger. This is in agreement with experiment, see Fig. 2a of the main text, for the region\n\u0000120 pm\u0014\u0001z\u00140 pm.\nIn Fig. S8b, we show the ZFS anisotropy parameters DandEderived from the model\nHamiltonian (Eq. S1) by comparing to the state energies calculated from Eq. S2, as a function\nof Fe-Cl distance. (For instance, according to the model Hamiltonian we have \u000f1= E 1-\n24a bFigure S8: (a) Zero-\feld splitted states as a function of Fe-Cl distance for the neutral, S=\n5=2 system calculated using the CASSCF/NEVPT2/MRCI methodology. (b) Anisotropic\nZFSDandEparameters extracted with the e\u000bective Hamiltonian method.\nE0\u00182Dand\u000f2= E 2-E1\u00184D, see above.) At around the equilibrium bond length, we \fnd\nD\u0018+0:15 meV and E\u00180.Dincreases with bond length.\nAnisotropy parameters for S= 2\nWe also extracted the anisotropy parameters DandEfor the anionic ground state, S= 2.\nThis is shown in Fig. S9. In this case DandEvalues show a more complex behavior, and\nboth being negative. However, we should also note that in this case there is some ambiguity\nwhen determining parameters, since di\u000berent data points belong to di\u000berent orientations of\nthe anisotropy axes.\nInteraction with a magnetic \feld\nIn addition to the simulation of zero-\feld splittings, the in\ruence of an additional magnetic\n\feldBhas been investigated, oriented along the z-axis (the Fe-Cl axis). In relation to the\nmeasurements presented in Fig. 3a and b of the main text at a distance of \u0001 z= -150 pm,\nwe performed simulations at the relaxed geometry of the negatively charged Fe(II) complex\n252.10 2.35 2.60 2.85 3.10\nFe-Cl distance [ ˚A]−0.5−0.4−0.3−0.2−0.10.0E [meV]\nD\nEFigure S9: Anisotropic ZFS parameters DandEas a function of Fe-Cl distance, calculated\nfor the anionic S= 2 manifold using the CASSCF/NEVPT2/MRCI methodology.\nwith a spin quantum number of S= 2. In Fig. S10 we show states arising from the \\exact\"\nHamiltonian (S2) including the Zeeman term.\nSince the degenerate ground states (no \feld) are composed of the eigenvectors of the\nspin projection operator ( z-axis) with eigenvalues of Ms=\u00062 (see Fig. 4a of the main\ntext), the energies of these states are heavily a\u000bected by the external magnetic \feld. The\nthird and fourth state ( Ms=\u00061) are less a\u000bected and the energy of the \ffth state does\nnot change at all ( Ms= 0). Since only transitions between states with \u0001 Ms= 0;\u00061 are\nallowed, an increasing excitation energy is the consequence (from the ground to the second\nexcited state). This is in agreement with the experiment, where an increasing \feld-dependent\nexcitation energy is observed.\nReferences\n(1) Heinrich, B. W.; Braun, L.; Pascual, J. I.; Franke, K. J. Nano Letters 2015 ,15,\n4024{4028\n(2) Heinrich, B. W.; Braun, L.; Pascual, J. I.; Franke, K. J. Nature Physics 2013 ,9,\n26Figure S10: Ground and excited states of the anionic (Fe2+) Fe-P-Cl model for S= 2, at its\nequilibrium geometry. The state energies are shown as a function of strength of an applied\nmagnetic \feld along the z-axis. Arrows indicate allowed excitations from the ground state.\n765{768\n(3) Atanasov, M.; Aravena, D.; Suturina, E.; Bill, E.; Maganas, D.; Neese, F. Coordi-\nnation Chemistry Reviews 2015 ,289, 177{214\n(4) Ternes, M. New Journal of Physics 2015 ,17, 063016\n(5) Neese, F. WIREs Computational Molecular Science 2011 ,2, 73{78\n(6) Becke, A. D. The Journal of Chemical Physics 1993 ,98, 5648{5652\n(7) Weigend, F.; Ahlrichs, R. Physical Chemistry Chemical Physics 2005 ,7, 3297\n(8) Neese, F.; Wennmohs, F.; Hansen, A.; Becker, U. Chemical Physics 2009 ,356,\n98{109\n(9) Stavretis, S. E.; Atanasov, M.; Podlesnyak, A. A.; Hunter, S. C.; Neese, F.; Xue, Z.-\nL.Inorganic Chemistry 2015 ,54, 9790{9801\n(10) Ganyushin, D.; Neese, F. The Journal of Chemical Physics 2006 ,125, 024103\n27(11) Malmqvist, P.- \u0017A.; Roos, B. O. Chemical Physics Letters 1989 ,155, 189{194\n(12) Angeli, C.; Cimiraglia, R.; Evangelisti, S.; Leininger, T.; Malrieu, J.-P. The Journal\nof Chemical Physics 2001 ,114, 10252{10264\n(13) Angeli, C.; Cimiraglia, R.; Malrieu, J.-P. Chemical Physics Letters 2001 ,350,\n297{305\n(14) Angeli, C.; Cimiraglia, R.; Malrieu, J.-P. The Journal of Chemical Physics 2002 ,\n117, 9138{9153\n(15) Angeli, C.; Bories, B.; Cavallini, A.; Cimiraglia, R. The Journal of Chemical\nPhysics 2006 ,124, 054108\n(16) Neese, F. The Journal of Chemical Physics 2005 ,122, 034107\n(17) Atanasov, M.; Ganyushin, D.; Sivalingam, K.; Neese, F. In Molecular Electronic\nStructures of Transition Metal Complexes II ; Mingos, D. M. P., Day, P., Dahl, J. P.,\nEds.; Springer Berlin Heidelberg: Berlin, Heidelberg, 2012; pp 149{220\n(18) Ganyushin, D.; Gilka, N.; Taylor, P. R.; Marian, C. M.; Neese, F. The Journal of\nChemical Physics 2010 ,132, 144111\n(19) Eichkorn, K.; Weigend, F.; Treutler, O.; Ahlrichs, R. Theoretical Chemistry Ac-\ncounts: Theory, Computation, and Modeling (Theoretica Chimica Acta) 1997 ,97,\n119{124\n(20) Reiher, M.; Salomon, O.; Hess, B. A. Theoretical Chemistry Accounts 2001 ,107,\n48{55\n28"    },    {        "title": "1801.06672v1.Magnetic_Behaviour_of_Assemblies_of_Interacting_Cobalt_Carbide_Nanoparticles.pdf",        "content": "Magnetic Behaviour of Assemblies of Interacting Cobalt -Carbide \nNanoparticles  \nPallabi Sutradhar a, Shiv N . Khannab, and Jayasimha Atulasimhaa \na. Dept. of Mechanical and Nuclear Engineering, Virginia Commonwealth Univ., Richmond, VA, USA  \nb. Dept. of Physics, Virginia Commonwealth Univ., Richmond, VA, USA  \nEmail:  sutradharp@vcu.edu  (Pallabi Sutradhar) snkhanna@vcu.edu (Shiv N. Khanna), \njatulasimha@vcu.edu (Jayasimha Atulasimha)  \n \nAbstract: Recent work [1] demonstrated high coercivity and magnetic moment in cobalt carbide \nnanoparticle assemblies and explained the high coercivity  from first principles  in terms of the high \nmagnetocrystalline anisotropy  of the cobalt carbide nanoparticle s. In this work, we \ncomprehensively model the interaction between the nanoparticles comprisi ng the assembly and \nsystematically understand the effect of particle size, distribution of the orientations of the \nnanoparticles’ magnetocrystalline anisotropy axis with respect to the applied magnetic field, and \ndipole coupling between nanoparticles on th e temperatur e dependent magnetic behavior of the  \nnanoparticle assembly. We show that magnetocrystalline anisotropy alone is not enough to explain \nthe large hysteresis over the 50K -400K temperature range and suggest that defects and \ninhomogeneties that pin the magnetization could a lso play a significant role on this temperature \ndependent magnetic behavior.  \n \nPermanent magnets are used in an increasing number o f applications and are typically  alloys that \nhave 4f elements (rare earth materials). These 4f elements lead to both high magnetic moment and \nhigh magnetocrystalline anisotropy that lead to properties desirable in permanent magnets: high \nremanence and high coercivity [2]. Ho wever , the mining process for  rare earth materials is \ndetrimental to the environment, their yield is  low and their extraction cost is high [3]. This \nmotivates research on  rare earth free  permanent magnets with  high coer civity and high magnetic \nmome nt. \nOne way to make such permanent magnets is by employing exchanged -coupled hard and soft \nmagnetic material, which is kn own as ‘exchange -spring magnet’ [4]. Exchange spring magnets \nhave high coercivity and high magnetic moment due to exchange interaction between a hard \nmagnetic material (whose moment may not be very high) and a soft magnetic material  with high \nmagnetic moment.  Thus, exchange spring magnets that have low rare earth material content \n(<15% hard magnetic layer) in combination with an exchange coupled high magnetic moment soft \nlayer can exhibit the desirable properties of hard magnets  [5]. Furthermore, by coupling an \nextremely hard FePt  phase  layer [6] with a sof t high moment Fe3Pt phase layer , rare earth free h ard \nmagnets can be synthesized [7]. \nAnother approach is the development of core-shell magnetic particles [8], [9] . The authors \nsynthesized Co/CoO nanoparticles where exchange bias between the central soft ferromag net Co  \nwith high magnetic moment  and the surrounding antiferromagnetic oxide CoO greatly enhances  the magnetic anisotropy of the  core-shell system while retaining  high magnetic moment . These \nnanoparticle s have a blocking temperature close to room temperatu re 290K  and coercivity of 0.59 \nT while exhibiting high  saturation magnetic moment [8] . \nRecently , El-Gendy et al   synthesized a phase of cobalt carbide (Co 3C) nanomagnets where the \ncobalt layers are far more separated than in bulk  or thin film cobalt  and embedded with intervening \nlayers of the carbon atoms allowing only partial mixing between C and Co states. These magnets \nexhibit thermal and stable long range ferromagnetic order up to 573±2K (the blocking \ntemperature). This high blocking temperature is due to the separate Co layers and mixing with \ncarbon states which contributes to the unusually high magnetocrystalline anisotropy energy  (MAE)  \n[5] [10]. The observed unusually high MAE in Co3C was explained using first principles \ntheoretical inv estigations  [5].  \nAlthough DFT correctly predicted the role of structure and composition of Co 3C on the high MAE \nand blocking temperature  of individual nanoparticles , studies on the collective behavior of an \nassembly of nanoparticles are needed to understand the behavior of the synthesized nanomagnets. \nIn this work, we have examined several unexplained aspects of the  magnetic  behavior of the Co 3C \nnanoparticle assembly  that have not been modeled  or understood . These include effect of particle \nsize, distribution in orientation of the nanoparticle crystal axis with the global magnetic field and \ndipolar  interaction  between the Co 3C nanoparticles , which is investigated in t his lett er. Such an \nunderstanding is particularly important when one scales f rom understanding the behavior o f \nindividual Co 3C nanoparticles to bulk materials produced from assemblies of such Co 3C \nnanoparticles.  \nOur model for the magnetization behavior  is described next , followed by a discuss ion on the  results \nand a final ly, a summary of  the understanding developed from this work.  \nWe consider a Co 3C system with 16 particles dispersed on a flat surface  as shown in Fig 1 a . In \nthis system , each particle is assumed to act  as a giant classical magnet due to the strong exchange \ncoupling between the spins, a reasonable assumption for d imensions < 100 nm, (see Ref [11]). The \nparticles are assumed to be spherica l in shape with diameter ~ 4 nm. Such a system is shown in Fig  \n1. The variation of magnetization of the entire system with time under the influence of an effective \nfield 𝐻𝑒𝑓𝑓 is described by the Landau -Lifshitz -Gilbert (LLG) equation  [12]:  \n         (1+𝛼2)𝑑𝑀⃗⃗ \n𝑑𝑡=−𝛾𝑀⃗⃗ ×𝐻𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗ −𝛼𝛾\n𝑀𝑠[𝑀⃗⃗ ×(𝑀⃗⃗ ×𝐻𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗ )]                          (1). \nHere 𝛾 is gyromagnetic ratio and 𝛼 is the damping factor . \n  \nFig.1: (a) 16 particle Co 3C system, each particle with radius 2  nm and (b) the co -ordinate system . \nThe total effective field ( 𝐻𝑒𝑓𝑓) on each Co 3C particle is the sum  of fields due to Zeeman  energy , \nmagnetocrystalline anisotropy  energy , dipole interaction  energy  and a random field that simulates \nthe effect of thermal noise at a give n temperature : \n𝐻𝑒𝑓𝑓= 𝐻𝑧𝑒𝑒𝑚𝑎𝑛+ 𝐻𝑚𝑎𝑔𝑛𝑒𝑡𝑜𝑐𝑟𝑦𝑠𝑡𝑎𝑙𝑙𝑖𝑛𝑒  𝑎𝑛𝑖𝑠𝑜𝑡𝑟𝑜𝑝𝑦+𝐻𝑑𝑖𝑝𝑜𝑙𝑒+𝐻𝑡ℎ𝑒𝑟𝑚𝑎𝑙       (2). \nThe field due to Zeeman energy i s the applied magnetic field in the y -direction as shown in Fig.  \n1. \n𝐻𝑧𝑒𝑒𝑚𝑎𝑛= 𝐻𝑎𝑝𝑝𝑙𝑖𝑒𝑑  𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐  𝑓𝑖𝑒𝑙𝑑 \nThe field due to magnetocrystalline anisotropy energy is given below:  \n        𝐻𝑚𝑎𝑔𝑛𝑒𝑡𝑜𝑐𝑟𝑦𝑠𝑡𝑎𝑙𝑙𝑖𝑛𝑒  𝑎𝑛𝑖𝑠𝑜𝑡𝑟𝑜𝑝𝑦 = 2𝐾𝑢\n𝜇0𝑀𝑠𝑎𝑡(𝑢⃗ .𝑚⃗⃗ )  𝑢⃗⃗⃗                                      (3). \nHere 𝐾𝑢 is the magnetocrystalline anisotropy energy constant  which is calculated by first principle \nDFT analysis (𝐾𝑢=7.5±1×105 J/m3), 𝑀𝑠𝑎𝑡 is saturation magnetization  estimated from experiment \n[5], 𝑢⃗  is the direction of the easy axis for each Co3C particle, 𝑚⃗⃗  is the magnetization direction  and \nµo is the permeability of free space . \nConsider   two  adjacent   particles  in a system    (labeled   as   the ith  and jth  elements),   whose \nmagnetizations have polar and azimuthal angles of 𝜃𝑖,𝜑𝑖 and 𝜃𝑗,𝜑𝑗, respectively then magnetic \ndipole energy is given by:  \n𝐸𝑑𝑖𝑝𝑜𝑙𝑒𝑖−𝑗= 𝜇0𝑀𝑠2𝛺2\n4𝜋𝑅3∑−2(sin𝜃𝑖cos𝜑𝑖)(sin𝜃𝑗cos𝜑𝑗)+(sin𝜃𝑖sin𝜑𝑖)(sin𝜃𝑗sin𝜑𝑗)+𝑖+1\n𝑖−1\n𝑗≠𝑖\ncos𝜃𝑖cos𝜃𝑗                    (4). \nwhere M s is the saturation magnetization, Ω is the volume  of each Co3C particle and R is the \nseparation  between their centers.  \nThe effective field due to dipole energy is:  \n                                       𝐻= −1\n𝜇0Ω 𝑑𝐸𝑑𝑖𝑝𝑜𝑙𝑒\n𝑑𝑀⃗⃗                                                      (5). \nThermal noise at a given temperature manifests itself as a random magnetic field given by:  \n                                                 𝐻𝑡ℎ𝑒𝑟𝑚𝑎𝑙=√2𝑘𝑏𝑇𝛼\nµ0𝑀𝑠𝛾𝑉ℎ                                                      (6). \nHere 𝑘𝑏 is Boltzman constant, T is the temperature and V is the volume of each Co 3C particle . \n \nAfter calculating the magnetization orientation as a function of  time under an effective magnetic \nfield using the LLG equation, w e calculated the magnetization in the y-direction  for each Co3C \nparticle for a particular applied magnetic field (in the y -direction) using the equation given \nbelow:  \n                                    𝑚=1\n𝑛∑𝑀⃗⃗  sin𝜃𝑖sin∅𝑖𝑛\n𝑖=1                                           (7). \nIn our study , we first calculated the response of the Co 3C system without dipole interaction \nbetween Co 3C particles  by considering  around 200 non-interacting Co particles  whose  \nmagnetization was considered to evolve  independently with time. In the next case  we considered \ndipole interaction  and chose a 16 particle system uniformly distri buted on a surface as shown in \nFig 1a. Here the magnetization of each particle was considered to evolve under the influence of \ndipole coupling of the nei ghboring  particles. We set the separation between the surfaces of \nneighboring spherical Cobalt Carbide nano particles to be 0.1nm  (separation distance assumed  to \nbe small to study  the effect of large dipole coupling ).While calculating the dipole interaction, we \nassumed that only the nearest neighbors have influence on the magnetization on each particle.  \nFinally, we note that after applying each increment in magnetic field, we allowe d sufficient time \n(typically ~10 to 100  nanoseconds) to allow the system to come to an equilibrium magnetic \norientation (though there is always some fluctuation due to thermal noise), as we are interested in \nsimulating quasistatic M -H curves.  \n \nWe have simulated  the effect of size of the particles, different distribution of the easy axis and \neffect of dipole interaction between particles.  \n1. Effect of Co 3C particle size on the overall response of the system :  Fig 2 shows the size \neffect of Co 3C parti cles. The easy axes of these Co 3C particles were  distributed  uniformly \nbetween 60° to 120°  in the co -ordinate axis as shown in Fig  3b. In other words, the easy \naxis of the magnetic particles are uniformly distributed within ±30o from the direction of \napplication of the magnetic field. The simulations were performed at  a tempe ratur e of 50K. \nThe figure  show s that as we decrease the size of the Co3C particles the area of  the hysteresis \nloop decreases as the effect of thermal noise at a given temperature  increases when the \nparticle size is reduced . If we continue to decrease size of the particles, they reach to \nsuperparamagnetic limit and the system would show  no hysteresis.  Fig. 2 shows that the \nparticle size with radius 1.7  nm best matches the experiment  at 50K . However , the \nexperiments were performed with  particles with a radius of around ~  3nm [1]. The discrepancy between theory and experime nt is potentially due to inactive outer carbon \nlayers  and it is likely that the magnetically active Co 3C are have a nominal radius ~ 2 nm . \n \nFig 2.  M-H curve for 1.7nm, 1.9nm, 2nm and 2.5nm radius Co 3C particles . Experimental data \nfrom Ref [9].  \n1. Effect of the  distribution of easy -axes of Co 3C particles for a specific size : In Fig  3a we \nchanged the distribution of easy axes while k eeping the size  of each particle  2nm and the \ntemperature 50K . We can see from the figure that changing the distribution of the easy \naxes of the particles affect s the shape of the hysteresis loop. When easy axe s of all the \nparticles are aligned along the applied magnetic field , the magnetization of all the particles \nrotate abruptly through 180o leading to a sudden change in magnetization. For the case of \nuniform distribution  of the orientation of the easy axis with respect to the magnetic field \ndirection , the easy axes  which are aligned closer to the applied field switch abruptly but \nneed smaller fields to drive the magnetiza tion to saturation while the easy axes which are \nfurther  away  from the applied field rotate gradually but need larger fields to drive the \nmagnetization to saturation . Therefore , there is no abrupt change in the magnetization for \nthe uniformly distributed  system as  is shown in Fig  3a. By evaluating different kinds of \ndistribution for our system we found out that the distribution for which the easy axes are \ndistributed uniformly between 60° to 120° best matches the experiment . While no magnetic \nfield was appl ied during the assembly of the Co 3C nanoparticles (a strong magnetic field \ncould potentially promote a preference for the magnetic easy axis of each particle aligning \nclose to the direction of this applied field thus enhancing the coercivity and remanence  of \nthe assembly) , the dipolar coupling between the  these nanoparticles  results in some \ntexturing in the nanoparticle assembly. This is evident from simulations in Fig 3 (a) as a \nrandom distribution in orientation of the magnetic easy axis leads to a vastly different M -\nH curve than observed experimentally while  magnetic nanop articles’ easy axes uniformly \ndistributed within ±30o (i.e. 60 ° to 120° ) from the direction of application of the magnetic \nfield provides a better qualitative fit to the experimental data.  Hence, in all other figures in \nthis lett er the simulations were per formed assuming the magnetic nanoparticles’ easy axes \nwas uniformly distributed within ±30o from the direction of the magnetic field.  \n \n \n \n  \nFig 3 . (a) M-H curve of Co 3C system with easy axis (i) distributed randomly (ii) uniformly \nbetween 60° to 120° and (iii)all pointing  in the same direction ; compared with (iv)experimental \ndata (b)  Coordinate system showing the e asy axis distribution between 60° and  120° . NOTE: Y -\naxis is the direction of application of the magnetic field.  Experimenta l data from Ref [9].  \n \n \n2. Effect of dipole interaction  on the Co 3C system : In Fig.  4, we consider dip ole interaction \nbetween Co 3C particles while keeping the size  of each particle 1.7nm, the temperature at \n50K and the di stribution of the easy axis between 60° to 120° . We can see from the figure \nthat the area of the hysteresis loop with dipole interaction is slightly smalle r than the loop \nwith no dipole. Consider the assembly modeled in Fig 1, with magnetic field applied along \nthe y -direction which orients t he magnetization of each nanoparticle to eventually align \nparallel to it.  The dipole coupling between nearest neighbors along the y -axis favor parallel \norientation of their magnetization making it easier to rotate the magnetization to point along \nthe y -axis, making the hysteresis loop smaller (compared with the simulations that do not \nconsider dipole effect). However, the dipole coupling between nearest neighbors along the \nx-axis favor anti -parallel orientation of their magnetization, making it harder to fu lly \nmagnetize the entire system, leading to slightly smaller net magn etization (compared to the \nsimulations that do not consider dipole effect) at higher magnetic fields. It is interesting to \nnote that  the dipolar coupling has a relatively weak effect on t he magnetization curves \nsuggesting that the nanoparticles can be packed very closely to increase the saturation \nmagnetization of the assembly without significantly degrading the coercivity . \n \n \nFig 4.  M-H curve of Co 3C system with and without dipole interaction between the particles . \nExperimental data from Ref [9].  \n1. Effect of temperature on the Co 3C system : In Fig  5, we simulate the e ffect of temperature \non Co 3C nano particles. F or 1.7  nm particles , while the hysteresis is modeled well at 50K \nwe get there is no hysteresis at 250K  which differs significantly from experimental results. \nMagnetocrystalline Anisotropy E nergy (MAE  = K uV, where K u is the magnetocrystalline \nanisotropy energy density and V is the volume of the nanoparticle ) is proportional to the \nvolume of the individual Co3C nanoparticles. From Fig  5b we can see that, at 250K , there \nis no hysteresis  for 1.7  nm radius particles  as the volume of these particles are small and \ntherefore their anisotropy energy is small and comparable with the thermal energy  at 250K . \nHowever , for particles with 2.3  nm radius at 250K we get a good match with the \nexperimental hysteresis loop , though in this case the hysteresis at 50K would be over \npredicted  (see Fig 5c) . Further, Fig 5c compares the simulated M -H curves of the 2.3 nm \nradius nanoparticle at 50K, 250K and 400K with experimental data and shows that while \nhysteresis is correctly modeled at 2 50K, it is significantly over -predicted at 50K and \nslightly under predicted at 400K. This indicates that the magnetocrystalline anisotropy \nalone cannot explain the temperature dependence of the M -H curves. D efects  and \ninhomogeneties  that pin the magnetizat ion can enhance the hysteresis over that is predicted \nfrom purely magnetocrystalline anisotropy considerations. Hence, it is likely that a smaller \nnanoparticle size (e.g. ~ 2 nm radius) may explain the hysteresis correctly at 50K but may \nnot show a rapid d ecrease in hysteresis at higher temperatures and model the hysteresis \ncorrectly at 250K and 400K if the effect of magnetization pinning is correctly modeled. \nHowever, i nclusion of such defects that pin the magnetization and modeling incoherent \nswitchin g in such assemblies of nanoparticles in the presence of thermal noise and defects \nis beyond the scope of this work .   But other research [13] suggests that such defects may be an important factor in modeling the behavior of magnetic nanoparticles  and patterne d \ndevices .  \n \n \n \n \n \nFig 5.  (a) M -H curve of 1.7nm Co 3C particles system at 50K (b) M -H curve of 1.7nm and 2.3nm \nCo3C particles system at 250K.  (c) Comparison of simulation vs. experimental data at 50K, \n250K and 400K for particles of  radius 2.3nm Experimental data from Ref [9].  \n \nWe have explored the effect of nanoparticle size, orientation with respect to the magnetic field and \ndipolar coupling b etween these nanoparticles on the magnetic behavior of the Co 3C nanoparticle \nassembly.  While the magnetic behavior is extremely sensitive to particle size and orientation, the \neffect of dipo le coupling is less significant.  This suggests that one could pack  nanoparticles with \nhigh densities in such assemblies to achieve high net magnetization without degrading the hard \nmagnetic behavior (coercivity) significantly. Furthermore, particle size and distribution of \norientation of the easy axis (of magneto crystal line anisotropy) can be used to optimize the \nmagnetic properties of such assemblies. Finally, the temperature dependent be havior is \nqualitatively predicted and hysteresis is explained  by magnetocrystalline anisotropy but the \nreduction in hysteresis with in creasing temperature is significantly over predicted by the model. \nThis suggests that magnetocrystalline anisotropy alone cannot describe the temperature dependent \nhysteresis. D efects that pin the magnetization could potentially enhance the hysteresis (and  \ntherefore coercivity) of these magnetic nanoparticles, making them retain favorable coercivity \neven at elevated temperatures.  \nReferences:  \n[1] A. A. El -Gendy, M. Qian, Z. J. Huba, S. N. Khanna, and E. E. Carpenter, “Enhanced \nmagnetic anisotropy in cobalt -carbide nanoparticles,” Appl. Phys. Lett. , vol. 104, no. 2, p. \n023111, Jan. 2014.  \n[2]  Schrefl,  Fidler, and  Kronmüller, “Remanence and coerciv ity in isotropic nanocrystalline \npermanent magnets,” Phys. Rev. B Condens. Matter , vol. 49, no. 9, pp. 6100 –6110, Mar. \n1994.  \n[3] S.-B. Cheng, C. Berkdemir, and A. W. Castleman, “Mimicking the magnetic properties of \nrare earth elements using superatoms,” Proc. Natl. Acad. Sci. U. S. A. , vol. 112, no. 16, pp. \n4941 –4945, Apr. 2015.  \n[4] E. F. Kneller and R. Hawig, “The exchange -spring magnet: a new material principle for \npermanent magnets,” IEEE Trans. Magn. , vol. 27, no. 4, pp. 3588 –3560, Jul. 1991.  \n[5] E. E. Fullerton, J. S. Jiang, and S. D. Bader, “Hard/soft magnetic heterostructures: model \nexchange -spring magnets,” J. Magn. Magn. Mater. , vol. 200, no. 1, pp. 392 –404, Oct. 1999.  \n[6] S. Sun, C. B. Murray, D. Weller, L. Folks, and A. Moser, “Monodisperse FePt N anoparticles \nand Ferromagnetic FePt Nanocrystal Superlattices,” Science , vol. 287, no. 5460, pp. 1989 –\n1992, Mar. 2000.  \n[7] H. Zeng, J. Li, J. P. Liu, Z. L. Wang, and S. Sun, “Exchange -coupled nanocomposite \nmagnets by nanoparticle self -assembly,” Nature , vol. 420, no. 6914, p. 395, Nov. 2002.  \n[8] D. Givord, G. Hadjipanayis, J. Nogués, S. Stoyanov, V. Skumryev, and Y. Zhang, “Beating \nthe superparamagnetic limit with exchange bias,” Nature , vol. 423, no. 6942, p. 850, Jun. \n2003.  \n[9] I. K. Schuller and J. Eisen menger, “Magnetic nanostructures: Overcoming thermal \nfluctuations,” Nat. Mater. , vol. 2, no. 7, p. 437, Jul. 2003.  \n[10] A. A. El -Gendy, M. Bertino, D. Clifford, M. Qian, S. N. Khanna, and E. E. Carpenter, \n“Experimental evidence for the formation of CoFe2C phase with colossal \nmagnetocrystalline -anisotropy,” Appl. Phys. Lett. , vol. 106, no. 21, p. 213109, May 2015.  \n[11] R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E. Welland, and D. M. Tricker, \n“Single -Domain Circular Nanomagnets,” Phys. Rev. Lett. , vol. 8 3, no. 5, pp. 1042 –1045, \nAug. 1999.  \n[12]  T. L. Gilbert, “A Phenomenological Theory of Damping in Ferromagnetic Materials”, IEEE \nTRANSACTIONS ON MAGNETICS, 40, 3443, 2004.   [13]       A. C. Chavez, W. -Y. Sun, J. Atulasimha , K. L. Wang, and G. P. Carman, “Voltage induced artificial \nferromagnetic -antiferromagnetic ordering in synthetic multiferroics”, Journal of Applied Physics 122, \n224102, 2017.  \n "    },    {        "title": "1801.08511v1.Heusler_compounds____how_to_tune_the_magnetocrystalline_anisotropy.pdf",        "content": "Heusler compounds { how to tune the magnetocrystalline\nanisotropy\nH. C. Herper\u0003\nDepartment of Physics and Astronomy,\nUppsala University, Box 516, 751 20 Uppsala, Sweden\nAbstract\nTailoring and controlling magnetic properties is an important factor for materials design. Here,\nwe present a case study for Ni-based Heusler compounds of the type Ni 2YZ with Y = Mn, Fe,\nCo and Z = B, Al, Ga, In, Si, Ge, Sn based on \frst principles electronic structure calculations.\nThese compounds are interesting since the materials properties can be quite easily tuned by com-\nposition and many of them possess a non-cubic ground state being a prerequisite for a \fnite\nmagnetocrystalline anisotropy (MAE). We discuss systematically the in\ruence of doping at the Y\nand Z sublattice as well of lattice deformation on the MAE. We show that in case of Ni 2CoZ the\nphase stability and the MAE can be improved using quaternary systems with elements from group\n13 and 14 on the Z sublattice whereas changing the Y sublattice occupation by adding Fe does not\nlead to an increase of the MAE. Furthermore, we studied the in\ruence of the lattice ratio on the\nMAE. Showing that small deviations can lead to a doubling of the MAE as in case of Ni 2FeGe.\nEven though we demonstrate this for a limited set of systems the \fndings may carry over to other\nrelated systems.\n1arXiv:1801.08511v1  [cond-mat.mtrl-sci]  25 Jan 2018I. INTRODUCTION\nThe demand for new magnetic materials is bigger than never before. Applications are\nvery diverse and comprise permanent magnets for cars and wind turbines, actuators, memory\ndevices, as well as for magnetic cooling where di\u000berent applications also demand for di\u000ber-\nent technical speci\fcations of the magnetic material. An eminent goal is to identify new\nmaterials for magnetic applications. A resource saving and often faster way compared to ex-\nperiments is computational materials design using ab initio methods or atomistic modeling.\nIn principle two di\u000berent routes exist: high throughput data mining to \fnd unknown phases\nor optimization and modi\fcation of known structures. In practice often high throughput\nstudies are carried out with certain constraints on the structure or other properties. A recent\nexample can be found in Ref. 1 where this technique has been used to \fnd new magnetic\nHeusler compounds. In the latter case the challenge is to \fnd out on which screw to turn\nto optimize all relevant properties. From \frst principles point of view the focus is on three\nproperties as depicted in Fig. 1: the stability of the phase, a ferromagnetic phase with suit-\nable high magnetization, and last but not least the magnetocrystalline anisotropy (MAE)\nwhich is crucial for the magnet. As depicted in Fig. 1 the properties are related to each\nT\t=\t0\tKT\t=\t0\tKT\t>\t0\tKCurie\ttemperatureTemperature\tdependentmagnetization\nFIG. 1: Computational design of magnetic materials: Basic properties which can be obtained\nfrom \frst principles calculations ( T= 0 K) and temperature dependent quantities which can be\ndetermined from additional atomistic modeling.\n2other, e.g. a large magnetization without a suitable large MAE will not result in a hard\nmagnet because the coercivity would be too small. Adding atomistic model calculations fur-\nther properties such as the Curie temperature can be predicted. However, here the focus is\non the basic properties shown on the left side of Fig. 1. Using Ni-based Heusler compounds\nas model system we discuss the in\ruence of mechanical deformation, alloying, and electronic\nstructure on the basic magnetic properties. Ni-based Heusler compounds are known to show\nin certain compositions a tetragonal instability which makes them an ideal test system even\nif the expected Curie temperatures are too low for high performance magnets. During the\nlast decades Heusler alloys have been discussed as possible candidates for di\u000berent magnetic\napplications, e.g. halfmetallic Co 2FeSi for spintronics applications2{4, as rare earth (RE) free\npermanent magnets5,6, or Ni-based actuators7,8and magneto-caloric materials9,10, because of\ntheir magnetic and electronic properties can be quite easily tuned by composition.11{14De-\npending on the application the key properties are a high spin polarization, a large magneto\ncrystalline anisotropy (MAE), a high Curie temperature or a large magnetic shape memory\ne\u000bect.15It has been shown that Co based Heusler alloys such as Co 2FeSi and Co 2MnSi are\nhalf metallic ferromagnets with magnetic moments following the Slater-Pauling curve and\nvery high Curie temperatures16,17, Ni based systems, e.g. Ni 2MnGa or o\u000b-stoichiometric\nNi-Mn-Z (Z = Sn, Sb, In) are well known for their shape memory behavior.13,18{20Recently\nthe MAE on Mn based Heusler alloys has been discussed in view of their suitability for spin\ntransfer torque applications12. These systems are ferrimagnetic with a small net moment\nof 1-2\u0016Bbut MAE values up to 1meV/f.u. Furthermore, the search for new rare earth\nfree or lean ferromagnets has become highly topical since permanent magnets with high\nMAE are needed en masse, such that cheap and abundant alternatives to the critical RE\nmagnets are needed. Here we focus on ferromagnetic Heusler systems and in particular on\ntheir magnetic properties. The goal is to reveal routes to improve the magnetic properties\n{ in particular the MAE. Even though the absolute values might not reach the MAE of\nRE or Pt containing materials and the Curie temperatures are expected to be lower than\nfor high performance magnets the properties might still be comparable to bonded magnets\nand ferrites but Heusler alloys have a huge potential due to the easy tuning by composition\nand they are comparably cheap. Heusler alloys X 2YZ usually crystallize in L2 1structure\nwith point group 225 (Fm \u00163m symmetry). It consists of four interpenetrating fcc lattices,\nsee Fig. 2. In ordinary magnetic Heusler alloys the sub lattices A and C are occupied by\n3the metal X whereas another transition metal Y sits on the B site. The last sub lattice\nD is occupied by a main group element Z. In some cases the occupation of the B and C\nsub lattice is interchanged which leads to a reduction in symmetry, i.e. point group 216\n(F\u001643m symmetry) and the so called inverse Heusler structure. Which structure is preferred\ndepends in general on the choice of the X and Y element. For systems with the X element\nhaving a higher atomic number than the Y element the normal L2 1structure is assumed\nto be the most stable structure whereas inverse ordered compounds appear for the opposite\ncase. However, we will show that the rule is not strictly followed by Ni 2YZ compounds.\nNi based Heusler alloys have attracted quite some interest in view of ferromagnetic shape\nmemory alloys (FSMA). They tend to possess a tetragonal instability, i.e. they undergo\na martensitic transformation from the high temperature cubic phase to a low temperature\ntetragonal distorted or in some cases to a modulated phase (e.g. 5M,14M)21{23. Since a\ntetragonal instability is fundamental for shape memory alloys and magneto caloric systems\nquite some e\u000bort has been done to \fnd tetragonally distorted Heusler systems. Furthermore,\nthe tetragonal distortion gives rise to a MAE and its dependence on the occupancy of the\nY lattice and the choice of the main group element is discussed here. The appearance and\nstability of the tetragonal state depends on the choice of the Y metal and the main group ele-\nment. Here we study N 2YZ Heusler alloys with Y = Mn, Fe, Co and Z being a III or IV main\ngroup element and the trends in the magnetic properties especially the MAE depending on\nthe Y and Z element within di\u000berent DFT methods. Even though the calculated MAE does\nnot reach the high values of Pt containing Heusler alloys we show that reasonable values due\nto lattice deformation and out-of plane orientation of the easy axis can be achieved without\n5d or RE elements and they bear certain potential for magnetic applications.\nAfter a brief description of the computational methods in Section II the stability of the\nHeusler compounds is discussed in Sec. III. Section IV focusses on the MAE and the mag-\nnetic moments pointing out trends and possibles routes to increase the MAE followed by\nconcluding remarks in Sec. V.\nII. METHODS\nThe electronic and magnetic structure of Ni 2YZ Heusler compounds has been investigated\nwithin the VASP code24,25employing the projected augmented wave (PAW) potentials26\n4and the approximations of Perdew, Burke, and Ernzerhof27for the exchange correlation\nfunctional. The calculations of the stoichiometric systems have been preformed within the 4\natomic primitive cell, see Fig. 2 (b). Systems have been relaxed to forces below 0.01 eV/ \u0017A.\nFor calculations within the primitive cell a 173k-point mesh has been used for structural\noptimization. Quaternary systems and systems with site disorder have been treated in a\n16 atomic cubic cell using a k-point mesh of 133. The c/a variation has been performed\nwith the same accuracy but the volume and the shape of the unit cell have been \fxed if not\nstated otherwise.\nThe MAE calculations have been performed within the full potential linearized augmented\nplane wave (LMTO) approach employing the RSPt code28whereby the optimized structures\nfrom the previous VASP investigations serve as input for the LMTO code and no further\nstructural optimization has been done. For consistency the same functional for exchange\nand correlation has been chosen as before. A mesh of 54 \u000254\u000254 points has been used to\ncalculate the MAE from the 4 atomic unit cells after a k-point convergence check. In case\nof o\u000b-stoichiometric systems and 16 atomic super cells slightly smaller meshes were used.\nWithin the RSPt code the wave functions are expanded to lmax= 8.\nFIG. 2: (a) Primitive cell of the cubic L2 1Heusler structure and the corresponding unit cell (b).\nIn ordinary Heusler alloys with space group 225 the sublattices A an C are occupied by material\nX, B hosts the Y metal, and the main group element Z is located on the D sub lattice. For the\ninverse Heusler structure (Hg 2CuTi) the occupation of the sub lattices B and C is inverted.\n5III. STRUCTURAL STABILITY\nA. Cubic vs. tetragonal\nHeusler alloys with Mn on the Y site and a main group element from group III or IV\non the Z site turned out to be regular ordered alloys with Fm \u00163m symmetry in the cubic\nphase. However besides the well known Ni 2MnGa only Ni 2MnB possesses a tetragonal\ninstability with c/a = 1.38 and a local minimum at c/a = 0.9 which was also found in\nliterature29, see Fig.3(a). All other investigated Ni 2MnZ alloys have a cubic ground state.\nThe only controversially discussed system is Ni 2MnGe which in agreement with experimental\n\fndings by Oksenenko et al.30turns out to be cubic from the present calculations using PAW\npotentials but Luo et al. observed tetragonal ground state from DFT calculations with US\npseudo potentials31. In addition calculations of the phonon spectra indicate a softening of\nthe TA 2mode which also hints to an instability of the cubic phase. A close look to E(c/a)\ncurve reveals that there is an indication of a shallow local minimum around c/a = 1.05 being\nonly 1.22 meV higher in energy than the ground state. Hence small distortions or di\u000berent\nchoice of potentials may be su\u000ecient to reverse the order if the local minima and stabilize\nthe tetragonal phase.\nIn agreement with previous investigations29,32,33a tetragonal ground state is observed for\nnearly all investigated systems if the Y site is occupied with Fe. Exceptions are Ni 2FeIn\nand Ni 2FeSn which remain cubic, see Fig.3(b). In case of Sn an indication for a saddle\npoint can be spotted at c/a = 1.2 but no real minimum appears for this system. The global\nminimum for the tetragonal systems occurs at c/a = 1.35, only for the B containing system\nit turns out to be larger thanp\n2. Nearly all investigated Ni 2FeZ prefer the ordinary Heusler\nstructure, however in some cases the energy di\u000berences between the ordinary and inverse\nphase are extremely small, see Fig.4(a) and for a summary of all c/a ratios including values\nfrom literature we refer to Tab. I. A prominent example is the Ni 2FeGe system in which the\ninverse ordered structure is only 0.2 meV in lower energy than the L2 1ordered phase. A\ninverse ordered ground state for Ni 2FeGe has also been observed in previous calculations33\nbut contradicts to the expectations from Burch's rule as discussed by Kreiner et al.33A\nmore detailed study revealed that if we release the constraint of volume conservation the\nL21ordered phase becomes the ground state being 1.86 meV lower in energy than the\n60.8 0.9 1 1.1 1.2 1.3 1.4 1.5\nc/a ratio-0.2-0.100.10.20.3 E(c/a)-E(c/a=1.0) (eV)Al\nGa\nIn\nSi\nGe\nSn\nB\n0.8 0.9 1 1.1 1.2 1.3 1.4 1.5\nc/a ratio-0.4-0.3-0.2-0.100.10.2 E(c/a)-E(c/a=1.0) (eV)B\nAl\nGa\nIn\nSi\nGe\nSn\n0.8 0.9 1 1.1 1.2 1.3 1.4 1.5\nc/a ratio-0.4-0.3-0.2-0.100.10.2 E(c/a)-E(c/a=1.0) (eV)B\nAl\nGa\nIn\nSi\nGe\nSn1.3 1.405101520∆E (meV/f.u.)inv.inv., shape rel.\nL21, shape rel.\nL21Ni2FeZNi2MnZ\nNi2CoZ(a)\n(b)\n(c)\nFIG. 3: Energy as function of the c/a ratio for Ni 2YZ Heusler alloys for Z elements with three (B,\nAl, Ga, In) and four (Si, Ge, Sn) valence electrons. Filled symbols denote normal Heusler structure\nX2YZ. Systems which crystallize in the inverse Hg 2CuTi structure are marked by open symbols.\nPlease denote the di\u000berent scales on the y axis for Y = Mn (a). The inset in (b) shows the E(c/a)\nfor L2 1and inverse ordered Ni 2FeGe in the vicinity of the global minimum.7e/a(Ni 2Y) Y Z\nB Al Ga In Si Ge Sn\n27 Mn here 1.38(r) 1.00(r) 1.25(r) 1.00(r) 1.00(r) 1.00(r) 1.00(r)\n27 other 1.38(r)291.00(r)351.27(r)291.00(r)361.18(r)311.00(r)37\n28 Fe here 1.44(r) 1.35(r) 1.35(r) 1.00(r) 1.36(r) 1.35(r) 1.00(r)\n28 other 1.35(r)291.00(r)31\n1.30(r)381.30(r)32\n(i)33\n29 Co here 1.42(r) 1.38(r) 1.38(r) 1.35(r) 1.30(i) 1.30(i) 1.30(i)\n29 other 1.40(r)381.38(r)39>1.35401.30(r)381.30(r)38\ne/a(Z) 3 3 3 3 4 4 4\nTABLE I: Calculated c/a ratios of the global minimum are given for Ni-based Heusler alloys Ni 2YZ.\nRegular phases are denoted by r, the occurrence on the inverse structure is named i. For comparison\nvalue from literature has been added. e/a(Ni 2Y) and e/a(Z) give the number of valence electrons\nfor the metals (Ni and Y) and the main group element on the Z site, respectively.\ninverse one, see inset of Fig. 3 (b). However, since the energy di\u000berence is so small it is hard\nto determine which structure has to be expected experimentally a since this tiny energy\nbarrier could be overcome at very low temperatures. So far the Heusler structure was not\nobserved for Ni 2FeGe. From recent high-temperature magnetocalorimetry experiments there\nis evidence that at high temperatures a cP4 (L1 2) structure can be stabilized.34However,\nthe Heusler structure might still be produced in thin \flms. For Co containing alloys the\nsymmetry depends on the choice of the Z element, i.e. Z elements from group IV prefer\ninverse order and the c/a ratio of the global minima is around 1.3 compared to 1.35 (1.38\nfor Ni 2CoGa) for the L2 1ordered alloys, cf. Fig. 3(c). It should be noted that in case of Y\n= Co the energy di\u000berence between the global minimum of the L2 1and inverse order can\nbe analogous to the Fe case quite small (see Fig. 4 (a)), i.e. especially for Ni 2CoGa the\ndi\u000berence is only -12.55 meV which corresponds to previous \fndings41moreover the local\nminimum of the inverse structure becomes more stable than the one of the L2 1ordered\nphase (Einv(c=a= 0:92)\u0000EL21(c=a= 0:865) =37.12 meV).\n8Moving from Mn via Fe to Co the energy di\u000berence between the cubic and tetragonal\nphase increases from about 20 meV for Ni 2MnGa to 200 meV for Ni 2CoGa. The Ni 2YB\nsystems are exceptional because the di\u000berence in energy between the tetragonal and cubic\nphase is much larger, i.e. between 240 (Mn) and 400 meV (Co) for the global minima and\n20 to 150 meV for the local minima with c/a <1. The di\u000berence of the total energies of the\nhigh temperature austenite (cubic) phase and the low temperature martensite phase (here:\ntetragonal) can be viewed as a rough measure for the martensitic transition temperature.\nBarman showed that { with certain restrictions { the \u0001 E=E(c=a)min\u0000E(c=a= 1:0)\nbasically proportional to the martensite temperature, i.e. kBTM42. Hence, the larger \u0001 E\nthe higher TMthe tetragonal Ni 2CoZ alloys and Ni 2FeZ (Z = Ge, Ga, Si) phases should\nbe more stable than the well known Ni 2MnGa which transforms at 206 K to the cubic L2 1\nstructure43. Accordingly, the Fe and Co containing systems might be more interesting from\nthe MAE point of view. However, one should keep in mind that relation between \u0001 Eand\nTMis a rough estimation and factors such as the number of valence electrons e=aand the\nstructure (inverse or regular) have some impact. Replacing one sublattice of Ni by Pd leads\nto a substantial shift of the c/a ratio to larger values (c/a = 1.42, see Tab.II) which is\nbelieved to improve the MAE. Besides the larger c/a ratio the energy di\u000berence between the\ncubic and the tetragonal phase in NiPdFeGe is with 0.16 eV/f.u. 3 times larger than the one\nof the isoelectronic ternary Ni 2FeGe system which hints to a higher martensite temperature\nand a larger stability range of the tetragonal phase.\nB. Phase stability\nIn the previous section the stability of the L2 1vs the inverse Heusler structure has been\ndiscussed. However, so far not all investigated systems have been synthesized and it is\nnot known whether the observed ground state structure will be likely to stabilise these\ncompounds experimentally or the system decomposes. In order to shed light on this the\nformation energy Eformhas been investigated. For a given compound Eformcan be obtained\nfrom\nEform=ENi2YZ\u0000(2ENi+EY+EZ) (1)\nwhereENi2YZis the total energy at T = 0 K for the ground state con\fguration of the\nNi2YZ compound. The other terms ENi,EY, andEZcorrespond to the ground state ener-\n9gies calculated for the elemental systems. This can be viewed as a lower boundary and was\nchosen to handle all systems on the same footing. However, in case of stable binaries the\nformation energies might change and systems with small negative formation energies might\nbecome unstable, too. No approximations have been made for the ground state structures\nof the elements, especially \u000b-Mn (I \u001643m structure) and \u000b-B (experimentally observed struc-\nture taken from structure data base ICSD44,45) have been used as reference states. Huge\ndi\u000berences in the stability of the Heusler compound have been observed for the investigated\nsystems depending on the Z element chosen, see Fig. 4(b). Independent from the element\non the Y site all three series show the same trend, compounds with Al and Si are most stable\nwhereas In and B containing compounds tend to decompose. In particular the B containing\nsystems turned out to be not very likely to occur in nature as a Heusler compound. To our\nknowledge all studies discussing N 2MnB for example in view of the high spin-polarization29\nand possible pressure behavior46are based on DFT investigations and so far no experimen-\ntal veri\fcation has been found which agrees with the tendency to decompose observed in\nFIG. 4: (a) Calculated energy di\u000berence between the L2 1and the inverse ordered phase for Ni 2FeZ\n(red squares) and Ni 2CoZ (blue triangles) compounds. For the Mn compounds (brown circles)\nonly the values for the systems with non cubic ground state are shown. (b) Energy of formation\nfor Ni 2YZ Heusler compounds in with Y = Fe (red squares) and Co (blue triangles) as obtained\nfrom Eq. (1). Negative values denote systems stable against decomposition. Lines should only be\nviewed as guide to the eye.\n10the present work. In case of In the trend is less unique. Ni 2MnIn is found to be stable with\na formation energy of -0.43 eV whereas Ni 2CoIn decomposes ( Eform= 0.23eV) and Fe is\nin-between with a formation energy being almost zero. In the last case a reliable conclusion\nabout the stability of this compound cannot be drawn. In case of Ni 2FeIn and Ni 2YSn the\nformation energies are small and the argument regarding elemental reference systems might\napply. Indeed these are the systems of the least importance for this paper because they are\ncubic or provide very small MAE. All other investigated systems turned out to be stable\nin the Heusler structure being consistent with previous theoretical \fndings.32,33In the next\nsection the magnetic properties are discussed focussing on the MAE of the stable systems\nwith non-cubic ground states.\nIV. MAGNETISM\nA. Magnetocrystalline anisotropy of Ni 2YZ\nWe have performed highly accurate MAE calculations within the full potential LMTO\nmethod using the optimized structures from previous VASP calculations. Although the\ninvestigated compounds have similar c/a ratios, number of valence electrons, and comparable\nelectronic structure, the spread in the MAE values turned out to be quite big. For the ground\nstate con\fgurations with c/a >1 it reaches from about 0.42 meV/f.u. to almost zero, see\nFig. 5. The highest MAE values are achieved for systems with Y=Co and a Z element from\ngroup III. In case of Ni 2CoGa and Ni 2CoIn MAE values of 1.30 MJ/m3(0.38 meV/f.u.) and\n1.26 MJ/m3(0.42 meV/f.u). The MAE for the two above mentioned Y = Co compounds\nis about 30% larger than the value obtained for the well known Ni 2MnGa system which\nshows an MAE value of about 0.34 meV/f.u., see Fig.5. Gruner et al. observed an even\nlarger value for c/a >1.25 (MAE = 0.6 meV/f.u.) in Ni 2MnGa47using the fully relativistic\nminimum basis set approach FLPO in the local density approximation.47For the Ni 2MnGa\nsystem numerous experimental studies exist which have studied the anisotropy. The values\nspread from Ku= 1:17\u0001106erg/cm3(0.117 MJ/m3) to 4-5\u0001106erg/cm3(0.4-0.5 MJ/m3),48{50\ndepending on the exact composition, and whether single or multi-variant samples have been\nused. In addition, temperature e\u000bects play a role and can reduce the size of the magneto\ncrystalline anisotropy49. Our calculated anisotropy energy for Ni 2MnGa is of the same\n11order of magnitude as the experimental values for single crystals51by a factor of 3-4 larger\nwhich is a quite good agreement regarding the fact that we consider an ideal system at zero\nKelvin. Furthermore, the comparison to calculated values in other works show a spreading\nin the theoretical data too for example due to the potential used47. The calculated values\ndepend partially on the volume, the computational method and the approximations made\nbut they give the right order of magnitude. Though the absolute values are sensitive to the\ncomputational method the sign is in our cases robust and it is possible to obtain trends within\na series of compounds or alloys calculated on the same footing which can be a guideline for\nthe design of new materials with even larger MAEs.\nAl Ga In Si Ge Sn\nZ element00.511.5|E[100]-E[001]| (MJ/m3)\n00.10.20.30.40.50.6\n|E[100]-E[001]| (meV/f.u.)Y = Mn\nY = Fe\nY = Co\nFIG. 5: Absolute values of the magnetic anisotropy energy per formula unit. The MAE in MJ/m3\n(scale on the left hand side) is given by \flled and hatched symbols where \flled (hatched) denotes\nregular (inverse) ordered Heusler systems. The open symbols show the MAE in atomic units e.g.\nmeV per formula unit (scale on the right hand side). For details see Sec. II. Arrows mark the\noverall trends.\nWhile Co has been proven to be advantageous if Z is taken from main group III (Z = Al,\nGa, In) Ni 2CoZ compounds with Z being Si, Ge, or Sn are less promising. The calculated\n12MAE values are in agreement with Ref. 52 signi\fcantly smaller, e.g. 0.42 MJ/m3in case\nof Ni 2CoGe. In contrast to Ni 2CoGa(In) the easy axis is out-of plane. However, even\nthough these systems are uniaxial ferromagnets the magnetic properties are less good due\nto the inverse order. One reason is the change in the local coordination of the magnetic\nCo atom which has 4 Ni and 4 Z atoms instead on 8 Ni atoms, see Sec. IV. Therefore, the\nmagnetization of the inverse ordered systems is smaller compared to the L2 1type systems.\nIn addition, this e\u000bects the lattice structure and leads to smaller c=avalues for the ground\nstate con\fguration.\nTaking everything into consideration i.e. MAE, magnetic moments, and phase stability\nNi2FeZ (Z = Si, Ge) seem to be advantageous. Both systems provide a tetragonal ground\nstate and an easy axis MAE. The largest MAE occurs for Ni 2FeGe 0.95 MJ/m3, see Fig. 5.\nThis is comparable to the \fndings for the Mn-based tetragonal Heusler systems.12Since for\nNi2FeGe the L2 1and inverse ordered structure are very close in energy (see discussion in\nSec. III) the MAE of the inverse ordered system has also been investigated. It turned out\nto be by a factor of 2 smaller then the one of the regular Heusler compound (c/a= 1.3,\nMAE=0.54 MJ/m3). This observation is in line with the \fndings for the inverse ordered\nNi2CoZ compounds. Summarizing, the largest values for the MAE have been predicted for\nNi2CoZ (Z = Ga, In) and Ni2FeGe which have the same number of valence electrons per\nf.u. (e/a = 32).\nFrom the c/a variation discussed in Sec.III it is obvious the tetragonal distorted systems\nhave not only a global minimum at c=a> 1 but also a local minimum for compressed systems\nwithc=a < 1. In some cases the global minimum is not reached and the local minimum\nc=a< 1 appears. The MAE values for the local minima at c=a< 1 tend to be smaller what\nis at least partially related to the smaller deviation from the cubic structure. For example\nin case of Ni 2CoGa, the system with the highest MAE for c=a > 1, the MAE for the local\nminimum (c=a= 0.87) is with -0.10 MJ/m3by a factor of 10 smaller, see Fig. 6. On the other\nhand moving from elongation to compression is accompanied by a change from easy plane\nto easy axis anisotropy. This has been reported before for Ni-based Heusler compounds\nsuch as Ni 2MnGa47and Ni 2Mn1:25In0:7519. For Ni 2MnGa a MAE of 0.186 meV/f.u. (c/a =\n0.94) is achieved which is in good agreement with the full potential augmented plane wave\ncalculations by Enkovaara et al. (MAE = 0.18 meV/f.u.).53Due to the smaller tetragonal\ndistortion the size of the MAE falls usually behind the one of the previously discussed case\n13forc=a> 1. An exception is Ni 2FeGe (MAE(c/a=0.85)=0.20 meV) system where the change\nis comparably smaller. This underlines that the c/a ratio is not the only determining factor.\nNo or very tiny MAE values are obtained for the inverse ordered Ni 2CoZ systems at c=a< 1\nsince the magnetic moment vanishes with decreasing c/a (Z = Si, Ge) and the tetragonal\ndistortion of the local minimum is small (Z = Sn, c/a= 0.92, MAE = -0.004 meV), see\nSupplement.\nB. Possible routes to tailor the magnetocrystalline anisotropy\nIt is fair to say that some Ni-based Heusler compounds reveal promising MAEs, but the\nmost interesting systems (in view of large MAE values) have positive formation energies,\nare easy-plane systems, or could not be synthesized as bulk systems even if predicted by\ntheory to be stable. Here we focus on possible ways how to tailor the size and the sign of\nthe MAE and simultaneously stabilize the tetragonal phase. Special focus will be on the\ne\u000bect of lattice deformation, forming quaternary compounds to turn easy-plane magnets\ninto easy-axis systems. Knowing that heavier materials provide a larger spin-orbit coupling\nare are therefore likely to possess also a larger anisotropy then 3d materials we consider also\nthe in\ruence of 4d element replacements of Ni. In the present paper we have studied in\nparticular Pd which is isoelectronic to Ni.\n1. Changing the lattice geometry\nChanging the lattice distortion from elongation to compression of the caxis is in many\nHeusler type compounds accompanied by a rotation of the easy axis from in-plane to out-\nof-plane orientation.19,47The variation is almost linear with c=ain some cases \rattening\naround the minima, one example is shown in Fig. 6 for Ni 2CoGa. another example for an\no\u000b-stoichiometric Ni-based Heusler system can be found in Ref. 19. Such a change in the\norientation of the easy axis is not observed for the systems which possess an out-of-plane\naxis forc=a > 1, see supplement and Fig.6. A strong non-linear c=adependence occurs in\ncase of Ni 2FeGe. Reducing the tetragonal distortion by about 10% ( c=a= 1:25) leads to\na 100% increase of the MAE compared to the ground state ( c=a= 1:35), cf Fig. 6. With\n1.96 MJ/m3the MAE is even larger than the values obtained for Ni 2CoZ and what is even\n140.8 0.9 1 1.1 1.2 1.3 1.4 1.5\nc/a-0.4-0.200.20.40.6MAE (meV/f.u.)Ni2FeGe\nNi2CoGaground\n  stateFIG. 6: Dependence of the magneto crystalline anisotropy on the lattice ratio c/a for Ni 2FeGe\n(red diamonds) and Ni 2CoGa (blue triangles) as obtained from RSPt. In case of Ni 2FeGe no sign\nchange i.e. rotation of the MAE from easy axis to planar is observed depending on the c/a ratio\nwhereas Ni 2CoGa undergoes the typical change from planar (negative values) to uniaxial when the\nc/a changes from c=a> 1 toc=a< 1.\nmore important the system is uniaxial. The large MAE and the possibility to tune by small\nlattice distortions might make this system interesting in view of applications. However, to\nour knowledge it has not been successfully synthesized as a bulk system in L2 1structure.\nHowever, one might think to stabilize a structure as a thin \flm where the c/a ratio could be\noptimized by dopants or deposition or the choice of the substrate. The remaining question\nis what drives the changes in the magnetic behavior and how is it related to changes in the\nelectronic structure? Comparing the total density of states (DOS) of the tetragonal ground\nstate (c/a = 1.35) to the DOS of the squeezed structure (c/a = 1.25) and the ground state\n(c/a = 1.3) of the inverse ordered system characteristic changes of the DOS at the Fermi\nlevel can be observed. Hence, the DOS at the Fermi level is dominated by the 3d states of\nthe transition metals (see also Fig. 10) the di\u000berences in Fig. 7 re\rect the changes in the 3d\n15-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6\nE-EF (eV)-6-4-202468Density of states (sts/eV)c/a = 1.35\nc/a = 1.25\nc/a = 1.35, inv.-0.2 0 0.2\nE-EF (eV)-6-4-202DOS (sts/eV)FIG. 7: (a) Calculated total density of states of Ni 2FeGe in the ground state with c/a = 1.35\n(red line, shaded area), for the con\fguration with the largest MAE (c/a =1.25) (dashed line),\nand the global minimum of the Hg 2CuTi ordered structure with c/a = 1.35 (thin blue line). The\ninset shows the region around the Fermi level (vertical line). Here the minority-spin DOS at EF\ndecreases from the inverse ordered structure over the global minimum to the arti\fcially distorted\nstructure with c/a=1.25 whereas the MAE increases at the same time.\nstates. Reducing the c/a ratio leads here to an increase of 2.4% (decrease of 0.9%) of the\nFe-Fe (Fe-Ni) distance and a decrease of the DOS in the minority channel at the Fermi level.\nHence, the DOS decreases from the inverse structure (c/a i= 1.35) over the ground state\nof the regular structure (c/a r=1.35) to the squeezed (c/a r= 1.25) structure the MAE of\nthese three systems behaves just opposite, i.e. it increases from (c/a i= 1.35) which has a\nnegligible MAE of about 0.09 MJ/m3over the ground state (about 1MJ/m3) to the distorted\ncon\fguration with a twice as large MAE. It should be noted that despite the huge changes\nin size the MAE remains always uniaxial for this system.\n162. Quaternary compounds\nWhile in the previous section structural changes within one system were discussed we will\nfocus here on alloying on one sublattice to improve the magnetic properties. Two examples\nhave been chosen. In the \frst case the Z sublattice is used to improve the MAE of Ni 2CoZ.\nOn the one hand from Fig. 5 it is conclusive that Ni 2CoGa and Ni 2CoIn have the largest\nMAE values, unfortunately is not an uniaxial anisotropy and as discussed in previously the\nIn compound is not expected to be stable at low temperatures (cf Sec.III). On the other\nhand the inverse ordered Ni 2CoGe has a much smaller MAE but it is uniaxial and the\ncompound is according to our survey stable in the Hg 2CuTi structure. Therefore, it seems\na natural choice to replace In partially by Ge which means increasing the e/a. Here, In has\nbeen chosen over Ga because of the larger atomic number the spin-orbit ( LS) coupling is\nexpected to be larger and this in turn should counteract the reduction which is expected due\nto Ge. We replaced 25 and 50% of In by Ge using a 16 atomic super cell, see Sec. II. The\nquaternary Ni 2Co(In 0:5Ge0:5) compound turns out to be stable (cf inset of Fig. 8) and as for\nNi2CoIn the inverse ordered structure is lower in energy. Compared to Ni 2CoIn or Ni 2CoGa\nthe MAE is with 0.61 MJ/m3by a factor of 2 smaller (50% Ge), but what is more important\nis that the partial replacement of In by Ge has led to an easy axis anisotropy, see Fig. 8.\nSmaller Ge concentrations would give smaller MAE values, but the magnetic anisotropy\nremains uniaxial as in the original compound Ni 2CoIn. Furthermore, for Ge concentrations\nequal to 25% or less the quaternary compound decomposes, see inset in Fig. 8. Summarizing,\nby mixing Ni 2CoGe and Ni 2CoIn one gets basically the best of both systems, i.e. a stable\nphase with uniaxial MAE. However, the magnetic moment of the quaternary compound is\nreduced compared to Ni 2CoIn which is a tribute to the inverse ordered structure in which\nthe moments are smaller compared to regular Heusler compounds, see also Sec. IV. Though,\nreplacing In partially by Ge stabilizes the compound and improves the magnetic properties\ni.e. turns the system in an uniaxial magnet similar behavior might be expected by using Si\ninstead of Ge since the preconditions are very similar but the ternary Ni 2CoSi compound\nshows an even larger uniaxial MAE. However, substituting 50% of In by Si leads to an inverse\nordered tetragonal structure with c/a <1, namely c/a = 0.905 and a local minimum at c/a\n= 1.2 (see supplement). In contrast to the previous case with Si the MAE remains planar if\n50% of Ge are replaced by Si, see Fig. 8. The MAE changes quasi linearly from Ni 2CoIn to\n17FIG. 8: Calculated dependence of the magneto crystalline anisotropy on the impurity concentration\ni.e. the valence electron number e/a for Ni 2CoIn 1\u0000xZxwith Z = Ge (squares) and Si (circles). The\nopen black symbols denote the corresponding magnetic moments per formula unit (f.u.), see scale\non the right hand side. The inset at the top right shows the formation energy and stabilization\nof the system with increasing Ge and Si content, respectively. The inset on the bottom right\nhand side shows the super cell (16 atoms) used for the quaternary systems, here 50% of In have\nbeen substituted by Ge. The hatched circle corresponds to the MAE of the local minima of\nNi2CoIn 0:5Si0:5(c/a = 1.2). The thick (green) arrows indicate the orientation of the MAE.\nNi2CoSi leading to a planar MAE of 0.31 MJ/m3for Ni 2CoIn 0:5Si0:5. This di\u000berence between\nSi and Ge seems to be related to the change of the neighbor distances in the quenched phase\n(c/a<1.0), hence for the local minimum (c/a = 1.2) the situation is the same as in the In\ncase. The MAE is uniaxial being slightly larger than for the ternary parent system Ni 2CoSi,\nsee hatched circle in Fig.8.\nIn the second case we followed the same line of argument but tailoring the occupation\nof the Y sublattice instead. In this case we keep the Z element \fxed. In our case we have\n18chosen Z = Al. Both ternary compounds are regular Heusler systems with a tetragonal\nground state and are stable according to Fig. 4. Aiming to increase the MAE of Ni 2FeAl Fe\nhas been partially replaced by Co, since Ni 2CoAl has a larger MAE which is unfortunately\nplanar, see Fig. 5. However, instead of improving the magnetic properties as in the previous\nexample the MAE almost vanishes. remains For Ni 2Fe0:75Co0:25Al calculated in a 16 atom\nsuper cell the MAE remains uniaxial but decreases to a value <0:1MJ/m3.\n3. Isoelectronic replacement of Ni\nIt has been pointed out in Bruno's model54the MAE is directly related to the spin-\norbit coupling strength \u0018of a system and knowing that \u0018is related to the atomic number\nby\u0018\u0018Z4, i.e. heavy elements are preferable aiming for a large MAE. Therefore we re-\nplaced Ni partially by Pd assuming that an isoelectronic exchange will improve the magnetic\nproperties, especially the MAE and leave the other properties such as the phase stability\nunchanged. As test system we have chosen Ni 2FeGe which has already a suitable MAE of\nabout 1 MJ/m3. Half of the Ni atoms in this compound have been replaced by Pd and\ncalculations have been performed using a 16 atomic supercell. For comparison we have\nalso studied the Ni-free system Pd 2FeGe. As expected the isoelectronic replacement has\nno signi\fcant in\ruence on the phase stability. The formation energies are with -0.676 eV\n(NiPd) 2FeGe and -0.881 eV (Pd 2FeGe) in the same range as for Ni 2FeGe, see stars in Fig. 4.\nInducing Pd in the Heusler compound to replace Ni should have only minor in\ruence on the\nelectronic structure (isoelectronic) and preserve to uniaxial MAE. This is indeed observed\nthe MAE remain uniaxial and the volume increases due to the larger Pd atom, see Tab. II.\nReplacing 50% of the Ni atoms (16 atom super cell) leads to an increase of the volume\nper f.u. by 12% but unfortunately the MAE does not change much. It basically remains\nconstant at 0.97 MJ/m3, cf Tab. II but would increase the prize by a factor of 1400. So the\nreplacement of Ni by Pd would not only be ine\u000ecient but also incredibly expensive.\nC. Magnetic moments\nThe magnetic moments of Heusler alloys show usually Slater-Pauling-type behavior, i.e.\ntheir total magnetic moments depend linearly on the number of valence electrons. First\n19TABLE II: Structure data and magnetic properties for a series of isoelectronic Heusler compounds\nNi1\u0000xPdxFeGe. Note the magnetic moments shown here are the total magnetic moments including\nthe orbital moment as obtained from full potential DFT calculations using RSPt28. Positive sign\nfor the MAE indicates uniaxial anisotropy.\nSystem c/a Volume m tot MAE\n(\u0017A3/f.u.) (\u0016B/f.u.) (MJ/m3)\nNi2FeGe 1.35 48.00 3.50 0.95\n(NiPd) 2FeGe 1.42 53.82 3.40 0.97\nPd2FeGe 1.38 58.86 3.24 0.61\ndemonstrated for L2 1ordered Co based half metallic ferromagnets similar behavior has\nbeen observed for related systems. In case of the halfmetallic Co-based compounds the\nmagnetic moments are given by M=Nv-24 withNvbeing the number of valence electrons\nper formula unit55which gives integer magnetic moments for stoichiometric ordered systems.\nIt has turned out that this rule can be generalized for many classes of Heusler alloys. Half-\nmetallic Heusler alloys X 2YZ with X being an early 3d transition metal obey depending on\nthe Y and Z constituent slightly di\u000berent rules namely M=Nv\u000018 andM=Nv\u00002856. For\nNi2Mn1\u0000xGaxalloys Dannenberg proposed a M= 34\u0000Nvbehavior of the total magnetic\nmoment.41As shown in Fig. 9 also all Ni-based Heusler alloys with a Z element from main\ngroup III obey this rule. Adding electrons to the system by occupying the Z sublattice with\nan element from main group IV increases the spin moments such that they follow the rule\nM= 35\u0000Nv. However, one should keep in mind that not the number of valence electrons\nNvdecides which rule the system obeys but the choice of the Z element. Taking for example\nNi2CoGa and Ni 2FeGe, both systems have Nv= 32 but obey di\u000berent rules, i.e. the Fe\ncompound has a higher magnetic moment, see Fig. 9. This related to the fact that the\nDOS at the Fermi level is mostly determined by the 3d states of the transition metals, cf\nFig. 10. In case of Ni 2CoGa the minority spin channel more occupied and therefore the spin\nmoment smaller compared to the Fe compound. A peculiarity occurs for the inverse ordered\nNi2CoZ with Z an element from group IV. We have argued in the previous sections that\ninverse ordered Heusler compounds for the same type of compound have smaller magnetic\n2029 30 31 32 33 34\nNumber of valence electrons per unit cell012345Magnetic moment ( µB/f.u.)Ni2YB\nNi2YAl\nNi2YGa\nNi2YIn\nNi2YSi\nNi2YGe\nNi2YSn\n29 30 31 32 33 34\nNumber of valence electrons per unit cell012345Magnetic moment ( µB/f.u.)\nNi2CoIn1-xGex\nNi2CoIn0.5Si0.5\nNi2Co0.25Fe0.75AlMtot = 34 - NvMtot = 35 - NvFIG. 9: Total magnetic moments of the ground state. The value of the cubic (c/a = 1.0) structure\nis given in brackets. All values are in \u0016B/f.u. The dashed line marks the 34- Nvrule for shape\nmemory alloys as suggested by Dannenberg et al.41whereas the magnetic moment for the inverse\nordered systems with the Z element being from group IV follows shows a 35- Nvbehavior (dashed-\ndotted line). Inverse ordered systems including the quaternary ones (hatched symbols) follow the\ndashed line since their moment is reduced due to the reversed nn positions.\nmoments due to di\u000berent local order. This observation can be quanti\fed as shown in Fig. 9.\nThese compounds obey the M= 34\u0000Nvrule instead. The same holds for the quaternary\ncompounds with inverse lattice structure, see hatched symbols in Fig. 9.\nV. CONCLUSION\nNi-based Heusler compounds Ni 2YZ have been used as an example to study di\u000berent\nroutes to improve the magnetic properties with special focus on the MAE. The magnetic\nmoments of the systems follow modi\fed Slater-Pauling laws. Showing clearly that the mag-\nnetic moment of the inverse ordered systems is systematically lower than for regular Heusler\n21-5 -4 -3 -2 -1 0 1 2 3 4-4-202DOS (sts./eV)Fe\nNi\nGe\n-5 -4 -3 -2 -1 0 1 2 3 4\nE-EF (eV)-4-202DOS (sts./eV)Co\nNi\nGaNi2FeGe, c/a = 1.35\nNi2CoGa, c/a = 1.38FIG. 10: Orbital resolved density of states for Ni 2FeGe (a) and Ni 2CoGa (b). Shaded areas denote\nthe Fe and Co 3d states, respectively. The blue solid line marks the 3d DOS of Ni and the Ge(Ga)\np-states are shown as dashed lines.\n22compounds.\nOut of the 21 studied systems 14 possess a non-cubic ground state being a prerequisite for\na \fnite MAE. From these candidate systems the ones with Co on the Y sublattice turned out\nto be most interesting. In Ni 2CoZ compounds the tetragonal phase turned out to be most\nstable, i.e. the transformation to the cubic austenite phase will occur at higher temperature\nas for example for Ni 2MnGa. However, the Ni 2CoZ (Z= In, Ga) systems with the largest\nMAE turned out to have a planar MAE and/or are even unstable. We could show that\nthis could be cured by combining them with inverse ordered Ni 2CoGe which has a uniaxial\nMAE. Using Si instead of Ge turned out not to be successful, because the tetragonal phase\nin Ni 2CoSi has c/a <1. For the local minimum at c/a = 1.2 the same e\u000bect as for Ge\nis observed. Hence, the phase can be stabilized by adding valence electrons (replacing In\npartially by Si or Ge) but to improve the magnetic properties also the lattice structure of\nthe ternary phases has to match. Aiming to increase the uniaxial MAE of Ni 2FeAl we used\nthe same strategy, but replacing partially Fe by Co since Ni 2CoAl as a larger (planar) MAE.\nHowever, it turned out that 25% Co of the Fe sublattice reduce the MAE drastically. This\nis not completely unexpected since changing the coordination and magnetism can lead to a\nreduction of the MAE.\nAnother way to improve the MAE could be the use of heavier elements, which possess\nlarger spin orbit coupling. To test this for our set of systems we selected the ternary system\nwith the largest uniaxial MAE, Ni 2FeGe. Partial isoelectronic replacement of Ni by Pd\nshowed only minor e\u000bect on the MAE, because due to Pd the volume increases and the\nmagnetic moments slightly decrease both facts counteract to an increase of the MAE.\nThe MAE also changes with the lattice ratio such that one can think to tailor the MAE\nby stress or strain. Basically this c/a dependence has been discussed for Ni 2MnGa and other\nsystems in literature before. Most systems show a quasi linear behavior with a sign change\nat c/a =1. However, in case of Ni 2FeGe the MAE remains uniaxial for all lattice ratios\nbetween 0.85 and 1.45. For this particular system small deviations from the equilibrium c/a\nboost the MAE from about 1 to 2 MJ/m3. Similar behavior is expected for other Ni 2FeZ\ncompounds since the orientation of the MAE for the ground state and the local minimum\natc=a< 1:0 are the same.\nConcluding for the systems under consideration we discussed di\u000berent routes to manipu-\nlate the MAE e.g. creating quaternary compound by doping or by lattice deformation. Both\n23routes turned out to be successful under certain conditions and MAE values of 1 MJ/m3\nand higher could be achieved. Limitations for practical use are the height of martensite\ntemperature and the Curie temperature. Especially the Curie temperature has recently be\npredicted to be quite low in some of the systems. 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Feng1, Manfred Reehuis2, Peter Adler1, Zhiwei Hu1, Michael Nicklas1, Andreas \nHoser2, Shih-Chang Weng3, Claudia Felser1, Martin Jansen1 \n \n1Max Planck Institute for Chemical Physics of Solids, Dresden,  D-01187, Germany \n2Helmholtz -Zentrum Berlin für Materialien und Energie, Berlin, D -14109, Germany  \n3National Synchrotron Radiation Research Center (NSRRC), Hsinchu, 30076, Taiwan  \n \n \nAbstract:  \nThe non -stoichiometric double perovskite oxide La 2Ni1.19Os0.81O6 was synt hesized by solid \nstate reaction  and its crystal and magnetic structure s were investigated  by powder x-ray \nand neutron diffraction. La 2Ni1.19Os0.81O6 crystallizes in the monoclinic doub le perovskite \nstructure  (general formula A2BB’O6) with space group P21/n, where the B site is fully \noccupied by Ni and the B’ site by 19 % Ni and 81 % Os atoms. Using x-ray absorption \nspectroscopy an Os4.5+ oxidation state  was established, suggesting  presence of about 50  % \nparamagnetic Os5+ (5d3, S = 3/2) and 50  % non -magnetic Os4+ (5d4, Jeff = 0) ions at the B’ \nsites. Magnetization and  neutron diffraction measurements on La 2Ni1.19Os0.81O6 provide \nevidence for a ferrimagnetic  transition at 125 K . The analysis of the neutron data suggests \na canted ferrimagnetic spin structure with collinear Ni2+ spin chains extending along the c \naxis but a non-collinear spin alignment within th e ab plane.  The magnetization curve of \nLa2Ni1.19Os0.81O6 features a hysteresis with a very high coercive field, HC = 41 kOe, at T = \n5 K, which is explained in terms of large  magnetocrystalline anisotropy due to the presence \nof Os  ions together with atomic disorder. Our  results are encouraging to  search for  rare \nearth free hard magnets in the class of double perovskite oxides.  \n \n \n \n   2 I. INTRODUCTION  \nDouble perovskite oxides A2BB’O6 containing  3d-5d(4d) transition metal ions  at the B \nand B’ sites are attracting great attention, due to the ir interesting physical properties . High-\ntemperature ferrimagnetic  half-metal s, ferro - or ferri magnetic  insulator s, as well as \nmaterials with large magnetoresistance and exchange bias, are found in this class of \ncompounds  [19]. They thus show prospects for spintronic applications. Most of these \ncompounds contain 5 d elements with 5 d2 and 5 d3 electronic configurations . Here,  the \ninteresting properties arise from competing  interactions within and between  the 3 d and 5 d \nsublattices. Studies on 5d4 system s are rare, probably  because of the expected nonmagnetic  \nground state for 5d4 ions. In the presence  of strong spin-orbit coupling (SOC), the orbital \nangular momentum associated with the three t2g orbitals in an octahedral crystal field  (l = \n1) entangle s with the spin  moments  of the electrons  which results in  an upper jeff = 1/2 \ndoublet  and a lower jeff = 3/2  quadruplet. The  formation of a SOC -assisted  Mott insulating \nstate in Sr2IrO 4 (Ir4+, t2g5) is understood within  this picture [10]. According to  this scenario, \nthe ground state of a 5d4 system in an octahedral crystal field is a trivial singlet with four \nelectrons filling the lower quadruplet, resulting in  a nonmagnetic state  with total angular \nmomentum  Jeff = 0.  Many studies focus on Ir5+ compounds, such as A2BIrO 6 (A = Sr, Ba; \nB = Sc, Y) [11 17]. In contra st to the expectations for Jeff = 0, long-range magnetic order  \nat 1.3 K was  reported for Sr 2YIrO 6 [11,12] and  considered as evidence for intermediate -\nstrength rather than strong spin -orbit coupling  which is usu ally anticipated for 5 d ions. \nThese results, however, have  been challenged and the observed magnetism in such \nmaterials may arise from  defects and /or nonst oichiometry  [13,18].   \nBeside Ir5+, also  Os4+ ions adopt  the 5 d4 electronic configuration . The singlet ground -\nstate magnetism was  discussed in AOsO 3 (A = Ca, Sr, Ba) [19]  and R2Os2O7 (R = Y, Ho) \n[20].  To the best of our knowledge, the magnetic  properties of double perovskite s \ncontaining Os4+ have not been reported yet. In this work, targeting at  an Os4+ double \nperovskite La 2NiOsO 6, we synthesize d a nonstoichiometric phase  La2Ni1.19Os0.81O6 with \nNi2+ ions at the B site and a mixture of Ni2+, Os4+, and Os5+ ions at the B’ site of the double \nperovskite structure . It was  verified by x-ray absorption spectroscopy that  the valence state \nof Os is about 4.5+.  Magnetization and neutron diffraction studies revealed that \nLa2Ni1.19Os0.81O6 shows a ferrimagnetic  transition at 125 K and features an extraordinary \nbroad hysteresis with a giant coercivity of 41  kOe at 5 K.  These results are encouraging  \nfor the search of  hard magnetic materials in the class of 3 d/5d double perovskites.  \n  3  \nII. EXPERIMENTAL  \nA polycrystalline sample  of La2Ni1.19Os0.81O6 was synthesized by solid -state reaction \nfrom La 2O3, NiO, and Os. La 2O3, NiO, and Os in a molar  ratio 1 : 1 : 1 were well ground \ntogether and pressed into a pellet. This was loaded into a corundum crucible which was \nplaced into a silica tube along with a second corundum  crucible containing MnO 2 (Alfa \n99.9%). The silica tube was then sealed under dynamic vacuum using a H 2/O2 torch, and \nheated at 1250 °C for 48 hours in a tube furnace. MnO 2 decomposes into ½ Mn 2O3 + ¼ O 2 \nat 550 °C and acts as an oxygen source for the reaction. The molar ratio of Os and MnO 2 is \n1 : 4. Small  pieces of La2Ni1.19Os0.81O6 were  cut from the synthesized pelle t and finely \nground  to a fine powder , which was characterized by powder  x-ray diffraction using a \nHuberG670 camera [Guinier technique, λ = 1.54056 Å (Cu -Kα1)]. A powder pattern was \ncollected in the 2 θ range between 10.9 and 100.2°.  A scanning electron microscope (SEM, \nPhilip s XL30) with an attached energ y dispersive x-ray spectrometer (EDX) was used for \nelemental analysis.  \nThe Os -L3 XAS spectra of  La2Ni1.19Os0.81O6, an Os5+ Sr2FeOsO 6 reference , and an Os4+ \nreference La 2MgOsO 6 were measured in transmission  geometry at the beamline BL07A  at \nthe National Synchrotron Radiation Research Center in Taiwan.  \n Neutron  powder diffraction experiments of La2Ni1.19Os0.81O6 were carried out on the \ninstruments E2, E6, and E9 at the BER II reactor of the Helmholtz -Zentrum Berlin. The \ninstrument E9 uses a Ge monochromator selecting the neutron wavelength λ = 1.309 Å, \nwhile the instrument s E2 and  E6 use  a pyrolytic graphite (PG) monochromator selecting \nthe neutron wavelength λ = 2.38 and 2.417  Å, respectively . In order to investigate in detail \nthe crystal structure of La2Ni1.19Os0.81O6 at 3.2 and 160 K, well below and above the \nmagnetic ordering temperature , neutron powder patterns were recorded on the instrument \nE9 between the diffraction angles  7.5 and 141 .8°. For a detailed analysis of the magnetic \nstructure  and its temperature dependence , we have  collected powder patterns  between 1.7 \nand 134 K on the instrument E6 between the diffraction angles 5 and 136.4°. In addition , \npowder patterns at 1.7 K and 150 K  with higher counting rate and better instrumental \nresolution in the 2  range between 11.8 and 87.3° were obtained on the instrument E2.  \nRietveld refinements of the powder diffraction data were carried out with the program \nFullProf [21]. In the case of the data analysis of x-ray diffraction data,  we used the atomic \nscattering form factors provided by the program. For the refinements of the neutron \npowder data the nuclear scattering lengths b(O) = 5.803 fm, b(La) = 8.24 fm, b(Ni) = 10.3  4 fm, and b(Os) = 10.7 fm were used [ 22]. The magnetic form factors of the Fe and Os atoms \nwere taken from Refs. [2 3,24].  \nUsing a polycrystalline piece,  the electrical resistivity ( ρ) was measured with direct \ncurrent (0.1 mA) in a four-point in -line arrangement (PPMS, Quantum Design). Electrical \ncontacts were made with Au wire s and silver paste. The temperature dependence of the \nmagnetic susceptibility was measured in a SQUID magnetometer (MPMS -5T, Quantum \nDesign). The measurements were conducted in warming after zero -field cooling (ZFC) and \nduring field -cooling (FC) in the temperature range 2  – 300 K under applied magnetic fields \nof 10 kOe. The high-temperature  magnetic susceptibility (300 – 565 K) was measured \nusing the same SQUID magnetometer with oven option. Isothermal magnetization curves \nwere initially  recorded for fields up to ±50 kOe at temperatures of 5 and 150 K using the \nsame SQUID magnetometer. As the magnetizat ion curves turned out to be unsat urated at 5 \nK and showed a broad hysteresis, M(H) curves  were  further measured for fields up to ±140 \nkOe at 5 K using a Physical Proper ty Measurement System (PPMS, Qu antum Design) . \n \n \nIII. RESULTS  \n \nA Crystal structure   \nThe room -temperature crystal structure of  La2Ni1.19Os0.81O6 was investigated by x-ray \npowder diffraction as shown in Figure 1  (top) . La2Ni1.19Os0.81O6 was successfully  refined  \nin the monoclinic space group P21/n (No. 14, standard setting P21/c). In this space group,  \nthe La and the three O atoms (O1, O2, and O3) occupy the Wyckoff position 4 e(x,y,z), \nwhile the Ni and Os atoms are at the positions 2 c(½,0,½) and 2 d(½,0,0), respectively. \nDuring the refinement, we used the constraint occ(Ni) + occ(Os) = 1 at the 2 c and 2 d sites. \nThe refinements indicated that the sample is nonstoichiometric  compared to the ideal \ncomposition La 2NiOsO 6 and the following occupancies were found: occ(Ni1)/ occ(Os1) = \n0.187(3)/0.813(3) at 2 c, and occ(Ni2)/ occ(Os2) = 1.001(2)/  0.001(2) at 2 d. This suggests  \nthat the 2 d site is fully occupied by Ni, and therefore in the final refinement,  the Ni \noccupancy at the 2 d site was fixed to be 1. This gives the  composition \nLa2Ni1.187(3) Os0.813(3) O6. The refinements resulted in a residual RF = 0.0180 (defined as RF = \n∑||Fobs|  |Fcalc||/∑|Fobs|). The sample contains a smaller amount of La 3OsO 7 (5.6 %) [25,26]. \nFor this compound,  only the overall scale factor and the lattice parameters were \nsimultaneously allowed to vary during the Rietveld refinements, whereas the structural  5 parameters, taken from Ref. 2 5, were fixe d. The loss of Os is presumably due to the \nformation of volatile OsO 4 during the synthesis. The EDX analysis revealed that the \nsample contains much more Ni than Os, Ni/Os = 1.17/0.83, which is close to the results \nfrom the Rietveld refinement.  \n \n \n \nFigure 1. (color online) Top: Rietveld refinement  of the x-ray powder diffraction data of  \nLa2Ni1.19Os0.81O6 collected at 295 K . The vertical bars indicate the positions of the nuclear  \nBragg reflections for  La2Ni1.19Os0.81O6 and impurity phase La 3OsO 7. Bottom: Rietveld \nrefinements of the powder neutron diffraction data of  La2Ni1.19Os0.81O6 collected at 160 K \nand 3.2 K (bottom). The strongest peaks of the impurity phase are marked with an asterisk.  \n \nDue to the fact that the x-ray scattering power of the O atoms is relatively weak in \ncomparison to those of the much heavier La and Os atom s, the crystal structure was also \ninvestigated by neutron diffraction at 3.2 and 160 K, well below and above the magnetic \n 6 ordering temperature of ab out 125 K . The crystal structure of La2Ni1.19Os0.81O6 could be \nsuccessfully refined in the space group P21/n with residuals RF = 0.0273 (3.2 K) and RF = \n0.0250 (160 K), respectively ( Figure 1 , bottom ). Duri ng the refinement, the occupancies  of \nthe Ni and Os atoms at the 2 c and 2 d site were fixed to the values obtained from the x-ray \ndata.  Concerning the impurity  phase  La3OsO 7, only the overall scale factor and the lattice \nparameters were simultaneously allowed to vary during the Rietveld  refinements, whereas \nthe structural parameters obtained at lower  temperature (taken from Ref. 26 ) were fixed.  \nThe obtained crystallographic data are summarized in Table I, and the atomic positions are \nsummarized in Table  II.  \n \nTABLE I . Crystallographic data obtai ned from refinements of powder x-ray and neutron \ndiffraction data of La 2Ni1.19Os0.81O6 at different temperatures.  \n 295 K  160 K  3.2 K  \nDiffraction source  x-ray Neutron  Neutron  \nSpace group  P21/n P21/n P21/n \nLattice parameters  \n \n \n a = 5.58129(8)  Å \nb = 5.61271(9)   Å \nc = 7.89614(12)  Å  \n β = 90.075(4) ° a = 5.5742(5)  Å \nb = 5.6116(5)  Å  \nc = 7.8907(7)  Å  \nβ = 90.01(3) ° a =  5.5700(4)  Å \nb = 5.6123(4)  Å  \nc = 7.8866(6)  Å  \nβ = 90.04(2) ° \nCell volume  247.356(7)  Å3 246.82(4)  Å3 246.54(3)  \nRF 0.0180  0.0250  0.0273  \n \n \nTABLE II . The atomic positions  and isotropic temperature factors ( Biso) of \nLa2Ni1.19Os0.81O6. The refinements were carried out in the monoclinic space group P21/n. \nThe 2c sites ( ½,0,½ ) are occupied by Ni1/ Os1 atoms while the 2d sites(0,0, ½) are occupied \nby Ni2 atoms with the occupanc y fixed to be 1.  The Biso were constrained to be equal for \nthe metal atoms and the oxygen atoms, respectively. All parameters marked with an \nasterisk were not allowed to vary during the refinements.  \nT (K) 295 160 3.2 \nDiffraction  source  x-ray Neutron  Neutron  \nx (La) 0.0060(5)  0.0157(7)  0.0162(6)  \ny (La) 0.0379(1)  0.0441(3)  0.0441(3)   7 z (La) 0.2521(3)  0.2508(19)  0.2496(18)  \nx (O1)  0.270(3)  0.2843(17)  0.2833(13)  \ny (O1)  0.295(3)  0.2809(13)  0.2850(12)  \nz (O1)  0.031(5)  0.0460(11)  0.0468(10)  \nx (O2)  0.213(3)  0.1970(17)  0.1961(13)  \ny (O2)  0.796(2)  0.8001(12)  0.7971(10)  \nz (O2)  0.026(5)  0.0347(11)  0.0370(10)  \nx (O3)  0.904(3)  0.9218 (11) 0.9218(10)  \ny (O3)  0.481(1) 0.4882(9)  0.4846(8)  \nz (O3)  0.245*  0.2478(12)  0.2475(10)  \nBiso (La/Ni/Os ) 0.16(2)  0.34(2)  0.25(2)  \nBiso (O) 1* 0.55(3)  0.46(3)  \nOcc(Ni1/Os1)   0.187(3)/0.813(3)  0.187/0.813*  0.187/0.813*  \n \n \n \nFigure 2. (color online) Crystal structure  of La2Ni1.19Os0.81O6. Blue octahedra  are fully \ncentered by Ni2+ ions, grey octahedra contain a majority fraction of Os (Os4+, Os5+) ions \nand a minor  fraction of Ni2+ ions.  \n \n       The inter -octahedral Ni OOs bond angles of  La2Ni1.19Os0.81O6 deviate  strongly  from \n180° (Table III), indicating strong octahedral tilting  as shown in Fig . 2. The bond distances \nin the NiO 6 and OsO 6 octahedra are summarized in Table 3. At 295 K, the average Ni O \nbond length, 2.06 Å, is comparable to the literature values for Ni2+ double perovskite \noxides A2NiOsO 6 (A = Ca, Sr, Ba) [6,2 7], which indicates that Ni is  divalent in this sample.  \n 8  \nTABLE  III. Selected bond lengths and bond angles of La2Ni1.19Os0.81O6 derived from x-ray \nand neutron powder  patterns at different temperatures.  \n \n \n \n \n \n \n \n \n \n \n \n \nB. X-ray absorption spectroscopy   \n \n \nFigure  3. (color online) The Os -L3 XAS spectra of La 2Ni1.19Os0.81O6 and of Sr 2FeOsO 6 as \nOs5+ reference and La 2MgOsO 6 as Os4+ reference.  \n \nTo explore the valence state of Os, the Os -L3 XAS spectrum of La 2Ni1.19Os0.81O6 was \nrecorded at room temperature. It is well known that XAS spectra are highly sensitive to the \nBond length (Å)  295 K  160 K  3.2 K  \nNiO1 2.109(16)  × 2 2.016(8) × 2 2.038(7)  × 2 \nNiO2 1.981(16) × 2 2.046(9)  × 2 2.061(7)  × 2 \nNi O3 2.086(5) × 2 2.038(13)  × 2 2.040(9) × 2 \nOsO1 1.912(16) × 2 2.038(9)  × 2 2.021(7) × 2 \nOsO2 2.052(15) × 2 2.029(8) × 2 2.014(7) × 2 \nOsO3 2.011(5)  × 2 2.004(13) × 2 2.002(9) × 2 \nNiO1Os 159.6(6)  154.6(3)  154.0(3)  \nNiO2Os 157.8(6)  152.1(3)  151.9(3)  \nNiO3Os 149.1(5)  154.8(2)  154.6(2)   9 valence state: an increase of the valence state of the metal ion by one causes a shift of the \nXAS L2,3 spectra by one or more eV toward higher energies [28,29].  Figure 3 shows the \nOs-L3 XAS spectrum of La 2Ni1.19Os0.81O6 together with spectra of Sr2FeOsO 6 and \nLa2MgOsO 6 as Os5+ and Os4+ references, respectively  [6]. The energy position of the \nLa2Ni1.19Os0.81O6 spectrum is located exactly in the middle between that of Sr 2FeOsO 6 and \nLa2MgOsO 6 demonstrating the oxidation state of Os4.5+. This is consistent with the \nexpected valence state Os4.46+ for the composition  La2Ni1.19Os0.81O6 obtained from the \nRietveld refi nements of the XRD data.  \n \nC. Electrical Transport  \nThe electrical resistivity of a sintered polycrystalline sample of La2Ni1.19Os0.81O6 is \ndisplayed  in Figure 4. The sample shows insulating behavior as the resistivity increases by \nseveral orders of magnitude as the temperature decreases  and exceeds the measurement \nlimit for temperatures lower than 100 K. The data was plotted on T1 and T1/4 scales, and \nthe plot is found to be roughly linear on a T1/4 scale (see the inset of Fig . 4), in accordance \nwith a three -dimensional  variable range hopping transport model.   \n \n \nFigure 4.  (color online) Temperature dependence of the electrical resistivity of \nLa2Ni1.19Os0.81O6. The inset shows the corresponding plots on T1 and T1/4 scales.  \n \nD. Magnetic properties   \n 10 The temperature dependence of the magnetic susceptibility χ of La2Ni1.19Os0.81O6 is \nshown in Figure 5. The sharp increase of χ below about 125 K with cooling indicates a \npossible ferro - or ferrimagnetic transition. Because the minor impurity La 3OsO 7 is \nantiferromagnetic with  a Néel temperature of 45 K [2 6], the sharp transition around 125 K \nis intrinsic for La2Ni1.19Os0.81O6. The  convex  χ1 vs T curve above the magnetic transition \ntemperature indicates  a ferrimagnetic  transi tion.  An attempt to fit the data above 400 K \nwith the Curie -Weiss law resulted in a Curie -Weiss temperature ( θCW) of 73 K and an \neffective  magnetic  moment ( µeff) per formula unit (f.u.) of 3.63 μB. The negative  θCW \nindicates that antiferromagnetic interactions  are dominant in this sample, supporting the \nferrimagnetic phase transition. The obtained µeff compares reasonably well with µeff = 3.69  \nμB/f.u., which is calculated from  the formula La2Ni1.192+Os0.434+Os0.385+O6 if one assumes a \nspin-only moment for Ni2+ (2.83 μB), and takes the average μeff = 3.28 μB for Os5+ as \nobtained for some Os5+ double perovskites [ 3033]. Any possible Van -Vleck type \nparamagnetic contribution to µeff of the Os4+ ions is neglected in this rough estimate . \n \n \nFigure 5. (color online) Temperature dependence of the magnetic susceptibility of \nLa2Ni1.19Os0.81O6. The inset shows the corresponding χ1 vs T plot. \n \nTo further characterize  the magnetic transition, the isothermal magnetization  curves of \nLa2Ni1.19Os0.81O6 were measured above and below the transition temperature after zero  \nfield cooling. The linear behavior of the M(H) data at 200 K is consistent with the \n 11 paramagnetic state. At 5 K, the M(H) curve collected between 50 kOe and +50 kOe \nshows  a large hysteresis , but the hysteresis loop cannot be closed . Thus, the M(H) curve \nwas measured between 140 kOe and +140 kOe , see Figure 6. The magnetization is about \n0.65 μB/f.u. at 5 K and 140 kOe, but still not saturated. A remarkable feature of the M(H) \ncurve of La2Ni1.19Os0.81O6 is the high coercive field HC = 41 kOe at T = 5 K. The M(H) \ncurve was also measured after field cooling (50 kOe) and found to be identical to that \nmeasured after zero field cooling, no exchange bias effect is observed.   \n \n \nFigure 6. (color online) The isothermal  magnetization of La2Ni1.19Os0.81O6 measured at 200 \nK and 5 K.  The small step near H = 0 may be due to a soft magnetic impurity.  \n \nE. Magnetic structure   \nIn order t o investigate the magnetic structure  of La2Ni1.19Os0.81O6, we have collected \nseveral neutron powder patterns on the instrument E6 of BER II ( λ = 2.42 Å) from 1.7 up \nto 134 K . In agreement with our magnetization  measurements  a magnetic contribution to \nthe powder patterns could be observed below 125 K. The strongest magnetic intensity was \nobserved at the position of the reflections 011 , 101 and 10 1 (Fig. 7). For the Ni and Os \natoms at the positions 2 d(½,0,0; 0,½,½) and 2 c(½,0,½; 0,½,0) magnetic intensity could be \ngenerated with a model, where the spins located at both sites are coupled antiparallel  with a \nspin sequence   . In the  pattern, shown in Fig. 7, addi tional weak intensities were  \nobserved at 2 θ = 16.7 and 24 .1 °, which are close  to the position of the reflection 001,  and \nthat of the reflection pair 100/010 (forbidden in P21/n), respectively.  The third reflection \nobserved at 21.5 ° can be indexed as (½½1) M belonging  to the impurity La 3OsO 7 [26]. In \n 12 order to characterize these reflections in more detail , we have collected powder patterns at \n1.7 and 150 K on the instrument E2 with a much higher counting rate and better \ninstrumental resolution. From the difference pattern shown in  the insert  of Fig. 7  magnetic \nintensity was clearly detected for  reflection 0 01, whereas for the reflection pair 100 and \n010 the magnetic contribution  was found to be negligible. Therefore the intensity observed \nat 24.1 ° should belo ng to another unknown impurity.  \nFrom our x-ray diffraction study , it was found that the osmium site 2 c site is partially \noccupied with Ni atoms giving a ratio occ(Ni1)/ occ(Os1) = 0.187(3)/0.813(3), while the 2 d \nsite is fully occupied by Ni2. For the determination of the moment values  from the \nrefinements of the magnetic structure we have assumed that Ni1 and Ni2, located at 2 c and \n2d sites, have the same magnetic moment, while initially the Os moments were kept at zero. \nFurther , we used the positional and thermal parameters as well as  the monoclinic angle β as \ndetermined more precisely from the analysis of the E9 data . In order to determine the \nmoment direction , we carried out Rietveld refinements using model structures . In the case \nof an ordering along the b and c axes, it could be seen that the magnetic peak significantly \nshifted to a higher 2θ value,  while a satisfactory fit was obtained  when the moments are \naligned to the a axis. In fact, due to the lattice distortions the reflection pair 101/10 1 is \nshifted to a high er 2θ value in comparison to the 011 (see inset of Fig. 7). From our \ncalculations , we obtained the following intensity ratios I(011/011)/I(101/10 1): ~3/1 (μ // \na), ~1/3 (μ // b), ~1/1 (μ // c). This shows that the best calculated  peak position is reached  \nwhen the strongest magnetic intensity appears on the reflection 011  generated with a  \nmoment direction  parallel to the a axis. In the next step, we allowed varying  the magnetic \nmoment of the Os atoms at the site 2 c. Independently from the models used above the \nrefined Os moment was found to be 0.00(6) μ B. Therefore in the final refinement , the Os \nmoment was fixed to be zero.  \n       In order to generate magnetic intensity at the position of the reflection 001 two models \nwere considered, where the Ni and mixed Os/Ni sites  (½,0,0 ), (0,½,½) , (½,0,½) and  \n(0,½,0) ha ve an additional b spin component with the equences   and , \nrespectively. The refinements showed that the spin sequence  results in similar ly \ncalculated intensities for the reflections 100 and 001. But the inset of Fig. 7 shows that the \nmagnetic intensity of the 100 is much weaker than that of the 001 reflection . A rea sonable \nfit was  obtained by using the spin sequence  . Together with the spin component \nalong the a direction  with the spin sequence   one obtains a spin structure which is \nnoncollinear in the ab plane,  but has collinear spin chains along the c direction (Fig. 8).  13 This spin arrangement is compatible with the crystal structure  symmetry  as shown by a \nrepresentation analysis [ 34]. For both atoms at the sites 2 d(½,0,0; 0,½,½) and 2 c(½,0,½; \n0,½,0) one finds an irreducible representation  with the spin sequence  along the b \ndirection  and a sequence along a and c. Finally for the 1.7 K data set collected on E2 \nwe obtained the moment values μ x(Ni)  = 1.77(5) μ B and μ y(Ni)  = 0.60(12) μ B resulting in \na total moment μ exp(Ni)  = 1.87(5) μ B and a residual RM = 0.073 (defined as RM = ∑|| Iobs|  \n|Icalc||/∑|Iobs). In this n oncollinear magnetic structure , the moments  form  a tilting angle to \nthe a axis of 19(1)  °. For the Rietveld refinements of the powder patterns collected on E6, \nwe have fixed the component μ y(Ni). Here we have obtained the component μ x(Ni)  = \n1.81(5) μ B and a total moment μ exp(Ni)  = 1.90(5) μ B, respectively. This value is only \nslightly smaller than the spin -only moment of 2.0 μ B expected for Ni2+ ions in octahedral \ncoordination environment ( t2g6eg2 electron configuration). The temperature dependence of \nthe obtained Ni moment, shown in Fig. 9, y ields an ordering temperature of 125(3) K. Due \nto the weakness of the magnetic reflection (001) M we only could follow the temperature \ndependence of the moment of the much stronger pronounced x component from E6 data.  \n \n \nFigure 7. (color online) Neutron powder pattern of La 2Ni1.19Os0.81O6 taken on E6 at 1.7 K. \nThe calculated patterns [nuclear contribution (blue), sum of the nuclear and the magnetic \n20 30 40 50 60 70400080001200016000\n15 20 25 300500010000\n28 29 30 31 32300040005000\n012 111 001\n100 / 010\n101 / 011\n110 / 002\n200 / 112 / 020\n120 / 210\n122 / 212113 022 / 202211 / 121\n103 / 013La2Ni1.19Os0.81O6 at 1.7 K, E6Intensity (counts)\n2 (deg)N\nM\nIm100 / 010\n101 / 011001Difference\n1.7 - 150 K\nE2\n*\n011\n101 14 contribution (red) ] are compared with the observations (black -filled circles). The positions \n(black bars) of  the nuclear ( N) and magnetic ( M) Bragg reflections of La2Ni1.19Os0.81O6, \nand those of the impurity phase La 3OsO 7 (Im) are shown. The region, where the stro ngest \nmagnetic  reflection  (011) M, and the pair (101) M and (101)M occur is shown  in the inset . \nThe second inset (left) shows the presence of the magnetic reflection (001) M as obtained \nfrom the difference pattern (1.7 – 150 K) from E2 data. The weak magnetic intensity \nobserved at 21.5 °  (marked with an asterisk) can be assigned to the magnetic reflection \n(½½1) M of the impurity La 3OsO 7 [26]. All prominent magnetic reflections are labeled in \nred color.  \n \nOur data analysis suggest s that the observed ferrimagnetic structure is driven by the \nantiferromagnetic interactions between Ni2+ ions on the 2 d sites and the additional Ni2+ \nions on the 2 c sites, which leads to a partial  cancellation of magnetic moments. It is noted, \nhowever, that an unambiguous refinement of Os  moments in double perovskites with two \nmagnetic ions at the B and B’ site is difficult  [5] and thus it cannot be ruled out that  \nordered Os5+ moments  contribute to the stabilization of the ferrimagnetic spin structure.   \n \n \n \n 15 Figure 8. (color online) Magnetic structure of La2Ni1.19Os0.81O6. For the Ni and mixed \nOs/Ni sites at 2 d(½,0,0; 0,½,½) and 2 c(½,0,½; 0,½,0) only the Ni atoms contribute to the \nmagnetic ordering. The Os site was found to be partially occupied with 19 % Ni.  The Ni \nmoments at both sites were constrained to be equal during the Rietveld refinements \nresulting in a moment μ exp(Ni)  = 1.90 (5) μ B. Therefore the total moment at the Os /Ni site \nonly reached the value 0.19 × μ exp(Ni) =  0.36 (1) μ B. For clarity , the mag nitude of the Ni \nmoment at the Os /Ni site is exaggerated. The magnetic atoms form ferrimagnetic chains \nalong the c axis, and within the ab plane one finds  a noncollinear spin arrangement.  \n \n \nFigure 9 . Temperature dependence of the magnetic moments of the Ni2+ ions in \nLa2Ni1.19Os0.81O6. The Wyckoff 2 c site is occupied with 81 % Os1 and 19  % Ni1 , while the \n2d site only contains  Ni2. During the refinements of the magnetic moments , we have \napplied the constraint μ(Ni1) = μ(Ni2), while the magnetic moment of Os  was set to be \nzero. The bold line is a guide for the eye.  \n \n \nIV. DISCUSSION  \n0 20 40 60 80 100 120 1400.00.40.81.21.62.0Magnetic moment / Ni2+ ion (B)\nTemperature (K)La2Ni1.19Os0.81O6\nTC = 125(3) K 16 We synthesized a non -stoichiometric double perovskite La 2Ni1.19Os0.81O6 containing \nmerely  Ni2+ ions at the 2 d sites but a mixed occupan cy of paramagnetic (Ni2+, Os5+) and \nnonmagnetic (Os4+) ions at the 2 c sites of the monoclinic crystal structure. Formally the \ncompound may be written as La2Ni2+(Ni 0.192+Os0.434+Os0.385+)O6. The most remarkable \nproperty of this insulating compound is that it features a ferrimagnetic  transition with an \nextraordinary broad hysteresis at 5 K.  The spin structure is noncollinear in the ab plane.   \nFor rationalizing the magnetic properties we consider  first the so far unknown \nstoichiometric  double perovskite La 2Ni2+Os4+O6 with the perfect  atomic order , where  the \nmagnetism should be  entirely determined by antiferromagnetic  exchange interactions \nbetween the half -filled eg orbitals of the Ni2+ ions as the Os4+ ions are expected to be \nnonmagnetic ( Jeff = 0).  Therefore, for La 2NiOsO 6 antiferromagnetic ordering is anticipated \nas is indeed observed for other A2NiB’O6 compounds with nonmagnetic ions on the B’site \n(B’ = W6+, Ti4+, Ir5+) [3538], c.f. Table IV. The adopted  antiferromagnetic spin structure \nwill be determined mainly by the balance between the nearest neighbor Ni2+OONi2+ \ninteractions and the next nearest neighbor Ni2+OOs4+ONi2+ exchange pathways. \nHowever, due to the non -stoichiometry the actual compound La 2Ni1.19Os0.81O6 contains a \nlarge fraction of paramagnetic Ni2+ and Os5+ ions at the 2 c sites, which is the origin for the \nobserved ferrimagnetism with TC = 125 K. Our analysis of the powder neutron diffraction \ndata indicates that the ferrimagnetism may be  solely driven by the antiferromagnetic \ninteractions between th e majority  of Ni2+ ions at the 2 d sites and the minority  of Ni2+ ions \nat the 2 c sites, which implies  that the Os5+ moments remain magnetically diso rdered and \npossibly freeze  at low temperatures. However, this analysis may not be unambiguous and \nalso antiferromagnetic Ni2+OOs5+ interactions may contribute to the stabilization of the \nnoncollinear ferrimagnetic spin structure. The latter  interactions  are a consequence of the \ndistorted crystal structure. Since  the t2g orbitals of Ni2+ are completely filled, the only \npossible antiferromagnetic nearest neighbor exchange pathway is via virtual  hopping \nbetween half -filled Ni -eg and Os -t2g orbitals. The Ni -eg and Os -t2g orbitals are orthogonal in \nthe cubic double perovskite structure, but  these interactions become  possible in the \nmonoclinic double perovskite where the Ni OOs bond angles are strongly reduced from \n18o°. This me chanism was invoked for explaining the ferrimagnetic state in \nCa2Ni2+Os6+O6 [42]. Antiferromagnetic interactions between Ni2+ (d8) and Os5+ (d3) ions \nappear to be in contradiction to the Goodenough -Kanamori rules which predict a \nferromagnetic interaction due to virtual hopping between the half -filled and empty eg \norbitals at the Ni2+ and Os5+ sites, respectively. However, compared to pure 3 d systems like  17 La2NiMnO 6 [43,44] the c rystal field splitting at the 5 d sites is much enhance d and the \nferromagnetic  coupling involving the σ-exchan ge pathway between the eg-orbital s becomes \nweak , in particular in the distorted double perovskite structure  [45,46]. \nMost remarkably the magnetization of the present powder sample of La 2Ni1.19Os0.81O6 \nfeatures a broad hysteresis with a coercive field as high as 41  kOe. Similar giant coercive \nfields have been report ed for diver se materials classes . Examples include rare earth \ntransition  metal films [ 47], ferrimagnetic transition metal – radical single chain  magnets \n[48], intermetallic systems  like Mn 1.5Ga films  [49] as well as ferrimagnetic Fe 3Se4 \nnanostructures [ 50]. In transition metal oxides  extremely high coercivities of 90 and 120 \nkOe were found for  single crystals of the ferrimagnet LuFe 2O4 [51] and weak ly \nferromagnetic Sr 5Ru5xO15 [52], respectively.  Such mater ials are of interest in the search  of \nrare earth free hard magnets. The detailed mechanism of giant coercivity is not generally \nunderstood. However, large magnetocrystalline  anisotropy in combination with frustration \nand/or defects and  disorder appears to favor huge  coercivities  [51]. A large \nmagnetocrystalline anisotropy is indeed expected for La 2Ni1.19Os0.81O6 due to the low -\nsymmetry crystal structure and the  presence of  the heavy  Os atoms . The non -collinear spin \narrangement in the ab plane can be attributed to  the Dzyaloshinskii -Moriya interaction \nwhich induces spin -canting, whereas the high coercivity  is possibly associated with a \npeculiar microstructure and competing exchange interactions introduced by  the atomic  \ndisorder.   \n \nTABLE  IV. Comparison of the magnetic properties of La 2Ni1.19Os0.81O6 with those of \nsome Ni2+ based double perovskite oxides. The ions on the B’ site of the double perovskite \nstructure are either non -magnetic (W6+, Ti4+, Ir5+) or magnetic (Os5+, Ir6+). \nMagnetic ions  composition  Space \ngroup  Properties  Tm (K) θw (K) µeff (µB) Ref. \nNi2+ Ca2NiWO 6 P21/n AFM  52.5 75 2.85 35 \nNi2+ Sr2NiWO 6 I4/m AFM  54 175 3.05 36 \nNi2+ La2NiTiO 6 P21/n AFM  25 60 3.12 37 \nNi2+ SrLaNiIrO 6 P21/n AFM  74  3.3 38 \nNi2+ + Os4.5+ La2Ni1.19Os0.81O6 P21/n FIM 125 73 3.63 This work  \nNi2+ + Os5+ SrLaNiOsO 6 P21/n AFM  60 23 4.13 39 \nNi2+ + Os5+ CaLaNiOsO 6 P21/n AFM  30 83 3.87 40  18 Ni2+ + Ir6+ Sr2NiIrO 6 P21/n AFM  58 - - 41 \n \n \nConclusions  \nTargeting  the Os4+ (Jeff = 0) compound  La2NiOsO 6, we actually obtained  \nnonstoichiometric polycrystalline samples of La 2Ni1.19Os0.81O6 by solid st ate reaction . As \nmany other double perovskite oxides , La2Ni1.19Os0.81O6 crystallizes in the monoclinic \nspace group P21/n. Here, the  B sites are fully occupied by Ni2+ ions whereas the  B’ sites \nfeature a mixed occupancy with paramagnetic Ni2+ and Os5+ as well as  non-magnetic Os4+ \nions.  Whereas stoichiometric La 2NiOsO 6 is expected to be an antiferromagnet, \nLa2Ni1.19Os0.81O6 reveals a  ferrimagnetic transition at 125 K  and adopts a ferrimagnetic \nspin arrangement wi th collinear chains along the c axis, but spin canting in the ab plane.  \nOur data analysis indicates  that only Ni2+ magnetic moments are long -range ordered, but \nthe behavior of the Os5+ moments is not entirely  clear . The magnetization curve of  \nLa2Ni1.19Os0.81O6 is characterized by a  huge  coercive field, HC = 41 kOe at T = 5 K  which \nis attribute d to the combined influence of large magnetocrystalline anisotropy and atomic \ndisorder . Our results suggest that careful tuning of the chemical composition may lead to \nthe discovery of new  rare earth free hard magnets  in the class of double perovskite oxides . \n \nAcknowledgement  \nThe work in Dresden was partially supported by the Deutsche Forschungsgemeinschaft \nthrough SFB 1143.  We thank R. Koban, H. Borrmann, and U. Burkhardt fo r performing \nmagnetization , x-ray, and EDX measurements.  \n \nReferences  \n1. K. L. Kobayashi, T. Kimura, H. Sawada, K. Terakura,  and Y. Tokura, Nature, 395, \n677 (1998) . \n2. H. Kato, T. Okuda, Y. Okimoto, Y. Tomioka, Y. Takenoya, A. Ohkubo, M. \nKawasaki,  and Y. Tokura, Appl. Phys. Lett. 81, 328  (2002) . \n3. Y. Krockenberger, K. Mogare, M. Reehuis, M. Tovar, M. Jansen, G. \nVaitheeswaran, V. Kanchana, F. Bultmark, A. Delin, F. Wilhelm, A. Rogalev, A. \nWinkler,  and L. Alff, Phys. Rev. B 75, 020404  (2007) .  19 4. H. L. Feng, M. Arai, Y. Matsush ita, Y. Tsujimoto, Y. F. Guo, C. I. Sathish, X. \nWang, Y. H. Yuan, M. 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Mater. 22, 5712  (2010) . \n "    },    {        "title": "1802.05685v2.Origin_of_spin_reorientation_transitions_in_antiferromagnetic_MnPt_based_alloys.pdf",        "content": "Origin of spin reorientation transitions in antiferromagnetic MnPt-based alloys\nP.-H. Chang, I. A. Zhuravlev, and K. D. Belashchenko\nDepartment of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,\nUniversity of Nebraska-Lincoln, Lincoln, NE 68588, USA\nAntiferromagnetic MnPt exhibits a spin reorientation transition (SRT) as a function of temper-\nature, and o\u000b-stoichiometric Mn-Pt alloys also display SRTs as a function of concentration. The\nmagnetocrystalline anisotropy in these alloys is studied using \frst-principles calculations based on\nthe coherent potential approximation and the disordered local moment method. The anisotropy is\nfairly small and sensitive to the variations in composition and temperature due to the cancellation\nof large contributions from di\u000berent parts of the Brillouin zone. Concentration and temperature-\ndriven SRTs are found in reasonable agreement with experimental data. Contributions from speci\fc\nband-structure features are identi\fed and used to explain the origin of the SRTs.\nAntiferromagnetic materials are of interest for mag-\nnetoelectronic applications thanks to their insensitivity\nto stray \felds and the accessibility of ultrafast dynamics.\nIn particular, memory cells controlled by current-induced\nspin-orbit torque [1], terahertz spin-Hall nano-oscillators\n[2], and magnetoelectric memory cells [3, 4] have been\nproposed. These features also make antiferromagnets at-\ntractive for magnonic applications [5, 6].\nMagnetocrystalline anisotropy energy (MAE) is an im-\nportant parameter for antiferromagnonic devices, be-\ncause it controls the spin wave spectrum at long wave-\nlengths. Resonant parametric excitation by utilizing\nvoltage-controlled MAE, like it was demonstrated for fer-\nromagnets [7, 8], could be used to generate spin wave\npackets in ultrathin antiferromagnetic nanostrips. Anti-\nferromagnets with a small but tunable MAE are desirable\nto take advantage of the linear magnon spectrum for low-\ndistortion signal transmission, while allowing for e\u000ecient\nspin wave generation, manipulation, and detection.\nMnPt and o\u000b-stoichiometric alloys based on this\ntetragonal compound exhibit spin reorientation transi-\ntions (SRT) driven by both temperature and composi-\ntion [9, 10], suggesting that they may be suitable for\nmagnonic applications. While most measurements ob-\ntained easy-axis anisotropy [9{12] at room temperature,\nin-plane anisotropy has also been reported [13]. Although\nthe magnetic moments on the Pt atoms vanish by symme-\ntry, spin-orbit coupling on Pt can strongly in\ruence MAE\nthrough hybridization with Mn, as in similar L1 0-ordered\nantiferromagnets [14]. Nevertheless, \frst-principles cal-\nculations \fnd a small MAE ( K\u00180:1 meV/f.u.), which\nis more than an order of magnitude smaller compared\nto FePt, and di\u000berent computational methods disagree\nin its sign [15, 16]. It was also found that MAE is very\nsensitive to band \flling in the rigid-band approximation\n[16]. All of these experimental and theoretical results\nclearly indicate a small and easily tunable MAE.\nIn itinerant magnets, anomalies in the temperature de-\npendence of MAE may occur due to a variety of band-\nstructure e\u000bects, such as the variation in band \flling\nand band broadening induced by thermal spin \ructua-\ntions [17]. Understanding of these e\u000bects calls for a \frst-principles analysis. In this paper, we examine the concen-\ntration and temperature dependence of MAE in MnPt-\nbased alloys, obtaining the phase diagram in reasonable\nagreement with experimental data. Similar to the case\nof ferromagnetic (Fe 1\u0000xCox)2B alloys [17], we \fnd that\nthe temperature-induced SRTs observed in antiferromag-\nnetic MnPt-based alloys are attributable to the e\u000bects of\nthermal spin disorder on the electronic structure.\nCalculations were performed using the Green's\nfunction-based formulation of the tight-binding linear\nmu\u000en-tin orbital (GF-LMTO) method and the coher-\nent potential approximation (CPA) to describe substi-\ntutional disorder [18]. A series of Mn 1\u0000xPt1+xalloys\nwas considered, where x= 0 corresponds to the L1 0-\nordered stoichiometric compound MnPt, while \fnite x\ncorresponds to excess Pt or Mn substituting randomly\non the other sublattice. Concentration-dependent room-\ntemperature lattice constants [9] were smoothly inter-\npolated and used in all calculations. We have veri-\n\fed that the results are not strongly a\u000bected by using\ntemperature-dependent lattice constants for stoichiomet-\nric MnPt [9].\nThermal spin \ructuations were included on the same\nfooting with substitutional disorder using the disordered\nlocal moment (DLM) method [17, 19{21]. Integration\nover the orientations of Mn spins was performed using\na 122-point quadrature including 12 vertices, 20 face\ncenters, and 30 edge centers of an icosahedron plus 60\nvertices of a truncated icosahedron (buckyball). The\nquadrature weights were chosen so that any linear com-\nbination of angular harmonics with l\u001415 is integrated\nexactly. The statistical probability distribution for the\nspin orientations was taken from the mean-\feld approx-\nimation for the classical Heisenberg model at the given\nT=T Nratio, where TNis the N\u0013 eel temperature.\nSpin-orbit coupling (SOC) was included as a pertur-\nbation to the LMTO potential parameters [22, 23], and\nthe generalized gradient approximation [24] was used\nfor exchange and correlation. The anisotropy energy\nKis calculated as the single-particle energy di\u000berence\nbetween the in-plane (100) and out-of-plane (001) ori-\nentations of the spins, taking the charge density fromarXiv:1802.05685v2  [cond-mat.mtrl-sci]  1 May 20182\nthe self-consistent calculation without SOC. A uniform\n32\u000232\u000232k-space mesh provided su\u000ecient convergence\nfor the Brillouin zone integration. The computational de-\ntails are similar to Refs. 17 and 22. In the analysis of\nk-resolved MAE, the data are symmetrized with respect\nto the C4rotation.\nFirst, we study the in\ruence of o\u000b-stoichiometry xin\nMn 1\u0000xPt1+xalloys on MAE at zero temperature. At\nx < 0, the excess Mn atoms occupy the sites on the\nPt sublattice, and their net interaction with the spins\non the Mn sublattice vanishes by symmetry. Therefore,\nthe spins of excess Mn atoms are expected to remain\ndisordered if their concentration is small and the tem-\nperature is not very low. At larger concentrations, the\ninteractions among the excess Mn spins could promote\ntheir ordering. To estimate its importance for MAE, we\nconsidered three hypothetical cases for excess Mn spins:\nfully ordered ferromagnetic, fully ordered antiferromag-\nnetic, and fully disordered.\nFig. 1(a) shows the MAE calculated in CPA as a func-\ntion of concentration. We see that MAE exhibits a qual-\nitatively similar behavior for all three descriptions of ex-\ncess Mn. Therefore, we did not attempt to determine\nthe ground state but simply considered the disordered\ncon\fguration in all calculations that follow.\nIn agreement with experimental data [9, 10], Fig. 1(a)\nshows easy-axis anisotropy at T= 0 in stoichiometric\nMnPt and SRTs to easy-plane anisotropy at x\u0019 \u00000:26\nandx\u00190:02 [25]. In addition, we \fnd another SRT\nback to easy-plane anisotropy at x\u00190:08, which, to our\nknowledge, has not been experimentally reported.\nFig. 1(b) shows the MAE as a function of the electron\ncount in the rigid-band approximation, which agrees well\nwith the calculations of Refs. 16. However, the compari-\nson with CPA calculations in Fig. 1(a) clearly shows that\nthe rigid-band approximation fails to describe the behav-\nior of MAE in Mn 1\u0000xPt1+xalloys even on the qualitative\nlevel. In particular, it predicts a qualitatively wrong be-\nhavior of MAE on the Pt-rich side and overestimates it\nby an order of magnitude. Full CPA calculations are,\ntherefore, essential for the description of MAE in this\nsystem.\nIf SOC is treated as a perturbation to the Hamiltonian,\nthe second-order approximation for MAE can be repre-\nsented as a sum of pairwise contributions corresponding\nto the pairs of sites on which the two SOC operators\nare applied in the perturbative expansion [26]. This de-\nscription is approximate, because second-order pertur-\nbation theory can fail in metals for band crossings near\nthe Fermi level, especially in antiferromagnets where all\nbands are degenerate by spin. In FePt and CoPt the\nMAE is dominated by single-site terms on Pt [26]. To\nestimate the role of di\u000berent terms for antiferromagnetic\nMnPt, we performed two auxiliary calculations with SOC\nsuppressed on Mn or Pt atoms. The MAE is negligibly\nsmall if SOC on Pt is suppressed, but it is large and pos-\n-0.3 -0.2 -0.1 0.0 0.1 0.2-0.10-0.050.000.050.100.15\nExcess Mn:\nAFM\nFM\nDisordered\nxK(meV/f.u.)\n33 34 35-2-1012\nElectron countK(meV/f.u.)(a)\n( (b)FIG. 1. (a) Calculated MAE of Mn 1\u0000xPt1+xat zero tem-\nperature. Three curves at x < 0 correspond to di\u000berent or-\nderings of excess Mn spins (see text). The experimental SRTs\n[9] are shown by vertical dashed lines. (b) Calculated MAE\nas a function of electron count per unit cell in the rigid-band\nmodel. The range of the electron count matches the range of\nxin panel (a).\nitive (1.1 meV/f.u.) if SOC is suppressed on Mn. This\nindicates that, in contrast to FePt and CoPt, in MnPt\nthe large negative Mn-Pt cross-term nearly cancels the\nlarge positive term coming solely from SOC on Pt.\nTo obtain insight into the origin of the SRTs, let us ex-\namine the k-resolved MAE displayed in Fig. 2. Panel (b)\nfor stoichiometric MnPt shows large positive and nega-\ntive contributions from di\u000berent regions of the Brillouin\nzone, which largely cancel each other out. For compari-\nson, panel (d) for ferromagnetic FePt shows that positive\ncontributions dominate over most of the Brillouin zone in\nthat material, adding up to a large easy-axis anisotropy of\nabout 2.6 meV/f.u., in agreement with other calculations\n[15, 27, 28]. Note that a very large easy-axis anisotropy\nof 4.7 meV/f.u. was obtained for the hypothetical ferro-\nmagnetic phase of MnPt [29] at the experimental lattice\nparameters. Strong dependence of MAE on the magnetic\nstate was also found in other similar compounds [14].\nOne can identify three distinct features in Fig. 2(a)-\n(c): the strong spherical \\hot spot\" giving mostly posi-\ntive contribution around the Apoint, the cylindrical re-\ngion around the \u0000 Zline giving a negative contribution,3\n(e) (f)(a) (b)(c)\n(d)\nFIG. 2. C4-symmetrized k-resolved MAE in Mn 1\u0000xPt1+xat\n(a)x=\u00000:12, (b) x= 0, (c) x= 0:04, (d) in stoichiometric\nFePt, (e) in pure MnPt at T=T N= 0:25 (near the SRT) and\n(f)T=T N= 0:75 (near the minimum of MAE). Red (blue)\ncolor shows positive (negative) values; the range of values in\npanels (e)-(f) is 2.5 narrower compared to panels (a)-(d).\nand the slowly varying background. The comparison of\npanels (a), (b), and (c), corresponding to di\u000berent con-\ncentrations, suggests that the Ahot spot and the \u0000 Z\ncylinder are sensitive to o\u000b-stoichiometry.\nSharp features in k-resolved MAE come from the pairs\nof occupied and unoccupied bands near the Fermi level\nthat are strongly mixed by SOC [26, 30]. Fig. 3 shows\nBloch spectral functions, for the same concentrations as\nin Fig. 2(a)-(c), at T= 0, along several high-symmetry\nlines in reciprocal space, for two orientations of the an-\ntiferromagnetic order parameter: (100) for panels (a)-\n(c) and (001) for (d)-(f). Centered around the Apoint,\nwe \fnd two bands with conical dispersions crossing the\nFermi level. These two bands are degenerate in the ab-\nsence of SOC, and their splitting depends on the orien-\ntation of the order parameter, producing the hot spot in\nZ R A Z-2-10 \n1 2 \nZ R\nA Z Z R\nA ZZ R A Z-2-10 \n1 2 \nZ R\nA Z Z R A Zx=-0.12 x=0 x=0.04\n(e) (f) (d)(c) (b) (a)E (eV) E (eV)FIG. 3. Bloch spectral functions in Mn 1\u0000xPt1+x. Red and\nblue color densities represent the spectral weights of the ma-\njority and minority-spin states of Mn, and green represents all\nstates of Pt. The order parameter is oriented along the (100)\ndirection in panels (a)-(c) and along (001) in panels (d)-(f).\nThe concentrations xare indicated above the panels.\nk-resolved MAE around the Apoint in Fig. 2.\nAt the Zpoint, SOC also splits two otherwise degen-\nerate bands, pushing one of them above and the other\nbelow the Fermi level. This results in a hot spot at Z,\nwhich makes a negative contribution to MAE. The mix-\ning of other bands along the \u0000 Zline also results in a\npronounced negative contribution to MAE. The sensitiv-\nity of the sharp features seen in Fig. 2 to o\u000b-stoichiometry\ncan be traced to the changing occupations, hybridization,\nand broadening of the bands in Fig. 3.\nThe smooth background in Fig. 2(a)-(c) comes from\nthe mixing of the occupied and unoccupied states that\nare far away from the Fermi energy.\nIt is interesting to note that the bands dominated by\nPt states (seen as green in Fig. 3) are insensitive to the\norientation of the order parameter, which is because Pt\natoms carry no magnetic moments. However, spin-orbit\ncoupling on Pt contributes to MAE through its e\u000bect on\nthe hybridized bands carrying both Mn and Pt character.\nThis is a reciprocal-space counterpart of the argument of\nRef. [14] for similar L1 0-ordered MnX antiferromagnets,\nwhich interpreted the contribution of SOC on X atoms to\nMAE in terms of the non-trivial real-space spin-density\ndistribution on X. In contrast, the Pt atoms are magnet-\nically polarized in ferro magnetic compounds like FePt\nand CoPt, whereby all bands can contribute to MAE.4\nFig. 4 shows the concentration dependence of the con-\ntributions to MAE integrated over three regions dis-\ncussed above: the sphere enclosing the hot spot near A,\nthe cylinder containing the features near the \u0000 Zline, and\nthe rest of the Brillouin zone (background). All three\ncontributions are fairly large and of the same order of\nmagnitude, but there is a strong cancellation of positive\nand negative contributions. All contributions are reduced\nby o\u000b-stiochiometry (both excess Mn and excess Pt) but\nat di\u000berent rates, leading to large relative variations in\nthe total MAE and to the SRTs. The strong cancellation\nalso makes MAE at small xsensitive to the variations in\nthec=aratio [15, 16, 31] and to temperature changes, as\nwe show below.\n-0.3 -0.2 -0.1 0.0 0.1 0.2-0.6-0.4-0.20.00.20.4 Mn-rich\n Sphere\n Cylinder\n Background\n Total\nxK (meV/f.u.)Pt-rich\nFIG. 4. Contributions to MAE from di\u000berent parts of the\nBrillouin zone as a function of o\u000b-stoichiometry.\nThe temperature dependence of MAE, calculated for\ndi\u000berent concentrations using the CPA-DLM method, is\ndisplayed in Fig. 5, where we used the experimental N\u0013 eel\ntemperature [9] for each concentration. The temperature\ndependence of MAE is anomalous (non-monotonic) at all\nconcentrations: the MAE decreases at low temperatures\nbut then passes through a minimum and increases back\nto zero as the temperature tends to the N\u0013 eel point. Thus,\nat those concentrations where Kis positive (easy-axis)\natT= 0, we always \fnd a SRT.\nAs we saw in Figs. 2-4 above, the electronic bands\ncrossing or approaching the Fermi level in di\u000berent re-\ngions of the Brillouin zone produce large contributions of\nopposite signs to MAE, which are sensitive to the shifts\nand broadening of those bands. For stoichiometric MnPt,\nthe temperature dependences of the k-resolved MAE, in-\ntegrated contributions from di\u000berent parts of the Bril-\nlouin zone, and spectral functions are displayed in Figs.\n2(e)-(f), 6, and 7, respectively.\nFigs. 2(b),(e)-(f) and 6 show that the positive back-\nground contribution [smooth red regions in Fig. 2(b),(e)-\n(f)] decreases the fastest with increasing temperature, to\nthe extent that the entire background contribution turns\n0 200 400 600 800 1000-0.2-0.10.00.10.2Pt-rich\nx= 0\nx= 0.04\nx= 0.08\nx= 0.12\nx= 0.16\nT(K)K(meV/f.u.)\n0 200 400 600 800 1000-0.10-0.050.000.050.10 x= 0\nx= -0.04\nx= -0.12\nx= -0.24\nx= -0.28\nT(K)K(meV/f.u.)Mn-rich\n(a)\n(b)FIG. 5. Temperature dependence of MAE for (a) Mn-rich\nand (b) Pt-rich Mn-Pt alloys.\nnegative at T=T N\u00190:6. Note that the color map in\npanels (e)-(f) has been rescaled to emphasize this back-\nground. In contrast, the contribution from the vicinity\nof the Apoint decreases relatively slowly with tempera-\nture. The competition of large contributions declining at\ndi\u000berent rates results in the anomalous behavior of the\ntotal MAE and leads to a SRT.\nFig. 7 shows that some of the bands, including those\nthat are split by SOC near Z, are strongly broadened\nalready at T=T N= 0:25, and most bands are completely\nsmeared out at T=T N= 0:75. On the other hand, the\nconical bands around the Apoint are visible even at\nT=T N= 0:75, which explains the slow decline of their\ncontribution to MAE.\nThe phase diagram based on our results is plotted in\nFig. 8, which also shows the experimental data [9]. Note\nthat we did not attempt to determine the N\u0013 eel temper-\nature TN, because our focus is on understanding the be-\nhavior of the MAE. The CPA-DLM calculations (Fig. 5)5\n0.0 0.2 0.4 0.6 0.8 1.0-0.6-0.4-0.20.00.20.4\nSphere\nCylinder\nBackground\nTotal\nT/TNK(meV/f.u.)\nFIG. 6. Temperature dependence of the contributions from\ndi\u000berent parts of the Brillouin zone in stoichiometric MnPt.\nZ R A Z-2-10 1 \n2 \nZ R A ZE (eV)(a) (b)\nFIG. 7. Spectral functions of stoichiometric MnPt at (a)\nT=T N= 0:25 and (b) T=T N= 0:75.\nproduce Ts=TN, where Tsis the temperature of the SRT.\nAs discussed above in connection with Fig. 1(a), the\nSRTs predicted at x\u0019 \u0000 0:26 and x\u00190:02 at zero\ntemperature agree with experimental results. Our cal-\nculation predicts a thermal SRT in the entire range\n\u00000:26< x < 0:02. In experiment [9, 25]; a thermal\nSRT was found at x= 0, x=\u00000:04, and x=\u00000:24,\nbut not at x=\u00000:13. In addition, the SRTs predicted\nby CPA-DLM occur at considerably lower temperatures\ncompared to experiment. These quantitative di\u000berences\nare not surprising in view of the strong cancellations of\ndi\u000berent contributions to MAE. In particular, an under-\nestimated MAE at T= 0 would also lead to an underesti-\nmated SRT temperature Ts. In addition, we note that the\nobserved SRT (detected using the [101]/[100] Bragg peak\nintensity ratio) occurs in a fairly wide temperature range\n[9, 11]. A careful study using a stoichiometric MnPt sin-\ngle crystal [11] reported a gradual transition between 580\nMnPtPt-rich\n0 -0.2 xTsTNT\n0.2Mn-richFIG. 8. Predicted phase diagram using the calculated Ts=TN\nratios and the experimental data for TN(black \flled circles\nguided by the solid red line) [9]. Black \flled squares: ex-\nperimental SRT Ts[9]; dashed blue line: sketch of the ex-\nperimental Ts(x). Open circles connected by solid green line:\ntheoretical Ts.\nand 770 K. Based on the analysis of the magnon spec-\ntrum, it was concluded that the transition involves the\nchanging volume ratios of the easy-axis and easy-plane\nregions. This \fnding suggests that the gradual character\nof the SRT is associated with the spatial inhomogeneity\nof the L1 0order parameter or concentration. Such varia-\ntions are likely to be even larger in the o\u000b-stoichiometric\npowder samples [9]. On the other hand, our CPA-DLM\ncalculations assume a perfectly homogeneous alloy with\nthe maximal order parameter allowed at the given con-\ncentration.\nTo conclude, we have studied the concentration\nand temperature dependence of magnetocrystalline\nanisotropy in L1 0-ordered Mn-Pt alloys using \frst-\nprinciples calculations. The strong cancellation of con-\ntributions from di\u000berent regions in the Brillouin zone\nexplains the small magnitude of the anisotropy en-\nergy and its sensitivity to o\u000b-stoichiometry and tem-\nperature changes, which gives rise to concentration and\ntemperature-driven spin reorientation transitions.\nWe are grateful to Sergii Khmelevskyi and Ilya Kriv-\norotov for useful discussions. This work was supported\nby the Nanoelectronics Research Corporation (NERC), a\nwholly-owned subsidiary of the Semiconductor Research\nCorporation (SRC), through the Center for Nanoferroic\nDevices (CNFD), a SRC-NRI Nanoelectronics Research\nInitiative Center (Task ID 2398.003), and by the Na-\ntional Science Foundation through the Nebraska MR-\nSEC (Grant No. DMR-1420645) and Grant No. DMR-\n1609776. Calculations were performed utilizing the Hol-\nland Computing Center of the University of Nebraska,\nwhich receives support from the Nebraska Research Ini-\ntiative.6\n[1] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich,\nNature Nanotech. 11, 231 (2016).\n[2] R. Cheng, D. Xiao, and A. Brataas, Phys. Rev. Lett.\n116, 207603 (2016).\n[3] X. He, Y. Wang, N. Wu, A. N. Caruso, E. Vescovo, K.\nD. Belashchenko, P. A. Dowben, and C. Binek, Nature\nMater. 9, 579 (2010).\n[4] T. Kosub, M. Kopte, R. H uhne, P. Appel, B. Shields, P.\nMaletinsky, R. H ubner, M. O. Liedke, J. Fassbender, O.\nG. 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Sakuma, Mater.\nTrans., JIM 47, 2 (2006)."    },    {        "title": "1802.05817v1.Role_of_typical_elements_in_Nd___2__Fe___14___X____X____B__C__N__O__F_.pdf",        "content": "arXiv:1802.05817v1  [cond-mat.mtrl-sci]  16 Feb 2018Role of typical elements in Nd 2Fe14X(X= B, C, N, O, F)\nYasutomi Tatetsu,1,2,∗Yosuke Harashima,2,3,†Takashi Miyake,2,3,4and Yoshihiro Gohda1,2\n1Department of Materials Science and Engineering,\nTokyo Institute of Technology, Yokohama 226-8502, Japan\n2ESICMM, National Institute for Materials Science, Tsukuba 305-0047, Japan.\n3CD-FMat, National Institute of Advanced Industrial Scienc e and Technology, Tsukuba 305-8568, Japan\n4CMI2, MaDIS, National Institute for Materials Science, Tsukuba 305-0047, Japan\n(Dated: February 19, 2018)\nThe magnetic properties and structural stability of Nd 2Fe14X(X= B, C, N, O, F) are theoreti-\ncally studied by first-principles calculations focusing on the role of X. We find that B reduces the\nmagnetic moment (per formula unit) and magnetization (per v olume) in Nd 2Fe14B. The crystal-field\nparameter A0\n2/angbracketleftr2/angbracketrightof Nd is not enhanced either, suggesting that B has minor role s in the uniaxial\nmagnetocrystalline anisotropy of Nd. These findings are in c ontrast to the long-held belief that B\nworks positively for the magnetic properties of Nd 2Fe14B. AsXchanges from B to C, N, O and F,\nboth the magnetic properties and stability vary significant ly. The formation energies of Nd 2Fe14X\nandα-Fe relative to that of Nd 2Fe17Xare negative for X= B and C, whereas they are positive\nwhenX= N, O and F. This indicates that B plays an important role in st abilizing the Nd 2Fe14B\nphase.\nI. INTRODUCTION\nBack in the 1970’s, Sm 2Co17was the strongest known\nhard magneticcompound. As Cois an expensiveelement\nwhereasFe is relativelycheap, developingiron-basedper-\nmanent magnets was required. Sm 2Fe17, more generally\nknownas R 2Fe17whereR representsrare-earthelements,\nwas a natural choice. However, the Curie temperature\nTcof R2Fe17is too low for practical applications. For\nexample, Tcof Nd2Fe17is about 330 K [1–3]. Sagawa\ncame up with the following idea in order to raise the Tc\nof R2Fe17: The introduction of light elements, such as\nB and C, at the interstitial region in R 2Fe17, would ex-\npand the volume of R 2Fe17[4, 5]. This volume expansion\ncould make the ferromagnetism stronger by the magne-\ntovolume effect, consequently Tcwould go up. He added\nB to Nd 2Fe17based on this idea, and successfully de-\nveloped Nd-Fe-B magnets having a Tcof 585 K [4, 6].\nIt is important to note that the main phase of Nd-Fe-B\nmagnets is Nd 2Fe14B, not Nd 2Fe17B.\nAfter the Nd-Fe-B magnet was developed, the role of\nB on the magnetism of Nd 2Fe14B was studied theoreti-\ncally [7–9]. Kanamori pointed out the importance of the\np-dhybridization between B and Fe sites. The magnetic\nmoment of Fe in the vicinity of B should be suppressed.\nIt is called cobaltization. The cobaltized Fe (pseudo Co)\ncan enhance the magnetic moment of the surrounding\nFe due to the d-dhybridization between pseudo Co and\nFe, as is in Fe-Co alloys [10, 11]. It has been believed\nfordecadesthatwhenthesechemicaleffects, owingtohy-\nbridizations, are summed up, the total magnetic moment\nof Nd2Fe14B increases by B.\nAsano and Yamaguchi studied the magnetic proper-\nties of Y 2Fe14B and the hypothetical compound Y 2Fe14\n∗tatetsu.y.aa@m.titech.ac.jp\n†yosuke-harashima@aist.go.jpthrough first-principles calculations [12, 13]. Their re-\nsults show (i) the magnetic moment of Fe, at the near-\nest neighbor of B in Y 2Fe14B, is reduced by the cobal-\ntization, (ii) the magnetic moment of Fe at the nearest\nneighbor of pseudo Co is enhanced, and (iii) the effect of\nB is slightly negative on the magnetization in Y 2Fe14B.\nTherefore, Y 2Fe14B has smaller total magnetic moment\nthan Y 2Fe14. However, they optimized only lattice pa-\nrameters within the atomic-sphere approximation, and\ninternal coordinates were fixed. Moreover, magnetovol-\nume effects and chemical effects were not distinguished.\nTherefore, the effects of B on the magnetism in R 2Fe14B\nstill needs to be clarified.\nIn this study, we systematically investigate the influ-\nence of B on the electronic states and magnetism of\nNd2Fe14B through first-principles calculations. We find\nthat the total magnetic moment (per formula unit) and\nmagnetization (per volume) of Nd 2Fe14B decrease by the\naddition of B . This result is analyzed in terms of the\nchemical effect and magnetovolume effect caused by B.\nThe local magnetic moment at each Fe site is computed\nand is discussed in connection with cobaltization. We\nalso study Nd 2Fe14X, whereXrepresents C, N, O and\nF, in order to investigate the effects of the Xelement\non the magnetization, magnetocrystallineanisotropyand\nstability of Nd 2Fe14X.\nII. COMPUTATIONAL METHODS\nWe perform first-principles calculations of Nd 2Fe14X\nwithin density functional theory. We use the computa-\ntional package OpenMX, which is based on pseudopo-\ntentials and pseudo-atomic-orbital basis functions [14].\nThe basis set of Nd, Fe and Xiss2p2d2, and the cutoff\nradii of these atoms are 8.0, 6.0 and 7.0 a.u., respec-\ntively. For semicore states, we treat 3 sand 3porbitals as\nvalence electrons in Fe, as well as 5 sand 5pin Nd. An2\nopen-core pseudopotential is used for Nd atoms, where\n4felectrons are treated as spin-polarized core electrons.\nThe calculation for valence electronic states does not in-\nclude the spin-orbit interaction. We adopt the Perdew-\nBurke-Ernzerhof exchange-correlation functional in the\ngeneralized gradient approximation [15]. The lattice pa-\nrameters and atomic positions of Nd 2Fe14Xare all re-\nlaxed. For the convergence criteria, the maximum force\non each atom and the total energy are 10−4hartree/bohr\nand 10−7hartree, respectively. Spin collinear structures\nare assumed in all the calculations for Nd 2Fe14X. The\ncutoff energy is 500 Ry, and 8 ×8×6k-point meshes\nare adopted for all the calculations.\nThe magnetic moment is estimated from the spin mo-\nment of valence electrons and the contribution of Nd-4 f\nelectrons gJJ= 3.273µB, wheregJis the Lande g-factor\nandJis the total angularmomentum ofNd-4 felectrons.\nThe magnetization is calculated from the total magnetic\nmoment of Nd 2Fe14Xby dividing volume V(see Ap-\npendix A) and multiplying the Bohr magneton µB. The\ncrystal-fieldparameters Am\nlofNd arecalculatedbyusing\nthe computational packageQMAS, which is based on the\nprojectoraugmented-wavemethod [16]. SeeRefs.[17,18]\nfor more details.\nWe note that there are different notations for the\nWyckoff positions of Nd 2Fe14B [19–22]. Throughout this\npaper, the notation given in Refs. 19 and 22 is used.\nIII. RESULTS AND DISCUSSION\nA. Roles of B in Nd 2Fe14B\nWe calculate the magnetic moment and magnetization\nof Nd2Fe14B and Nd 2Fe14in order to investigate the ef-\nfects of B on Nd 2Fe14B. Nd 2Fe14is a hypothetical com-\npound in which the Xsite is empty and all the atomic\npositions and lattice parameters are optimized. For com-\nparison, we also study Nd 2Fe14B0, in which the lattice\nparameters and atomic positions are the same as those\nof Nd2Fe14B, but B has been removed. These hypothet-\nical compounds are introduced in order to distinguish\nthe magnetovolume and chemical effects on the mag-\nnetic moment and magnetization of Nd 2Fe14B. The re-\nsults are listed in Table I. From the comparison between\nNd2Fe14B and Nd 2Fe14, we find that the introduction of\nB decreases both the magnetic moment and magnetiza-\ntion of Nd 2Fe14B. The magnetovolume effect of B can\nTABLE I. The magnetic moment m[µB/f.u.] and magneti-\nzationµ0M[T] of Nd 2Fe14B, Nd 2Fe14B0, and Nd 2Fe14.\nm[µB/f.u.]µ0M[T]\nNd2Fe14B 37.42 1.86\nNd2Fe14B038.87 1.93\nNd2Fe14 38.43 1.9522.22.42.62.8\n16k116k28j18j24e4cmFe [µB]\nFe siteNd2Fe14B\nNd2Fe14B0\nFIG. 1. (Color online) The local magnetic moments of Fe at\neach site mFein Nd2Fe14B (closed red circles) and Nd 2Fe14B0\n(open black circles). The site notation follows Refs. 19 and\n22. Lines are provided as a visual guide.\nbe evaluated by comparing magnetization of Nd 2Fe14B0\nwith that of Nd 2Fe14. The difference between the two\ncomes solely from the structural change. The magnetic\nmoment of Nd 2Fe14B0is enhanced by 1% compared to\nthat of Nd 2Fe14. When it is converted to magnetization,\nhowever, the magnetization is reduced by 0.017 T due\nto the volume expansion. By comparing Nd 2Fe14B and\nNd2Fe14B0, we can see that the chemical effect of B is\nnegative on both the magnetic moment and magnetiza-\ntion. The magnitude is larger in the chemical effect than\nin the magnetovolume effect. This leads to a decrease\nof the magnetic moment caused by B. This result gives\nus a different insight from that of the current belief that\nB enhances the magnetic moment and magnetization of\nNd2Fe14B.\nThe local magnetic moments of Fe at each site in\nNd2Fe14B and Nd 2Fe14B0are shown in Fig.1. By fol-\nlowing the concept of cobaltization, one can expect the\ndecrease of the Fe magnetic moment at the neighbors of\nB. The local magnetic moment at the Fe sites near the\npseudo Co, in turn, can be expected to be enhanced due\nto the presence of pseudo Co. The Fe atoms at the 16k 1\nand 4e sites are the first and second neighbors of B, and\nthe distances between these Fe atoms and B are about\n2.1˚A. Therefore, these Fe atoms become pseudo Co. In\nfact, the decreaseofthe Fe localmoments isactuallyseen\nat these two sites in Fig.1. All the other Fe sites are near\nthe pseudo Co, thus, we can see the small enhancement\nof the magnetic moment at these sites. The reduction at\nthe pseudo Co sites are larger than the increase at the\nneighbors of the pseudo Co sites in magnitude. To sum\nup, B reduces the magnetic moment and magnetization\nof Nd2Fe14B.\nNext, we examine how the magnetocrystalline\nanisotropy of Nd 2Fe14B is influenced by the added B.\nTo do this, we analyze the crystal-field parameter A0\n2/an}bracketle{tr2/an}bracketri}ht\nof Nd. The magnetocrystalline anisotropy energy EMAE3\ncan be expressed as,\nEMAE=K1sin2θ+K2sin4θ+···.(1)\nIn crystal-field theory, K1is expressed as,\nK1=−3J/parenleftbigg\nJ−1\n2/parenrightbigg\nΘ0\n2A0\n2/an}bracketle{tr2/an}bracketri}ht, (2)\nwhereA0\n2/an}bracketle{tr2/an}bracketri}htis a crystal-field parameter. The crystal-\nfield parameter of Nd can be calculated from the ef-\nfective potential obtained through first principles cal-\nculations. Although the magnetocrystalline anisotropy\nenergy includes higher-order contributions, the magne-\ntocrystalline anisotropy of Nd at high temperatures can\nbe discussed by the lowest order contribution [23, 24].\nThe higher-order crystal-field parameters are discussed\nin Appendix B. There are two different Wyckoff posi-\ntions for Nd, labeled as Nd(4f) and Nd(4g). The calcu-\nlatedA0\n2/an}bracketle{tr2/an}bracketri}htare 220 K at Nd(4f) and 330 K at Nd(4g)\nfor Nd 2Fe14B. Those for Nd 2Fe14are 206 K at Nd(4f)\nand 434 K at Nd(4g). This result indicates that Nd 2Fe14\nalready shows uniaxial anisotropy, and the presence of B\nhasaminorimpactonthemagnetocrystallineanisotropy.\nFromtheaboveresults, theadvantageofaddingBcan-\nnot be seen, so far, since its addition does not seem to\nplay an important role in terms of enhancing the magne-\ntization and magnetocrystalline anisotropy of Nd 2Fe14B.\nWhat isthe benefit ofaddingBin Nd 2Fe14B? Aprobable\nrole is to stabilize the structure of Nd 2Fe14B. In order to\nconfirm that, we examine the stability of Nd 2Fe14B. We\nchoose Nd 2Fe17B as a reference system from the histori-\ncal reason that Sagawa intended to insert B to Nd 2Fe17,\nas explained in Sec. I. We define the formation energy\n∆Eas follows;\n∆E≡(E[Nd2Fe14B]+3µ[Fe])−E[Nd2Fe17B] (3)\n={(E[Nd2Fe14B]+3µ[Fe])−(E[Nd2Fe17]+µ[B])}\n− {E[Nd2Fe17B]−(E[Nd2Fe17]+µ[B])}.(4)\nE[Nd2Fe14B],E[Nd2Fe17B],E[Nd2Fe17] denote the to-\ntal energy of Nd 2Fe14B, Nd2Fe17B and Nd 2Fe17per for-\nmula unit, respectively, while µ[Fe] and µ[B] represent\nchemical potentials corresponding to the total energy\nofα-Fe and α-B per atom, respectively. Equation (4)\nshows the difference between the formation energies of\nNd2Fe14B and the system in which one B atom is added\nin Nd2Fe17per formula unit. The resultant formation\nenergy is ∆ E=−1.208 eV/f.u. From this negative for-\nmationenergy, weconcludethatBinNd 2Fe14Bstabilizes\nthe structure of Nd 2Fe14B.\nB. Comparison with Nd 2Fe14X(X= C, N, O, F)\nIn this section, we systematically compare the\nmagnetization, magnetic moments, magnetocrystalline\nanisotropy and stability of Nd 2Fe14Xwith the results of\nNd2Fe14B whenXis C, N, O and F. Figure 2 illustrates37383940\nB C N O FMagneto−\nvolume\neffectChemical\neffectm [µB/f.u.]\nXNd2Fe14X\nNd2Fe14X0\nNd2Fe14\nFIG.2. (Color online)The Xdependenceofthemagnetic mo-\nmentm[µB]. The closed red circles are data for Nd 2Fe14X\nand the open black circles are data for Nd 2Fe14X0. The bro-\nken red line illustrates the magnetic moment of Nd 2Fe14.\nthe magnetic moment of Nd 2Fe14X. By comparing the\nbroken red line of Nd 2Fe14with the black open circles\nof Nd2Fe14X0, we can see that the magnetovolume ef-\nfect works positively to increase the magnetic moment in\nall the cases studied here. This tendency is proportional\nto the volume expansion caused by the Xelements (see\nTable III in Appendix A). Similarly, we can evaluate the\nchemical effects of the Xelements on the magnetic mo-\nment by comparing the black circles with the red circles\nof Nd2Fe14X, because the structures of Nd 2Fe14X0are\nfixedtothoseofNd 2Fe14X. Thedifferencebetweenthese\ntwo systems is entirely the absence/presence of the Xel-\nements. We can see that the chemical effect strongly\ndepends on X. When X= B, C and N, the chemical\neffect is negative, while it is positive when X= O and F.\nSimilar trends for the magnetovolume and chemical ef-\nfects have been reported before in NdFe 11TiX[18] (also\nin a simpler system Fe 4X[25]). A difference between\nNd2Fe14Xand NdFe 11TiXis seen in the total magnetic\nmoment for X= N. The total magnetic moment is sup-\npressedintheformer,whereasitisenhancedinthelatter;\nnamely NdFe 11TiN has a larger magnetic moment than\nNdFe11Ti.\nThe above results are converted to the magnetization\nin Fig. 3. The magnetization is lower in Nd 2Fe14Xthan\nin Nd 2Fe14, except when X= O. From the compari-\nson between Nd 2Fe14and Nd 2Fe14X0, we can see that\nthe magnetovolume effect is negative in all cases, which\nis in contrast to the positive magnetovolume effect for\nNdFe11TiX[18]. Especially, the negative effect is large\nwhenXis O or F. On the other hand, the chemical effect\nis negative when Xis B, C or N, but positive when X\nis O or F. The chemical effect plays an important role in\nenhancing the magnetization of Nd 2Fe14O.\nFigure4illustratesthelocalmagneticmomentsateach\nFe site in Nd 2Fe14Xand Nd 2Fe14X0. We can see the in-\ncreaseofthemagneticmomentsat16k 1and4esiteswhen4\nXchanges from N to O. This enhancement arises from\nthep-dhybridization between O and Fe [12, 13, 18]. Fig-\nure5showsthe projecteddensityofstatesonNandO.In\nthe case of X= N, there is a peak above the Fermi level\nin the majority-spin channel. It comes from the anti-\nbonding state between N-2 pand Fe-3dorbitals. The cor-\nresponding peak in the minority-spin channel is observed\nat higher energy (2–3 eV above the Fermi level). In the\ncase ofX= O, the state is pulled down and is occupied\nin the majority-spin channel. This spin-dependent occu-\npancy makes a distinction in magnetization between the\ncases of N and O. A similar increase is observed in the\nlocal moment of the Fe(8j) sites in NdFe 11TiXwhenX\n= C and N [18]. The difference can be explained as fol-\nlows: In NdFe 11TiX, the anti-bonding state can be con-\nstituted from πbonds with surrounding Fe atoms due to\nthe highly symmetric environment at X(even though Ti\nslightly breaks the symmetry). In Nd 2Fe14X, oneX-Fe\nπbond hybridizes with other Fe atoms via σbond. This\nσbond is stronger than πbond and makes the level of\nthe anti-bonding state higher. Consequently, the anti-\nbonding state is shifted down to the Fermi level and oc-\ncupied for an element having deeper 2 plevel, that is, O\nin Nd2Fe14X.\nFigure 6 illustrates the dependence of the crystal-field\nparameter A0\n2/an}bracketle{tr2/an}bracketri}htofNd on X. The values ofboth Nd(4f)\nand (4g) sites are shown. They are positive in all cases\nandA0\n2/an}bracketle{tr2/an}bracketri}htincreases as the atomic number of Xin-\ncreases (except for Nd(4g) at X= O). This implies that\nNd2Fe14Xhas uniaxial anisotropy at high temperatures\nand that the anisotropy is enhanced as the atomic num-\nber increases. A0\n2/an}bracketle{tr2/an}bracketri}htof Nd(4f) in Nd 2Fe14X0also in-\ncreases from X= C to F. The chemical effect on Nd(4f)\nis large in magnitude and seems to be similar among X\n= C, N, O and F. For Nd(4g) in Nd 2Fe14X0,A0\n2/an}bracketle{tr2/an}bracketri}htis\nalmost constant and Xdependence in A0\n2/an}bracketle{tr2/an}bracketri}htof Nd(4g)\n1.841.881.921.96\n B C N O FMagneto−\nvolume\neffect\nChemical\neffectµ0M [T]\nXNd2Fe14X\nNd2Fe14X0\nNd2Fe14\nFIG. 3. (Color online) The Xdependence of magnetization\nµ0M[T]. The closed red circles and the open black circles\nrepresent the magnetization of Nd 2Fe14Xand Nd 2Fe14X0, re-\nspectively. Lines are a visual guide. For reference, the bro ken\nred line is the magnetization of Nd 2Fe14.is mainly due to the chemical effect.\nFigure 7 shows the formation energies of Nd 2Fe14Xas\na function of the Xelements calculated with Eq. (3), but\nB is replaced with C, N, O and F. Nd 2Fe14Xis stable\nwhenXis B or C, but not when Xis N, O or F. In\nother words, Nd 2Fe17Xis more stable when Xis N, O\nor F. These results are in good agreement with the fact\nthat Nd 2Fe14B and Nd 2Fe17Nx(0< x/lessorsimilar3) can be stably\nsynthesized in experiments.\nIV. CONCLUSIONS\nThe roles of added B in Nd 2Fe14B have been carefully\ninvestigated by first-principles calculations. As reported\nin previous studies, cobaltized Fe atoms can be seen in\nNd2Fe14B. However, magnetization is not enhanced by\nthe added B. We clarify that both the chemical and\nmagnetovolume effects work negatively on the magneti-\nzation of Nd 2Fe14B. The magnetocrystalline anisotropy\ndiscussed by the crystal-field parameter A0\n2/an}bracketle{tr2/an}bracketri}hthas a mi-\nnor effect from the addition of B. We also systematically\n22.5\n     Fe (16k1)Nd2Fe14X Nd2Fe14X0\n22.5\n     Fe (16k2)\n22.5\n     Fe (8j1)\n22.5\n     Fe (8j2)\n22.5\n     Fe (4e)\n22.5\nB C N O FmFe [µB]\nFe (4c)\nX\nFIG. 4. (Color online) The Xdependence of the local mag-\nneticmomentofNd 2Fe14X. Theclosed redcircles aredatafor\nNd2Fe14Xand the open black circles are data for Nd 2Fe14X0.\nThe site notation follows Refs. 19 and 22.5\n-0.4-0.20.00.20.4pDOS of X [arb.u.]N\nO\n-0.4-0.20.00.20.4\n-4-3-2-101234pDOS of X [arb.u.]N\nO\n[eV]\nFIG. 5. (Color online) The local density of states of N in\nNd2Fe14N and O in Nd 2Fe14O.\nexamined the effects ofthe Xelements on the magnetism\nin Nd2Fe14Xthrough first-principles calculations. The\nenhancement in the magnetization is observed only for\nX= O. The crystal-field parameter A0\n2/an}bracketle{tr2/an}bracketri}hthas a ten-\ndency to increase as the atomic number of Xincreases.\nThe stability of Nd 2Fe14Xwas also calculated by com-\nparing the formation energies to that of Nd 2Fe17X. The\nformation energies of Nd 2Fe14B and Nd 2Fe14C are neg-\native relative to Nd 2Fe17X, while Nd 2Fe14N, Nd 2Fe14O\nand Nd 2Fe14F are not energetically stable. This result is\nconsistent with experimental results.\nACKNOWLEDGMENTS\nWe thank Dr. Taro Fukazawa for his fruitful and stim-\nulating discussion. This work was supported by the El-\n100200300400500600\nB C N O FA20 <r2> [K]\nXNd2Fe14X, 4f\nNd2Fe14X0, 4f\nNd2Fe14, 4fNd2Fe14X, 4g\nNd2Fe14X0, 4g\nNd2Fe14, 4g\nFIG. 6. (Color online) A0\n2/angbracketleftr2/angbracketrightof Nd 2Fe14Xand Nd 2Fe14X0.\nThe closed red circles are data for Nd 2Fe14Xand open black\ncircles for Nd 2Fe14X0. The horizontal broken red line is data\nfor Nd 2Fe14. Lines are a guide to the eye.−1.2−0.8−0.400.4\nB C N O FΔE [eV]\nX\nFIG. 7. (Color online) The formation energy of Nd 2Fe14X\ndefined by Eq.(3) with replacement of B with C, N, O, and\nF.\nements Strategy Initiative Project under the auspices of\nMEXT, by “Materials research by Information Integra-\ntion” Initiative (MI2I) project of the Support Program\nfor Starting Up Innovation Hub from Japan Science and\nTechnology Agency (JST), and also by MEXT as a so-\ncial and scientific priority issue (Creation of new func-\ntional Devices and high-performance Materials to Sup-\nport next-generation Industries; CDMSI) to be tackled\nby using post-K computer, as well as JSPS KAKENHI\nGrant No. 17K04978. The calculations were partly car-\nried out by using supercomputers at ISSP, The Univer-\nsity of Tokyo, and TSUBAME, Tokyo Institute of Tech-\nnology, the supercomputer of ACCMS, Kyoto Univer-\nsity, and also by the K computer, RIKEN (Project No.\nhp160227, No. hp170100, and No. hp170269).\nAppendix A: Lattice constants\nThe calculated inner coordinates and lattice constants\nof Nd2Fe14X(X= B, C, N, O, F) and Nd 2Fe14are listed\nin Table II and Tables III, respectively. Volume expan-\nsion due to added elements can be seen in all Nd 2Fe14X\nsystems compared to Nd 2Fe14. The lattice parameters c\nbecome shorter when B is replaced by C and N, and it\nbecomes longer as X= O and F. This tendency might\nbe related to the p-dhybridization between Xand the\nneighboringFe atthe 16k 1and 4esites. Xand the neigh-\nboringFealignalongthe caxisratherthan the aaxisand\nccan reflect the bond length between Xand Fe. From\nX= B to N, the atomic radius decreases [26], thus, the\nbond length becomes shorter.6\nTABLE II. The inner coordinates for Nd 2Fe14Xand Nd 2Fe14.\ncompound atom site x y z\nNd 4f 0.266 0.266 0.0\n4g 0.142 −0.142 0.0\nFe 16k 10.225 0.567 0.127\n16k20.037 0.360 0.177\nNd2Fe14B 8j 10.098 0.098 0.205\n8j20.317 0.317 0.246\n4e 0.5 0.5 0.114\n4c 0.0 0.5 0.0\nB 4g 0.375 −0.375 0.0\nNd 4f 0.257 0.257 0.0\n4g 0.142 −0.142 0.0\nFe 16k 10.225 0.566 0.120\n16k20.036 0.359 0.177\nNd2Fe14C 8j 10.098 0.098 0.209\n8j20.317 0.317 0.245\n4e 0.5 0.5 0.107\n4c 0.0 0.5 0.0\nC 4g 0.374 −0.374 0.0\nNd 4f 0.254 0.254 0.0\n4g 0.142 −0.142 0.0\nFe 16k 10.225 0.565 0.120\n16k20.036 0.359 0.178\nNd2Fe14N 8j 10.098 0.098 0.209\n8j20.317 0.317 0.244\n4e 0.5 0.5 0.106\n4c 0.0 0.5 0.0\nN 4g 0.371 −0.371 0.0\nNd 4f 0.252 0.252 0.0\n4g 0.146 −0.146 0.0\nFe 16k 10.224 0.562 0.129\n16k20.035 0.360 0.177\nNd2Fe14O 8j 10.098 0.098 0.204\n8j20.316 0.316 0.244\n4e 0.5 0.5 0.117\n4c 0.0 0.5 0.0\nO 4g 0.363 −0.363 0.0\nNd 4f 0.258 0.258 0.0\n4g 0.146 −0.146 0.0\nFe 16k 10.224 0.563 0.141\n16k20.036 0.362 0.174\nNd2Fe14F 8j 10.098 0.098 0.197\n8j20.317 0.317 0.246\n4e 0.5 0.5 0.135\n4c 0.0 0.5 0.0\nF 4g 0.369 −0.369 0.0\nNd 4f 0.256 0.256 0.0\n4g 0.139 −0.139 0.0\nFe 16k 10.229 0.566 0.120\nNd2Fe14 16k20.035 0.360 0.177\n8j10.098 0.098 0.210\n8j20.316 0.316 0.245\n4e 0.5 0.5 0.103\n4c 0.0 0.5 0.0TABLE III. The calculated lattice constants and volumes of\nNd2Fe14Xand Nd 2Fe14(denoted as E). The lattice constants\naandcare for a conventional unit cell described in ˚A. The\nvolumes Vare per formula unit described in ˚A3.\nX= B C N O F E\na[˚A] 8.791 8.805 8.816 8.858 8.833 8.760\nc[˚A] 12.141 11.934 11.925 12.154 12.561 11.989\nV[˚A3/f.u.] 234.6 231.3 231.7 238.4 245.0 230.0\nAtX= O, the anti-bonding state starts to be oc-\ncupied and the occupation weakens the covalent bond\nbetween Xand Fe. As a result, cin Nd 2Fe14O and\nNd2Fe14F become longer. Analogous trend is also ob-\nserved in NdFe 11TiX[18].\nAppendix B: Crystal-field parameters\nIn this appendix, we discuss the crystal-field pa-\nrameters Am\nl/an}bracketle{trl/an}bracketri}htof Nd. The large magnetocrystalline\nanisotropy of Nd 2Fe14B is attributed to well-localized\nNd-4felectrons with large orbital moments. In crystal-\nfield theory, the magnetocrystalline anisotropy energy\nEMAE(θ,ϕ) of Nd in Nd 2Fe14Xreads from the crystal-\nfield hamiltonian HCEF;\nEMAE(θ,ϕ) =/an}bracketle{tJ,M=−J|HCEF|J,M=−J/an}bracketri}ht(B1)\n=/summationdisplay\nlmΘlAm\nl/an}bracketle{trl/an}bracketri}htfl(J)Zm\nl(θ,ϕ)\nalm,(B2)\nwhere\nf2=J/parenleftbigg\nJ−1\n2/parenrightbigg\n,\nf4=J/parenleftbigg\nJ−1\n2/parenrightbigg\n(J−1)/parenleftbigg\nJ−3\n2/parenrightbigg\n, (B3)\nf6=J/parenleftbigg\nJ−1\n2/parenrightbigg\n(J−1)/parenleftbigg\nJ−3\n2/parenrightbigg\n(J−2)/parenleftbigg\nJ−5\n2/parenrightbigg\n,\nΘ2=−7\n32·112,\nΘ4=−23·17\n33·113·13, (B4)\nΘ6=−5·17·19\n33·7·113·132,7\na2−2=a22=1\n4/radicalbigg\n15\nπ,\na20=1\n4/radicalbigg\n5\nπ,\na4−4=a44=3\n16/radicalbigg\n35\nπ,\na4−2=a42=3\n8/radicalbigg\n5\nπ,\na40=3\n16/radicalbigg\n1\nπ, (B5)\na6−6=a66=231\n64/radicalbigg\n26\n231π,\na6−4=a64=21\n32/radicalbigg\n13\n7π,\na6−2=a62=1\n64/radicalbigg\n2730\nπ,\na60=1\n32/radicalbigg\n13\nπ.\nThis formulation is referred to Ref. 27. Constant param-\neters are the Stevens factor Θ landflin which the total\nangular momentum J(=L+S) is 9/2 ( L= 6,S= 3/2)\nfor Nd. Zm\nlis a real spherical harmonics. Am\nl/an}bracketle{trl/an}bracketri}htare\ncrystal-field parameters at the Nd site. Here ltakes 2, 4,\nor 6, and mis even number from −ltol.\nAm\nl/an}bracketle{trl/an}bracketri}ht=alm/integraldisplayrc\n0dr r2|R4f(r)|2Vlm(r).(B6)\nWe use the effective potential Vlm(r) of Nd atoms\nobtained by solving the Kohn-Sham equation self-\nconsistently. rcis a cutoff radius which is determined\nby Bader analysis as in Ref. 18, and R4f(r) is an atomic\nradial function of the Nd-4 felectrons. The results for\nAm\nl/an}bracketle{trl/an}bracketri}htof Nd at the 4f and 4g sites in Nd 2Fe14Xand\nNd2Fe14are summarized in Table IV. The comparison\nwith the previous calculation [28] and experiment [27] is\nalso shown for Nd 2Fe14B. The order of magnitude for l=\n2 and 4 agrees with the previous calculated values, and\nthe sign for them also agrees, except for ( l,m) = (4,−2)\nof Nd(4g).0\nπ/6\nπ/3\nπ/20π/4π/23π/4π 0 2 4 6 8E [MJ/m3]Nd(4f)\nθϕ 0 2 4 6 8\n2\n46\n0\n42\n0\nπ/6\nπ/3\nπ/20π/4π/23π/4π 0 2 4 6 8E [MJ/m3]Nd(4g)\nθϕ 0 2 4 6 8\n2468\n−0.0200.020.040.060.08\n0 π/6 π/3 π/2E (ϕ = π/4) [MJ/m3]\nθNd(4f), Nd2Fe14B(a)\n(b)\n(c)\nFIG. 8. (Color online) The spin angle dependence of MAE\nfor (a) Nd(4f) and (b) Nd(4g) in Nd 2Fe14B. The solid lines\nwith numbers in the color map represent the contour lines of\nMAE. The dashed line is a guide to the eye for the data in (c).\n(c) The angle dependence of MAE for Nd(4f) in Nd 2Fe14B at\nϕ=π/4.\nFigures 8 (a) and (b) show the angle dependence of\nEq. (B2) for Nd(4f) and Nd(4g) in Nd 2Fe14B. Strong\ndependence on the azimuthal angle ϕcan be seen in the\nMAEforbothNd(4f) andNd(4g). Figure8(c) showsthe\nMAEatϕ=π/4forNd(4f). Thenegativeregionappears\naroundθ= 0. We here note that the exchange coupling\nwith Fe is not taken into account and that such small\ncontribution of MAE may not survive in the presence\nof the strong exchange coupling. Therefore, we cannot8\nconclude the existence of the spin canting at low temper-\natures from the present result [29]. However, it indicatesthe importance of the azimuthal angle dependence of the\nMAE to describe the spin configuration at low tempera-\nture.\n[1] P. Gubbens, A. van der Kraan, and K. Buschow,\nPhysica B+C 86-88, 199 (1977).\n[2] H. Bo-Ping and J. M. D. Coey,\nJournal of the Less Common Metals 142, 295 (1988).\n[3] F. Weitzer, K. Hiebl, and P. Rogl,\nJ. Appl. Phys. 65, 4963 (1989).\n[4] M. Sagawa, S. Fujimura, N. Togawa, H. Yamamoto, and\nY. Matsuura, J. Appl. Phys. 55, 2083 (1984).\n[5]http://www.japanprize.jp/data/prize/commemorative_l ec_2012_e.pdf .\n[6] M. Sagawa, S. Fujimura, H. Yamamoto, Y. Matsuura,\nand S. Hirosawa, J. Appl. Phys. 57, 4094 (1985).\n[7] J. Kanamori, Prog. Theor. Phys. Suppl. 101, 1 (1990).\n[8] J. Kanamori, Trans. Magn. Soc. Jpn. 1, 1 (2001).\n[9] J.Kanamori,J. Alloys Compd. Proceedings of Rare Earths ’04 in Nara, Japan, 408, 2 (2006).\n[10] H. Hasegawa and J. Kanamori,\nJ. Phys. Soc. Jpn. 33, 1607 (1972).\n[11] N. Hamada, J. Phys. Soc. Jpn. 46, 1759 (1979).\n[12] S. Asano and M. Yamaguchi,\nPhysica B: Condensed Matter Proceedings of the Yamada Confe rence XLV, the International Conference on the Physics of Tr ansition Metals, 237, 541 (1997).\n[13] M. Yamaguchi and S. Asano,\nPhysica B: Condensed Matter 254, 73 (1998).\n[14] T. Ozaki, Phys. Rev. B 67, 155108 (2003).\n[15] J. P. Perdew, K. Burke, and M. Ernzerhof,\nPhys. Rev. Lett. 77, 3865 (1996).\n[16]http://qmas.jp/ .\n[17] T. Miyake, K. Terakura, Y. Harashima, H. Kino, andS. Ishibashi, J. Phys. Soc. Jpn. 83, 043702 (2014).\n[18] Y. Harashima, K. Terakura, H. Kino, S. Ishibashi, and\nT. Miyake, Phys. Rev. B 92, 184426 (2015).\n[19] J. F. Herbst, J. J. Croat, F. E. Pinkerton, and W. B.\nYelon, Phys. Rev. B 29, 4176 (1984).\n[20] C. B. Shoemaker, D. P. Shoemaker, and R. Fruchart,\nActa Crystallogr. C 40, 1665 (1984).\n[21] D. Givord, H. Li, and J. Moreau,\nSolid State Commun. 50, 497 (1984).\n[22] J. F. Herbst, Rev. Mod. Phys. 63, 819 (1991).\n[23] K. H. J. Buschow, Rep. Prog. Phys. 54, 1123 (1991).\n[24] R. Sasaki, D. Miura, and A. Sakuma,\nAppl. Phys. Express 8, 043004 (2015).\n[25] H. Akai, M. Takeda, M. Takahashi, and J. Kanamori,\nSolid State Commun. 94, 509 (1995).\n[26] P. Pyykk¨ o, Phys. Rev. B 85, 024115 (2012).\n[27] M. Yamada, H. Kato, H. Yamamoto, and Y. Nakagawa,\nPhys. Rev. B 38, 620 (1988).\n[28] K. Hummler and M. F¨ ahnle,\nPhys. Rev. B 53, 3290 (1996).\n[29] S. Hirosawa, Y. Matsuura, H. Yamamoto,\nS. Fujimura, M. Sagawa, and H. Yamauchi,\nJ. Appl. Phys. 59, 873 (1986).\n[30] T. Yoshioka, H. Tsuchiura, and P. Nov´ ak,\nMater. Res. Innovat. 19, S4 (2015).\n[31] A. J. Freeman and R. E. Watson,\nPhys. Rev. 127, 2058 (1962).9\nTABLE IV. The calculated Am\nl/angbracketleftrl/angbracketrightof Nd at the 4f and 4g sites in Nd 2Fe14X(= B, C, N, O, F) and Nd 2Fe14. They are\ndescribed in Kelvin.\nA0\n2/angbracketleftr2/angbracketrightA−2\n2/angbracketleftr2/angbracketrightA0\n4/angbracketleftr4/angbracketrightA−2\n4/angbracketleftr4/angbracketrightA4\n4/angbracketleftr4/angbracketrightA0\n6/angbracketleftr6/angbracketrightA−2\n6/angbracketleftr6/angbracketrightA4\n6/angbracketleftr6/angbracketrightA−6\n6/angbracketleftr6/angbracketright\nNd(4f)\nNd2Fe14B 220 442 -21 -17 -18 2.6 1.5 -3.0 0.0\nNd2Fe14C 292 579 -23 -54 -37 4.1 7.9 2.9 4.7\nNd2Fe14N 336 627 -26 -73 -50 4.2 10.0 1.9 10.1\nNd2Fe14O 405 652 -28 -46 -59 4.5 6.5 7.4 15.3\nNd2Fe14F 459 691 -23 -12 -45 4.0 -1.1 10.6 15.9\nNd2Fe14206 372 -19 -71 -34 4.3 7.9 -1.0 11.4\nNd(4g)\nNd2Fe14B 331 -327 -30 -28 34 3.4 -3.2 3.0 11.9\nNd2Fe14C 387 -431 -33 -34 47 3.5 1.5 10.2 0.8\nNd2Fe14N 444 -532 -36 -36 58 2.9 4.2 14.1 -4.0\nNd2Fe14O 435 -562 -32 -70 -13 1.9 5.1 6.4 0.7\nNd2Fe14F 477 -658 -28 -68 33 2.0 6.0 7.7 6.8\nNd2Fe14434 -308 -34 23 16 3.2 2.6 8.1 -0.3\nTABLE V. The comparison of the calculated Am\nl/angbracketleftrl/angbracketrightwith previous theoretical [28, 30] and experimental [27] st udies for\nNd2Fe14B. They are described in Kelvin. In Ref. 28, the notation of th e Nd sites, 4f and 4g, are opposite from the present\nnotation, and here it is alternated to compare with our data. The data for Nd1 and Nd2 on the middle layer described in\nRef. 30 correspond to Nd(4f) and Nd(4g), respectively. The d ata in Ref. 27 is shown as Am\nl[Ka−l\nB] itself, and we multiply /angbracketleftr2/angbracketright\n= 1.001 a2\nB,/angbracketleftr4/angbracketright= 2.401 a4\nB, and/angbracketleftr6/angbracketright= 12.396 a6\nB, to the values of Am\nl, respectively [31]. Nd(4f) and Nd(4g) sites are not\ndistinguished in Ref. 27. For A−2\n2/angbracketleftr2/angbracketrightandA−6\n6/angbracketleftr6/angbracketright, the values at Nd(4f) and Nd(4g) on one plane is the same magni tude of\neach Nd on the plane shifted by 0.5 cwith opposite signs.\nA0\n2/angbracketleftr2/angbracketrightA−2\n2/angbracketleftr2/angbracketrightA0\n4/angbracketleftr4/angbracketrightA−2\n4/angbracketleftr4/angbracketrightA4\n4/angbracketleftr4/angbracketrightA0\n6/angbracketleftr6/angbracketrightA−2\n6/angbracketleftr6/angbracketrightA4\n6/angbracketleftr6/angbracketrightA−6\n6/angbracketleftr6/angbracketright\npresent Nd(4f) 220 442 -21 -17 -18 2.6 1.5 -3.0 0.0\nNd(4g) 331 -327 -30 -28 34 3.4 -3.2 3.0 11.9\ncalc. [28] Nd(4f) 323 807 -50 -85 -117 -2.7 3.9 -24 0.1\nNd(4g) 540 -727 -54 82 125 -2.2 0.3 -15 5\ncalc. [30] Nd(4f) 442 777 -28 -69 -106 17 -2 156 57\nNd(4g) 605 -286 -35 84 96 13 2 79 11\nexp. [27] 295 -454 -29.5 -22.8 121 -197"    },    {        "title": "1802.07959v2.Power_law_analysis_for_temperature_dependence_of_magnetocrystalline_anisotropy_constants_of_Nd__2_Fe___14__B_magnets.pdf",        "content": "Power law analysis for temperature dependence of magnetocrystalline anisotropy\nconstants of Nd 2Fe14B magnets\nDaisuke Miura and Akimasa Sakuma\nDepartment of Applied Physics, Tohoku University, Sendai 980-8579, Japan∗\n(Dated: July 14, 2018)\nWe perform phenomenological analysis of the temperature dependence of magnetocrystalline\nanisotropy (MA) in rare-earth magnets. We define the phenomenological power laws applicable to\ncompound magnets using the Zener theory, and we apply these laws to study the magnetocrystalline\nanisotropy constants (MACs) of Nd 2Fe14B magnets. The results indicate that the MACs closely\nobey the power law, and further, our analysis yields a better understanding of the temperature-\ndependent MA in rare-earth magnets. Furthermore, to examine the validity of the power law, we\ndiscuss the temperature dependence of the MACs in Dy 2Fe14B and Y 2Fe14B magnets as examples\nof cases wherein it is difficult to interpret the MA using the power law.\nI. INTRODUCTION\nThe complex behaviors of magnetic anisotropy (MA)\nin rare-earth permanent magnets have attracted consid-\nerable research attention, since such magnets are widely\nused as high-performance magnets that combine high\nmagnetocrystalline anisotropy with reasonable magneti-\nzation and Curie temperature.1–4Typical examples of\nsuch magnets include R2Fe14B magnets (where Rde-\nnotes a rare-earth element), which form the focus of the\npresent study. The MA of such magnets strongly de-\npends on temperature and is strongly influenced by the\nchoice ofR.1,5–7From previous studies, it has been found\nthat the localized-moment picture is suitable to under-\nstand the magnetism of R2Fe14B magnets. For example,\nYamada et al. showed through their systematic analysis\nof the magnetization process for the R2Fe14B series that\nparameters representing crystalline electric fields (CEFs)\nand effective exchange fields (EXFs) at Rsites do not\ndepend significantly on R.8Furthermore, we have previ-\nously confirmed that an effective Hamiltonian using the\nparameters suitably describes the experimental temper-\nature dependence of the magnetocrystalline anisotropy\nconstants (MACs) of Nd 2Fe14B magnets.9,10However,\nthere remains a gap between our microscopic picture and\nthe macroscopic aspects of MA since previous theoreti-\ncal studies on the temperature dependence of MA for\nR2Fe14B magnets are based on complete diagonalization\nof the Hamiltonian9,10or numerical methods such as the\nMonte Carlo method.4,11,12\nAn effective means of bridging this gap is to develop\na power-law theory5,6,13–18to establish the relationship\nbetween MACs and magnetization. The history of in-\nvestigations on MA based on the power-law theory has\nbeen reviewed by Callen and Callen.16In their review\narticle, they provided a quantum-mechanical foundation\nto the power-law theory and extended the foundation to\nmore general cases such as the interaction between two\nmoments. In deriving these power laws, the important\nassumptions are that a local magnetic moment picture\nis valid and that the magnetic structure is spatially uni-\nform as assumed in Zener’s original work14(in addition,a low temperature assumption is required from a mean\nfield perspective16,19). However, the total magnetic mo-\nment ofR2Fe14B magnets is the sum of the Rmoments\nand the Fe moments; thus, in the case that the directions\nof both moments are mutually non-collinear, the temper-\nature dependence of the MACs will not obey the power\nlaw, and we can no longer regard the magnetic struc-\nture as a uniform one. In this regard, Dy 2Fe14B magnets\nare typical examples exhibiting high non-collinearity be-\ncause of the antiferromagnetic coupling existing between\nthe Dy and Fe moments,1,18and further, a wide plateau\nhas been found in the temperature-dependence curve of\ntheir first-order MAC in the low-temperature range.20,21\nIn contrast, in Nd 2Fe14B magnets, it is possible to as-\nsume that the Nd moments are collinear with the Fe\nmoments.18,22\nIn this article, we aim to provide a simple and gen-\neral understanding of the temperature dependence of\nthe MA of rare-earth permanent magnets, considering\nNd2Fe14B magnets as an example, using the power-law\ntheory for MACs as developed by Zener.14Here, we note\nthat the original theory targets single-element magnets\nrather than compounds magnets; therefore, it is required\nto suitably correct the power law to apply it to Nd 2Fe14B\nmagnets. In addition, we emphasize that the presence of\na large higher–order MAC at the zero temperature con-\nsiderably effects on the temperature dependence of the\nlower–order MACs than it; the Nd 2Fe14B magnet is in\nthis case. Subsequently, we first present an explicit form\nof the extended power law picture within the framework\nof the Zener theory; next, we prove the validity of the\nextended power law and discuss the limitations of our\napproach. In addition, we briefly discuss the applicabil-\nity of this law to Y 2Fe14B and Dy 2Fe14B magnets.\nII. POWER LAW FOR MACS IN\nTETRAGONAL SYMMETRY\nR2Fe14B magnets have tetragonal symmetry, and thus,\ntheir free-energy density, F(Θ,Φ;T), can be expressed as/summationtext∞\nm=0/summationtext⌊m/2⌋\nn=0K(/prime)n\nm(T) sin2mΘ cos 4nΦ,whereK(/prime)n\nm(T)\narXiv:1802.07959v2  [cond-mat.mtrl-sci]  14 Jul 20182\ndenotes the mth-order MAC at temperature T, and Θ\nand Φ denote the zenithal and azimuthal angles of the\ntotal magnetization direction, respectively.1–3This ex-\npansion is an infinite series in a purely mathematical\nanalysis,23but a finite-order form is usually assumed for\npractical purposes. In this study, we consider the third-\norder form as follows:\nF(Θ,Φ;T) =K1(T) sin2Θ\n+ [K2(T) +K/prime\n2(T) cos 4Φ] sin4Θ\n+ [K3(T) +K/prime\n3(T) cos 4Φ] sin6Θ,(1)\nwhere the angle-independent term, K0(T), is omitted.\nIn general,{K(/prime)n\nm(T)}includes not only the R- and the\nFe-sublattice contributions but also the cross effect be-\ntween them. However, if the directions of the Rand Fe\nmagnetizations are approximately collinear for arbitrary\nvalues of Θ and Φ, then one can regard Θ and Φ as the\nangles of the Ror/and Fe magnetizations, and therefore,\nwe can divide the total MA into the R- and Fe-sublattice\ncontributions as\nK(/prime)n\nm(T) =KR(/prime)n\nm(T) +KFe(/prime)n\nm (T). (2)\nHere, by assuming that the localized Rmoments are\nestablished by an effective exchange field induced by\nthe Fe magnetization and that the origin of the R\nanisotropy is local at each Rion site,KR(/prime)n\nm(T)\ncan be phenomenologically described by means of the\nZener theory. Namely, when the free-energy density\nof theRsublattice is expressed as FR(Θ,Φ;T)≡/summationtext\nk,qAq\nk(T)Zq\nk(Θ,Φ) in terms of a phenomenological real\nparameterAq\nk(T) and the tesseral harmonics Zq\nk(Θ,Φ),\nthe Zener theory14,15yields the Akulov–Zener power law,\nAq\nk(T) =Aq\nk(0)µR(T)k(k+1)/2, whereµR(T) denotes the\nnormalized Rmagnetization as µR(0) = 1. On the other\nhand,{Aq\nk(T)}is related to{KR(/prime)n\nm(T)}by another\nform corresponding to Eq. (1):\nFR(Θ,Φ;T) =3/summationdisplay\nm=1⌊m/2⌋/summationdisplay\nn=0KR(/prime)n\nm(T) sin2mΘ cos 4nΦ,\n(3)\nand thus, we obtain the below equations of interest:\nKR\n1(T) =KR\n1(0)µR(T)3\n+8\n7KR\n2(0)/bracketleftbig\nµR(T)3−µR(T)10/bracketrightbig\n+8\n7KR\n3(0)/parenleftbigg\nµR(T)3−18\n11µR(T)10+7\n11µR(T)21/parenrightbigg\n,\n(4a)\nKR\n2(T) =KR\n2(0)µR(T)10\n+18\n11KR\n3(0)/bracketleftbig\nµR(T)10−µR(T)21/bracketrightbig\n, (4b)\nKR\n2/prime(T) =KR\n2/prime(0)µR(T)10+10\n11KR\n3/prime(0)/bracketleftbig\nµR(T)10−µR(T)21/bracketrightbig\n, (4c)\nKR\n3(T) =KR\n3(0)µR(T)21, (4d)\nKR\n3/prime(T) =KR\n3/prime(0)µR(T)21, (4e)\nin which the first term represents the Akulov–Zener–\nCallen–Callen power law (or today simply called the\nCallen–Callen law),13,14,16and the second and/or the\nthird terms represent the corrections to the Callen–\nCallen law due to the presence of the higher–order MAC\n(s) at the zero temperature.\nHere, we should notice a connection between Zener’s\nphenomenological theory and a microscopic theory. The\nZener theory has been discussed well by using a mean\nfield theory (MFT),16,19,24–26and the present phe-\nnomenological expressions (4) can be supported within\na linear theory for CEFs. Yamada et al. explicitly rep-\nresentedKR(/prime)n\nm(0) in terms of the CEF parameters by\nusing the perturbative theory with respect to the CEF.8\nRecently, on the basis of the expression, an MA has been\ndiscussed from first principles4,27–35although in general,\nthe quantitative validity of the perturbative treatment\nfor the MA of rare-earth magnets is not trivial.8In\nfinite temperatures, Kuz’min represented KR(/prime)n\nm(T) in\nterms of the CEF and the EXF within the same frame-\nwork, where the temperature dependence of the MACs\nis related to Kuz’min’s generalized Brillouin functions\n(GBFs),24,26,36,37and furthermore, Magnani et al. in-\nvestigated the effect of a finite spin–orbit interaction on\nKR(/prime)n\nm(T).38As pointed out by Keffer,19the tempera-\nture dependence of MA predicted by the MFT tends to\nthe Akulov–Zener power law in a low temperature range,\nand in fact, it has been demonstrated that the GBFs and\nthe power law are in quantitative agreement in the low\ntemperature range.18,26Although the MFT does not sup-\nport the power law quantitatively in a high temperature\nrange, it is worth enough to employ the power law in the\nwhole temperature range if limiting the discussion to a\nqualitative level.\nTo understand the temperature dependence of the MA\nrepresented by the present power law (4), let us consider\na simple case wherein KR\n2/prime(0) =KR\n3(0) =KR\n3/prime(0) = 0:\nKR\n1(T) =KR\n1(0)µR(T)3+8\n7KR\n2(0)/bracketleftbig\nµR(T)3−µR(T)10/bracketrightbig\n,\n(5a)\nKR\n2(T) =KR\n2(0)µR(T)10. (5b)\nA schematic of µR(T) and the resultant µR(T)3,\nµR(T)10, and8\n7/bracketleftbig\nµR(T)3−µR(T)10/bracketrightbig\nare shown in Fig.\n1. In the case that/vextendsingle/vextendsingleKR\n2(0)/KR\n1(0)/vextendsingle/vextendsingle/lessmuch1, the correction\nresulting from the presence of KR\n2(0) in Eq. (5a) is small,\nand we confirm the applicability of the Callen–Callen law\nKR\n1(T)/similarequalKR\n1(0)µR(T)3. An interesting feature is ob-\nserved in the case where KR\n2(0) is comparable to KR\n1(0).\nIn this case, the correction to KR\n1(T) fromKR\n2(0) im-\nmediately occurs on increasing the temperature from ab-\nsolute zero, as indicated by the dashed line in Fig. 1;3\n0.0 0.2 0.4 0.6 0.8 1.0\nT/T C0.00.20.40.60.81.0\n1\n3\n108\n7/bracketleftbig\nµR(T)3−µR(T)10/bracketrightbigµR(T)n\nFIG. 1. Schematic of the normalized Rmagnetization as a\nfunction of T, and the resultant of each component of the\npower law. The solid lines indicate µR(T)nwherenis rep-\nresented by the number on each solid line. The dashed line\nindicates the correction component of the power law. Further,\nTCrepresents the Curie temperature.\n0.0 0.2 0.4 0.6 0.8 1.0\nT/T C−20−1001020KR\n1(0) =KR\n2(0)/2\nKR\n1(0) =−KR\n2(0)/2\nKR\n1(0) =−8\n7KR\n2(0)SRTKR\n1(T)\nKR\n2(T)\nFIG. 2. Possible forms of KR\n1(T) for a fixed KR\n2(T) with\nK2(0)>0.\nthus, the Callen–Callen law for KR\n1(T) is no longer valid\nin the low-temperature range. In the high-temperature\nrange satisfying µR(T)3/greatermuchµR(T)10, we can determine\nthe power law as KR\n1(T)/similarequal/bracketleftbig\nKR\n1(0) +8\n7KR\n2(0)/bracketrightbig\nµR(T)3\nagain; however, the proportional factor differs from that\nof the Callen–Callen law. Figure 2 illustrates the pos-\nsible forms of KR\n1(T) for a fixed KR\n2(T) in Eqs. (5),\nwherein we fixed KR\n2(T) asKR\n2(0) = 20 MJ/m3(dashed\nline) and determined KR\n1(T), indicated by solid lines, for\ncases where KR\n1(0) =KR\n2(0)/2,KR\n1(0) =−KR\n2(0)/2,\nandKR\n1(0) =−8KR\n2(0)/7. From the figure, we can ob-\nserve a uniaxial magnetic anisotropy over the whole tem-\nperature range in the case of KR\n1(0)>0 and an in-plane\nmagnetic anisotropy in the case of KR\n1(0)<−8KR\n2(0)/7.\nHowever, in the case where KR\n1(0) takes on an inter-\nmediate value with −8KR\n2(0)/7< KR\n1(0)<0, we can\nobserve that the MA changes from in-plane to uniaxial\nat a particular temperature; that is, spin reorientationtransition (SRT) occurs at the temperature indicated by\nthe solid line corresponding to KR\n1(0) =−KR\n2(0)/2 in\nFig. 2. Therefore, we can immediately predict from the\nvalues ofKR\n1(0) andKR\n2(0) as to whether the SRT oc-\ncurs, although the actual SRT temperature cannot be\ndetermined because it depends on µR(T). As the form of\nµR(T) is approximately the same for rare-earth perma-\nnent magnets,20the above analysis is valid for magnets\nin which non-collinearity is negligible.\nThe application of the Zener theory to the Fe\nanisotropy (i.e., KFe(/prime)n\nm (T) in Eq. (2)) requires caution,\nbecause it is not trivial that a local-moment picture for\nthe Fe sublattice is appropriate. This issue is referred in\na discussion on the anisotropy of Y 2Fe14B magnets in the\nnext section.\nIII. TEMPERATURE DEPENDENCE OF MACS\nINR2Fe14BMAGNETS\nLet us apply the power law (4) to an experiment\nto study the temperature dependence of the MACs of\nNd2Fe14B magnets. One way to estimate µR(T) of an\nR2Fe14B magnet experimentally is to subtract the mag-\nnetization of Y 2Fe14B magnets from that of the R2Fe14B\nmagnet. In this regard, Hirosawa et al. have shown that\nµR(T) estimated by this method is reproduced suitably\nwithin the mean field theory.20Although one can directly\nuse the experimental µR(T), here, we use the theoretical\nµR(T) based on the mean field theory as follows. Assum-\ning an effective exchange interaction between the total\nangular momentum of the Rions and the Fe spins to plot\ntheµR(T) curve, we have the conventional result:20,26\nµR(T) :=BJ/parenleftbigg2|gLSJ−1|JHRµFe(T)\nkBT/parenrightbigg\n, (6)\nBJ(x) :=2J+ 1\n2Jcoth/parenleftbigg2J+ 1\n2Jx/parenrightbigg\n−1\n2Jcoth/parenleftBigx\n2J/parenrightBig\n,\n(7)\ngLSJ:= 1 +J(J+ 1) +S(S+ 1)−L(L+ 1)\n2J(J+ 1), (8)\nwhereJdenotes the quantum number with respect to the\ntotal angular momentum of the Rion,HRthe effective\nexchange field, kBthe Boltzmann constant, and TCthe\nCurie temperature of the R2Fe14B magnet. The temper-\nature dependent Fe magnetization is phenomenologically\nexpressed by the Kuz’min formula:39,40\nµFe(T) =/bracketleftBigg\n1−0.5/parenleftbiggT\nTC/parenrightbigg3/2\n−0.5/parenleftbiggT\nTC/parenrightbigg5/2/bracketrightBigg1/3\n,(9)\nwhere, the shape parameters have been determined by\nfitting with the temperature-dependent magnetization of\nthe Y 2Fe14B magnet reported in Ref20,21,41. The other\nparameters for Nd 2Fe14B magnets are given as L= 6,\nS= 3/2,J= 9/2,HNd= 350 K,8andTC= 5864\n0.0 0.2 0.4 0.6 0.8 1.0\nT/T C0.00.20.40.60.81.0\nµFe(T)\nµNd(T)\nFIG. 3. Temperature dependences of the normalized magne-\ntizations of Fe and Nd sublattices. The closed circles denote\nthe experimental results for the Y 2Fe14B magnet ( TC= 571\nK),20,21,41the solid line represents µFe(T) calculated by the\nKuz’min formula, and the dashed line represents µNd(T) cal-\nculated within the framework of the molecular field theory\nwhere the effective exchange field is given by 350 µFe(T) [K]\nandTC= 586 K.\n0 100 200 300 400 500 600\nT[K]−10−50510152025MACs [MJ/m3]KNd\n1(T)\nKNd\n2(T)\nFIG. 4. Comparison of the calculated results with those of\nthe experiment conducted by Durst et al.,42for the MACs of\nthe Nd 2Fe14B magnet. The closed and open circles, respec-\ntively, are the first-order and the second-order MACs in the\nexperimental results, and the lines represent results of this\nstudy.\nK.20,21The experimental results, µFe(T), and the resul-\ntantµNd(T) are shown in Fig. 3. Using the above pa-\nrameters, we demonstrate comparisons of our extended\npower law with two experiments. Figure 4 compares the\ncalculated result with the results of the experiment con-\nducted by Durst et al.,42who determined the first-order\nand the second-order MACs for Nd 2Fe14B magnets. In\nour study, the solid lines in the figure were obtained us-\ning Eqs. (5) and (6) where KNd\n1(0) =−8.86 MJ/m3and\nKNd\n2(0) = 23.85 MJ/m3; we note that the experimental\nresults obey the present power law qualitatively. An-\nother example corroborating the validity of our approach\nis shown in Fig. 5. Cadogan et al. reported experi-\n0 50 100 150 200 250 300\nT[K]−400−20002004006008001000MACsKNd\n1(T)\nKNd\n2(T)\nKNd\n3(T)\n0 50 100 150 200 250 300\nT[K]−200−150−100−50050100150200MACsKNd\n2/prime(T)\nKNd\n3/prime(T)FIG. 5. Comparison of the calculated results with those of the\nexperiment conducted by Cadogan et al.,22for the MACs of\nthe Nd 2Fe14B magnet. In the upper figure, the closed circles,\nopen circles, and closed rectangles represent the experimental\nK1(T),K2(T), andK3(T) values, respectively. In the lower\nfigure, the closed and open circles represent the experimental\nK/prime\n2(T) andK/prime\n3(T) values, respectively. The lines represent\nthe results of this study.\nmental results up to third-order MACs for a Nd 2Fe14B\nmagnet.22With the use of these results, the solid lines\nin the figure were plotted utilizing Eqs. (4) and (6)\nwhereKNd\n1(0) =−279,KNd\n2(0) = 813, KNd\n2/prime(0) = 151,\nKNd\n3(0) =−368, andKNd\n3/prime(0) =−155 (the unit is un-\nknown). Further, in this case, it can be observed that the\npresent power law correlates well with the experimental\nresults. From the results of both experiments, we can\nconclude that the proposed predictions and experimental\nresults are in good agreement. Here, we emphasize that\nthe common parameters in the two comparisons are used\ninµNd(T), and particularly that the value of HNd= 350\nK is reasonable because the strength of the EXF strongly\ninfluences the plateau of the MACs appearing in the low-\ntemperature range.9In addition, we must consider the\nnon-collinearity between the Rand Fe moments. Al-\nthough the MACs of R=Nd obey the present power law\nwell, this is not the case for magnets with a high non-\ncollinearity. For example, Dy 2Fe14B magnets show a high\nnon-collinearity between Dy and Fe moments.18In fact,\nthe wide plateau observed in K1(T) up toT∼0.5TCfrom5\nT= 020,21cannot be explained using the present power\nlaw forHDy= 145 K, as estimated by Yamada et al.8\nThis violation is attributed to the high non-collinearity,\nnot meaning that 145 K is a very small value. Therefore,\na special treatment beyond the present one is necessary\nin such a case. A detailed analysis on Dy 2Fe14B magnets\nwill be presented in a separate paper.\nFinally, let us consider an interesting example in the\ncontext of the above discussion, wherein we have ig-\nnored the contribution to the MA from the Fe sublattice,\nKFe(/prime)n\nm (T), which appeared in Eq. (2). This contribution\nis, often assuming that the Fe sublattice has an intrinsic\nMA, estimated from the MA measured in M2Fe14B mag-\nnets with no Mmoment (M=Y, La, Ce, and so on).1\nHere, let us consider the Y 2Fe14B magnet referred to in\nthe above discussion. The history of phenomenological\ninvestigations on the temperature-dependence of the MA\nof Y 2Fe14B magnets was reviewed by Cadogan et al.43\nIn a similar manner, we attempted to apply the power-\nlaw-like expression to the temperature-dependent MACs\nof Y 2Fe14B magnets. As an analogy to Eq. (4a), we\nassume the following expression (previously assumed by\nCarr,15the same formulae were used in the study on the\nMA of hexagonal Co), as the Fe contribution to K1(T)\nin Y 2Fe14B magnets,\nKFe\n1(T) =KFe\n1(0)µFe(T)3\n+8\n7KFe\n2(0)/bracketleftbig\nµFe(T)3−µFe(T)10/bracketrightbig\n+8\n7KFe\n3(0)/bracketleftbigg\nµFe(T)3−18\n11µFe(T)10+7\n11µFe(T)21/bracketrightbigg\n,\n(10)\nwhere we use the same parameters as the correspond-\ning ones in Fig. 3 for µFe(T); thus, the remaining ad-\njustable parameters are KFe\n1(0),KFe\n2(0), andKFe\n3(0).\nFitting the form (10) to the experimental data,20,21we\nobtainKFe\n1(0) = 0.77 MJ/m3,KFe\n2(0) = 1.21 MJ/m3,\nandKFe\n3(0) = 0.11 MJ/m3. The comparison of the ex-\npression (10) with the experimental data is shown in Fig.\n6, wherein we note that a good fit is obtained. Here,\nwe should emphasize that assuming the presence of fi-\nniteKFe\n2(0) andKFe\n3(0) leads us to a considerably large\nKFe\n2(T) value (and, a small but not negligible KFe\n3(T)\nvalue) consistent with the treatment analogy using Eqs.\n(4b) and (4d). However, the presence of such a higher-\norder MAC has not thus far been reported, as mentioned\nin an earlier review.43We also plot these ghost MACs\nin the form of dashed and dotted lines in Fig. 6. This\nsituation is the same as that encountered by Zener in\nthe study on the MA of nickel,14which suggests that the\nassumptions of the Zener theory are invalid for Y 2Fe14Bmagnets. Therefore, the expression (10) along with Eq.\n(9) should be used only for obtaining the continuous val-\nues ofKFe\n1(T), and not for explaining the mechanisms\nof the MA in the Fe sublattice; this remains an unre-\nsolved problem in itinerant electron systems such as that\nin the case of the temperature-dependence of the MA of\n0 100 200 300 400 500 600\nT[K]0.00.20.40.60.81.01.21.4MACs [MJ/m3]KFe\n1(T)\nKFe\n2(T)\nKFe\n3(T)×10\nFIG. 6. Comparison of the experimental results (closed\ncircles) with the phenomenological expression based on the\nZener theory (lines). The solid lines are obtained by fitting the\nlinear combination of µFe(T)3,µFe(T)10, andµFe(T)21to the\nexperimental first-order MAC of the Y 2Fe14B magnet.20,21\nThe dashed and dotted lines, respectively, represent the\nsecond-order and the third-order MACs that should be\npresent for consistency of the Zener theory.\nMnBi.44,45\nIV. SUMMARY\nIn summary, we have shown explicit forms of the power\nlaw applicable to rare-earth magnets within the frame-\nwork of the Zener theory, and we have demonstrated\nthat the power law suitably describes the experimen-\ntal MACs of Nd 2Fe14B magnets. Thus, we obtained a\nsimple and general understanding of the complex tem-\nperature dependence of the MA often observed in rare-\nearth magnets. Furthermore, we considered Dy 2Fe14B\nand Y 2Fe14B magnets as examples regarding the inter-\npretation of MA when the power law failed. We believe\nthat our findings can further contribute to studies on the\nmagnetocrystalline anisotropy of rare-earth magnets.\nACKNOWLEDGMENTS\nThis work was supported by JSPS KAKENHI Grant\nNo. 16K06702 and 17K14800 in Japan.\n∗dmiura@solid.apph.tohoku.ac.jp\n1J. F. Herbst, Rev. Mod. Phys. 63, 819 (1991).2R. Skomski and J. M. D. Coey, Permanent Magnetism\n(CRC Press, 1999).6\n3J. M. D. Coey, “Magnetism and Magnetic Materials,”\n(2010).\n4T. Miyake and H. 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Ke 1* \n1Department of Physics and Astronomy, Michigan State  University, East Lansing, Michigan \n48824, USA \n2Department of Physics and Engineering Physics, Tula ne University, New Orleans, Louisiana \n70118, USA \n3Neutron Scattering Division, Oak Ridge National Lab oratory, Oak Ridge, Tennessee 37831, \nUSA \n* Corresponding author: ke@pa.msu.edu \n \nSr 4Ru 3O10 , the n = 3 member of the Ruddlesden-Popper type ruthenate  Sr n+1 Ru nO3n+1 , is \nknown to exhibit a peculiar metamagnetic transition  in an in-plane magnetic field.   However, the \nnature of both the temperature- and field-dependent  phase transitions remains as a topic of debate. \nHere, we have investigated the magnetic transitions  of Sr 4Ru 3O10  via single-crystal neutron \ndiffraction measurements. At zero field, we find th at the system undergoes a ferromagnetic \ntransition with both in-plane and out-of-plane magn etic components at Tc ≈ 100 K. Below T* = \n50 K, the magnetic moments incline continuously tow ard the out-of-plane direction. At T = 1.5 \nK, where the spins are nearly aligned along the c axis, a spin reorientation occurs above a critical  \nfield Bc, giving rise to a spin component perpendicular to the plane defined by the field direction \nand the c axis. We suggest that both the temperature- and fi eld-driven spin reorientations are \nassociated with a change in the magnetocrystalline anisotropy, which is strongly coupled to the \nlattice degrees of freedom. This study elucidates t he long-standing puzzles on the zero-field \nmagnetic orders of Sr 4Ru 3O10 and provides new insights into the nature of the fi eld-induced \nmetamagnetic transition.   2 \n Introduction \nRuddlesden-Popper type perovskite ruthenates (Sr,Ca ) n+1 Ru nO3n+1 are prototypical 4 d \ntransition-metal oxides which exhibit a variety of intriguing phenomena. The magnetic and \nelectronic properties of these materials are sensit ively dependent on the layer number n and the \nstructural distortions induced by substituting Ca f or Sr. For instance, the single-layer Sr 2RuO 4 (n \n= 1) shows an unconventional superconducting state [ 1], whereas the ground state of the double-\nlayer Sr 3Ru 2O7 (n = 2) is a Fermi liquid and is close to a ferromagn etic instability [ 2]. The three-\ndimensional SrRuO 3 (n = ∞) is a ferromagnetic metal with a Curie tempera ture Tc = 160 K [ 3,4]. \nOn the other hand, Ca 2RuO 4 (n = 1) is an antiferromagnetic Mott insulator with a  Neel \ntemperature TN ~ 110 K [ 5], while Ca 3Ru 2O7 (n = 2) exhibits a quasi-two-dimensional metallic \nbehavior and becomes antiferromagnetic below TN ~ 56 K [ 6,7], and CaRuO 3 is a paramagnetic \nmetal [ 8,9]. To date, compared to the intense efforts investe d on the single-, double-layer, and \nthree-dimensional ( n = 1, 2, and ∞) ruthenates, there have been much fe wer studies on the triple-\nlayer ( n = 3) compounds.  \nSr 4Ru 3O10  crystallizes in an orthorhombic space group Pbam [10 ], as shown in Fig. 1(a), \nwhich displays interesting but perplexing magnetic properties. From the magnetic susceptibility \nmeasurements, it has been reported that Sr 4Ru 3O10  undergoes a paramagnetic-ferromagnetic \ntransition at Tc = 105 K, below which the easy axis is along the c direction [ 10 ]. Intriguingly, \nwhile an additional transition has been found in th e magnetic susceptibility at T* = 50 K [10 ,11 ], \nno anomaly is revealed in the specific heat measure ments [ 12 ]. Although several distinct \nscenarios have been proposed to account for the fea ture at T*, its intrinsic character remains an \nopen question. On the one hand, below T* a canted antiferromagnetic structure with a \nferromagnetic component along the c axis and an antiferromagnetic component in the ab  plane 3 \n has been proposed [ 11 ,13 ]. Nevertheless, no evidence of antiferromagnetism has been observed \nvia neutron diffraction experiments [ 14 ,15 ]. On the other hand, previous neutron diffraction \nmeasurements have yielded contradictory results reg arding the direction of the ferromagnetic \nmoments in Sr 4Ru 3O10 [14 ,15 ]. It is initially argued that spins are aligned in  the ab  plane based on \nthe observation of (0 0 L)-type magnetic Bragg peaks and no anomaly has been  found at T* [14 ]. \nAs a result, the feature at T* in magnetic susceptibility measurements is ascribe d to a magnetic \ndomain process [ 14 ]. In contrast, a distinct magnetic easy axis has b een proposed more recently, \nwhere the magnetic moments are determined to be alo ng the c axis since the (0 0 L)-type \nmagnetic Bragg peaks are found absent. In addition,  a kink-like feature has been reported in the \ntemperature dependence of the Bragg reflection (2 0  0) at T*, which is also claimed to be \nobserved in all other reflections [ 15 ]. Therefore, the easy axis of the magnetic moments  and the \nnature of the anomaly at T* of Sr 4Ru 3O10  remain elusive.  \nAnother intriguing property of Sr 4Ru 3O10  is the field-induced metamagnetic transition. Whil e \na typical magnetic hysteresis loop of ferromagnets is observed as the magnetic field is applied \nalong the c axis, a first-order metamagnetic transition is see n below T* for the field applied in the \nab  plane [ 10 ,11 ]. Experimental signatures of the electronic phase separation during the phase \ntransition have been observed [ 16 ]. Nevertheless, previous neutron diffraction studi es reveal \ncontroversial features regarding this metamagnetic transition. Across the critical field, while no \nanomaly has been observed in the field dependence o f the magnetic reflection (0 0 8) in Ref. [ 14 ], \na field-induced transition at the same wave vector is reported in Ref. [ 15 ]. In addition, it has been \nreported that the metamagnetic transition is accomp anied by a change in the lattice, as evidenced \nin Raman scattering [ 13 ], neutron diffraction [ 15 ], and magnetostriction studies [ 17 ], which 4 \n suggests the presence of strong spin-lattice coupli ng in this system. To date, the nature and the \norigin of this metamagnetic transition are yet to b e resolved . \nIn this paper, we present a single-crystal neutron diffraction study of the magnetic order and \nthe metamagnetic transition in Sr 4Ru3O10 . We find that the material orders ferromagneticall y at \nTc ≈ 100 K without any signature of antiferromagnetic components, in agreement with the \nprevious studies [ 14 ,15 ]. However, the magnetic moments are found to posse ss both in-plane and \nout-of-plane components below Tc, in contrast to the previous neutron diffraction s tudies where \nthe spins are proposed to align either in the ab  plane [ 14 ] or along the c axis [ 15 ]. In addition, we \nhave observed spin reorientations below T* ≈ 50 K, where the magnetic moments incline toward \nthe out-of-plane direction. Furthermore, we show th at while the magnetic moments are nearly \nalong the c axis at T = 1.5 K, a magnetic field applied along the in-pla ne [1 –1 0] direction gives \nrise to a spin reorientation at a critical field Bc, above which there exists a spin component \nperpendicular to the plane defined by the field dir ection and the c axis, in accordance to the \nmetamagnetic transition. This observation is distin ct from the previous studies, where the \nmetamagnetic transition is ascribed to magnetic dom ain processes [14 ] or a coexistence of zero-\nfield c-axis component and the field-induced in-plane spin  polarization [ 15 ]. Both temperature- \nand magnetic-field-driven spin reorientations are p resumably ascribable to the change in the \nmagnetocrystalline anisotropy that is strongly corr elated with the RuO 6 octahedral distortion. \nOur results reconcile the inconsistency among previ ous neutron diffraction studies on the \nmagnetic easy axis of Sr 4Ru 3O10 , resolve the puzzling character of the transition at T*, and \nelucidate the nature of the field-induced metamagne tic transition.  \nResults 5 \n We start with the zero-field neutron diffraction da ta collected at the HB-3A four-circle \ndiffractometer. Figure 2(a) and 2(b) present the ro cking curves of Bragg reflections (0 0 2) and \n(1 1 1) at T = 8, 50 and 100 K, respectively. Note that both re flections meet the requirements of \nnuclear Bragg diffraction. The intensity of (0 0 2)  shows a nonmonotonic behavior, which is \nsignificantly enhanced at 50 K compared to that mea sured at 8 K and 100 K; in contrast, the (1 1 \n1) intensity becomes stronger with decreasing tempe rature. These observations suggest that there \nis magnetic intensity superimposed on the nuclear B ragg peaks. To further elucidate the \nnonmonotonic temperature dependence of the magnetic  intensity, Figure 2(c) and 2(d) present \nthe peak intensity of (0 0 2) and (1 1 1) Bragg ref lections as a function of temperature, \nrespectively. Upon cooling, additional reflection i ntensity in both peaks emerges below 100 K. \nSimilar enhancement has been observed in other nucl ear Bragg peaks including (0 0 6) and (0 0 \n8), indicating the development of a ferromagnetic o rder below Tc = 100 K which is in line with \nthe magnetic susceptibility measurements [ 10 ]. Consistent with Fig. 2(a), the intensity of (0 0  L)-\ntype magnetic reflections, L = 2, 6 and 8, also exhibits nonmonotonic temperatu re dependence, \nwith a maximum at T* ≈ 50 K that is reminiscent of the anomaly at T* observed in the magnetic \nsusceptibility data [ 10 ,11 ]. In accordance to Fig. 2(b), the magnetic intensi ty at (1 1 1) emerges at \nTc and continues to increase below T*, which suggests a spin reorientation at T*. On the one hand, \nthe observation of (0 0 L)-type magnetic reflections is in agreement with th e previous neutron \ndiffraction experiments reported in Ref. [14 ], but is different from that in Ref. [15 ] where they are \nfound to be absent. On the other hand, we have reve aled distinct temperature-dependent \nbehaviors of the intensity of (0 0 L) and (1 1 1), which is in sharp contrast to the pr evious studies \nwhere either no anomaly was observed [ 14 ], or a kink feature was claimed to exist in the \ntemperature dependence of all reflections [ 15 ]. Note that the HB-3A diffractometer is equipped 6 \n with a two-dimensional neutron detector, from which  the change in the scattering angle 2 θ of the \ndiffracted neutrons at different temperatures is fo und too small to be resolved due to the \ninstrumental resolution limitation. This is in agre ement with a change in the lattice constants of \nonly ~0.04% upon cooling observed in the previous st udies [ 15 ]. Therefore, the variation in the \npeak intensity of these Bragg peaks cannot be ascri bed to the structural change, but is magnetic \nin origin.  \nIn order to explore the possible magnetic configura tions in Sr 4Ru 3O10 , we carried out the \nmagnetic representational analysis using the progra m SARAh [18 ]. The space group of the crystal \nstructure is No. 55 Pbam  and the propagation vector is k = (0 0 0). There are four inequivalent \nRu atoms in a chemical unit cell which are located at Ru1 (0 0 0), Ru2 (0 0 0.1402), Ru3 (0.5 0 \n0.3598) and Ru4 (0.5 0 0.5). The analysis shows tha t an in-plane antiferromagnetic component \nwould give rise to strong magnetic reflections at ( 1 0 0), (1 0 1) or (0 1 1), which, nevertheless, \nare all absent in our measurements. Therefore, the feature at T* in the magnetic susceptibility \ncannot be ascribed to the formation of antiferromag netic components in the ab  plane that was \nclaimed previously [ 11 ,13 ]. As a result, the only symmetry-allowed magnetic configurations in \nSr 4Ru 3O10  is the ferromagnetic structure with the magnetic m oments either along the c axis or in \nthe ab  plane. Since neutrons couple to the magnetic momen t perpendicular to the momentum \ntransfer q, the observation of magnetic intensities at (0 0 L), L = 2, 6, and 8, at T* < T < Tc \nsuggests the development of a magnetic order with s pins aligned in the ab  plane. This, combined \nwith the magnetic susceptibility measurements which  show an easy axis closer to the c axis, \nsuggests that the magnetic moments have both in-pla ne and out-of-plane components in this \ntemperature regime. Note that at T* < T < Tc, the intensity of (0 0 2) increases more rapidly t han \nthat of (1 1 1), which implies that the magnetic mo ments incline toward the ab  plane, though the 7 \n easy axis is always closer to the c axis. Below T*, the decrease in the intensity of (0 0 L)-type \nmagnetic peaks suggests that the in-plane ferromagn etic moment reduces, and the spins incline \ncontinuously toward the out-of-plane direction upon  cooling. This speculation is supported by \nthe observation of an increase in the reflection in tensity of the (1 1 1) Bragg peak below T*, as \nthe direction of the (1 1 1) wave vector is almost perpendicular to the c axis (note that a* ≈ b* ≈ \n1.137 Å-1, c* ≈ 0.219 Å-1). We have collected the nuclear and magnetic Bragg  peaks at 8 K, 50 K \nand 100 K, and attempted to perform Rietveld analys is using the program FullProf [ 19 ]. Note that \nthe magnetic moments of the outer-layer Ru ions (Ru 1 and Ru4) and the central-layer ones (Ru2 \nand Ru3) can be different, owing to the difference in the direction and magnitude of the RuO 6 \noctahedral rotation [ 10 ]. At T = 8 K, due to the very weak intensity of the (0 0 L) peaks, the spins \nare nearly aligned along the c axis. Since only the (1 1 1) peak is associated wi th this spin \ncomponent, the magnitude of the magnetic moments of  outer- and central-layer Ru ions cannot \nbe exclusively determined at this temperature. Simi larly, more magnetic reflections are required \nin order to determine the moment sizes and the cant ing angles quantitatively at T*. The schematic \nof the magnetic structure at zero field is plotted in Fig. 1(b).   \nNext, we discuss the neutron diffraction measuremen ts done on the HB-1A triple-axis \nspectrometer with the magnetic field applied along the in-plane [1 -1 0] direction, where a first-\norder metamagnetic transition is observed in the ma gnetic susceptibility measurements [ 10 ]. \nFigure 3(a) and 3(b) display the peak intensity of (0 0 2) and (1 1 1) Bragg peaks as a function of \nmagnetic field at T = 1.5 K. It is remarkable that the (0 0 2) Bragg p eak intensity increases \nsignificantly at Bc = 1.5 T, whereas that of the (1 1 1) decreases. Th ese observations are distinct \nfrom the previous studies where no anomaly has been  observed in the intensity of magnetic \nreflections as a function of magnetic field [14 ]. It is worth noting that in another neutron 8 \n diffraction study a similar trend to (0 0 2) [Fig. 3(a)] has been observed in the field dependence \nof the intensity of (0 0 8) Bragg peak [ 15 ]. However, a slight reduction in the scattering in tensity \nof ( H K 0) reflections was claimed and suggested to be ass ociated with the magnetic moment \nalong the c axis, which led the authors to argue that the fiel d-induced transition is due to a \ncoexistence of the initial zero-field ferromagnetic  order with a field-induced in-plane spin \npolarization [ 15 ]. If this were the case, one would expect a slight  increase in the magnetic \nintensity of (1 1 1) reflection above the critical field Bc, as (1 1 1) is perpendicular to [1 -1 0] but \nis about ~82.2° with respect to the [0 0 1] directi on. This is not in line with our observation of the  \nfield-induced decrease in the intensity of (1 1 1) reflection, which suggests that the applied \nmagnetic field gives rise to a spin component perpe ndicular to the plane defined by the field \ndirection and the c axis. This is supported by a recent study which, b y measuring the magnetic \nmoment vectors, reports the existence of a magnetic  moment component that is perpendicular to \nthe field rotation plane [ 20 ]. Note that the slight difference in the critical field compared with the \nmagnetic susceptibility data may arise from differe nt applied magnetic field directions in the ab  \nplane. To further support the variation of the magn etic scattering intensity below and above Bc, \nFigure 3(c) and 3(d) plot the θ-2θ scans over (0 0 2) and (1 1 1) Bragg peaks measure d at T = 1.5 \nK with B = 0 and 3.5 T, respectively. We can clearly see th at above Bc the intensity of (0 0 2) \npeak becomes enhanced while that of the (1 1 1) pea k is suppressed. Furthermore, there is no \nnoticeable change in the 2 θ values of both peak positions across the field-ind uced magnetic \ntransition, which is consistent with the previous s tudy that the lattice constants a and c change \nonly by ~0.04% at the critical field that is beyond the instrumental resolution in our study [15 ]. \nFor comparison, in the inset of Fig. 3(d), we prese nt the θ-2θ scan of a high-| Q| nuclear peak (0 0 \n16), where the magnetic intensity is expected to be  very weak due to the small magnetic form 9 \n factor. The very little difference in the intensity  of (0 0 16) at 0 and 3.5 T magnetic field further \nsubstantiates that the field-induced intensity vari ation at low-| Q| peaks (0 0 2) and (1 1 1) are \nmagnetic in origin, rather than a structural change . Figure 4(a) and 4(b) show the θ-2θ scans of \n(0 0 2) and (1 1 1) Bragg peaks measured at T = 50 K with B = 0 and 3.5 T, respectively. Both (1 \n1 1) and (0 0 2) scattering intensity displays negl igible changes upon applying the magnetic field. \nThese results, combined with the features observed at low temperature shown in Figure 3, are in \nagreement with the magnetic susceptibility measurem ents that the metamagnetic transition only \noccurs below T* = 50 K. \nDiscussions \nThe observation of a spin reorientation at T* in Sr 4Ru 3O10 at zero field has elucidated the \nlong-standing puzzle on the anomaly in the magnetic  susceptibility measurements [ 10 ]. This spin \nreorientation suggests a change in the magnetic eas y axis as a function of temperature. This \nfeature is reminiscent of the widely studied spin r eorientation transitions in rare earth magnets \nand orthoferrites [ 21], which are ascribed to the change in the magnetic  anisotropy constants as a \nfunction of external parameters, such as temperatur e, magnetic field and pressure, etc [ 22]. Our \nfinding raises an intriguing question: What is the underlying mechanism responsible for the \nchange in the magnetic easy axis in Sr 4Ru 3O10 ? It is known that in Ruddlesden-Popper type \nruthenates, the magnetic anisotropy is strongly cou pled to the structural distortions of the RuO 6 \noctahedra. For instance, magnetocrystalline anisotr opy has been extensively investigated in \nSrRuO 3 thin films, where the magnetic anisotropy and the saturated magnetic moments can be \nreadily tuned by controlling the RuO 6 octahedral distortion via the epitaxial strain imp osed by \nsubstrates [ 23]. Considering the fact that the ferromagnetic stat e with Tc ≈ 100 K in triple-layer ( n \n= 3) Sr 4Ru 3O10  bridges the physics of the double-layer ( n = 2) Sr 3Ru 2O7 (Fermi liquid, 10 \n ferromagnetic instability) [ 2] and the three-dimensional ( n = ∞) SrRuO 3 (ferromagnetic, Tc = 160 \nK) [ 24], the change in the magnetocrystalline anisotropy in Sr 4Ru 3O10  across T* may arise from \nthe corresponding changes in lattice structures. In deed, anomalies in the lattice constants with c \nexpanding while a contracting below T* at zero field have been observed previously [ 15,17 ]. These \nchanges in lattice parameters are expected to alter  the occupancy of the Ru t2g  orbitals in this \nmultiband system, which consequently affects the ma gnetic anisotropy through spin-orbit \ncoupling [ 25]. \nThe nature of the in-plane field-induced metamagnet ic transition in Sr 4Ru 3O10  below T*, \nwhich has been an unresolved puzzle, can now be rea dily understood. It is also due to a spin \nreorientation where the magnetic moments reorient f rom the nearly c axis toward the ab  plane. \nIntriguingly, we find that there exists a magnetic moment component perpendicular to the plane \ndefined by the field direction and the c axis above the critical field, which is consistent  with the \nobservation of a recent study where it has been asc ribed to the Dzyaloshinskii-Moriya interaction \nassuming that the easy axis is along the c axis [20 ]. On the other hand, it is worth noting that \nacross the field-induced metamagnetic transition be low T* the lattice parameter a expands while \nc shrinks [ 15 ], a trend which is opposite to that observed upon cooling through T* at zero field \ndiscussed above [ 15 ,17 ]. This suggests that the magnetic anisotropy is al tered above the critical \nfield due to the strong spin-lattice coupling, whic h needs to be taken into account in the future \ntheoretical modeling. \nIn summary, the magnetic structure and the metamagn etic transition in the triple-layer \nruthentate Sr 4Ru 3O10  (n = 3) are revisited by neutron diffraction measurem ents. The magnetic \norder below Tc is found to be ferromagnetic with both in-plane an d out-of-plane spin components, \nwhile it evolves toward the c axis below T* = 50 K. This finding elucidates the long-standing 11 \n puzzle of the nature of the magnetic transition at T* in Sr 4Ru 3O10 . Below T*, upon the \nmetamagnetic transition with a magnetic field appli ed along the in-plane [1 -1 0] direction, the \nmagnetic moments undergo a spin reorientation from the c axis toward the ab  plane with a spin \ncomponent perpendicular to the plane defined by the  magnetic field direction and the c axis. \nBoth temperature- and field-induced spin reorientat ions can be attributed to the change of \nmagnetocrystalline anisotropy via spin-orbit coupli ng due to the change in the RuO 6 octahedral \ndistortion.  \nMethods \nThe single-crystal Sr 4Ru 3O10  was grown by the floating zone method [ 26]. The lattice \nconstants are a = 5.5280 Å, b = 5.5260 Å and c = 28.651 Å. The phase and quality of the crystal \nhave been verified by x-ray diffraction measurements. The magnetization as  a function of both \ntemperature and magnetic field has been measured us ing SQUID, and the results are in \nagreement with previous studies [10, 11 ]. A very small amount of intergrowth of SrRuO 3 has been \nrevealed in the magnetic susceptibility measurement s, which is easy to be separated in the \nneutron diffraction measurements due to its very di fferent lattice parameter c (c = 7.8446 Å in \northorhombic unit cell). Zero-field and field-depen dent neutron diffraction measurements were \nperformed using the HB-3A four-circle diffractomete r (λ = 1.5426 Å) and the HB-1A triple-axis \nspectrometer (λ = 2.36 Å) respectively at High Flux  Isotope Reactor in Oak Ridge National \nLaboratory. At HB-3A, the sample was mounted on an aluminum stick and loaded into a closed-\ncycle Helium displex. At HB-1A, the sample was orie nted in the horizontal ( H H L) scattering \nplane and loaded into a vertical-field cryomagnet s uch that the magnetic field is applied along \nthe [1 -1 0] direction. The Bragg reflections ( H K L) were in the reciprocal lattice units 2 π/a, \n2π/b and 2 π/c of the orthorhombic space group Pbam  (No. 55) [ 10 ].  12 \n Data availability \nThe data that support the findings of this study ar e available from the corresponding author \nupon reasonable request. \nReferences \n1 Maeno, Y.  et al.  Two-dimensional Fermi liquid behavior of the super conductor Sr 2RuO 4. \nJournal of the Physical Society of Japan  66 , 1405-1408 (1997). \n2 Ikeda, S. -I., Maeno, Y., Nakatsuji, S., Kosaka, M. & Uwatoko, Y. Ground state in \nSr 3Ru 2O7: Fermi liquid close to a ferromagnetic instability . Physical Review B  62 , \nR6089-R6092 (2000). \n3 Kanbayasi, A. Magnetocrystalline Anisotropy of Sr RuO 3. Journal of the Physical Society \nof Japan  41 , 1879-1883 (1976). \n4 Allen, P. B.  et al.  Transport properties, thermodynamic properties, an d electronic \nstructure of SrRuO 3. Physical Review B  53 , 4393-4398 (1996). \n5 Nakatsuji, S., Ikeda, S. -I. & Maeno, Y. Ca 2RuO 4: New Mott insulators of layered \nruthenate. Journal of the Physical Society of Japan  66 , 1868 (1997). \n6 Cao, G., McCall, S., Crow, J. E. & Guertin, R. P.  Observation of a metallic \nantiferromagnetic phase and metal to nonmetal trans ition in Ca 3Ru 2O7. Physical Review \nLetters  78 , 1751-1754 (1997). \n7 Yoshida, Y.  et al.  Quasi-two-dimensional metallic ground state of Ca 3Ru 2O7. Physical \nReview B  69 , 220411 (2004). \n8 Shepard, M., Cao, G., McCall, S., Freibert, F. & Crow, J. E. Magnetic and transport \nproperties of Na doped SrRuO 3 and CaRuO 3. Journal of Applied Physics  79 , 4821-4823 \n(1996). 13 \n 9 Cao, G., Freibert, F. & Crow, J. E. Itinerant-to- localized electron transition in perovskite \nCaRu 1−x Rh xO3. Journal of Applied Physics  81 , 3884-3886 (1997). \n10 Crawford, M. K.  et al.  Structure and magnetism of single crystal Sr 4Ru 3O10 : A \nferromagnetic triple-layer ruthenate. Physical Review B  65 , 214412 (2002). \n11 Cao, G.  et al.  Competing ground states in triple-layered Sr 4Ru 3O10 : Verging on itinerant \nferromagnetism with critical fluctuations. Physical Review B  68 , 174409 (2003). \n12 Lin, X. N., Bondarenko, V. A., Cao, G. & Brill, J. W. Specific heat of Sr 4Ru 3O10 . Solid \nState Communications  130 , 151-154 (2004). \n13 Gupta, R., Kim, M., Barath, H., Cooper, S. L. & Cao, G. Field- and pressure-induced \nphases in Sr 4Ru 3O10 : A Spectroscopic Investigation. Physical Review Letters  96 , 067004 \n(2006). \n14 Bao, W. et al.  arXiv:cond-mat/0607428v1  (2006). \n15 Granata, V., Capogna, L., Reehuis, M., Fittipald i, R. & Ouladdiaf, B. Neutron diffraction \nstudy of triple-layered Sr 4Ru 3O10 . Journal of physics. Condensed matter  25 , 056004 \n(2013). \n16 Mao, Z. Q., Zhou, M., Hooper, J., Golub, V. & O’ Connor, C. J. Phase separation in the \nitinerant metamagnetic transition of Sr 4Ru 3O10 . Physical Review Letters  96 , 077205 \n(2006). \n17 Schottenhamel, W.  et al.  Dilatometric study of the metamagnetic and ferroma gnetic \nphases in the triple-layered Sr 4Ru 3O10  system. Physical Review B  94 , 155154 (2016). \n18 Wills, A. S. A new protocol for the determinatio n of magnetic structures using simulated \nannealing and representational analysis (SARAh). Physica B: Condensed Matter  276 , \n680-681 (2000). 14 \n 19 Rodríguez-Carvajal, J. Recent advances in magnet ic structure determination by neutron \npowder diffraction. Physica B: Condensed Matter  192 , 55-69  (1993). \n20 Weickert, F.  et al.  Missing magnetism in Sr 4Ru 3O10 : Indication for antisymmetric \nexchange interaction. Scientific Reports  7, 3867  (2017). \n21 Konstantin, P. B., Anatolii, K. Z., Antonina, M.  K. & Levitin, R. Z. Spin-reorientation \ntransitions in rare-earth magnets. Soviet Physics Uspekhi  19 , 574 (1976). \n22 Belov, K. P.  et al.  Spin-flip transitions in cubic magnets. Magnetic p hase diagram of \nterbium-yttrium iron garnets. Zh. Eksp. Teor. Fiz.  68 , 1189 (1974). \n23 Koster, G.  et al.  Structure, physical properties, and applications o f SrRuO 3 thin films. \nReviews of Modern Physics  84 , 253-298 (2012). \n24 Kanbayasi, A. Magnetic properties of SrRuO 3 single crystal. II. Journal of the Physical \nSociety of Japan  44 , 89-95  (1978). \n25 Cuoco, M., Forte, F. & Noce, C. Probing spin-orb ital-lattice correlations in 4d 4 systems. \nPhysical Review B  73 , 094428 (2006). \n26 Zhou, M.  et al.  Electronic and magnetic properties of triple-layer ed ruthenate Sr 4Ru 3O10  \nsingle crystals grown by a floating-zone method. Materials Research Bulletin  40 , 942-\n950  (2005). \nAcknowledgements \nWork at Michigan State University was supported by the National Science Foundation under \nAward No. DMR-1608752 and the start-up funds from M ichigan State University. Work at \nTulane University was supported by the U.S. Departm ent of Energy (DOE) under EPSCOR \nGrant No. DE-SC0012432 with additional support from  the Louisiana Board of Regents (support 15 \n for crystal growth). A portion of this research use d resources at the High Flux Isotope Reactor, a \nDOE Office of Science User Facility operated by the  Oak Ridge National Laboratory. \nAuthor Contributions \nX. K. conceived the project. P. G. L., Y. W., and Z . Q. M. grew the single-crystal sample. M. \nZ., H. B. C., W. T., X. K., H. D. Z., and B. D. P. performed neutron diffraction experiments. M. \nZ., H. B. C., W. T., and X. K. analyzed the data. M . Z. and X. K. wrote the manuscript. All \nauthors commented on the manuscript. \nAdditional information \nCompeting Interest: The authors declare that they h ave no competing financial interests. \n  16 \n Figure Captions \nFig. 1. (a) The crystal structure of Sr 4Ru 3O10 . Sr, Ru and O atoms are represented by the green, \ngray and red balls, respectively. (b) The magnetic structure of Sr 4Ru 3O10 . Only the magnetic Ru \nions are shown. Note that both moment size and spin  direction cannot be uniquely determined \ndue to the lack of enough Bragg peaks with reasonab ly good magnetic intensity in the \nmeasurements. \n \nFig. 2. (a),(b) Rocking curve scans across (0 0 2) and (1 1 1) Bragg reflections at T = 8, 50 and \n100 K, respectively. (c),(d) Temperature dependence  of the peak intensity of (0 0 2) and (1 1 1) \nreflections at B = 0 T, respectively. The red and green dashed line s denote Tc and T*, respectively. \n \nFig. 3. (a),(b) Field dependence of the peak intens ity of (0 0 2) and (1 1 1) reflections at T = 1.5 \nK. (c),(d) θ-2θ scans across (0 0 2) and (1 1 1) Bragg reflections  at B = 0 and 3.5 T, T = 1.5 K. \nInset shows the θ-2θ scan over (0 0 16) peak at B = 0 and 3.5 T, T = 1.5 K. \n \nFig. 4. θ-2θ scans across (0 0 2) and (1 1 1) Bragg reflections  at B = 0 and 3.5 T, T = 50 K.  \n \n  17 \n Figure 1 \nM. Zhu et al. \n  \n \n  \n18 \n Figure 2 \nM. Zhu et al. \n \n \n \n  \n19 \n Figure 3 \nM. Zhu et al. \n \n \n \n  \n20 \n Figure 4 \nM. Zhu et al.  \n \n \n \n \n"    },    {        "title": "1802.09420v1.Multiferroic_Micro_Motors_with_Deterministic_Single_Input_Control.pdf",        "content": "arXiv:1802.09420v1  [physics.app-ph]  26 Feb 2018-\nMultiferroic Micro-Motors With Deterministic Single Inpu t Control\nJohn P. Domann∗\nDepartment of Biomedical Engineering and Mechanics,\nVirginia Polytechnic and State University\nCai Chen, Abdon E. Sepulveda, and Greg P. Carman\nDepartment of Mechanical and Aerospace Engineering,\nUniversity of California, Los Angeles\nRob N. Candler\nDepartment of Electrical and Computer Engineering,\nUniversity of California, Los Angeles\nDepartment of Mechanical and Aerospace Engineering,\nUniversity of California, Los Angeles and\nCalifornia NanoSystems Institute, Los Angeles\n(Dated: February 27, 2018)\nAbstract\nAbstract:\nThis paper describes a method for achieving continuous dete rministic 360◦magnetic moment rotations\nin single domain magnetoelastic discs, and examines the per formance bounds for a mechanically lossless\nmultiferroic bead-on-a-disc motor based on dipole couplin g these discs to small magnetic nanobeads. The\ncontinuous magnetic rotations are attained by controlling the relative orientation of a four-fold anisotropy\n(e.g., cubic magnetocrystalline anisotropy) with respect to the two-fold magnetoelastic anisotropy. This\napproach produces continuous rotations from the quasi-sta tic regime up through operational frequencies of\nseveral GHz. Driving strains of only ≈90 to 180 ppm are required for operation of motors using exist ing\nmaterials. The large operational frequencies and small siz es, with lateral dimensions of ≈100s of nanome-\nters, produce large power densities for the rotary bead-on- a-disc motor, and a newly proposed linear variant,\nin a size range where power dense alternative technologies d o not currently exist.\nKeywords: motor, multiferroic, piezoelectric, magnetoel astic, magnetostriction, anisotropy, deterministic cont rol\n∗jpdomann@vt.edu\n1I. BACKGROUND / LITERATURE REVIEW\nIn his 1842 lecture On a new Class of Magnetic Forces James Joule describes how follow-\ning the suggestion of an ”ingeneous gentleman,” he initiate d the first ever measurements of\nmagnetoelasticity.[1] From the outset, Joule was focused o n determining not just if a bar of\niron would elongate in the presence of a magnetic field, but if ”power could be advantageously\nemployed for the movement of machinery.” In other words, Jou le was interested in making mag-\nnetoelastic motors. However, after measuring a magnetic fie ld induced strain of only ε≈1.4\nppm, Joule concluded that ”With regard to the application of the new force to the movement of\nmachinery, I have nothing favourable to advance.”\nOf course, in that very same lecture Joule was doubtful that e lectromagnetic motors had any\nfuture use, showing a slight lack of foresight in an otherwis e exemplary career. While many dif-\nferent types of motors have been created, macro-scale combu stion and electromagnetic motors are\nthe most widely used. However, poor scaling laws rapidly deg rade the performance of these tradi-\ntional motor technologies at the microscale, and new approa ches need to be considered. Figure 1a\nhighlights the available power density for a variety of diff erent micro and nanoscale motor tech-\nnologies. For reference, the combustion engine of a 2017 For d Mustang can generate around 400\nhorsepower (300 kW) and is sized on the order of 1 m3, generating a power density of 300 kWm−3\n(i.e., moderate to high power density but at significantly la rger sizes than shown on Figure 1a).\nThe larger sizes on Figure 1a are representative of MEMS moto rs with volumes ≈1 mm3. The\ndata points come from numerous different technological app roaches, including electrostatic,[2–\n8] magnetostatic,[9, 10] electromagnetic,[11] electrohy drodynamic,[12] piezoelectric,[13–22]\npneumatic,[23–27] and plasmonic[28, 29] technology bases . These motors can advantageously\nbe controlled with conventional electronic systems; howev er the power density of existing MEMS\napproaches clearly doesn’t scale well to sizes significantl y smaller than a cubic millimeter.\nAt volumes near 1 nm3, molecular motors like rotaxane are extremely power dense. These\nmolecules generate forces near 30 pN, with strokes of 5 nm, at rates of up to 3 kHz. This produces\npower densities on the order of 100 GWm−3.[30, 31] In addition to rotaxane, molecular motors\nbased on nanocrystal actuators,[32, 33] and ATP synthase[3 5, 36] have been created. However,\nmolecular motors are exceedingly difficult to locally contr ol, leading to poor array / scaling prop-\nerties.\nFigure 1a illustrates that there are currently no power dens e motor technologies at the µm3\n2Volume (m3)Power Density (W/m3)\n10-25 10-20 10-15 10-10 10-51012\n1010\n108\n106\n104\n100\n10-2\n10-4mm3\nnm3\nRBC\nSize\nMolecular\nMotorsμm3\na) b)\nFIG. 1. a) Power density of micro and nanoscale motor technol ogies, indicating no power dense technolo-\ngies exist at the µm3size scale.[2–36] Shown for scale are several red blood cell s (RBCs). b) The proposed\nmultiferroic bead-on-a-disc motor uses dipole coupling to drag a small magnetic bead around a stationary\nmagnetic disc with rotating magnetic moment.\nsize scale. A µm3motor would enable localized control on the same size scale a s biological\ncells, facilitating diagnostic and therapeutic applicati ons in addition to opening new avenues for\nfundamental research. As an example, Di Carlo et al. have rec ently demonstrated magnetic cell\nsorting techniques near the microscale, and improved contr ol could enable the study of individual\ncellular components instead of relying on traditional cell lysis based techniques.[37]\nMultiferroic heterostructures have recently been propose d as highly power dense µm3motors.[38]\nThese motors manipulate large magnetic forces by controlli ng the dipole fields of single do-\nmain magnetic heterostructures or domain walls with strain . Multiferroic control is energy\nefficient[39, 40] even at high frequency operation.[41, 42] These effects should combine to pro-\nduce large power densities. Figure 1b shows a hypothetical b ead-on-a-disk motor,[38] where the\nmagnetic moment of the large disk rotates (the disk itself is stationary), and drags around the\nsmaller bead through dipole coupling. The B-field generated by the large disk applies a torque on\nthe magnetic bead to keep it aligned with the B-field, while ∇Bapplies a force on the bead and\ncauses it to rotate around the large disk.\n3A key impediment to the creation of multiferroic motors has b een achieving deterministic con-\ntrol of continuous 360◦magnetic rotations. Initial studies on magnetoelastic con trol used single\nelectrodes to cause 90◦non-deterministic rotations in ellipse shaped structures .[38, 43, 44] How-\never, the use of a single electrode and elliptical shape is re stricted to a maximum rotation of 90◦,\nsince it combines two uniaxial (i.e., two-fold) anisotropi es (see the Supplementary Information\nfor a mathematical explanation). This restriction motivat ed subsequent studies with multiple elec-\ntrodes that generated a rotating biaxial plane strain state .[45, 46] Several modeling efforts have\nlooked at this challenge, and observed that the use of 4 or mor e electrodes can deterministically\ncontrol rotations.[47, 48] The multi-electrode approach h as resulted in experimental observation of\ndeterministic rotations in increments of 45◦confirmed up to 180◦.[49] Recent work has also shown\ndynamic effects can lead to 180◦rotations (i.e., ballistic switching with PMA).[50] Howev er, fabri-\ncating motor arrays with multiple electrodes is undesirabl e, and ballistic switching requires precise\ntiming with very narrow-band performance limiting variabl e frequency use.\nA path to single electrode deterministic control with broad band operation can be found in three\ndifferent studies analyzing the effects of rotating the rel ative orientation of the magnetoelastic\nanisotropy with respect to the shape of nanomagnetic struct ures. This approach has led to model\nbased predictions of deterministic rotations in shapes lik e four leaf clovers,[51] squares,[52] cat\neyes,[53] and peanuts.[53] However, most of these shapes re quire the use of complex fabrication\ntechniques to resolve fine features, and are prone to the crea tion of pinning sites that have likely\nprevented rotation in previous work.[46] To overcome these issues, the present study analyzes\na single electrode control method for deterministic 360◦rotations, avoiding complex geometries\nor electrode patterns. A bead-on-a-disk motor is then analy zed and predictions provided for a\nmechanically lossless motor’s performance (i.e., only mag netic and inertial forces are considered\nin the limit of zero fluidic drag).\nII. MODEL\nWhile previous studies have combined magnetoelasticity wi th an additional two-fold anisotropy\n(i.e., elliptical shapes), this work combines a four-fold a nisotropy (e.g., cubic MCA) with the mag-\nnetoelastic anisotropy. This work demonstrates the key to g enerating deterministic 360◦rotations\nis to rotate the principal orientation of the anisotropies w ith respect to one another. In the following\nsection the static and dynamic rotational characteristics of the stator element (large magnetic disk)\n4with rotated anisotropies will be analyzed, then a second se ction analyzing the upper performance\nbounds of a bead-on-a-disk motor will be presented.\nA. Quasi-Static Energy Analysis\nFigure 2 shows the a) dipole field, b) stator geometry, and c) d ipole forces for the motor being\nanalyzed. This figure is a top down view of the motor shown in Fi gure 1b. A circular thin-film\nFIG. 2. a) Bead-on-a-disk motor with surrounding magnetic d ipole field. The large disk (stator) remains\nstationary, but its magnetic moment rotates in response to a n applied strain. Dipole coupling drags the bead\n(rotor) around the stator. b) Orientation of the magnetic mo ment and principal strains. c) Dipolar forces on\na bead near the stator element.\nstator is studied, with in-plane magnetic orientation θ. The stator is subject to a voltage induced\nplane strain ( ε13=ε23=ε33=0), with principal strains components ε1andε2rotated ϕεfrom\nthe x-axis ([100] direction). While the magnitude and sign o f the applied strain may change as\na function of time, the orientation is assumed fixed during mo tor operation (i.e., applied using a\nsingle fixed electrode). Further assuming the out-of-plane magnetization m3is small due to shape\nanisotropy, the relevant stator energy density reduces to\nUtot=UMCA+UME (1)\n=KMCA\n1/bracketleftbig\nm2\n1m2\n2/bracketrightbig\n+B1/bracketleftbig\nε11(m2\n1−1/3)+ε22(m2\n2−1/3)/bracketrightbig\n+2B2[ε12m1m2] (2)\nwhere Utot,UMCA, and UMEare the total, magnetocrystalline, and magnetoelastic ene rgies, re-\nspectively, Kmca\n1is the first order cubic MCA constant, B1andB2are the cubic magnetoelastic\n5anisotropy coefficients, the miterms are the magnetization direction cosines, and εi jare the ap-\nplied strain components. Note that a factor of 2 has been incl uded on the B 2term, as tensorial\nstrain components have been used in place of the engineering strain components commonly used\nin this expression (i.e., ei j=2εi jfori/negationslash=j).[54, 55] The tensorial strain components are now\nwritten in terms of the principal strains using a rank 2 tenso r transformation\nε11=εavg+1\n2εbcos(2ϕε) (3)\nε22=εavg−1\n2εbcos(2ϕε) (4)\nε12=1\n2εbsin(2ϕε) (5)\nwhere ϕεis the principal strain orientation shown on Figure 2b, εavg=(ε1+ε2)/2 is the average\nstrain, and εb=ε1−ε2is the biaxial strain .\nUsing equations 3 to 5 in equation 2 and converting to polar co ordinates ( m1=cosθ, and\nm2=sinθ), the energy expression becomes\nUtot=K2cos(2θ−δs)+K4cos(4θ) (6)\nK2=1\n2εb/radicalBig\nB2\n1cos2(2ϕε)+B2\n2sin2(2ϕε) (7)\nK4=−1\n8KMCA\n1 (8)\ntan(δs)=B2\nB1tan(2ϕε) (9)\nwhere K2andK4are the total second and fourth order anisotropy coefficient s, and δsis the\nstatic rotation of the second order anisotropy with respect to the x-direction. Equation 6 is in-\ndicative of the fact that a conservative anisotropy energy c an be expanded in a Fourier series\n(Utot=Re[∑\nnKnejnθ]), and highlights the two and four fold rotational symmetry in herent to the\ncubic magnetoelastic and magnetocrystalline anisotropie s, respectively. It should be noted that\nin these equations the second order anisotropy K2→K2(t)is a time dependent function of the\nvoltage induced biaxial strain εb, and independent of the average strain εavg. For an isotropic mag-\nnetoelastic material ( λ100=λ111=⇒B1=B2) these expressions simplify to K2=εb|B1|/2 and\ntan(δs)=tan(2ϕε).\nTo locate the equilibrium configuration for an arbitrary inp ut strain equation 6 is first converted\n6to a complex polynomial\nUtot=Re/bracketleftBig\nK2ej(2θ−δs)+K4e4jθ/bracketrightBig\n(10)\n=1\n2K2/parenleftbig\npz2+pz2/parenrightbig\n+1\n2K4/parenleftbig\nz4+z4/parenrightbig\n(11)\nwhere p=e−jδsandz=ejθ. The extrema of this expression are found using the Lagrange multi-\nplier method to enforce the constraint that zhas unit magnitude.\nL(z,¯z,λ)=Utot(z,¯z)−λ(z¯z−1) (12)\nCombing the partial derivatives of equation 12 and making th e substitution R=K2/K4, yields the\n8thorder complex polynomial\nz8+1\n2Rpz6−1\n2R¯pz2−1=0 (13)\nwhich can be converted into a 4thorder polynomial with the substitution w=z2\nw4+1\n2Rpw3−1\n2R¯pw−1=0 (14)\nThe roots of this equation provide the local extrema of the en ergy landscape.\nNoting that Ris the ratio of the second and fourth order anisotropies, it i s easy to see when\nno strain is applied (R=0)energy extrema are located at the 8throots of unity (w4=1=⇒\nz8=1=⇒ϕ=±nπ/4). Since the roots provide the location of both the maxima and m inima\nof the fourth order cubic magnetocrystalline anisotropy, t he solution for zhas 8 angles per 2 π\nradians when R=0. The closed form solution for an arbitrary plane strain sta te is provided in the\nSupplementary Information.\nB. Dynamic Analysis\nThis section analyzes the dynamics of the proposed motor usi ng the Landau-Lifshitz-Gilbert\n(LLG) equation. The LLG equations is\n˙/vectorm=−γ/vectorm×/vectorHe f f−α/vectorm×˙/vectorm (15)\n/vectorHe f f=−1\nµ0Ms∂Utot\n∂/vectorm(16)\n7where the gyromagnetic ratio γ, Gilbert damping factor α, saturation magnetization Ms, and ef-\nfective magnetic field /vectorHe f fhave been used. The effective field has one component for each of the\nenergy expressions in Equation 1, with components\nH1=−2\nµ0Ms/bracketleftbig\nKmca\n1m1m2\n2+B1m1ε11+B2m2ε12/bracketrightbig\n(17)\nH2=−2\nµ0Ms/bracketleftbig\nKmca\n1m2\n1m2+B1m2ε22+B2m1ε12/bracketrightbig\n(18)\nH3=−2\nµ0Ms/bracketleftbigg\nKmca\n1m3(m2\n1+m2\n2)+Kmca\n2m2\n1m2\n2m3+1\n2µ0M2\nsm3/bracketrightbigg\n(19)\nwhere fields linear in m3have been retained, but all higher order contributions drop ped.\nThe following analysis determines the strain amplitude and frequency required to cause ro-\ntational motion with constant angular velocity. It is assum edm1(t) =cosω0t,m2(t) =sinω0t,\nand that the biaxial strain follows a temporal dependence of εb∝cos(ωεt+δd), where ωεis the\nmechanical driving frequency, and δdis dynamic phase difference between εbandmi. It is also\nassumed that the mechanical strain is applied at twice the fr equency of rotation ωε=2ω0, with one\nstrain cycle causing a maximum of 180◦rotation. This assumption will be justified after looking\nat the results of the quasi-static analysis.\nUsing the fact that the magnitude of the magnetic moment is co nserved\n/vectorm·˙/vectorm≈m1˙m1+m2˙m2=0 (20)\nwhere ˙/vectormis calculated by inserting Equations 17 to 19 into Equation 1 5. This results in an expres-\nsion of the form\nAsin(4ωεt)+Bcos(4ωεt)+C=0 (21)\nwhere A, B, and C are functions of the material properties and operating conditions but indepen-\ndent of time. For the equality to hold for all time t, the coeffi cients A, B, and C must be identically\nzero, resulting in the operating conditions to achieve unif orm rotation.\nω0=γKmca\n1B1B2\nαµ0Ms(B2\n1cot(2ϕε)+B2\n2tan(2ϕε))(22)\nεb=±Kmca\n1\nB1sec(2ϕε)/radicalbig\n1+B2\n2/B2\n1tan2(2ϕε)(23)\ntanδd=B2\nB1tan(2ϕε) (24)\n8From equations 22 and 23, it should be noted that the strain am plitude and frequency required\nfor constant precession depend on both the material’s aniso tropy coefficients and orientation of\nthe the principal strain. The dynamic phase difference betw een the strain and magnetic moment\nin equation 24 is equal to the static rotational offset of the magnetoelastic anisotropy provided\nin equation 9 (i.e. tan δd=tanδs). While equations 22 - 24 provide the conditions for constan t\nangular velocity, applying a larger strain than εbwill produce faster rotation, albeit with non-\nconstant angular velocity. For isotropic magnetoelastic m aterials, these equations simplify to\nωiso\n0=γKmca\n1\n2αµ0Mssin(4ϕε) (25)\nεiso\nb=±Kmca\n1\nB1(26)\ntanδiso\nd=tan(2ϕε) (27)\nwhere the resulting frequency ωiso\n0increases with large crytalline anisotropies, and decreas es with\nlarger Gilbert damping and saturation magnetization. Addi tionally, the required strain amplitude\nεiso\nbdepends exclusively on the ratio of the crystalline and magn etoelastic anisotropy coefficients.\nIn addition to determining the constant precession conditi ons, the LLG equation was numeri-\ncally simulated using Matlab, with a macrospin model based o n the dynamic equations of motion.\nThis was done to verify the calculations performed in this se ction, and to study the behavior of the\nmotor at frequencies above and below ω0.\nC. Motor Analysis\nThe analysis outlined in the preceding sections controls th e rotational characteristics of the sta-\ntor’s magnetic moment. Assuming the uniformly magnetized r otor / stator can be treated as point\ndipoles, the stator’s dipole field (due to magnetic moment /vectorµ1) and force on the rotor (magnetic\nmoment/vectorµ2) due to the rotating moment of the stator is provided by\n/vectorB1(/vectorµ1,/vectorr)=µ0\n4πr3(3(/vectorµ1·ˆr)ˆr−/vectorµ1) (28)\n/vectorF(/vectorµ1,/vectorµ2,/vectorr)=∇/parenleftBig\n/vectorµ2·/vectorB1(/vectorµ1,/vectorr)/parenrightBig\n(29)\n∴/vectorF(/vectorµ1,/vectorµ2,/vectorr)=3µ0\n4πr5/bracketleftbigg\n(/vectorµ1·/vectorr)/vectorµ2+(/vectorµ2·/vectorr)/vectorµ1+...\n(/vectorµ1·/vectorµ2)/vectorr−5/vectorr\nr2(/vectorµ1·/vectorr)(/vectorµ2·/vectorr)/bracketrightbigg\n(30)\n9where/vectorµα=Vα/vectormαMsis the vector net magnetic moment of the rotor / stator, Vαis the volume,\nMsthe saturation magnetization, and /vectormαis the direction cosine unit vector. Figure 2c shows the\nresulting force vectors at several locations around the sta tor. For a bead with magnetic moment\nof amplitude |/vectorµ2|located a distance rfrom the stator with moment amplitude |/vectorµ1|, the maximum\nradial and tangential forces are given by Equations 31 and 32 . The radial force is maximized when\n/vectorr·/vectorµ1=0. The maximum tangential force occurs when the angle betwee n the radius and /vectorµ1is\nnπ±asin/parenleftBig√\n2/parenrightBig\n.\n/vextendsingle/vextendsingle/vextendsingle/vectorFr/vextendsingle/vextendsingle/vextendsingle=3µ0|/vectorµ1||/vectorµ2|\n2πr4(31)\n/vextendsingle/vextendsingle/vextendsingle/vectorFϕ/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vectorFr/vextendsingle/vextendsingle/vextendsingle/6 (32)\nFor both of these equations is has been assumed that /vectorµ2/bardbl/vectorB1.\nIII. RESULTS\nThis section presents the quasi-static energy equilibrium states, followed by results from the\nanalytic and numerical treatment of the dynamic LLG respons e. Lastly, the relation between the\ndynamic response and overall motor power density will be dis cussed.\nA. Quasi-Static Energy Analysis Results\nFigure 3 shows the energy landscape for two different princi pal strain orientations ( ϕε). The x-\naxis represents the orientation of the in-plane magnetic mo ment θ, the y-axis shows the anisotropy\nratio K2/K4. Using Equations 7 and 8 positive K2/K4values corresponds to tension and negative to\ncompression. Surface color represents the energy for each g iven state calculated from Equation 6.\nThe solid black lines trace the locations of energy minima, a nd dashed blue lines trace the locations\nof energy maxima. Both sets of lines correspond to the roots o f the characteristic polynomial\nprovided by Equation 14. Yellow stars indicate inflection po ints in the energy landscape which are\nlocations where the system irreversibly flips from one energ y minima to another.\nFor example, suppose the system in Figure 3a starts unstrain ed (K2/K4=0) at position 1, in\nthe energy well at θ=0 radians. If tensile strain is applied (moving up the y-axis ), then the\nsystem moves into a deeper energy well and doesn’t rotate. If compressive strain is applied, the\n10system moves down the black line until K2/K4=−4, at which point the magnetic moment non-\ndeterministically flips to energy wells at θ=±π/2 corresponding to positions 2a and 2b.\nFIG. 3. Energy landscapes for two different principal strai n orientations ( ϕε). Solid black lines track energy\nminima and dotted blue lines track energy maxima. Numbered b lack arrows correspond to loading pathways\ndescribed in the text. a) Non-deterministic switching occu rs when the ϕε=0. b) Deterministic switching\noccurs when ϕε=−π/8\nThe case of deterministic switching is shown in Figure 3b, wh ereϕε=−π/8=−22.5◦. In\nthis case starting unstrained in the energy well at position 1 (θ=0), compression again moves\nthe system down the y-axis, but also slightly rotates it in th e counter-clockwise direction until it\nreaches position 2. When K2/K4=−2 an inflection point is encountered and the system deter-\nministically flips to position 3, the energy well at θ≈π/2. Applying tension to the system until\nK2/K4=+2 causes counter-clockwise rotation to the inflection point at position 4, and the system\nthen jumps to position 5 ( θ≈π), and removing the strain moves it to position 6. This exampl e\ndemonstrates how a single cycle of biaxial compression / ten sion causes the magnetic moment to\ndeterministically rotate 180◦. Therefore, two compression / tension cycles are needed to r otate\na full 360◦, and the system rotates at half the driving frequency (i.e., ωε=2ω0). This indicates\nthat it is possible to deterministically rotate a magnetic m oment using a magnetoelastic anisotropy\nwith a fixed direction (i.e., allowing single electrode cont rol).\nFigure 4 shows the minimum anisotropy ratio required to rota te the magnetic moment. The\nupper solid blue line corresponds to the minimum ratio K2/K4required to rotate, while the surface\n11color and insets indicate the direction of magnetic rotatio n. When ϕε=±nπ/4 the magnetic\nmoment changes rotation directions. As ϕε=±nπ/4 corresponds the /angbracketleft100/angbracketrightand/angbracketleft110/angbracketrightfamily of\ndirections, it becomes clear that non-deterministic rotat ion results when strain is applied along a\ncrystallographic direction with large degree of symmetry. When ϕε=π/8±nπ/4, the system is\nrotated with the minimum possible anisotropy ratio ( |K2/K4|=2). For isotropic magnetoelasticity,\nthe minimum biaxial strain required to generate quasi-stat ic rotation is\nmin(/vextendsingle/vextendsingleεiso\nb/vextendsingle/vextendsingle)=\n\nKMCA\n1\n2B1forϕε=π\n8±nπ\n4\nKMCA\n1\nB1forϕε=±nπ\n4(33)\nFIG. 4. Minimum anisotropy ratio required for rotation to oc cur. Deterministic rotation occurs as long as\nϕε/negationslash=nπ/4. The minimum flipping strain occurs when ϕε=π/8±nπ/4\nApplying strain at the appropriate angle reduces the flippin g strain amplitude by a factor of 2,\nand therefore lowers the strain energy by a factor of 4 compar ed to the non-deterministic case.\nBased on Figure 4, a two input system generating principal st rain states oriented at ±π/8 will\ncontrol bipolar operation. Lastly, while the fourfold anis otropy has been assumed to be crystalline\nin nature, shape anisotropy can also produce cubic anisotro pies, as experimentally demonstrated\nby Lambson et al. [56].\nB. Dynamic Results\nRecall that the biaxial strain and frequency required to obt ain constant precession were previ-\nously presented in Equations 22-24 for a general material, a nd Equations 25-27 for an isotropic\nmaterial. For isotropic magnetoelasticity, the strain amp litude required for rotation at constant\nangular velocity matches the strain required for non-deter ministic flipping to occur in the quasi-\nstatic analysis ( εiso\nb=±K1\nB1). Furthermore, the rotational frequency depends on the ori entation of\n12the applied strain ( ωiso\n0∝sin(4ϕε)). When the strain is applied at an angle ϕε=±nπ/4, the an-\ngular velocity ω0=0. Therefore, when ϕε=±nπ/4, both the quasi-static and dynamic analyses\nconverge to predicting εiso\nb=±K1\nB1, with zero angular frequency (i.e., random non-determinis tic\nrotations). This demonstrates consistency between the sta tic and dynamic solutions. Consistency\nwas further checked by numerically simulating the motor sta tor element operating at several strain\namplitudes and frequencies.\nFigure 5 shows the frequency response of this motor stator fo r quasi static and dynamic op-\neration at 7 different frequencies for two full cycles of ten sion / compression. Parts a)-c) show\nthe impact of using three different biaxial strain amplitud es. At quasi-static rates (i.e., calculated\ntracking energy equilibrium orientations), the magnetic m oment exhibits ratchet-like behavior,\nwith abrupt switches every time the minimum switching strai n is reached ( εb=±KMCA\n1\n2B1). If a\nlarger strain is applied (shown in part b and c) the magnet und ergoes a small period of clockwise\nmotion (i.e., temporarily changes direction), as would be e xpected from Figure 3.\nFIG. 5. Magnetic revolutions vs time for a motor with isotrop ic magnetoelasticity and ϕε=−π/8 at several\ndriving frequencies. Results are shown temporally normali zed with respect to the period of the strain signal\n(T=2π/ωε). Simulations are shown for Gilbert damping α=0.01. Specific material properties and\nrelevant model terms are presented in Table I.\nFigure 5a indicates that the minimum predicted strain gener ates continuous rotation only for\nquasi-static behavior. Operating at ωε/2ω0≥0.01 when εb=±KMCA\n1\n2B1results in minor fluctuations\nabout a fixed orientation. While not depicted in Figure 5a, it was observed that increasing the\n13strain to εb=1.01·KMCA\n1\n2B1generated successful rotation for ωε/2ω0≤0.01, indicating the minimum\nflipping strain is valid at frequencies that are small compar ed to ω0. The under damped yellow line\nin Figure 5b indicates that if εb=±KMCA\n1\nB1, quasi-static performance is maintained up to ωε/2ω0≤\n0.1. Furthermore, the case where εb=±KMCA\n1\nB1andωε=2ω0resulted in constant angular velocity\nas analytically predicted by Equation 25 (shown with the str aight purple line). If the frequency ωε\nis further increased by 10%, the motor fails to deterministi cally rotate.\nFigure 5c shows that when larger strains are applied, larger operational frequencies can be\nattained. Doubling the strain from part b results in constan t precession with a 50% increase in\nrotational frequency. It should be noted that this increase is not directly proportional, as operating\nwith double the strain amplitude did not enable operation at twice the frequency. The data in\nFigure 5 was reproduced for a variety of Gilbert damping coef ficients from 0.01 to 0.1, with\nFigure 5 showing results for α=0.01. They were found to provide qualitatively similar result s,\nalbeit with the highest damping leading to smaller overshoo t and smoother overall behavior.\nTable I provides a list of magnetic materials along with thei r relevant properties, predicted min-\nimum strain for quasi-static operation, and operational fr equency for constant precession. This\ntable makes it abundantly clear that the ratio Kmca\n1/B1is the key metric to reduce the applied strain\namplitude. Moderately magnetoelastic materials like Ni an d Ni 55Fe45require the smallest strains\n(90≤εb≤360ppm) due to their low crystalline anisotropies. On the ot her hand, an epitaxially\ngrown thin film of Terfenol-D (Tb 0.3Dy0.7Fe2) has the largest magnetoelastic anisotropy coeffi-\ncient B1, but due to a large crystalline anisotropy K1would require εb≈3,300 ppm to rotate,\nwhich would be challenging for most ferroelectric material s even at quasi-static frequencies.\nThe frequencies listed in Table I range from 1 GHz to 1 THz, and indicate the magnetic dy-\nnamics of these motors will have minimal effect once mechani cal losses are accounted for. Also\nrecall that quasi-static behavior was seen to persist up to 1 % of ω0in Figure 5b. That means these\nmotors are magnetically quasi-static at frequencies up to 1 0s of MHz (i.e., in energy equilibrium),\nproviding broadband operation. It should be noted that the G ilbert damping parameter was as-\nsumed to be α=0.01 for all the listed materials, while larger values may be en countered. At the\nfrequencies predicted in this paper, the coupled mechanica l behavior of the motor will be impor-\ntant, and needs to be incorporated for more thorough and accu rate predictions (i.e., using a fully\ncoupled modeling approach), however that is consider outsi de the scope of the present article.\n14TABLE I. Comparison of the minimum strain required for deter ministic control of a variety of magnetoe-\nlastic materials.\nµ0Ms K1 B1a bεmin\nb=K1\n2B1ω0/2π=γKmca\n1\n4παµ0Msc\nMaterial (T) (kJ /m3) (MJ /m3) (ppm) (GHz) Source\nNi-FCC 0.6 -4.5 6.2 363 13 [55]\nCo-FCC 1.8 -120 -16 3,750 117 [55]\nCo-HCP 1.8 350 6 29,167 342 [55]\nFe-BCC 2.1 48 -2.9 8,276 40 [55]\nFe81Ga19d1.4 17.5 22.4 391 22 [57]\nTb0.3Dy0.7Fe2e0.9 -525 -80 3,280 1,027 [58]\nNi55Fe45 1.6 1 5.5 90 1.1 [55]\naB1=−3\n2λ100(C11−C12)\nbFor isotropic materials (C11−C12)=E/(1+2ν)\ncAssumed α=0.01 for all materials\ndAssuming E=75 GPa, ν=0.3\neEpitaxial thin film\nC. Multiferroic Motor Power Density\nThe last focus of this paper is to determine approximate perf ormance bounds of a mechanically\nlossless bead-on-on-a-disk motor using the developed cont rol scheme. The material from Table I\nrequiring the smallest strain is Ni 55Fe45, which is the focus of this section. A motor with stator\nradius r1=100 nm, thickness t=10 nm and rotor (bead) radius of r2=10 nm was simulated, with\ndimensions chosen to keep the stator a single domain.\nIn the absence of mechanical losses, the upper bound for the r otary motor’s frequency is dic-\ntated by inertial forces. In other words, if the rotary motor is operated at too large a frequency, the\nbead will be flung away from the stator due to its own momentum. While this may be useful in\nsome circumstances, it does limit the maximum frequency to\n/vextendsingle/vextendsingle/vextendsingle/vectorFr/vextendsingle/vextendsingle/vextendsingle=mω2r (34)\n∴ fmax=1\n2π/radicalbigg\n3µ0|/vectorµ1||/vectorµ2|\n2πmr5(35)\nwhere Equation 31 has been used. For the specified motor, fmax≈9.7 MHz, which corresponds to\n15a linear velocity v≈6.1 ms−1. Based on the results from Table I, this is roughly 100 times s lower\nthan ωiso\n0/2π. However, this low frequency / velocity is a limit of this spe cific rotary motor design,\nand not an intrinsic limit for multiferroic motors. For inst ance, a linear motor can be made that\novercomes these inertial limitations.\nThe linear motor shown in Figure 6a overcomes the frequency l imitations of the rotary motor\nby not forcing the bead to change directions (i.e., it works w ith the bead’s inertia). This motor can\nbe fabricated by patterning rotary motors adjacent to each o ther, and optionally etching a trough,\nor depositing a wall / barrier on top of them to physically gui de the bead. As inertial effects are\nnow advantageous, and help pass the bead from one disc to the n ext, in the limit of zero fluidic\ndrag the bead can be propelled as fast as the stator’s magneti c moment can rotate. For a Ni 55Fe45\nmm3nm3\nMolecular\nMotorsMEMS\nMotorsμm3\nvmmmm\u0002 \u0000 \u0001 \u0003P=F v a)\nb)Linear Motor Array\nFIG. 6. Updated power density chart now showing approximate upper bounds for both rotary and linear\nmultiferroic bead-on-a-disk motors.[2–36]\nmotor with the dimensions listed above, the magnetic moment is able to precess at a frequency of\nf≈1.1 GHz, which corresponds to a linear velocity of v≈700 ms−1.\n16Based on the attainable speed, and dipole forces driving the motion, approximate bounds on the\nmaximum power density of the rotary and linear motors can be c alculated. Assuming the vibrating\npiezoelectric substrate is 100 µm thick, the rotary motor vo lume is on the order of 10−18m3. The\nmaximum rotary power density is predicted to be 4 .4 MW/m3, and the maximum linear power\ndensity is predicted to be over two orders of magnitude highe r at 470 MW /m3. The increase in\npower density is due entirely to the higher operational freq uency of the linear motor. These pre-\ndictions provide approximate upper bounds for multiferroi c bead-on-a-disc motors with negligible\nmechanical losses.\nFigure 6 shows the power density chart now updated with predi ctions for the rotary and linear\nmultiferroic motors. The multiferroic motor concept fills t he void left by previous technologies,\nproviding a power dense motor with µm3dimensions. The upper limit to this boundary corre-\nsponds to motors operating at GHz frequencies, with lower po wer densities achieved by decreas-\ning the operational frequency. Furthermore, the linear mot or is able to scale up in size while\nstill maintaining the same power density. This is accomplis hed in practice by creating particle\nconveyor belts with the linear motor that have relatively la rge lengths.\nTo close, we comment on a key simplifying assumption used in t his analysis, namely the ab-\nsence of mechanical losses including friction, stiction, a nd fluidic drag. Operation anywhere close\nto these large frequencies will involve mechanical losses, which need to be a key focus of future\nstudies. This analysis highlights the key fact that the intr insic magnetic dynamics are sufficiently\nfast that the stator element is likely to always be in a state o f energy equilibrium during actual\nuse. This suggests two very important motor characteristic s. One, since stator speed is not a fun-\ndamental limit of these devices, future researchers would b e prudent to focus on designs moving\nmagnetic structures like domain walls and onion states with large magnetic gradients that amplify\ndipolar forces. Such a structure has been analyzed by Sohn et al. [38], who dragged a 1 µm di-\nameter magnetic bead at speeds near 1 mms−1. That motion likely achieved a power density on\nthe order of 10 kW /m3, a point right in the middle of the power densities predicted in Figure 6.\nSecond, the expected quasi-static behavior of the stator al so implies it’s rotational behavior will be\nsimilar to a stepper motor, with abrupt transitions. This ch aracteristic would complicate the use of\na rotary motor design, with rapid changes shaking the beads l oose. However a linear motor should\nstill work even with abrupt transitions, as the bead’s own mo mentum becomes an aid to continued\nmotion.\n17IV . CONCLUSION\nThis work has demonstrated how controlling the relative ori entation of a two-fold anisotropy\n(magnetoelasticity) with a four-fold anisotropy (e.g., cu bic MCA) can achieve single electrode\ncontrol of deterministic 360 degree magnetic rotations. Bo unds were determined for the strain re-\nquired to generate continuous rotation for quasi-static an d dynamic operation, and the conditions\nfor achieving constant angular precession were determined . For currently available materials,\nstrains as low as 90 ppm are predicted to enable the operation of these motors, and large frequen-\ncies are attainable in the absence of fluidic damping. A novel linear motor was also proposed, that\navoids some limitations of the rotary bead-on-a-disk motor . The proposed motors are predicted to\nhave large power densities at a size scale where no power dens e alternative technologies currently\nexist, and offer a promising new technology for further expl oration.\nACKNOWLEDGMENTS\nThis research was supported by the NSF Nanosystems Engineer ing Research Center for\nTranslational Applications of Nanoscale Multiferroic Sys tems (TANMS) Cooperative Agreement\nAward (No. EEC-1160504).\n[1] J. P. 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Lynch,\n“Nanoscale magnetic ratchets based on shape anisotropy,” 28(), 10.1088/1361-6528/aa56d4.\n[54] Soshin Chikazumi, Physics of Ferromagnetism 2e , 94 (Oxford University Press, 2009).\n[55] Robert C OHandley, Modern magnetic materials: principles and applications , V ol. 1 (Wiley).\n22[56] Brian Lambson, Zheng Gu, Morgan Monroe, Scott Dhuey, An dreas Scholl, and Jeffrey Bokor, “Con-\ncave nanomagnets: Investigation of anisotropy properties and applications to nanomagnetic logic,”\nApplied Physics A: Materials Science and Processing 111, 413–421 (2013).\n[57] Jayasimha Atulasimha and Alison B Flatau, “A review of m agnetostrictive iron-gallium alloys,”\nSmart Materials and Structures 20, 043001 (2011).\n[58] C de la Fuente, J I Arnaudas, L Benito, M Ciria, A del Moral , C Dufour, and\nK Dumesnil, “Magnetocrystalline anisotropy in a (110) (tb 0 .27 dy 0 .73 )fe 2 thin-film,”\nJournal of Physics: Condensed Matter 16, 2959 (2004).\nV . SUPPLEMENTARY INFORMATION\nA. Proof Ellipsoidal Magnets Can Not Enable 360◦Deterministic Single Input Control\nSeveral previous modeling and experimental studies have fo cused on controlling magnetic\nstructures with elliptical shapes. However, the use of elli psoidal structures alone can not result\nin deterministic 360◦rotations if the principal strain axis is stationary (i.e., for single input con-\ntrol). While it should be clear from physical arguments that combining magnetoelasticity and\na uniaxial shape anisotropy (i.e., using an ellipse) is limi ted to≤90◦rotations, we provide the\nfollowing proof for completeness.\nAssuming without loss of generality that the ellipse has a lo ng axis in the y-direction, and that\nthe magnetization is constrained to lie in the xy plane, the d emagnetization energy is\nUdemag=Kucos(2θ) (36)\nwhere Ku=−(Ny−Nx)/2 is the uniaxial demagnetization energy coefficient which d epends on\nthe x and y values of the demagnetization tensor NxandNy. Combining this energy with the\narguments that lead to Equation 6, magnetoelastic rotation s in an elliptical shape are governed by\nthe total energy expression.\nUtot=Kucos(2θ)+1\n2εbB1cos(2ϕε)cos(2θ)+1\n2εbB2sin(2ϕε)sin(2θ) (37)\nwith ϕεstill the principal strain orientation.\nEquation 37 is simply the addition of sin and cos terms of the s ame frequency. Of course,\nAcos(nθ)+Bsin(nθ)=Ccos(nθ−δ), where C=sgn(A)√\nA2+B2, and δ=tan(B/A). Therefore\n23the energy can be written as a single uniaxial anisotropy\nUtot=Ktotcos(2θ−δ) (38)\nKtot=sgn/parenleftbigg1\n2εbB1cos(2ϕε)+Ku/parenrightbigg\n...\n/radicalBigg/parenleftbigg1\n2εbB1cos(2ϕε)+Ku/parenrightbigg2\n+/parenleftbigg1\n2εbB2sin(2ϕε)/parenrightbigg2\n(39)\ntan(δ)=εbB2sin(2ϕε)/2\nKu+εbB1cos(2ϕε)/2(40)\nThis energy has local extrema when θ=±nπ/4+δ/2, which is controlled by δ,, while the sign\nofKtotdetermines whether the locations is a maxima or minima. The e quation for tan (δ)can be\nsimplified to\ntan(δ)=B2sin(2ϕε)\n2Ku/εb+B1cos(2ϕε)(41)\nIn this equation, the only dynamic variable is the biaxial st rain εb. Examining tan (δ)asεbap-\nproaches several limiting values reveals\nlim\nεb→0(tan(δ))= 0 (42)\nlim\nεb→±∞(tan(δ))=B2\nB1tan(2ϕε) (43)\nThe changes in the total anisotropy orientation are constra ined to 0≤|tan(δ)|≤|B2/B1tan(2ϕε)|,\nwith the sign controlled by the material properties and orie ntation of the principal strains. While\nthe total anisotropy has the same orientation for large ±εb, the sign of Ktotwill change. In one case\nδspecifies an energy minimum, while in the other δspecifies an energy maximum, with minimum\nrotated 90◦. Therefore, a maximum repeatable rotation of 90◦can be achieved by combining\nan elliptical element with a magnetoelastic anisotropy whe re the principal strain orientation is\nconstant.\n24B. Static Energy Analysis: Complex Roots\nThe roots of equation 14 were determined using Mathematica, and are provided with the closed\nform expression:\nwi(s1,s2,s3)=−A\n4+s11\n2/radicalbigg\nA2\n4+d+...\ns21\n2\nA2\n2+/parenleftbigg2\n3c/parenrightbigg1/3\na−/parenleftBigc\n18/parenrightBig1/3\n+s3A3+8B\n4/radicalBig\nA2\n4+d\n (44)\nA=Rp\n2(45)\nB=−R¯p\n2(46)\na=4+AB (47)\nb=A2−B2(48)\nc=−9b+/radicalbig\n12a3+81b2 (49)\nd=/parenleftbigg1\n36c/parenrightbigg1/3\n(−231/3a+(2c2)1/3) (50)\nwhere the sirepresent sign changes from one root to the next. The roots ar e provided by\nw1=wi(−1,−1,+1) (51)\nw2=wi(−1,+1,+1) (52)\nw3=wi(+1,−1,−1) (53)\nw4=wi(+1,+1,−1) (54)\nzi=±√wi (55)\n25"    },    {        "title": "1803.00235v1.Calculating_the_Magnetic_Anisotropy_of_Rare_Earth_Transition_Metal_Ferrimagnets.pdf",        "content": "Calculating the Magnetic Anisotropy of\nRare-Earth|Transition-Metal Ferrimagnets\nChristopher E. Patrick,1,\u0003Santosh Kumar,1Geetha Balakrishnan,1Rachel\nS. Edwards,1Martin R. Lees,1Leon Petit,2and Julie B. Staunton1\n1Department of Physics, University of Warwick,\nCoventry CV4 7AL, United Kingdom\n2Daresbury Laboratory, Daresbury, Warrington WA4 4AD, United Kingdom\n(Dated: March 2, 2018)\nAbstract\nMagnetocrystalline anisotropy, the microscopic origin of permanent magnetism, is often ex-\nplained in terms of ferromagnets. However, the best performing permanent magnets based on rare\nearths and transition metals (RE-TM) are in fact ferrimagnets, consisting of a number of mag-\nnetic sublattices. Here we show how a naive calculation of the magnetocrystalline anisotropy of\nthe classic RE-TM ferrimagnet GdCo 5gives numbers which are too large at 0 K and exhibit the\nwrong temperature dependence. We solve this problem by introducing a \frst-principles approach\nto calculate temperature-dependent magnetization vs. \feld (FPMVB) curves, mirroring the exper-\niments actually used to determine the anisotropy. We pair our calculations with measurements on\na recently-grown single crystal of GdCo 5, and \fnd excellent agreement. The FPMVB approach\ndemonstrates a new level of sophistication in the use of \frst-principles calculations to understand\nRE-TM magnets.\n1arXiv:1803.00235v1  [cond-mat.mtrl-sci]  1 Mar 2018High-performance permanent magnets, as found in generators, sensors and actuators, are\ncharacterized by a large volume magnetization and a high coercivity [1]. The coercivity |\nwhich measures the resistance to demagnetization by external \felds | is upper-bounded\nby the material's magnetic anisotropy [2], which in qualitative terms describes a preference\nfor magnetization in particular directions. Magnetic anisotropy may be partitioned into\ntwo contributions: the shape anisotropy, determined by the macroscopic dimensions of the\nsample, and the magnetocrystalline anisotropy (MCA), which depends only on the mate-\nrial's crystal structure and chemical composition. Horseshoe magnets provide a practical\ndemonstration of shape anisotropy, but the MCA is less intuitive, arising from the relativistic\nquantum mechanical coupling of spin and orbital degrees of freedom [3].\nPermanent magnet technology was revolutionized with the discovery of the rare-earth/transition-\nmetal (RE-TM) magnet class, beginning with Sm-Co magnets in 1967 [4] (whose high-\ntemperature performance is still unmatched [5]), followed by the world-leading workhorse\nmagnets based on Nd-Fe-B [6, 7]. With the TM providing the large volume magnetization,\ncareful choice of RE yields MCA values which massively exceed the shape anisotropy con-\ntribution [8]. RE-TM magnets are now indispensable to everyday life, but their signi\fcant\neconomic and environmental cost has inspired a global research e\u000bort aimed at replacing\nthe critical materials required in their manufacture [9].\nIn order to perform a targeted search for new materials it is necessary to fully understand\nthe huge MCA of existing RE-TM magnets. An impressive body of theoretical work based\non crystal \feld theory has been built up over decades [10], where model parameters are\ndetermined from experiment (e.g. Ref. [11]) or electronic structure calculations [12{14]. An\nalternative and increasingly more common approach is to use these electronic structure\ncalculations, usually based on density-functional theory (DFT), to calculate the material's\nmagnetic properties directly without recourse to the crystal \feld picture [15{19].\nCalculating the MCA of RE-TM magnets presents a number of challenges to electronic\nstructure theory. The interaction of localized RE-4 felectrons with their itinerant TM\ncounterparts is poorly described within the most widely-used \frst-principles methodology,\nthe local spin-density approximation (LSDA) [12]. Indeed, the MCA is inextricably linked to\norbital magnetism whose contribution to the exchange-correlation energy is missing in spin-\nonly DFT [20, 21]. MCA energies are generally a few meV per formula unit, necessitating\na very high degree of numerical convergence [22]. Finally, the MCA depends strongly on\n2temperature, so a practical theory of RE-TM magnets must go beyond zero-temperature\nDFT and include thermal disorder [23].\nEven when these signi\fcant challenges have been overcome, there is a more fundamental\nproblem. Experiments access the MCA indirectly, measuring the change in magnetization of\na material when an external \feld is applied in di\u000berent directions. By contrast, calculations\nusually access the MCA directly by evaluating the change in energy when the material is\nmagnetized in di\u000berent directions, with no reference to an external \feld. These experimental\nand computational approaches arrive at the same MCA energy provided one is studying a\nferro magnet. However, the majority of RE-TM magnets (and many other technologically-\nimportant magnetic materials) are ferrimagnets, i.e. they are composed of sublattices with\nmagnetic moments of distinct magnitudes and orientations. Crucially the application of\nan external \feld may introduce canting between these sublattices, a\u000becting the measured\nmagnetization. Thus the standard theoretical approach of ignoring the external \feld is hard\nto reconcile with real experiments on ferrimagnets.\nIn this Letter, through a combination of calculations and experiments, we provide the\nhitherto missing link between electronic structure theory and practical measurements of the\nMCA. Speci\fcally, we show how to directly simulate experiments by calculating, from \frst\nprinciples (FP), how the measured magnetization ( M) varies as a function of \feld ( B) applied\nalong di\u000berent directions and at di\u000berent temperatures. We apply our \\FPMVB\" approach\nto the RE-TM ferro and ferrimagnets YCo 5and GdCo 5, which are isostructural to the\ntechnologically-important SmCo 5[24] and, in the case of GdCo 5, a source of controversy in\nthe literature [25{35]. Pairing FPMVB with new measurements of the MCA of GdCo 5allows\nus to resolve this controversy. More generally, FPMVB enables a new level of collaboration\nbetween theory and experiment in understanding the magnetic anisotropy of ferrimagnetic\nmaterials.\nThe electronic structure theory behind FPMVB treats magnetic disorder at a \fnite tem-\nperatureTwithin the disordered local moment (DLM) picture [36, 37]. The methodology\nallows the calculation of the magnetization of each sublattice i,M i(T) =Mi(T)^M i, and the\ntorque quantity @F(T)=@^M i, whereFis an approximation to the temperature dependent\nfree energy. @F(T)=@^M iaccounts for the anisotropy arising from the spin-orbit interaction,\nwhile the contribution from the classical magnetic dipole interaction is computed numer-\nically [38]. Many of the technical details of the DFT-DLM calculations [36, 39{43] were\n3FIG. 1. Data points and \fts of dF=d\u0012 calculated for GdCo 5(blue, empty symbols; Gd and Co\nmoments held antiparallel) and YCo 5(green, \flled symbols), at 0 and 300 K.\ndescribed in our recent study of the magnetization of the same compounds [44]; the exten-\nsions to calculate the torques are described in Ref. [37]. The Gd-4 felectrons are treated\nwith the local self-interaction correction [43], and we have also implemented the orbital\npolarization correction [20] following Refs. [45, 46] using reported Racah parameters [47].\nDetails are given as Supplemental Material (SM) [48].\nYCo 5and GdCo 5crystallize in the CaCu 5structure, consisting of alternating hexagonal\nRCo 2c/Co 3glayers [24]. Y is nonmagnetic, while in GdCo 5the large spin moment of Gd\n(originating mainly from its half-\flled 4 fshell) aligns antiferromagnetically with the Co\nmoments. We now consider a \\standard\" calculation of the MCA based on a rigid rotation\nof the magnetization. If the Gd and Co moments are held antiparallel, GdCo 5is e\u000bectively\na ferromagnet with reduced moment MCo\u0000MGd. Then, from the hexagonal symmetry we\nexpect the angular dependence of the free energy to follow \u00141sin2\u0012+\u00142sin4\u0012+O(sin6\u0012),\nwhere\u0012is the polar angle between the crystallographic caxis and the magnetization direc-\ntion. The constants \u00141;\u00142determine the change in free energy \u0001 F, calculated e.g. from the\nforce theorem [49] or the torque dF=d\u0012 [50].\nIn Fig. 1 we show dF=d\u0012 calculated for ferromagnetic YCo 5and GdCo 5at 0 and 300 K.\nFitting the data to the derivative of the textbook expression, sin 2 \u0012(\u00141+ 2\u00142sin2\u0012), \fnds\u00141\nand\u00142to be positive (easy caxis) with\u00141an order of magnitude larger than \u00142. Considering\nexperimentally measured anisotropy constants in the literature, for YCo 5our\u00141value of\n3.67 meV (all energies are per formula unit, f.u.) at 0 K compares favorably to the values of\n3.6 and 3.9 meV reported in Refs. [28] and [51]. At 300 K, our value of 2.19 meV exhibits a\nslightly faster decay with temperature compared to experiment (2.6 and 3.0 meV), which we\nattribute to our use of a classical spin hamiltonian in the DLM picture [36, 44]. However, for\nGdCo 5our calculated values of \u00141show very poor agreement with experiments [26, 29]. First,\n4at 0 K we \fnd \u00141to be larger than YCo 5(4.26 meV), while experimentally the anisotropy\nconstant is much smaller (1.5, 2.1 meV). Second, we \fnd \u00141decreases with temperature\n(2.39 meV at 300 K) while experimentally the anisotropy constant increases (2.7, 2.8 meV).\nTo understand these discrepancies we must ask how the anisotropy energies were actually\nmeasured. Torque magnetometry provides an accurate method of accessing the MCA [52],\nbut is technically challenging in RE-TM magnets, which require very high \felds to reach\nsaturation [53]. Singular point detection [54] and ferromagnetic resonance [55] has also\nbeen used to investigate the MCA of polycrystalline and thin-\flm samples. However, the\nmost commonly-used method for RE-TM magnets, employed in Refs. [26, 29], is based on\nthe seminal 1954 work by Sucksmith and Thompson [56] on the anisotropy of hexagonal\nferromagnets. This work provides a relation between the measured magnetization Mab\nand \feldBapplied in the hard plane in terms of \u00141,\u00142and the easy axis magnetization\nM0[48, 56]:\n(BM 0=2)=(Mab=M0)\u0011\u0011=\u00141+ 2\u00142(Mab=M0)2: (1)\nFurther introducing m= (Mab=M0), equation 1 shows that a plot of \u0011againstm2should\nyield a straight line with \u00141as the intercept. Even though this \\Sucksmith-Thompson\nmethod\" was derived for ferromagnets, the technical procedure of plotting \u0011againstm2can\nbe performed also for ferrimagnets like GdCo 5[26, 29]. In this case, the quantity extracted\nfrom the intercept is an e\u000bective anisotropy constant Ke\u000bso, unlike YCo 5, the anisotropy\nconstants reported in Refs. [26, 29] are distinct from the \u00141values extracted from Fig. 1. As\nrecognized at the time of the original experiments [27{30], the reduced value of Ke\u000bwith\nrespect to\u00141of YCo 5is a \fngerprint of canting between the Gd and Co sublattices.\nMaking contact with previous experiments thus requires we obtain Ke\u000b. To this end\nwe have developed a scheme of calculating \frst-principles hard-plane magnetization vs. \feld\n(FPMVB) curves, on which we perform the Sucksmith-Thompson analysis to directly mirror\nthe experiments. The central concept of FPMVB is that at equilibrium, the torques from\nthe exchange, spin-orbit and dipole interactions must balance those arising from the external\n\feld. Then,\nB=@F(T)\n@\u0012i1\nMicos\u0012i+P\njsin\u0012j@M j\n@\u0012i: (2)\nThe magnetization at a given B;T is determined by the angle set f\u0012Gd;\u0012Co1;\u0012Co2;:::gwhich\nsatis\fes equation 2 for every magnetic sublattice. The spin-orbit interaction breaks the\n5FIG. 2. Magnetization of GdCo 5vs. applied magnetic \feld shown on a standard plot (left panel)\nor after the Sucksmith-Thompson analysis (eq. 1, right panel). Crosses/circles are calculated with\nmethods (i)/(ii) discussed in the text, and the area between them shaded as a guide to the eye.\nNote the two methods are e\u000bectively indistinguishable in the left panel. The dashed/solid lines are\ncalculated from the model free energies F1andF2. The right panel also shows the geometry of the\nmagnetization and \feld with respect to the crystallographic c-axis (thick gray arrow).\nsymmetry of the Co 3gatoms such that altogether there are four independent angles to vary\nfor GdCo 5. The second term in the denominator of equation 2 re\rects that the magnetic\nmoments themselves might depend on \u0012i(magnetization anisotropy). We have tested (i)\nneglecting this contribution and (ii) modeling the dependence as Mi(\u0012i) =M0i(1\u0000pisin2\u0012i),\nwhereM0iandpiare parameterized from our calculations.\nFigure 2 shows FPMVB curves of GdCo 5calculated using equation 2 with methods (i)\nand (ii), (crosses and circles) which yield virtually identical values of Ke\u000b. TheMvs.B\ncurves in the left panel resemble those of a ferromagnet where, as the temperature increases,\nit becomes easier to rotate the moments away from the easy axis so that a given B\feld\ninduces a larger magnetization. However, plotting \u0011againstm2in the right panel tells a\nmore interesting story. The e\u000bective anisotropy constant Ke\u000b(y-axis intercept) at 0 K is\n1.53 meV, much smaller than \u00141of YCo 5. Furthermore Ke\u000bincreases with temperature,\nto 1.74 meV at 300 K. Therefore, in contrast to the standard calculations of Fig. 1, the\nFPMVB approach reproduces the experimental behavior of Refs. [26, 29].\n6Our FPMVB calculations provide a microscopic insight into the magnetization process.\nFor instance at 0 K and 9 T, we calculate that the cobalt moments rotate away from the\neasy axis by 6.1\u000e. By contrast the Gd moments have rotated by only 3.9\u000e, i.e. the ideal\n180\u000eGd-Co alignment has reduced by 2.2\u000e(the geometry is shown in Fig. 2). We also \fnd\ncanting between the di\u000berent Co sublattices, but not by more than 0.1\u000eat both 0 and 300 K\n(the calculated angles as a function of \feld are shown in the SM [48]). This Co-Co canting\nis small thanks to the Co-Co ferromagnetic exchange interaction, which remains strong over\na wide temperature range [44]. The temperature dependence of Ke\u000bcan be traced to the\nfact that the easy axis magnetization M0of GdCo 5initially increases with temperature [44].\nEven ifMabincreases with temperature at a given \feld, a faster increase in M0can lead to\nan overall hardening in Ke\u000b(equation 1).\nWe assign the canting in GdCo 5to a delicate competition between the exchange interac-\ntion favoring antiparallel Co/Gd moments, uniaxial anisotropy favoring c-axis (anti)alignment,\nand the external \feld trying to rotate all moments into the hard plane. We can quantify\nthese interactions by looking for a model parameterization of the free energy F. Crucially\nwe can train the model with an arbitrarily large set of \frst-principles calculations exploring\nsublattice orientations not accessible experimentally, and test its performance against the\ntorque calculations of equation 2. Neglecting the 0.1\u000ecanting within the cobalt sublattices\ngives two free angles, \u0012Gdand\u0012Co. Including Gd-Co exchange A, uniaxial Co anisotropy\nK1;Coand a dipolar contribution S(\u0012Gd;\u0012Co) [31, 48] leads naturally to a two-sublattice\nmodel [30],\nF1(\u0012Gd;\u0012Co) =\u0000Acos(\u0012Gd\u0000\u0012Co) +K1;Cosin2\u0012Co\n+S(\u0012Gd;\u0012Co): (3)\nThe training calculations showed additional angular dependences not captured by F1, so we\nalso investigated:\nF2(\u0012Gd;\u0012Co) =F1(\u0012Gd;\u0012Co) +K2;Cosin4\u0012Co\n+K1;Gdsin2\u0012Gd: (4)\nAs discussed below the training calculations showed no strong evidence of Gd-Co exchange\nanisotropy [31{34].\nThe dashed (solid) lines in Fig. 2 are the calculated Mvs.Bcurves obtained by mini-\nmizingF1(2)\u0000P\niM i\u0001B. The second term includes magnetization anisotropy on the cobalt\n7FIG. 3. Anisotropy constants Ke\u000bvs. temperature of YCo 5(green) and GdCo 5(blue). The left\npanel shows calculations using equation 2 at 0 and 300 K (stars), or using parameterized model\nexpressions F1(diamonds) and F2(circles), and from Ref. [57] (YCo 5, squares). For GdCo 5we\nalso show in red \u00141extracted from \\standard\" calculations where the Gd and Co moments were\nheld rigidly antiparallel (cf. Fig. 1). The experimental data in the right panel was measured by\nus for GdCo 5(crosses, with shaded background) or taken from Refs. [26], [29] and [58] (squares,\ndashed lines, circles) and Refs [28] and [51] (green diamonds and dashed lines, YCo 5).\nmoments [48, 57]. On the scale of the left panel both F1andF2give excellent \fts to the\ntorque calculations, especially up to moderate \felds. The plot of \u0011againstm2reveals some\ndi\u000berences with F2giving a marginally improved description of the data, but F1already\ncaptures the most important physics.\nWe also applied the FPMVB approach to YCo 5, using equation 2 and the model for F\nintroduced in Ref. [57]. Then, parameterizing the models [48] over the temperature range\n0{400 K, calculating Mvs.Bcurves and extracting Ke\u000busing the Sucksmith-Thompson\nplots gives the results shown in the left panel of Fig. 3. We also show \u00141of GdCo 5to\nemphasize the di\u000berence between FPMVB calculations and the \\standard\" ones of Fig. 1.\nComparing Ke\u000bto previously-published experimental measurements on GdCo 5raises\nsome issues. First, the three studies in the literature report anisotropy constants which di\u000ber\nby as much as 1 meV [26, 29, 58]. Indeed there was controversy over whether the observed\nresults were evidence of an anisotropic exchange interaction between Gd and Co [31, 32] or\n8an artefact of poor sample stoichiometry [33, 34]. Furthermore the only study performed\nabove room temperature [26] reports without comment some peculiar behavior where Ke\u000b\nof GdCo 5exceeds that of YCo 5at high temperature [28], despite conventional wisdom that\nthe half-\flled 4 fshell of Gd does not contribute to the anisotropy.\nOur calculations do in fact show an excess in the rigid-moment anisotropy of GdCo 5of\n16% at 0 K (Fig. 1) compared to YCo 5. The authors of Refs. [29, 31] \ftted their experimental\ndata with a much larger excess of 50%, while the high-\feld study of Ref. [33] found (11\n\u000615)%, with the authors of that work attributing the di\u000berence to an improved sample\nstoichiometry [34]. Our calculated excess at 0 K is formed from two major contributions:\nthe dipole interaction energy, which accounts for 0.31 meV/f.u., and K1;Gd(equation 4) which\nwe found to be 24% the size of K1;Co. The nonzero value of K1;Gdis due to the 5 delectrons,\nwhose presence is evident from the Gd magnetization (7.47 \u0016Bat 0 K). We did not \fnd a\nsigni\fcant contribution from anisotropic exchange, which we tested in two ways: \frst by\nattempting to \ft a term A(1\u0000p0sin2\u0012Co) cos(\u0012Gd\u0000\u0012Co) to our training set of calculations,\nand also by computing Curie temperatures with the (rigidly antiparallel) magnetization\ndirected either along the coraaxes. We found the magnitude of the anisotropy ( p0) to be\nsmaller than 0.5% and negative at 0 K, and to decrease in magnitude as the temperature\nis raised. Consistently the Curie temperature was found to be only 1 K higher for aaxis\nalignment, which we do not consider signi\fcant.\nHowever, our calculations do not predict the Ke\u000bvalue of GdCo 5to exceed YCo 5. Indeed,\nin Fig. 3\u00141of GdCo 5approaches that of YCo 5at high temperatures, which is signi\fcant\nbecause\u00141provides an upper bound for Ke\u000b[32]. To resolve this \fnal puzzle we performed\nour own measurements of Ke\u000bon the single crystal whose growth we reported recently [44].\nHard and easy axis magnetization curves up to 7 T were measured in a Quantum Design\nsuperconducting quantum interference device (SQUID) magnetometer, and the anisotropy\nconstants extracted from Sucksmith-Thompson plots [48]. The right panel of Fig. 3 shows\nour newly measured data as crosses. Previously reported measurements are shown in faint\nblue/green for GdCo 5[26, 29, 58]/YCo 5[28, 51].\nUp to 200 K, there is close agreement between the experiments of Ref. [26], our own ex-\nperiments, and the FPMVB calculations. Above this temperature our new experiments show\nthe expected drop in Ke\u000b, while the previously reported data show a continued rise [26]. We\nrepeated our measurements using di\u000berent protocols and found a reasonably large variation\n9in the extracted Ke\u000b[48]. Even taking this variation into account as the shaded area in\nFig. 3, the drop is still observed.\nWe therefore do not believe the high temperature behavior reported in Ref. [26] has\nan intrinsic origin. Possible extrinsic factors include the method of sample preparation,\ndegradation of the RCo 5phase at elevated temperatures [59], and potential systematic error\nwhen extracting Ke\u000b. We note that even the idealized theoretical curves in Fig. 2 show\ncurvature at higher temperature, making it more di\u000ecult to \fnd the intercept.\nIn conclusion, we have introduced the FPMVB approach to interpret experiments mea-\nsuring anisotropy of ferrimagnets, particularly RE-TM permanent magnets. We presented\nthe method in the context of our DLM formalism, but any electronic structure theory ca-\npable of calculating magnetic couplings relativistically [60{64] should be able to produce\nFPMVB curves, at least at zero temperature. However standard calculations which neglect\nthe external \feld should be used with care when comparing to experiments on ferrimagnets.\nSimilarly, the prototype GdCo 5serves as a reminder that a simple view of the anisotropy\nenergy does not fully describe the magnetization processes in ferrimagnets, which might have\nimplications in understanding e.g. magnetization reversal in nano-magnetic assemblies [65].\nOverall our work demonstrates the bene\ft of interconnected computational and experimen-\ntal research in this key area.\nThe present work forms part of the PRETAMAG project, funded by the UK Engineer-\ning and Physical Sciences Research Council (EPSRC), Grant no. EP/M028941/1. 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Lett. 81, 2029\n(2002).\n13"    },    {        "title": "1803.02038v1.Gate_tunable_Room_temperature_Ferromagnetism_in_Two_dimensional_Fe__3_GeTe__2_.pdf",        "content": "Page 1 of 17 \n Gate -tunable Room -temperature  Ferromagnetism in Two -\ndimensional Fe 3GeTe 2  \n \nYujun Deng1,2,3†, Yijun Yu1,2,3†, Yichen Song1,2,3, Jingzhao Zhang4, Nai Zhou Wang2,5,6, \nYi Zheng Wu1,2, Junyi  Zhu4, Jing Wang1,2, Xian Hui Chen2,5,6 and Yuanbo Zhang1,2,3* \n \n1State Key Laboratory of Surface Physics and Department of Physics, Fudan \nUniversity, Shanghai 200438, China  \n2Collaborative Innovation Center of Advanced Microstructures, Nanjing University, \nNanjing, 210093, China  \n3Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, \nShanghai 200433, China  \n4Department of Physics, the Chinese University of Hong Kong, Shatin, New \nTerritories, Hong Kong, China  \n5Hefei National Laboratory f or Physical Science at Microscale and Department of \nPhysics, University of Science and Technology of China, Hefei, Anhui 230026, China  \n6Key Laboratory of Strongly Coupled Quantum Matter Physics, University of Science \nand Technology of China, Hefei, Anhui 230026, China  \n \n \n† These authors contributed equally to this work.  \n \n* Email: zhyb@fudan.edu.cn  \n \n  Page 2 of 17 \n Material research has been a major driving force in the development of \nmodern nano -electronic devices. In particular, research in magnetic thin films has \nrevolutionized the development of spintronic devices1–8; identifying new magnetic \nmaterials is key to better device performance and new device paradigm. The \nadvent of two-dimensional  van der Waals  crystals  creates  new possibilities. This \nfamily of  materials retain their chemical stability and structural integr ity down to \nmonolayers and, being atomically thin, are readily tuned by various kinds of gate \nmodulation9–11. Recent experiments have demonstrated that it is possible to obtain  \ntwo-dimensional ferromagnetic order in insulating  Cr2Ge2Te6 (ref. 12) and CrI 3 \n(ref. 13) at low tempe ratures . Here, we developed a  new device fabrication \ntechnique , and successfully isolated mon olayers from  layered metallic magnet \nFe3GeTe 2 for magneto transport study . We found that the itinerant \nferromagnetism persists in Fe3GeTe 2 down to monolayer  with an out -of-plane \nmagnetocrystal line anisotropy . The ferromagnetic transition  temperature , 𝑻𝒄, is \nsuppressed  in pristine  Fe3GeTe 2 thin flakes. An ionic gate , however, dramatically \nraises  the 𝑻𝒄 up to room temperature, significantly higher than the bulk  𝑻𝒄 of \n205 Kelvin. The gate -tunable room -temperature ferromagnetism in two -\ndimensional Fe3GeTe 2 opens up opportunities for  potential  voltage -controlled \nmagnetoelectronics14,15 based on atomically thin van der Waals crystals . \n \nAtomically thin , layered  van der Waals (vdW) crystals  represent ideal two -\ndimensional (2D) material systems with exceptional electronic structures. Since the \ndiscovery of graphene16, such c rystals have been at the frontier of material research , \nwhere v ast opportunities arise from i) emerging physical properties as a result of \nreduced dimensionality , and ii) new gating capabilities to modulate the  material \nproperties now that the entire crystal is a surface. The former underpins  the rapid \npermeation of 2D materials in areas ranging from semiconductor s17–19 to highly \ncorrela ted material11,20,21 and superconductor22–24. The trend continues with recent \naddition of magnetic crystals in the 2D material family12,13. In particular, intrinsic \nferromagnetic order  has been obse rved in monolayer CrI 3 in the form of a 2D Ising \nferromagnet13.  \nHere we rep ort the discovery of 2D itinerant ferromagnetism in atomically thin  \nFe3GeTe 2 (FGT)  crystal s. As in the case of CrI 3, intrinsic magnetocrystalline  anisotropy \nin FGT monolayers counteracts thermal fluctuations and preserves the 2D long -rang Page 3 of 17 \n ferromagnetic order, which is otherwise precluded in an  isotropic magnetic  system \naccording to the Mermin -Wagner theorem25. FGT, however, offers a critical advantage: \nits metallic  nature enables the interplay  of both spin and char ge degree s of freedom that \nlies at  the heart of various spintronic architectures26. In this study  we focus on the  large  \nanomalous Hall effect (AHE)  resulting from  such interplay . The AHE  enable s us to \nextract the ferromagnetic transition  temperature 𝑇𝑐, and elucidate the evolution of the \nmagnetic order from bulk down to monolayer  FGT. Taking advantage of the tunability \nof atomically thin crystals, w e further show that extreme doping induced by an ionic \ngate dramatically elevates the 𝑇𝑐  up to room temperature, accompanied by large \nmodulation s in the  coercivity . These results establish  FGT as a new itinerant \nferromagnetic 2D material  potentially suitable for electrically controlled \nmagnetoelectronic  devices operating at room temperature.  \n A variety of metallic layered  compo unds exhibit magnetism27. Among them FGT  \nstands out with a  relatively high 𝑇𝑐  (ref. 28, 29, ranging from 150 K to 220 K  \ndepending on Fe occupancy30,31). The a tomic structure of a monolayer FGT  is shown \nin Fig. 1b . In each of FGT monolayer, covalently -bonded Fe3Ge heterometallic slab is \nsandwiched between  two Tellurium (Te)  layers . The structure and valence states of the \ncompound can be written as (Te2-)(FeI3+)[(FeII2+)(Ge4-)](FeI3+)(Te2-) per formula with \ntwo inequivalent Fe sites , FeI3+ and Fe II2+, within the Fe 3Ge slab29. Partially filled Fe d \norbitals dominate the band structure around the Fermi level, and give rise to itinerant  \nferromagnetism in bulk FGT32. Adjacent monolayers  are separated by a 2.95 Å van der \nWaals gap in the bulk crystal.  As a result of the reduced crystal symmetry of the layered \nstructure, bulk FGT exhibit s a strong magneto crystalline  anisotropy32,33. Such  \nanisotropy is expected to lift the restriction imposed by Mermin -Wagner theorem  and \nstabilize the long -range  ferromagnetic order in FGT monolayers . \n Experimentally isolating atomically thin FGT crystals from the bulk, however, \nposes a  challenge . Although bulk FGT cleaves along the v dW gap, the intra -layer \nbonding is not strong enough for thin flakes with reasonable size (e.g. 5 m) to survive \nconventional  mechanical exfoliation process . Similar issues plague the exfoliation of \nmany other layered materials (only a small fraction of known layered crystal s is \ncleavable  down to monolayer s), and have largely impeded the research on 2D materials \nin general.  \n To this end, we  developed  an Al2O3-assisted exfoliation method that enable s Page 4 of 17 \n isolation of monolayer s from bulk layered  crystal s that are otherwise difficult to \nexfoliate with conventional method. This method was  inspired by the gold -mediated \nexfoliation of layered transition -metal  chalcogenide compounds34,35. The fabrication \nprocess is illustrated step -by-step in Fig. 1a. We start ed by covering the freshly  cleaved \nsurface of the  bulk crystal with  Al2O3 thin film with thickness ranging from 50 nm to \n200 nm . The film was deposited by thermally evaporating Al under an oxygen partial \npressure of 10−4 mBar. We then  used a thermal release tape  to pick up  the Al2O3 film, \nalong with pieces of  FGT micro -crystals separated from the bulk . The Al2O3/FGT stack \nwas subsequently transferred onto a  piece of polydimethyl siloxane  (PDMS ), with the \nFGT side  in contact with PDMS  surface . Next , we quickly peel ed away  PDMS , leaving \nAl2O3 film covered with freshly cleaved FGT flakes on a substrate.  Fig. 1 c displays an \noptical image of atomically thin FGT flakes on Al2O3 film supported on a 285-nm \nSiO 2/Si substrate . Mono - to tri -layer  regions of FGT are clearly resolved  from  the \noptical contrast.  Such layer  identification was corroborated by  direct topography \nmeasurement with an atomic force microscope  (AFM ); the 0.8 nm steps in the height \nprofile exactly matc h FGT monolayer thickness  (Fig. 1 d and e). We then fabricated \nelectrodes on FGT thin flakes , either with direct metal deposition through stencil masks \nor with indium  cold welding , for subsequent transport measurements.  The entire  device \nfabrication  process was  performed in an argon atmosphere with O 2 and H 2O co ntent  \nkept below 0.5 ppm to avoid sample degradation . Details of the sample exfoliation and \nelectrode fabrication are presented in Supplementary Section II. Finally, we point out  \nthat even though we only focus on FGT  in this study , the method should be  applicable \nto a wide variety of vdW crystals . We attribute the drastically enhanced exfoliation \ncapability to higher  affinity , as well as increased  contact  area, between evaporated  \nAl2O3 film and freshly -cleaved FGT  surface . \n We study  the magnetism  in FGT by probing the  Hall resistance, 𝑅𝑥𝑦, under an \nexternal magnetic field , 𝜇0𝐻, applied perpendicular to the v dW plane . In our samples \nthat typically has a  van der Pauw configuration, the fact that 𝑅𝑥𝑦 is antisymmetric with \nrespect to the polarity of 𝜇0𝐻, whereas  magne toresistance  is symm etric, enables  us to  \nseparate  the contribution of  magnetoresistance from t he raw Hall voltage data . For a \nmagnetic material, 𝑅𝑥𝑦 can be decomposed into two parts36: \n 𝑅𝑥𝑦=𝑅𝑁𝐻+𝑅𝐴𝐻 (1) \nwhere 𝑅𝑁𝐻=𝑅0𝜇0𝐻  is the  normal  Hall resistance , and 𝑅𝐴𝐻=𝑅𝑆𝑀  is the Page 5 of 17 \n anomalous Hall resistance ; 𝑅0 and 𝑅𝑆 are constant coefficients  characterizing the \nstrength of 𝑅𝑁𝐻  and 𝑅𝐴𝐻 , respectively.  Crucial information on the long -range \nmagnetic order, which is contain ed in 𝑀 , can therefore be extracted from the \nmeasurement of the anomalous Hall resistance.  \n Ferromagnetism persists in atomically thin FGT flakes  down to monolayer.  \nUnambiguous evidence comes from the clear hysteresis in 𝑅𝑥𝑦 for all FGT  thin flakes \n(with thickness ranging from monolayer to 50 layers in the bulk limit; Fig. 2a) under \ninvestigation at low temperature s. Such hysteresis reflects the hysteresis  in 𝑀 ; the \nremnant 𝑀 at 𝜇0𝐻=0 signifies  spontaneous magnetization , and thus the long -range \nferromagnetic order , in two -dimensional FGT.  Raising the temperature introduce s \nthermal fluctuations to the ferromagnetic order , and ferromagnetism eventually \ndisappear s above 𝑇𝑐. In Fig. 2b, m easurement s of 𝑅𝑥𝑦 show that ferromagnetism in \nsamples with various layer numbers respond  differently to thermal fluctuations : at an \nelevated temperature of 𝑇=100  K hysteresis vanishes in mono - and bilayer samples, \nbut survives  in thicker crystals. The observation clearly indicates a strong \ndimensionality effect on  ferromagnetism  in atomically thin FGT . \n To fully elucidate  the effect of reduced dimensionality on ferromagnetism in 2D \nFGT , we precisely determine 𝑇𝑐  as a function of the number of layers. For each \nsample thickness, we examine t he remnant Hall resistance  at zero external magnetic \nfield , 𝑅𝑥𝑦𝑟≡𝑅𝑥𝑦|𝜇0𝐻=0 , as a function of  temperature. According to equation (1), \n𝑅𝑥𝑦𝑟 is directly proportional to the zero-field spontaneous magnetization 𝑀|𝜇0𝐻=0. \nSo the onset of nonzero 𝑅𝑥𝑦𝑟 indicates  the emergence of spontaneous magnetization . \nThe temperature at the onset  therefore marks  the 𝑇𝑐, where long -range ferromagnetic \nordering start to set in  (Fig. 2c ; ref. 37). Analysis for various sample thicknesses reveals  \nthat 𝑇𝑐 decreases monotonically as the samples are thinned down, from ~ 200 K in the \nbulk limit to 20 K in a monolayer . The one -order -of-magnitude decrease in 𝑇𝑐 implies \na dramatic reduction in the energy scale of magnetic ordering in 2D FGT , which we \nshall discuss next.  \nThe strong dimensionality effect on the ferromagnetism in atomically thin FGT \nstems from the fundamental role  of thermal fluctuation in 2D. The itinerant \nferromagnetism in FGT is, in principle, described by Stoner model  for metallic \nmagnetic systems29,32,33. It has, however, been established that such itinerant \nferromagnetism  can be mapped on to a classical Heisenberg mode l with Ruderman -Page 6 of 17 \n Kittel -Kasuya -Yosida (RKKY)  exchange38. Armed with this insight , we adopt the \nfollowing anisotropic Heisenberg Hamiltonian  in the absence of external magnetic \nfield :  \n 𝐻=∑𝐽𝑖𝑗𝑺𝑖\n𝑖,𝑗⋅𝑺𝑗+∑𝐴(𝑆𝑖𝑧)2\n𝑖 (2) \nwhere 𝑺𝑖 is the spin operator on site  i, and 𝐽𝑖𝑗 the exchange coupling between  spins \non site i and j; 𝐴  is the single -ion perpendicular magnetocrystalline  anisotropy  \narising from spin -orbital coupling . In the three -dimensional (3D) bulk limit, the \ndensity of states per spin for the magnon modes  is strongly suppressed, so thermal \nfluctuations destroy  the long -range magnetic order only at a finite 𝑇𝑐 determined \nprimarily by exchange interactions  𝐽𝑖𝑗 (ref. 12). The bulk 𝑇𝑐 of ~ 200 K implies that \nthe ene rgy scale of 𝐽𝑖𝑗 (summed over nearest neighbours) in FGT is on the order of ~ \n10 meV . In monolayer FGT, however, the largely isotropic exchange couplings 𝐽𝑖𝑗 (see \nSupplementary Section IV for detailed ab initio  calculation s) alone  will not be able \nto sustain  magnetic order at finite temperatures because of thermal fluctuations of \nthe long -wavelength gapless magnon modes in 2D.  In this case, t he \nmagneto crystalline anisotropy 𝐴  gives rise to  an energy  gap in the magnon \ndispersion . The gap suppresses low-frequency , long -wavelength  magnon excitations, \nand protects the magnetic order below a finite 𝑇𝑐 determined primarily by 𝐴. The \nmonolayer 𝑇𝑐 of 20 K implies an  𝐴 on the order of 0.6 meV. The estimations of both  \nexchange interactions and  magneto crystalline anisotropy are in reasonable  agreement \nwith results from ab initio  density functional  calculations ( ref. 32, 39; see also \nSuppl ementary Section IV). The magneto crystalline anisotropy  corresponds to an  \nenergy density 𝐾𝑢  on the order of 106 Jm−3 , which is one ord er of  magnitude \nlarger than interfacial  magnetic anisotropy  in a 1.3-nm-thick  CoFeB film40. Such a \nlarge intrinsic perpendicular anisotropy is important for potential stable, compact \nspintronic devices based on atomically thin FGT.  \n The evolution from 3D to 2D magnetism in FGT is linked with the critical \nbehaviour at the paramagnet (PM) to ferromagnet (FM) phase transition. Because 𝑇𝑐 \nis the critical temperature where spin-spin correlation  length  diverges, a finite sample \nthickness limits the divergence of the correlation length , and therefore depresses 𝑇𝑐. \nAnalyses of the critical behavio ur41–43 show that layer -number -dependent 𝑇𝑐(𝑁) Page 7 of 17 \n follows  a universal scaling law as the sample thickness approaches the 2D limit: \n[𝑇𝑐(∞)−𝑇𝑐(𝑁)]/𝑇𝑐(∞)=((𝑁0+1)/2𝑁)𝜆 , where  𝑇𝑐(∞)  denotes 𝑇𝑐  of the bulk \ncrystal. The critical exponent 𝜆=1/𝜈 reflects the universality class  of the transition ; \n𝑁0  is the critical layer number defined by the mean spin -spin interaction range . \nAnalyse s of thin ferromagnetic metal  films43–46 also reveals  that the power  law is \nreplaced by a linear relation when the film is thinner than the critical value, i.e. 𝑁≤\n𝑁0. 𝑁0, therefore , marks  the boundary that separates 2D magnetism and  3D magnetism \nin ultra-thin ferromagnet s. The scaling law fits well to the 𝑇𝑐(𝑁) measured in FGT as \nshown in Fig. 2d  (here we fitted the highest 𝑇𝑐 achieved for each sample thickness) . \nThe fit yields  a critical layer number of 𝑁0=3.2±0.6 and a critical exponent  of 𝜆=\n1.7±0.6 . We also confirmed that the linear relation well describes our data in the \nthinnest samples ( 𝑁≤3; see Fig. 2d ). We note that 𝑁0 obtained in  FGT is comparable \nto that in the magnetic thin films43,44,47. Meanwhile,  𝜆 lies between the values expected \nfrom  mean field model ( 𝜆=2) and  3D Ising model ( 𝜆=1.587), although it ’s also \nconsistent  with 3D Heisenberg model ( 𝜆=1.414); more accurate determination of 𝑇𝑐 \nis required to resolve the universality class of bulk FGT.  \nEven though 𝑇𝑐 is suppressed in atomically thin FGT , we now demonstrate that \nan ionic gate could drastically modulate  the ferromagnetism in FGT thin flakes, and \nboost the 𝑇𝑐 up to room temperature. We employ an ionic field -effect transistor setup, \nin which solid electrolyte  (LiClO 4 dissolved in polyethylene oxide (PEO) matrix ) \ncovers both the FGT thin flake  and a side gate  (ref 11; Fig. 3a inset) . A positive gate \nvoltage, 𝑉𝑔, intercalates lithium ions  into the FGT thin flake , much like the charging \nprocess in a lithium -ion battery. The charge transfer betwe en lithium ion s and the host \ncrystal induces electron  doping up to  the order of  1014cm−2  per layer11. The \nunprecedented  doping level effects profound change in the ferromagnetism in FGT thin \nflake. Fig. 3b and 3c display the 𝑅𝑥𝑦 of a tri -layer sample obtained at 𝑇=10 K and \n𝑇=240  K, respectively. A moderate 𝑉𝑔 of a few volts induces significant variations \nin the coercivity  of the ferromagnetic FGT, and  the ferromagnetism, manifested as  a \nfinite 𝑅𝑥𝑦𝑟, persists at high temperatures under certain gate doping. Indeed, detailed \nanalysis reveals that 𝑇𝑐 is dramatically modulated by the ionic gate  with the highest \n𝑇𝑐  reaching  room temperature ( 𝑇=300  K ; Fig. 3 d). The gate-induced room -Page 8 of 17 \n temperature ferromagnetism is also directly observed in a four-layer sample  as shown \nin Fig. 4.  𝑇𝑐 as a function of the gate voltage  exhibits a complex pattern : after an \ninitial drop, 𝑇𝑐  increases sharply to room temperature at 𝑉𝑔=1.75 V . 𝑇𝑐  peaks \nagain at 𝑉𝑔=2.39 V on top of an otherwise slow ly declining  trend as the doping \nlevel increases  (Fig. 3d; see Supplementary Section III for details) . The variations  in \n𝑇𝑐  is accompanied by  changes in the sample conductance, 𝜎𝑥𝑥 , obtained at 𝑇=\n330  K (Fig. 3a). In particular, the two peaks in 𝑇𝑐 coincide  with sudden increases  in \n𝜎𝑥𝑥, suggesting chan ges in the electronic structure as the origin of the magnetism \nmodulation. Finally , we note that the  coercivity measured at 𝑇=10 K also roughly \nfollows the variations in  𝑇𝑐, as shown in Fig. 3e. \nThe gate -tuned ferromagnetism in atomically thin FGT is consistent with Stoner \nmodel  that describes itinerant magnetic systems . The Stoner criterion infers  that the \nformation of ferromagnetic order  is dictated by the density of states (DOS) at the \nFermi level48. The extreme electron doping induced by the ionic gate causes \nsubstantial  shift of FGT ’s electronic bands. Consequently, t he large variation in  the \nDOS at the Fermi level leads to appreciable modulation in the ferromagnetism. \nDetailed calculations support such a picture.  Specifically , our calculation shows that \ngate-induced electrons sequentially fill the sub-bands originated from Fe 𝑑z2, 𝑑xz \nand 𝑑yz orbitals , leading to sharp peaks in DOS  that corresponds to Fermi level ’s \npassing through the  flat band edges of the sub -bands  (see Supplementary Section IV). \nSuch DOS peaks may be responsible for the sharp increase of  𝑇𝑐  and the \ncorresponding  sudden jump in 𝜎𝑥𝑥. Meanwhile , calculations indicate that change in \nthe average magnet moment per Fe atom is mini mal (< 3%) under charge doping  \n(Supplementary Section IV). This result suggest s that the gate-tuned  coercivity  shown \nin Fig. 3e may i n fact reflect  the large modulation in the  anisotropy energy , although \nwe cannot rule out  effect s from  domain nucleation that complicates the correlation \nbetween coercivity and anisotropy . \nTo conclude, we develop a new fabrication method and successfully isolated \natomically thin FGT from the layered bulk crystal . Itinerant ferromagnetism persists in  \n2D FGT down to monolayer . The ferromagnetism  is prote cted by a perpendicular \nmagneto crystalline  anisotropy against thermal fluctuations in the 2D limit, and exhibits \nstrong dimensionality effect as the thickness increases. The atomically thin FGT \nprovides a rare opportunity for  electric al modulation of its magnetic properties. W e Page 9 of 17 \n achieve unprecedent ed control of its ferromagnetism with an ionic gate. 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S. & Fisher, M. E. Finite -Size and Surface Effects in Heisenberg Films. \nPhys. Rev. B  7, 480–494 (1973).  \n43. Zhang, R. & Willis, R. F. Thickness -Dependent Curie Temperatures of Ultrathin \nMagnetic Films: Effect of the Range of Spin -Spin Interactions. Phys. Rev. Lett.  86, \n2665 –2668 (2001).  \n44. Henkel, M., Andrieu, S., Bauer, P. & Piecuch, M. Finite -size scaling in thin Fe/Ir \n(100) layers. Phys. Rev. Lett.  80, 4783 (1998).  \n45. Huang, F., Mankey, G. J., Kief, M. T. & Willis, R. F. Finite ‐size scaling behavior \nof ferromagnetic thin films. J. Appl. Phys.  73, 6760 –6762 (199 3). \n46. Ambrose, T. & Chien, C. L. Finite -size scaling in thin antiferromagnetic CoO \nlayers. J. Appl. Phys.  79, 5920 (1996).  \n47. Huang, F., Kief, M. T., Mankey, G. J. & Willis, R. F. Magnetism in the few -\nmonolayers limit: A surface magneto -optic Kerr -effect study of the magnetic \nbehavior of ultrathin films of Co, Ni, and Co -Ni alloys on Cu(100) and Cu(111). \nPhys. R ev. B  49, 3962 –3971 (1994).  \n48. Stoner, E. C. & S, F. R. Collective electron ferronmagnetism. Proc R Soc Lond A  \n165, 372–414 (1938).  Page 13 of 17 \n 49. Ueno, K. et al.  Anomalous Hall effect in anatase Ti1 −xCoxO2 −δ above room \ntemperature. J. Appl. Phys.  103, 07D114 (2008) . \n \nAcknowledgements  \nWe thank Xiaofeng Jin , Biao Lian and Gang Chen for helpful discussions. Part of \nthe sample fabrication was conducted at Fudan Nano -fabrication Laboratory. D.Y ., Y .Y ., \nS.Y ., J.W. and Y .Z.  acknowledge financial support from National Key Research \nProgram of China ( grant no. 2016YFA0300703), and NSF of China  (grant nos. \nU1732274, 11527805 and 11425415 ). N.Z.W.  and X.H.C. acknowledge support from \nthe ‘Strategic Priority Research Program ’ of th e Chinese Academy of Sciences (grant \nno. XDB04040100) and the National Basic Research Program of China (973 Program; \ngrant no. 2012CB922002). X.H.C. also acknowledges support from the National \nNatural Science Foundation of China (grant no. 11534010) and th e Key Research \nProgram of Frontier Sciences, CAS (grant No. QYZDY -SSW -SLH021).  J.Y .Z \nacknowledge financial support from  Chinese University of Hong Kong (CUHK) under \nGrant No 4053084,  from  Universit y Grants Committee of Hong Kong under Grant No \n24300814, an d the Start -up Funding of CUHK.  J.W. also acknowledges support from \nthe NSF of China  (grant no. 11774065).  \n \nAuthor contributions  \nY .Z. conceived the project. N.Z.W.  and X.H.C. grew bulk  FGT crystal. Y . D., Y .S . \nand Y .Y . developed device fabrication method . Y.D. fabricated devices , and performed \nelectric measurements  with the help of Y .Y .. Y .D., Y .Y . and Y .Z. analyzed the data. J.Z.Z \nand J.Y .Z carried out DFT calculations; J.W. carried out theoretical  calculations  and \nmodeling; and all three performed theoretical analysis . Y .D., Y .Y ., J.W.  and Y .Z. wrote \nthe paper and all authors commented on it.  \n  1L 2L 3Lbulk\nSubstrateAl2O3\nFeIFeIIGe\n1L\n2L3L5 nm\n4\n3\n2\n1\n0\n234\nheight (nm)\nlateral displacement (µm)3L\n2L\n1L\n0      1       2        3       4       5\n0 100 2000.80.91.01.11.21.31.41.5f\n1L\n2L\n3L\n4L\n6L\n50L\nBulkR/R(T=240K)\nT (K)Te\ncrystal Evaporated Al2O3 \nthin flake\non Al2O3\nHeatinga\nb\nc\ned\nFigure 1 | Fabrication and characterization of atomically-thin FGT devices. a , Schematic of a \nnew Al2O3-assisted mechanical exfoliation method. The new method uses an Al2O3 film evaporated \nonto the bulk crystal to cleave layered materials. The strong adhesion between crystal and Al2O3 film \nmakes it possible to exfoliate layered crystals that are otherwise difficult to cleave with SiO2 surface \nin conventional methods. b, Atomic structure of monolayer FG T. Left: view along [001]; right: view \nalong [010]. Bulk FGT  is a layered crystal with interlayer vdW  gap of 2.95 Å (ref. 29). FeI and FeII \nrepresent the two inequivalent Fe sites in +3 and +2 oxidation state, respectivel y. c, Optical image of \ntypical few-layer FGT  flakes exfoliated on top of Al2O3 thin film. The Al2O3 film is supported on Si \nwafer covered with 285 nm SiO2. d, AFM  image of the area marked by the square in c. Mono- and \nfew-layer flakes of FGT  are clearly visible. Scale bar: 2 mm. e, Cross-sectional profile of the FGT \nflakes along the white line shown in d. The steps are 0.8 nm in height, consistent with the thickness of \nmonolayer FGT (0.8 nm). f, Temperature-dependent sample resistance of FGT with varying number of \nlayers. Resistances are normalized to their values at T=240 K. Typical behaviour of bulk FGT \n(purple) is also shown here for reference.\nPage 14 of 17Figure 2 | Ferromagnetism in atomically-thin FG T. a,b, Rxy at low temperature (a) and 100 \nK (b) obtained in FGT thin-flake samples with varying number of layers. Ferromagnetism, ev -\nidenced by the hysteresis in low-temperature Rxy, is observed in all atomically-thin FGT down \nto monolayer. At  T = 100 K, hysteresis disappears in mono- and bilayer FGT, but persists in \nthicker specimens.  All the low temperature data were obtained at T = 1.5 K except for monolay -\ner data, which were recorded at T = 3 K. c, Remnant anomalous Hall resistance Rr\nxy as a function \nof temperature obtained from FGT  thin-flake samples with varying number of layers. Rr\nxy are \nnormalized to their values at T = 1.5 K except for the monolayer dataset, which is normalized to \nthe T = 3 K value. Arrows mark the ferromagnetic transition temperature Tc. d, Phase diagram \nof FGT as layer number and temperature are varied. Filled squares represent Tc of samples with \nCr/Au electrodes made by direct evaporation through stencil masks, and open squares represent \nTc of samples with cold-welded indium microelectrodes. Back line is a fit to Tc (N) (for N ≥ 3) by \nthe finite-size scaling formula described in the main text. Solid red line is line fit to Tc (N) for \nN ≤ 3. Here we fitted the highest Tc achieved for each sample thickness.\nPage 15 of 17FMPM\n-2 0 2 -2 0 2 1 2 3 4 5 6 7 8 9 50 550100200×2.5b\n×35×2.5×2\nµ0H (T)Rxy20Ωa\n×35×2.5×2\nµ0H (T)20Ω\n1L\n2L\n3L\n4L\n6L\n50L0 100 200011L\n2L\n3L\n4L\n6L\n50LNormalized Rr\nxy\nT (K)dc\nTcT (K)\nLayer numberFMPM\n-2 0 2 -2 0 2 1 2 3 4 5 6 7 8 9 50 550100200×2.5b\n×35×2.5×2\nµ0H (T)Rxy20Ωa\n×35×2.5×2\nµ0H (T)20Ω\n1L\n2L\n3L\n4L\n6L\n50L0 100 200011L\n2L\n3L\n4L\n6L\n50LNormalized Rr\nxy\nT (K)dc\nTcT (K)\nLayer number\nFigure 3 | Ferromagnetism in atomically-thin FGT flake modulated by ionic gate. a, Con-\nductance as a function of gate voltage, Vg, measured in a tri-layer FGT  device. Data were ob-\ntained at T = 330 K. Breaks in the curve were caused by temperature-dependent measurements \nat fixed gate voltages during the gate-sweep. Inset: schematic of the FGT  device structure and \nmeasurement setup. S and D label source and drain electrodes, respectivel y, and V1, V2, V3 and \nV4 label the voltage probes. Solid electrolyte LiClO4/PEO covers both the FGT flake and the side \ngate. b,c, Rxy as a function  of external magnetic field recorded at representative gate voltages. \nData obtained at  T = 10 K and T = 240 K are displayed in b and c, respectively. d, Phase dia -\ngram of the tri-layer FGT sample as the gate voltage and temperature are varied. We determine \nthe transition temperature from temperature-dependent  anomalous Hall resistance extrapolates \nto zero (see Supplementary Section III). e, Coercive field as a function of the gate voltage. Data \nwere obtained at T = 10 K. \nPage 16 of 17\nVg\nDS\nV1\nV2V3V4Electrolyte\nPM\nFM3.24.04.8\n0100200300\n0 1 2 30.00.20.40.6σxx(mS)T=330K\n-2 0 2T=10KRxy\nµ0H (T)20Ω\n-2 0 22.39V2.31V2.21V4Ω\n3.20V2.48V2.00V1.75V1.73V1.72V1.48V1.27VT=240K\nµ0H (T)0.00V\nTcc cT (K)a\nT=10Kec b\ndµ0Hc(T)\nVg(V)a200 250 300 3500.00.51.01.5\n-0.2 0.0 0.2-2-1012b\nRr\nxy(Ω)\nT (K)Vg=2.1Va\n310KRxy(Ω)\nµ0H (T)Vg=2.1V200K\nFigure 4 | Direct observation of room-temperature ferromagnetism in atomically \nthin FGT. a, Rxy of a four-layer FGT under a gate voltage of Vg = 2.1 V. We have sub-\ntracted a background in Rxy resulting from Lithium ions ’ slow drifting in the electro -\nlyte under the gate bias above T = 250 K (ref. 49). Hysteresis in Rxy persists up to T = \n310 K, providing unambiguous evidence for room-temperature ferromagnetism in the \nsample. b, Remnant Hall resistance Rr\nxy as a function of temperature. Extrapolating Rr\nxy \nto zero yields a Tc higher than 310 K.\nPage 17 of 17"    },    {        "title": "1803.03022v1.Direct_imaging_of_antiferromagnetic_domains_in_Mn__2_Au_manipulated_by_high_magnetic_fields.pdf",        "content": "Direct imaging of antiferromagnetic domains in Mn 2Au\nmanipulated by high magnetic \felds\nA.A. Sapozhnik,1, 2M. Filianina,1, 2S.Yu. Bodnar,1A. Lamirand,3M. Mawass,4\nY. Skourski,5H.-J. Elmers,1, 2H. Zabel,1, 2M. Kl aui,1, 2and M. Jourdan1, 2\n1Institut f ur Physik, JGU Mainz, Staudingerweg 7, 55128 Mainz, Germany\n2Graduate School Materials Science in Mainz,\nStaudingerweg 9, 55128 Mainz, Germany\n3Diamond Light Source, Chilton, Didcot, Oxfordshire, OX11 0DE, UK\n4Helmholtz-Zentrum Berlin f ur Materialien und Energie,\nAlbert-Einstein Str. 15, 12489, Berlin, Germany\n5Hochfeld-Magnetlabor Dresden (HLD-EMFL),\nHelmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany\nAbstract\nIn the \feld of antiferromagnetic (AFM) spintronics, information about the N\u0013 eel vector, AFM\ndomain sizes, and spin-\rop \felds is a prerequisite for device applications but is not available\neasily. We have investigated AFM domains and spin-\rop induced changes of domain patterns\nin Mn 2Au(001) epitaxial thin \flms by X-ray magnetic linear dichroism photoemission electron\nmicroscopy (XMLD-PEEM) using magnetic \felds up to 70 T. As-prepared Mn 2Au \flms exhibit\nAFM domains with an average size \u00141µm. Application of a 30 T \feld, exceeding the spin-\rop\n\feld, along a magnetocrystalline easy axis dramatically increases the AFM domain size with N\u0013 eel\nvectors perpendicular to the applied \feld direction. The width of N\u0013 eel type domain walls (DW)\nis below the spatial resolution of the PEEM and therefore can only be estimated from an analysis\nof the DW pro\fle to be smaller than 80 nm. Furthermore, using the values for the DW width and\nthe spin-\rop \feld, we evaluate an in-plane anisotropy constant ranging between 1 and 17 µeV/f.u..\n1arXiv:1803.03022v1  [cond-mat.mtrl-sci]  8 Mar 2018I. INTRODUCTION\nIn antiferromagnetic (AFM) spintronics, ferromagnets (FM) are replaced by AFMs as\nactive device materials [1{5]. This novel approach takes advantage of the fast THz dynamics\nof antiferromagnets, which is driven by exchange interaction. This in principle enables\nwriting speeds superior to those in conventional spintronics based on FM. Additionally, the\nabsence of a net magnetization in AFMs results in vanishing dipolar interactions allowing\nfor an increased information density and a high stability against disturbing external \felds.\nIn AFM spintronics, information is encoded by the direction of the N\u0013 eel vector, which\nis de\fned by the vectorial di\u000berence of the sublattice magnetizations. For spintronic appli-\ncations, one requires e\u000ecient methods for reading and changing the N\u0013 eel vector orientation\n(writing). In case of AFMs with non-centrosymmetric magnetic sublattices, it was predicted\nthat a current induced spin-orbit torque can change the orientation of the N\u0013 eel vector [6].\nThis was indeed recently demonstrated for the compounds CuMnAs [7, 8] and Mn 2Au [9, 10].\nHowever, the necessary current densities are in general close to the destruction limit and fur-\nther materials optimization is required, speci\fcally aiming at a reduced DW pinning. To this\nend, the characterization of the AFM domain structure is of major importance. Whereas\nX-ray magnetic linear dichroism photoemission electron microscopy (XMLD-PEEM) was\nsuccessfully used to observe AFM domains and their manipulation in CuMnAs thin \flms\n[7, 11], no such experiments have been reported for Mn 2Au yet. Moreover, information\nabout the magnetocrystalline anisotropy determining switching current [6] is required for\nlowering its threshold density.\nMagnetic domains in di\u000berent AFMs were widely studied by XMLD-PEEM during the\nlast two decades [7, 12{15]. The common procedure developed for obtaining XMLD-contrast\nin oxides is to calculate the asymmetry from images taken at two energies corresponding to\nmultiplet peaks of a magnetic atom. Typical domain sizes observed in oxide AFM are in\nthe micrometer regime [12{14]. However, conductive materials, like Mn 2Au, exhibit broader\nX-ray absorption spectra (XAS) with no multiplet structure [16]. This renders the selection\nof the appropriate X-ray energies for obtaining su\u000ecient magnetic contrast challenging [17].\nIn order to understand Mn 2Au and use this novel material for future devices, one needs to\nbe able to visualize the magnetic domain con\fguration and obtain key magnetic properties,\nsuch as the anisotropies, which are currently unknown. We report on the visualization\n2of AFM domains in Mn 2Au epitaxial thin \flms using high resolution XMLD-PEEM. We\ndemonstrate the alignment of the N\u0013 eel vector by the application of a high magnetic \feld\nresulting in a spin-\rop transition. This allows us based on an analysis of the AFM domain\nwall width and on the spin-\rop \feld to evaluate the magnetic in-plane anisotropy constant\nof Mn 2Au.\nII. METHODS\nEpitaxial thin \flm samples with a stacking sequence of Al 2O3(1\u0016102) substrate/Ta(001) 30 nm/\n/Mn 2Au(001) 240 nm/AlO x2 nm were grown by radio frequency magnetron sputtering. The\nin-plane epitaxial relation as determined by X-ray di\u000bractometry (XRD) is Ta(001)[100] k\nMn2Au(001)[100]. AlO xcapping was used as an oxidation protection. More details on the\nsample preparation and characterization can be found in Ref. [18].\nThe samples were exposed to high pulsed magnetic \felds targeting an alignment of the\nN\u0013 eel vector by a spin-\rop transition. Magnetic \felds of di\u000berent amplitudes ranging from\n30 T to 70 T were applied to the Mn 2Au samples along the [110] and [100]-directions at room\ntemperature at the High Magnetic Field Laboratory of the Helmholtz-Zentrum Dresden-\nRossendorf (HZDR).\nThe XMLD-PEEM studies were performed at beamline I06 at Diamond Light Source and\nwith the SPEEM setup at BESSY II (HZB). Both instruments are equipped with Elmitec\nphotoemission electron microscopes providing \u001850 nm spatial resolution and 0 :4 eV energy\nresolution. The X-ray beam linearly polarized in the sample plane was incident under an\nangle of 16\u000eto the sample surface. Controllable rotation procedures guaranteed measure-\nments within the same area on the surface at di\u000berent rotation angles around the sample\nnormal. The strongest in-plane magnetic contrast was achieved by calculating the asym-\nmetry of two images taken at energies corresponding to the maximum ( EMAX) and to the\nminimum (EMIN) of XMLD:\nIasym=I(EMAX)\u0000I(EMIN)\nI(EMAX) +I(EMIN): (1)\nThe XAS was determined from a set of images obtained in a range of energies close to\nthe L 3absorption edge of Mn. The absorption coe\u000ecient was calculated as the sum of\ngray-scale levels over a region of interest in the center of the \feld of view. In our previous\n3work we measured the absorption spectrum of Mn 2Au [16] and demonstrated that EMAX\nandEMIN are separated by 0 :8 eV and 0:0 eV from the L3absorption edge, respectively.\nThis information, in combination with the XAS determined from the images as discussed\nabove, was used for de\fning EMAX andEMIN for each sample.\nIII. EXPERIMENTAL RESULTS AND DISCUSSION\nFIG. 1. Asymmetry images of the as-prepared sample. The in-plane angle of the X-ray incidence is\n(a) 0\u000e, (b) 45\u000e, and (c) 90\u000ewith respect to the crystallographic [1 \u001610]-axis. The red double-headed\narrow indicates electric \feld vector E of the linearly polarized X-ray beam. The double box at\nthe bottom speci\fes the N\u0013 eel vector orientation in the AFM domains. The bright area on the left\nhand side of the image is caused by the marker, which was used to keep the image position during\nrotation of the sample.\nThe as-prepared sample not exposed to a high magnetic \feld exhibits small contrast\nfeatures with an average size of \u00181µm in the asymmetry image (Fig. 1 (a)). The contrast\ndisappears when the sample (corresponding to the direction of X-ray incidence) is rotated by\n45\u000e(Fig. 1 (b)) and reverses after 90\u000erotation of the sample (Fig. 1 (c)), which demonstrates\nthe magnetic origin of the observed asymmetry. From the vanishing contrast of Fig. 1 (b)\nand the appearance of basically two levels of gray in Fig. 1 (a) and (c), we conclude that\nthe N\u0013 eel vector is always oriented parallel to the h110i-directions, which is consistent with\nthe reported easy axes of Mn 2Au [19, 20]. Thus, an as-prepared Mn 2Au sample shows AFM\ndomain pattern as revealed in Fig. 1 with an average domain size of \u00181µm. Moreover, the\n4coverage of the sample with the two energetically equivalent domains with 90\u000edi\u000berent N\u0013 eel\nvector orientations are comparable.\nFIG. 2. Asymmetry images of the Mn 2Au sample after exposure to a magnetic \feld of 30 T along\nthe [110]-direction (green arrow). The in-plane angle of the X-ray incidence is (a) 0\u000e, (b) 45\u000e, and\n(c) 90\u000e. The red double-headed arrow indicates the polarization of the linearly polarized X-ray\nbeam. The double box at the bottom speci\fes the N\u0013 eel vector orientation in the AFM domains.\nHaving established the as-grown domain structure, the next step is to study changes\nin the domain structure when the AFM is exposed to magnetic \felds su\u000eciently high to\npotentially manipulate the AFM. Fig. 2 shows XMLD-PEEM images of a sample exposed to\na 30 T external \feld along the [110]-direction, which is an easy-axis of the material. A strong\nmagnetic contrast in the asymmetry image appears for the X-ray incidence direction (surface\nprojected) parallel to [1 \u001610] (0\u000e), which displays large bright (light gray) areas with minor\ndark inclusions (Fig. 2 (a)). Again, the contrast reverses upon rotation of the sample by\n90\u000ewith respect to the direction of X-ray incidence (Fig. 2 (c)) demonstrating the magnetic\norigin of the asymmetry. Please note that the magnetic contrast vanishes upon rotation\nby 45\u000eleaving only some morphology related features visible (Fig. 2 (b)). A \feld of 30 T\nsigni\fcantly increases the size of domains with the N\u0013 eel vector perpendicular to the \feld.\nThis phenomenon can be explained by a spin-\rop transition, which results in reorientation of\nthe N\u0013 eel vector perpendicular to the direction in which the magnetic \feld was applied. From\nthe image we see, that the majority of the AFM spin structure has been aligned with the\naxis favored by the spin-\rop. So from this we can deduce an upper bound for the spin-\rop\n\feld of our Mn 2Au thin \flms of 30 T.\n5FIG. 3. Asymmetry images of the Mn 2Au sample after exposure to a magnetic \feld of 50 T along\nthe [110]-direction (green arrow). The in-plane angle of the X-ray incidence is (a) 0\u000e, (b) 45\u000e, and\n(c) 90\u000e. The red double-headed arrow indicates the polarization of the linearly polarized X-ray\nbeam. The double box at the bottom speci\fes the N\u0013 eel vector orientation in the AFM domains.\nA higher magnetic \feld of 50 T applied along the [110]-direction causes a similar reori-\nentation of the domain structure (Fig. 3). The asymmetry image shows bright (light gray)\nareas separated by narrow worm-like dark lines (Fig. 3 (a)). They have an average width of\n\u0018100 nm, which requires a higher resolution for resolving the spin structure within them.\nThe dark lines can either be magnetic domains with the N\u0013 eel vector oriented along the [110]-\ndirection or can be considered 180\u000edomain walls. The presence of not closed lines can be\nseen as evidence against the domain wall hypothesis. However, the gaps in the black lines\ncan be caused by the surface morphology contributing to the asymmetry via e.g. residual\ndrifts in the images remaining even after corrections.\nFinally, we probe the e\u000bect of a 70 T external \feld applied along the [100]-axis of Mn 2Au,\nwhich is a hard magnetic direction (Fig. 4). The AFM domain structure is decomposed into\ndomains with an average size of \u00181 to 3 µm. However, the proportion of both types of AFM\ndomains is equal, indicating no preferred N\u0013 eel vector orientation on a large scale.\nThis observation is explained by the fact that a high enough external \feld applied along\nthe hard [100]-axis orients the N\u0013 eel vector along the perpendicular [010]-hard axis. When\nthe \feld is reduced, the moments redistribute themselves parallel to the easy axes creating\na new domain pattern. This leads to an increase of the average domain size in comparison\nto the as-prepared state shown in Fig. 1. These results again con\frm h110ito be the easy\n6FIG. 4. Asymmetry images of the Mn 2Au sample after exposure to a magnetic \feld of 70 T along\nthe [100]-direction (green arrow). The in-plane angle of the X-ray incidence is (a) 0\u000e, (b) 90\u000e. The\nred double-headed arrow indicates the polarization of the linearly polarized X-ray beam.\naxes in our Mn 2Au thin \flms [19, 20].\nThe upper boundary speci\fed for the \feld required to generate a spin-\rop transition\nallows us to determine the in-plane anisotropy constant ( HIP\na) of Mn 2Au(001) utilizing the\nfollowing expression for spin-\rop \feld HSF=p\n2HexHIP\na[21]. Using the exchange \feld from\nRef. [22] to be \u00160Hexch=1300 T, we \fnd an upper boundary of \u00160HIP\na= 0:35 T. Adopting\nthe expression for the in-plane anisotropy K4sin4\u0012cos(4\u001e) from [19], the derived anisotropy\n\feld corresponds to the maximal value of K4k\u001417µeV/f.u., which is in line with theoretical\npredictions of 10 µeV/f.u. [19].\nIV. DETERMINATION OF THE DOMAIN WALL WIDTH IN Mn 2Au\nDue to a high c-axis magnetocrystalline anisotropy in Mn 2Au [19], in-plane (N\u0013 eel type)\ndomain walls are energetically more favorable in our thin \flms than Bloch walls with an\nout-of-plane component. The intrinsic DW width can be determined by minimizing the\ntotal DW energy per unit area, which contains exchange interaction and magnetocrystalline\nanisotropy terms [23]:\nE[\u001e(x)] =Z1\n\u00001 \nJa2\u0012d\u001e\ndx\u00132\n+K4(1\u0000cos 4\u001e)!\nndx; (2)\n7FIG. 5. (a) The unit cell of Mn 2Au indicating the exchange constants. (b) Schematic representation\nof a N\u0013 eel type domain wall when viewed along the [001]-axis.\nwhereJ=1\n4(J1+ 2J3) (see Fig. 5 (a)) for [100]-domain walls (see Fig. 5 (b)), K4is the\nfour-fold anisotropy constant, nis the volume density of Mn atoms, and ais the Mn 2Au\nlattice constant. The choice of the anisotropy term corresponds to the angle \u001ebetween the\n[\u0016110]-easy axis and the staggered magnetization direction (Fig. 5 (b)). The solution of the\nvariational problem for the functional in Eq. (2) with the boundary conditions \u001e(\u00001)=0\nand\u001e(1)=\u0019/2 provides the DW pro\fle:\n\u001e[100](x) = arctan exp r\n8K4\nJa2x!\n: (3)\nThe DW width can be expressed by the slope of \u001e(x) atx= 0, i. e. in the center of the\nDW. Using Eq. (3), the 90\u000eDW width is derived as:\nw=\u0019\n21\nd\u001e[100](x)=dx(x= 0)=\u0019\n2r\nJ\n2K4a: (4)\nPlease note that similar considerations apply for determining the width of [110]-domain\nwalls, resulting in the same expression (4).\nSince the XMLD does not change sign upon N\u0013 eel vector inversion, the normalized XMLD-\nPEEM contrast across a DW is proportional to the cosine squared of the angle between the\n8N\u0013 eel vector and the X-ray electric \feld IDW(x)/cos2\u001e(x) [24]. Thus, in analogy with Eq.\n(4), the antiferromagnetic DW width observed in the experiment is:\nwexp=1\ndIDW(x)=dx(x= 0)=r\nJ\n2K4a: (5)\nThis result indicates that a DW appears \u0019=2 times more narrow in a XMLD-PEEM image\nin comparison with the actual width.\nHowever, in an XMLD-PEEM experiment, the DW image is broadened due to a \fnite\ninstrumental resolution, which can be represented as Gaussian function ( Res\u001b(x)), with the\nparameter 2 \u001bde\fning the resolution. For the determination of 2 \u001b, an intensity pro\fle was\nmeasured across the edge of the defect in the top right part of Fig. 4 (a) (blue line in Fig. 6\n(a)). Each point is averaged over 150 nm perpendicular to the line. The pro\fle was \ftted\nby the Gaussian error function (Fig. 6 (b)), which is a convolution of the step-function and\nthe Gaussian function. We obtain 2 \u001b'47 nm.\nFIG. 6. (a) Reproduction of Fig. 4 (a) with indicated paths used for measuring the line pro\fles. (b)\nLine pro\fle across the topographical structure indicated by the blue line in (a) and corresponding \ft\nwith a Gaussian error function. (c) Line pro\fle across the straight domain wall section indicated by\nthe green line in (a) and \fts corresponding to 2 K4=J= 2\u000210\u00004(solid line) and 2 K4=J= 0.5\u000210\u00004\n(dashed line).\nA domain wall pro\fle was determined across a straight section of a domain wall (green line\nin Fig. 6 (a)) with every point averaged over 300 nm perpendicular to the line, as depicted\nin Fig. 6 (c). The pro\fle was \ftted by a convolution of the instrumental resolution function\nand the determined domain wall pro\fle ( IDW\u0003Res\u001b) (x) with the \ft parameter 2 K4=J. The\nvalue of 2K4=Jproviding the best \ft is 2 \u000210\u00004. Additionally, we estimated a lower limit\nof this parameter of 0.5 \u000210\u00004(see Fig. 6 (c)). Based on Eq. (4), this value corresponds to\n9an upper limit for the DW width of 80 nm, which is of the same order of magnitude as the\ninstrumental resolution. Please note that our analysis relies on the assumption of a perfect\nstraight domain wall section. In FM thin \flms, straight DW sections are favored minimizing\nstray \felds, which are absent in AFM. Therefore, the apparently straight DW section in an\nAFM might show a variation of the position perpendicular to the pro\fle. Considering the\nexpected DW width, a much higher spatial resolution of better than 10 nm is necessary for\ndetailed investigations of DWs in Mn 2Au.\nFinally, the DW width provides an additional estimate of the anisotropy constant K4.\nUsingJ= 13:5 meV according to Ref. [23] we \fnd K4= 1µeV/f.u.. This value is the lower\nboundary for the in-plane anisotropy constant corresponding to an anisotropy \feld of 0 :02 T\nandHSF= 7 T.\nV. CONCLUSIONS\nUsing XMLD-PEEM we obtained images of AFM domains in Mn 2Au thin \flm samples.\nThe easy axis was experimentally determined to be parallel to the crystallographic h110i-\ndirections, in agreement with reports on bulk single crystals. A typical AFM domain size of\n'1µm was observed for as grown thin \flms.\nIt was possible to manipulate the AFM domains by a large magnetic \feld of 30 T gen-\nerating a spin-\rop transition. From the magnitude of this \feld we estimate the in-plane\nmagnetic anisotropy constant K4\u001417µeV/f.u. The samples exposed to a high external\n\feld applied along [110]-easy axis show large AFM domains with the N\u0013 eel vector oriented\nprimarily perpendicular to the \feld. A strong magnetic \feld directed along [100]-hard axis\nresults an increase of the domain size preserving almost equal proportion of in-plane Mn 2Au\ndomains oriented along the two in-plane easy axes.\nA detailed analysis of the measured domain wall pro\fles indicates a DW width smaller\nthan 80 nm, which is at the limit of the instrumental resolution. Nevertheless, this value\ncan be used to estimate a lower limit of K4to be 1 µeV/f.u.. Thus from the combination of\nboth limits we estimate anisotropy constant K4to be between 1 and 17 µeV/f.u..\nThe research was \fnancially supported by the German Research Foundation (Deutsche\n10Forschungsgemeinschaft) through the Transregional Collaborative Research Center 173\n\"Spin+X\", Project A05. A. A. S. also wishes to acknowledge the fellowship of the MAINZ\nGraduate School of Excellence. We thank the Diamond Light Source for the allocation of\nbeam time under Proposal No. SI17120-1 and HZB for the allocation of beam time under\nProposal No. 17206035-ST. We acknowledge the support of the HLD at HZDR, member of\nthe European Magnetic Field Laboratory (EMFL).\n[1] A. H. MacDonald and M. Tsoi, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 369, 3098\n(2011).\n[2] O. Gomonay, T. Jungwirth, and J. Sinova, Phys. Status Solidi RRL 11, 1700022 (2017).\n[3] E. V. Gomonay and V. M. Loktev, Low Temp. Phys. 40, 17 (2014).\n[4] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nature Nanotechnol. 11, 231 (2016).\n[5] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Rev. Mod. Phys.\n90, 015005 (2018).\n[6] J. \u0014Zelezn\u0013 y, H. Gao, K. V\u0013 yborn\u0013 y, J. Zemen, J. Ma\u0014 sek, A. Manchon, J. Wunderlich, J. Sinova,\nand T. Jungwirth, Phys. Rev. Lett. 113, 157201 (2014).\n[7] P. Wadley, B. Howells, J. \u0014Zelezn\u0013 y, C. Andrews, V. 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Thole, and G. A. Sawatzky, EPL (Europhysics Letters) 32, 259 (1995).\n12"    },    {        "title": "1803.08219v1.Large_Perpendicular_Magnetocrystalline_Anisotropy_at_Fe_Pb_001__interface.pdf",        "content": "Large Perpendicular Magnetocrystalline Anisotropy at Fe/Pb(001) interface\nXiaoxuan Ma and Jun Hu\u0003\nCollege of Physics, Optoelectronics and Energy, Soochow University, Suzhou, Jiangsu 215006, China\nJiangsu Key Laboratory of Thin Films, Soochow University, Suzhou, Jiangsu 215006, China.\nSearch for ultrathin magnetic \flm with large perpendicular magnetocrystalline anisotropy (PMA)\nhas been inspired for years by the continuous miniaturization of magnetic units in spintronics devices.\nThe common magnetic materials used in research and applications are based on Fe because the pure\nFe metal is the best yet simple magnetic material from nature. Through systematic \frst-principles\ncalculations, we explored the possibility to produce large PMA with ultrathin Fe on non-noble and\nnon-magnetic Pb(001) substrate. Interestingly, huge magnetocrystalline anisotropy energy (MAE)\nof 7.6 meV was found in Pb/Fe/Pb(001) sandwich structure with only half monolayer Fe. Analysis\nof electronic structures reveals that the magnetic proximity e\u000bect at the interface is responsible for\nthis signi\fcant enhancement of MAE. The MAE further increases to 13.6 meV with triply repeated\ncapping Pb and intermediate Fe layers. Furthermore, the MAE can be tuned conveniently by charge\ninjection.\nKeywords: Magnetocrystalline anisotropy, Ultrathin \flm, Magnetic/nonmagnetic interface, Spin-orbit cou-\npling, Charge injection\nI. INTRODUCTION\nMagnetic tunnel junctions (MTJs) are the key build-\ning blocks in spintronics devices for the applications in\nmodern technologies such as data storage and spin valve.\n[1{6] A typical MTJ consists of two ferromagnetic Fe thin\n\flms separated by ultrathin MgO intermediate layer. [7{\n9] As the spintronics devices keep miniaturizing rapidly\nin the recent decade, MTJs with sizes scaling down to\na few nanometers are desired. [3, 4] To this end, the\nthickness of the ferromagnetic thin \flms in MTJs down\nto a few atomic layers is preferred. In this realm, large\nperpendicular magnetocrystalline anisotropy (PMA) is\nthe most critical prerequisite, because it overcomes the\nthermal \ructuation of spins and in-plane shape magnetic\nanisotropy to maintain perpendicular magnetization at\nhigh temperature. [10]\nGenerally, the magnetocrystalline anisotropy (MA) de-\nrives from the spin-orbit coupling (SOC). However, the\nstrong magnetic materials such as 3d transition metals\n(TMs) which are commonly used in the MTJs usually\ndisplay weak SOC. Therefore, heavy rare-earth and noble\nTM elements (such as Gd, Tb, Pd and Pt) were alloyed\nwith 3d TMs (especially Fe and Co), in order to enhance\nthe SOC through the interaction between the light and\nheavy TM atoms. [11{14] Interestingly, large PMA were\nsuccessfully achieved in a series of such alloys as expected.\nFor example, the magnetocrystalline anisotropy energies\n(MAEs) in L1 0FePt and CoPt crystals are respectively\nenhanced to 2.7 and 1.0 meV per formula unit, hundreds\nof times of the intrinsic MAEs in bulk Fe and Co ( \u001810{3\nmeV per atom). [15{17] The MAEs are even larger in ul-\ntrathin \flms, due to the more localized surface states and\nquantum con\fnement. It was found that the MAE can be\n\u0003E-mail: jhu@suda.edu.cnas large as 5.2 meV/Fe atom in the sandwich structure\nPt/Fe/Pt(001) where the thicknesses of both the capping\nPt layer and ferromagnetic Fe layer are only one mono-\nlayer (ML). [18, 19] However, less expensive but more\nabundant elements such as Mo and W are more attrac-\ntive for real applications. Unfortunately, the Fe/Mo(110)\nand Fe/W(110) systems show only in-plane MA. [20, 21]\nParallelly, alloys and multilayers with only 3d TM ele-\nments were also investigated, but their MAEs are much\nsmaller than that of Pt/Fe/Pt(001). [22{28] Therefore,\nit is of great interest to search and design new magnetic\nmaterials without noble elements but with large PMA.\nBesides TM elements, the heavy p{valent elements\nsuch as Pb possess strong SOC. The SOC constant of\nPb{6 porbitals is\u00180.9 eV [30], much larger than that of\nPt{5 dorbitals (\u00180.6 eV) [31]. Intuitively, the SOC e\u000bect\nin Fe may be enhanced signi\fcantly through alloying Fe\nwith Pb or building Fe/Pb multilayers. Note that bulk\nPb crystallizes face-centered cubic (FCC) phase with lat-\ntice constant of 4.95 \u0017A, so its (001) surface has favorable\nlattice constant ( a= 3.50 \u0017A) to match thep\n2\u0002p\n2\r{\nFe(001) surface ( a0= 3.59 \u0017A). In this paper, we inves-\ntigated the magnetic properties of Fe/Pb(001) multilay-\ners, based on \frst-principles calculations. Interestingly,\nalternately placing two MLs Pb and half ML Fe by three\ntimes on Pb(001) possesses huge MAE of 13.6 meV, and\nthe MAE can be further enhanced to 16.5 meV by inject-\ning hole charge. Analysis of electronic properties reveals\nthat the magnetic proximity e\u000bect at the Fe/Pb interface\nplays an important role in the enhancement of the MAE.\nII. COMPUTATIONAL DETAILS\nWe \frstly studied the Fe/Pb(001) bilayer system, with\nsix-layer Pb slab to model the Pb(001) substrate and a\nfew ML(s)p\n2\u0002p\n2\r{Fe(001) on it. The combined bi-\nlayer system is denoted as Fe n/Pb, where nis the num-arXiv:1803.08219v1  [cond-mat.mtrl-sci]  22 Mar 20182\n123456789102.42.62.83.0Fe1/PbFe2/PbFe3/PbFe4/PbFe6/PbFe8/PbFe10/PbSpin moment (µB)\nLabel of Fe atoms(a)\n1234567891001234Number of Fe atoms\nFe2/PbFe1/PbFe3/PbFe4/PbMAE (meV/slab)(b)Fe1Pb5Fe2Pb6Pb1123456789102.42.62.83.0Fe1/PbFe2/PbFe3/PbFe4/PbFe6/PbFe8/PbFe10/Pb123456789102.42.62.83.0Fe1/PbFe2/PbFe3/PbFe4/PbFe6/PbFe8/PbFe10/Pb123456789102.42.62.83.0Fe1/PbFe2/PbFe3/PbFe4/PbFe6/PbFe8/PbFe10/Pb\nFIG. 1: (Color online) Magnetic properties of Fe n/Pb slabs.\n(a) Local spin moments on the Fe atoms. (b) Magnetocrys-\ntalline anisotropy energy (MAE) of each slab. The green\ndashed line indicates the MAE of Fe monolayer (0.8 meV)\nas a benchmark. The insets show the relaxed structures for\nn = 1\u00184. The grey and purple spheres stand for Pb and Fe\natoms, respectively.\nber of Fe atoms in the unit cell of the slab. Density\nfunctional theory (DFT) calculations were carried out\nwith the Vienna ab-initio simulation package. [32, 33]\nThe interaction between valence electrons and ionic cores\nwas described within the framework of the projector aug-\nmented wave (PAW) method. [34, 35] The spin-polarized\ngeneralized-gradient approximation (GGA) was used for\nthe exchange-correlation potentials. [36] The energy cut-\no\u000b for the plane wave basis expansion was set to 350 eV.\nFor all slabs a vacuum of about 15 \u0017A along the surface\nnormal was inserted between neighboring slabs to mimic\nthe two-dimensional periodicity. The two-dimensional\nBrillouin zone was sampled by a 45 \u000245 k-grid mesh.\nThe atomic positions except the bottom two layers were\nfully relaxed using the conjugated gradient method until\nthe force on each atom is smaller than 0.01 eV/ \u0017A. We\nhave checked the results using 13-layer Pb slab, which\nshows that six-layer Pb slab is good enough in this work.\nIII. RESULTS AND DISCUSSIONS\nAfter relaxation of atomic positions, the interfacial Fe\nlayer is signi\fcantly buckled, due to the large di\u000berence\nof atomic sizes between Fe and Pb atoms. The Fe-Pb\nbond length between the closest Fe and Pb atoms at the\ninterface is around 2.75 \u0017A, manifesting strong interactionbetween the Fe and Pb atoms. For example, the binding\nenergy of Fe 1/Pb is 1.7 eV/Fe. All the Fe atoms are\nspin polarized. As shown in Fig. 1(a), the local spin\nmoment on an Fe atom (M S,Fe) depends on its location\nand can be sorted into three groups. The outermost Fe\natoms in Fe 2/Pb and Fe 3/Pb have the largest M S,Feof\n2.94\u0016B, while the outermost Fe atoms in other slabs\nwith n>4 are 2.73\u00060.03\u0016B. In addition, the M S,Fe\non the interfacial Fe atom [denoted as Fe1in Fig. 1(b)]\nwhich is also the outermost Fe atom in Fe 1/Pb are almost\nthe same for all slabs (2.79 \u00060.03\u0016B). Finally, the Fe\natoms of the intermediate layers possess smaller M S,Fe\nof 2.47\u00060.05\u0016B. This situation is similar to the pure\nFe thin \flm where the M S,Feof the inner and surface Fe\natoms are 2.25 and 2.95 \u0016B, respectively. On the other\nhand, spin polarization is induced on the interfacial Pb\natom [Pb6as indicated in Fig. 1(b)], which results in\nthe local spin moment of {0.07 \u0016B. Nevertheless, the\nmagnetic proximity e\u000bect in Fe n/Pb is much weaker than\nthat in L1 0FePt alloy where the induced spin moment\non the Pt atom is as large as 0.35 \u0016B. [15]\nThen we calculated the MAEs of the Fe n/Pb slabs.\nConventionally, the MAEs is evaluated by the change\nof the total energy for the magnetization switching be-\ntween orientations parallel and perpendicular to the sur-\nface plane [18, 19], MAE = Ek{E?, where Ekand E?de-\nnote the corresponding total energies, respectively. This\nmethod require extremely \fne sampling meshes for the\nk-space integrations, which leads to bad convergence of\nMAE to the number of k points. On the contrary, the\ntorque method proposed by Wang et al. [37] is much less\nsensitive to the number of k points. Instead of directly\ncalculating the total energy di\u000berence, the torque method\nintegrates the torque of the SOC Hamiltonian (H SO)\nMAE =occX\nnk\u001c\n\tnk\f\f\f\f@HSO\n@\u0012\f\f\f\f\tnk\u001d\n\u0012=45\u000e. (1)\nHere, \t nkis the nth relativistic eigenvector at kpoint,\nand\u0012is the angle between the spin orientation and sur-\nface normal. In the spin space, H SOcan be expressed\nas a 2\u00022 matrix. Therefore, we can extract the con-\ntributions from the diagonal and o\u000b-diagonal elements\nof H SOmatrix and divide the MAE into three parts:\nMAE(uu), MAE(dd) and MAE(ud+du). Here, `uu', `dd'\nand `ud+du' represent the contributions from the cou-\npling between majority spin states, minority spin states,\nand cross-spin states, respectively. Recently, we im-\nplemented this method in the framework of PAW and\nshowed that the torque method is quite e\u000ecient for cal-\nculating MAEs of magnetic materials. [38, 39] From Fig.\n1(b) it can be seen that the MAEs of Fe 1/Pb and Fe 2/Pb\nare signi\fcantly enhanced to 2.5 and 4.1 meV, respec-\ntively, compared to the pure Fe thin \flm (0.8 meV).\n[40] In contrast, the MAE of Fe 3/Pb becomes -0.3 meV,\nimplying in-plane magnetocrystalline anisotropy. The\nMAEs of the other slabs vary between 0.65 and 1.75 meV.\nClearly, the di\u000berent MAEs are associated with the dif-3\n-2.00.02.04.0dxydx2− y2dxz+ dyzdz2\n-4-3-2-1012-0.20.00.2spx+ pypzPDOS (states/eV)E - EF (eV)(b)(a)Fe1 /Pb\n(c)\n-0.030.03-0.500.5510152025\n510152025-0.4-0.200.20.4z (Å)Δρz (e/Å)(d)\nPb1Fe1Pb6\nFIG. 2: (Color online) Electronic properties of Fe 1/Pb. (a)\nand (b) Projected density of states (PDOS) of Fe1and Pb6,\nrespectively. The positive and negative signs stand for ma-\njority and minority spins, respectively. (c) Top panel: top\nview of the atomic structure; middle panel: electron charge\nredistribution caused by the interaction between the Fe and\nPb atoms [\u0001 \u001a=\u001a(Fe1/Pb) {\u001a(Pb) {\u001a(Fe)]; bottom panel:\nspin density ( \u001as=\u001a\"{\u001a#). Both \u0001\u001aand\u001asare on (110)\nplane indicated by the red line in top panel, with cuto\u000b of\n\u00060.03 e/ \u0017A3. The grey and purple spheres stand for Pb and Fe\natoms, respectively. (d) Planar summation of electron charge\nredistribution (\u0001 \u001az) along the surface normal ( z).\nferent structures of the Fe thin \flms. For Fe n/Pb thin\n\flms with n>4, the interaction between the Fe atoms\ndominate their magnetic properties, hence the MAEs os-\ncillate near the intrinsic value of pure Fe thin \flms. On\nthe contrary, the interactions between the interfacial Fe\nand Pb atoms are dominant in the MAEs for Fe 1/Pb and\nFe2/Pb.\nTo illustrate the e\u000bect of the interfacial interactions on\nthe MAE, we calculated the projected density of states\n(PDOS) of Fe 1/Pb as shown in Fig. 2. Here the Fe atom\nlocates above the hollow site composed of four Pb atoms\nwith Fe{Pb bond length of 2.70 \u0017A, which imposes local\nsymmetry of C 4varound the Fe atom. Therefore, the\nFe { 3d orbitals split into four group: d xy, dx2{y2, dxz/yz\nand dz2, with the d xzand d yzkeeping degenerate. From\nFig. 2(a), it can be seen that the dxz/yz orbitals have\nsharp and localized PDOS, which indicates that they do\nnot interact with Pb atoms notably. In contrast, the\nother Fe { 3d orbitals are more delocalized, because they\ninteract with the broadly dispersive 6p orbitals of Pb6as\nseen in Fig.2(b). Moreover, the electron charge redistri-\nbution in Fig. 2(c) and 2(d) indicates that the Fe atoms\ngain electron charge from Pb6and the electron charge\naccumulates at the Fe layer. Integrating the PDOS of\neach orbital of the Fe atom up to the Fermi energy (E F)\nyields the electron occupancies of 1.6 e, 1.2e, 2.2eand 1.5 e\non the d xy, dx2{y2, dxz/yz and dz2orbitals, respectively.\nTherefore, the total electron charge on the Fe { 3d or-\nbitals is 6.5 e, which indicates the accumulated electron\ncharge on each Fe atom is 0.5 e. Furthermore, the dxz/yz\norbital is nearly completely spin-polarized with local spin\n-2024\n-6-30369\nuuddud+du-1.5-1.0-0.50.00.51.01.5-202Fe1/PbFe1/PbMAE (meV/slab)\nE - EF (eV)-2024\n-6-30369uuddud+du-1.5-1.0-0.50.00.51.01.5-202Fe1/Pb-2024\n-6-30369\nuuddud+du-1.5-1.0-0.50.00.51.01.5-202Fe1/Pb-2024\n-6-30369uuddud+du-1.5-1.0-0.50.00.51.01.5-202Fe1/PbFe1    Pb6   Pb5(b) Pb6(a)\n(c) Fe1-2024\n-6-30369\nuuddud+du-1.5-1.0-0.50.00.51.01.5-202Fe1/PbFIG. 3: (Color online) Atom resolved MAEs in Fe 1/Pb. (a)\nThe total (grey area) MAE of the slab and local MAEs on the\ninterfacial atoms. (b) and (c) The spin resolved MAEs of Pb6\nand Fe1. Here the grey area stands for the total local MAE.\n`uu', `dd' and `ud+du' notate the contributions from the cou-\npling between majority spin states, minority spin states, and\ncross spin states, respectively.\nmoment of\u00181.7\u0016B, contributing about 60% to the total\nspin moment. The local spin moments contributed by the\nother Fe{3d orbitals are much smaller, about 0.4, 0.7 and\n0.4\u0016Bby d xy, dx2{y2and dz2, respectively. Similarly, in\nFe2/Pb the Fe{3d and Pb{6p orbitals of hybridize with\neach other strong, as indicated by the PDOS in Fig. S1\nin the Supporting Information.\nApparently, the interfacial Pb atom plays an impor-\ntant role in the enhancement of the MAEs of Fe 1/Pb\nand Fe 2/Pb, even though the induced spin moments on\nthe Pb atom are small. Therefore, it is instructive to\nestimate the contributions of individual atoms quantita-\ntively. For this purpose, we rewrite Eq. (1) by inserting\natomic orbitals\nMAE =occX\nnkX\nijCij2\u001c\n'i\f\f\f\f@HSO\n@\u0012\f\f\f\f'i\u001d\n\u0012=45\u000e, (2)\nwhere Ci=h'ij\tnkiis the projected coe\u000ecient of an\natomic orbital ( 'i) on the eigenvector of the system\n(\tnk). Summing over all the atomic orbitals of a se-\nlected atom yields the atomic resolved MAE.\nInterestingly, we found that the interfacial Pb atoms\nhave much larger contribution to the total MAE than\nthe Fe atom. For Fe 1/Pb, Pb6contributes MAE of 2.20\nmeV which is 88% of the total MAE [see Fig. 3(a)],4\n-6-4-202468101214Fe1Fe2MAE (meV/slab)\nFen/Pb(Pb1/Fen)2/Pb(Pb1/Fen)1/Pb(Pb1/Fen)3/Pb(Pb2/Fen)1/Pb(Pb2/Fen)2/Pb(Pb2/Fen)3/Pb\n(Pbm/Fe1)k/Pb(Pbm/Fe2)k/Pb\nFIG. 4: (Color online) MAEs of slabs (Pb m/Fen)k/Pb. The\nMAEs of Fe 1/Pb and Fe 2/Pb are plotted for reference. The\ninsets show the atomic structures.\nwhile the MAE contributed by Fe1is only 0.26 meV,\nand the MAEs from other Pb atoms are negligible. For\nFe2/Pb, it can be seen from Fig. S1 in the Supporting\nInformation that, both Pb6and Pb5(interfacial Fe and\nPb atoms) have large contributions to MAE, 1.75 and\n1.44 meV, respectively, accounting for 78% of the total\nMAE. The outer Fe atom (Fe2) also contributes to MAE\nlargely (0.98 meV), but the MAE from Fe1is negligible\n(0.08 meV). This is di\u000berent from Pt/Fe/Pt(001) where\nthe Fe layer has dominant contribution (5.50 meV/Fe) to\nthe total MAE (5.21 meV/slab), while the Pt layer has\nnegative and smaller contribution ({0.85 meV). [18]\nTo get a deeper insight into the large MAEs from\nthe interfacial Pb atoms, we extracted the spin-resolved\nMAEs and plotted the E F-dependent curves in Fig.\n3(b). It can be seen that the total MAE near the E F\nmainly originates from the competition between negative\nMAE(uu) and positive MAE(ud+du). However, it is dif-\n\fcult to further distinguish the contributions of Pb { 6p\norbitals to MAE(uu) and MAE(ud+du), because their\nenergy bands quite dispersive as shown in Fig. S2 in the\nSupporting Information. As for Fe1, Fig. 3(c) shows\nthat the MAE comes from the competition between neg-\native MAE(dd) and positive MAE(ud+du), while the\nMAE(uu) contributes little because the majority spin\nstates are fully occupied [Fig. 2(a)]. Clearly, the neg-\native MAE(dd) and positive MAE(ud+du) near the E F\nhave comparable amplitudes, so they cancel each other\nand result in small contribution to the total MAE.\nThis unexpected result indicates that the magnetic\nproximity e\u000bect which induces small amount of spin po-\nlarization on the interfacial Pb atoms is the essential\nfactor for the signi\fcant enhancement of the MAEs in\nFe1/Pb and Fe 2/Pb. Therefore, large MAE is expected\nif Pb layers exist on both sides of the Fe layer. We\nconsidered a series of slab models with repeating struc-\nture of Pb capping layer on the Fe layer, denoted as(Pbm/Fen)k/Pb. Here, mandnare the thicknesses of\nthe capping Pb layer and Fe layer, and kdenotes the re-\npeating times. From Fig. 4, it can be seen that capping\nextra Pb layer on Fe 2/Pb turns the large positive MAE\ninto negative, implying that the (Pb m/Fe2/)k/Pb prefers\nin-plane magnetization. On the contrary, the capping Pb\nlayer boosts the MAE dramatically for (Pb m/Fe1/)k/Pb.\nCapping 1 ML Pb on Fe 1/Pb makes the MAE increase\nfrom 2.49 meV to 4.79 meV. Repeating the Pb 1/Fe1bi-\nlayer by 2 and 3 times further augment the MAEs to\n7.11 and 7.50 meV, respectively. More dramastically,\ncapping 2 MLs Pb on Fe 1/Pb (i.e. Pb 2/Fe1/Pb) re-\nsults in MAE of 7.59 meV, which is greatly larger than\nthat of Pt/Fe/Pt (5.21 meV). The atom-resolved MAEs\nin Fig. S3 in Supporting Information shows that the\ntwo Pb layers adjacent to the Fe layer have compara-\nble and dominant contributions (totally 5.75 meV) to\nthe total MAE. Repeating the Pb 2/Fe1trilayer by 2 and\n3 times further enhances the MAEs to 8.59 and 13.60\nmeV, respectively. Here, we took the sandwich structure\nPb2/Fe1/Pb to test the convergence of MAE to the num-\nber of k points. By calculating the MAE with di\u000berent\nk-grid meshes through both the torque method and to-\ntal energy di\u000berences, we found that the MAE from the\ntorque method converges around k-grid mesh of 40 \u000240,\nbut that from the total energy di\u000berence is still far from\nconvergence up to 45 \u000245, as seen in Fig. S4 in the Sup-\nporting Information. This bad convergence behavior is\nmainly caused by the situation that there are many bands\ncrossing the Fermi level, as shown in Fig. S5 in Support-\ning Information. In fact, the convergence problem is a\nlong-term issue, especially for the calculations of proper-\nties related to the SOC e\u000bect. [41]\nInterestingly, from Fig. 5(a), we can see that the\nPb atoms adjacent to the Fe layer contribute most part\nof the MAE for (Pb 2/Fe1/)k/Pb, similar with the sit-\nuation in Fe 1/Pb. Furthermore, we can see that the\nMAEs change rapidly near the E Ffor (Pb 2/Fe1/)2/Pb\nand (Pb 2/Fe1/)3/Pb, which is associated with the sig-\nni\fcant change of the electron occupancies on the or-\nbitals near the E F. [38, 42] Therefore, if the E Fcan\nbe controlled to shift upwards or downwards, then the\nMAEs will be reduced or enhanced. Usually, this can\nbe achieved by applying external electric \feld or inject-\ning charge. [18, 19, 42] In our calculations, an external\nelectric \feld is introduced by adding the corresponding\nelectrostatic potential to the slab with planar dipole layer\nmethod [43] and the dipolar correction is included. An\nelectric \feld is assumed to be positive when it is an-\ntiparallel to the surface normal. The charge doping is\ncontrolled by the number of valence electrons and a uni-\nformly charged background is used to neutralize the pe-\nriodic slab. Typically, by adding (removing) small part\nof the valence electrons such as 1 e per unit cell, one can\nsimulate a negative (positive) charge doping.\nUnfortunately, the electric \feld has little in\ruence on\nthe MAE as seen in Fig. 5(b). This is because the\nscreening e\u000bect strongly suppresses the penetration of5\n-1.5-1.0-0.50.00.51.01.5-20-1001020(Pb2/Fe1)1/Pb(Pb2/Fe1)2/Pb(Pb2/Fe1)3/Pb\n-1.0-0.50.00.51.011121314151617-20-1001020\n-1.5-1.0-0.50.00.51.01.5-2-1012\nPb2/Fe1/Pb and (Pb2/Fe1)3/Pb(Pb2/Fe1)1/Pb(Pb2/Fe1)2/Pb(Pb2/Fe1)3/PbE - EF (eV)\n Electric Field (V/Å) or Injected Electron (e)MAE (meV/slab)MAE (meV/slab)(a)\n(b)-1.5-1.0-0.50.00.51.01.5-30-20-100102030(Pb2/Fe1)1/Pb(Pb2/Fe1)2/Pb(Pb2/Fe1)3/Pb\n-1.0-0.50.00.51.011121314151617Electric FieldInjected Charge010203040-0.0500.05z (Å)\nFIG. 5: (Color online) (a) The E F{dependent MAEs of\n(Pb2/Fe1)k/Pb with k = 1 \u00183. The dashed curves are the\nMAE contributed by the Pb atoms adjacent to the Fe layer\nof each slab. (b) MAEs of (Pb 2/Fe1)3/Pb under electric \feld\n(diamond) and injected electrons (circles). Positive electric\n\feld goes anti-parallelly to the surface normal of the slabs.\nThe inset shows the electron charge redistribution along the\nsurface normal ( z) caused by the positive (red curve) and neg-\native (blue curve) electric \felds [\u0001 \u001a=\u001a(\"=\u00061 V/\u0017A){\u001a(\"=\n0)].\nthe electric \feld into the slab, so that only the outermost\ncharge is disturbed [see the inset in Fig. 5(b)]. Accord-\ningly, the electron occupancies do not change visibly un-\nder external electric \feld. In contrast, charge injection\nthrough gate voltage is not restricted by the screening\ne\u000bect, so it has much more remarkable impact on theMAEs of (Pb 2/Fe1/)2/Pb and (Pb 2/Fe1/)3/Pb, as dis-\nplayed in Fig. 5(b). It can be seen that injecting holes\ninto (or extracting electrons from) the slab makes the\nMAE of (Pb 2/Fe1/)3/Pb increase signi\fcantly and the\nMAE reaches 16.5 meV when one hole is injected into\nthe slab, while injections of electrons reduces the MAE.\nObviously, the response of the MAE to the charge in-\njection follows the trend predicted by the E Fdependent\nMAE curve in Fig. 5(a). Therefore, we can expect that\nthe MAE can be even larger than 20 meV per slab when\nmore holes are injected.\nIV. CONCLUSIONS\nIn summary, we studied the electronic and magnetic\nproperties of ultra-thin Fe \flm on Pb(001), based on \frst-\nprinciples calculations. Huge perpendicular MAE can be\nproduced in Pb 2/Fe1/Pb, with MAE as large as 7.59\nmeV per e\u000bective Fe atom. Electronic structure analy-\nsis reveals that small amount of local spin moments are\ninduced on the interfacial Pb atoms. Yet this small spin\npolarization results in huge MAE due to the large SOC\nconstant of Pb { 6p orbitals. Repeating the Pb capping\nlayers and Fe layer by three times [(Pb 2/Fe1)3/Pb] can\nlead to even larger MAE of 13.6 meV and it can be tuned\nby injecting charge. Furthermore, these structures may\nbe fabricated under delicately controlled conditions, so\nthey are of great potential in applications in spintronics\ndevices at high temperature.\nAcknowledgements\nThis work is supported by the National Natural Sci-\nence Foundation of China (11574223), the Natural Sci-\nence Foundation of Jiangsu Province (BK20150303) and\nthe Jiangsu Specially-Appointed Professor Program of\nJiangsu Province.\nREFERENCES\n[1] Wolf,S. A.; Awschalom, D. D.; Buhrman, R. A.;\nDaughton, J. M.; von Moln\u0013 ar, S.; Roukes, M. L.;\nChtchelkanova, A. Y.; Treger, D. M. Spintronics: A Spin-\nBased Electronics Vision for the Future. 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Rev.B2017 , 96, 014435.\n[43] Neugebauer, J.; Sche\u000fer, M. Adsorbate-substrate and\nadsorbate-adsorbate interactions of Na and K adlayers\non Al(111). Phys. Rev. B 1992 , 46, 16067."    },    {        "title": "1803.08292v1.Tuning_magnetocrystalline_anisotropy_of_Fe___3__Sn_by_alloying.pdf",        "content": "Tuning magnetocrystalline anisotropy of Fe 3Sn by alloying\nOlga Yu. Vekilova\u0003\nDepartment of Physics and Astronomy,\nUppsala University, Box 516, 75121 Uppsala, Sweden\nBahar Fayyazi, Konstantin P. Skokov, and Oliver Gut\reisch\nMaterials Science, TU Darmstadt, Alarich-Weiss-Str. 16, 64287 Darmstadt, Germany\nCristina Echevarria-Bonet and Jos\u0013 e Manuel Barandiar\u0013 an\nBCMaterials, UPV/EHU Science Park, 48940 Leioa, Spain\nAlexander Kovacs, Johann Fischbacher, and Thomas Schre\r\nDepartment for Integrated Sensor Systems, Danube University Krems,\nViktor Kaplan Str. 2/E, 2700 Wiener Neustadt, Austria\nOlle Eriksson\nDepartment of Physics and Astronomy,\nUppsala University, Box 516, 75121 Uppsala, Sweden and\nSchool of Science and Technology, Orebro University, SE-701 82 Orebro, Sweden\nHeike C. Herper\nDepartment of Physics and Astronomy,\nUppsala University, Box 516, 75121 Uppsala, Sweden\n(Dated: March 23, 2018)\n1arXiv:1803.08292v1  [cond-mat.mtrl-sci]  22 Mar 2018Abstract\nThe electronic structure, magnetic properties and phase formation of hexagonal ferromagnetic\nFe3Sn-based alloys have been studied from \frst principles and by experiment. The pristine Fe 3Sn\ncompound is known to ful\fll all the requirements for a good permanent magnet, except for the\nmagnetocrystalline anisotropy energy (MAE). The latter is large, but planar, i.e. the easy mag-\nnetization axis is not along the hexagonal c direction, whereas a good permanent magnet requires\nthe MAE to be uniaxial. Here we consider Fe 3Sn0:75M0:25, where M= Si, P, Ga, Ge, As, Se, In,\nSb, Te and Bi, and show how di\u000berent dopants on the Sn sublattice a\u000bect the MAE and can alter\nit from planar to uniaxial. The stability of the doped Fe 3Sn phases is elucidated theoretically\nvia the calculations of their formation enthalpies. A micromagnetic model is developed in order\nto estimate the energy density product ( BH)maxand coercive \feld \u00160Hcof a potential magnet\nmade of Fe 3Sn0:75Sb0:25, the most promising candidate from theoretical studies. The phase sta-\nbility and magnetic properties of the Fe 3Sn compound doped with Sb and Mn has been checked\nexperimentally on the samples synthesised using the reactive crucible melting technique as well as\nby solid state reaction. The Fe 3Sn-Sb compound is found to be stable when alloyed with Mn. It is\nshown that even small structural changes, such as a change of the c/a ratio or volume, that can be\ninduced by, e.g., alloying with Mn, can in\ruence anisotropy and reverse it from planar to uniaxial\nand back.\nPACS numbers: 75.50.Ww, 75.30.Gw, 75.20.En\n\u0003olga.vekilova@physics.uu.se\n2I. INTRODUCTION\nStrong permanent magnets creating high magnetic \feld are of ultimate importance for\nmany technological applications, from magnetic resonance imaging to magnetic hard disk\ndrives in information storage. They are also used in a number of green energy applications,\nlike motors for hybrid and electric cars and direct-drive wind turbines [1]. The strongest\nknown permanent magnets typically contain rare earth elements [2]. For the past few decades\nthe demand for such magnets has substantially increased. As the cost of the rare-earth based\nmaterials is high, the search for magnets, that are cheaper and contain smaller amounts of\nrare earth elements, has become an important \feld of research [1, 3, 4].\nFerromagnets with rather high Curie temperature (T C) above 400 K and high saturation\nmagnetization as well as high magnetocrystalline anisotropy energy (MAE) are considered\nas good candidates for permanent magnet applications. Furthermore, for such materials\nthe axis corresponding to the longest lattice constant, i.e. the c axis in most hexagonal\nstructures, should be the unique easy magnetization direction [1, 3, 4]. These properties can\nbe found in particular in Fe-rich materials with non-cubic uniaxial crystal structures. The\nhexagonal Fe 3Sn compound satis\fes these conditions to a large degree. Among \fve existing\nintermetallic compounds containing Fe and Sn the Fe 3Sn phase is one of the most attractive\nones, due to the highest concentration of iron and therefore highest magnetic moment. The\nother advantages of the rare-earth free Fe 3Sn system are its relatively low price and rather\nhigh T Cof about 743 K [5]. However, as it has recently been shown both experimentally\nand theoretically, the magnetocrystalline anisotropy of Fe 3Sn is planar, which is undesirable\nfor a permanent magnet [6]. It has also been suggested by Sales et. al that alloying with\nSb might change the anisotropy to uniaxial [6]. Motivated by this research we studied\nthe in\ruence of di\u000berent dopants on the MAE of the Fe 3Sn compound. The electronic\nstructure and magnetic properties, as well as the e\u000bect of the hexagonal c/a ratio on the\nmagnetocrystalline anisotropy in the Fe 3Sn compound and its alloys were addressed from\n\frst principles by means of highly accurate full-potential linear mu\u000en-tin orbital (LMTO)\nmethod implemented in the RSPt code. The stability of the doped phases were elucidated\nby the calculation of formation enthalpies by means of the VASP code.\nTheoretical calculations were combined with an experimental study in order to verify\nthe phase stability as well as the magnetic properties of Fe 3Sn compounds. Combinatorial\n3materials science and high-throughput screening methods, such as Reactive Crucible Melting\n(RCM) technique, o\u000ber an e\u000ecient strategy for the discovery of new materials with promising\nproperties. Using the knowledge of the required synthesis conditions for the formation of\nthe Fe 3Sn phase [7], we performed the RCM method in order to search for the (Fe) 3(SnM)\nphase. To get more insight into the magnetic properties of the (Fe) 3(SnM) phase, additional\nexperiments were performed using the Solid State Reaction (SSR) method, which was used\nfor the preparation of the desirable compound from the mixture of the starting elements\nby means of atomic di\u000busion. For the most promising system a micromagnetic model was\ndeveloped to calculate the magnetic induction as a function of the internal \feld.\nThe paper is organized as follows. Section II A describes theoretical methods used for the\n\frst-principles calculations. Section II B provides the experimental details. In sections III A,\nIII B, III C and III D the results of theoretical simulations as well as experimental results are\ngiven. Section III E addresses the results of the micromagnetic simulations. Conclusions are\ngiven in section IV.\nII. METHODS\nA. Theory\nThe high temperature phase of Fe 3Sn has a hexagonal crystal structure with space group\nP63/mmc (#194) and contains 8 atoms per unit cell (see Fig. 1). For calculations of\nphase stability Vienna Ab Initio Simulation Package (VASP) [8{10] was used within the\nprojector augmented wave (PAW) method [11]. The electronic exchange and correlation\neffects were treated by the generalized gradient approximation (GGA) in the Perdew, Burke,\nand Ernzerhof (PBE) form [12] in all the used methods. A 64 atom supercell of Fe 3Sn\ncomprising of 46 Fe and 16 Sn atoms was considered. The plane-wave energy cut-o\u000b was set\nto 350 eV. The converged k-point mesh was found to be 8 \u00028\u00028 k-points. The obtained\nmagnetic moment on iron was \u00182:4\u0016Bper atom, in agreement with previous studies [6].\nThe Sn atoms of the Fe 3Sn compound were partially replaced by alloying elements. In or-\nder to examine the phase stability of considered doped systems for each impurity, i.e. M=Sb,\nGa, Ge, and Hf, we calculated 3 di\u000berent distributions of impurity atoms, namely ordered,\nrandom (i.e. mimicking a disordered alloy), and phase separated (i.e. mimicking clusteriza-\n4FIG. 1. (Color online) 1 \u00021\u00022 Fe 3Sn hexagonal cell with one impurity atom on the tin sublattice.\nIron atoms are shown with brown spheres, Sn atoms with grey and M (M=Si, P, Ga, Ge, As, In,\nSb, Te and Bi) impurity atom is shown with the green sphere.\ntion of dopants) (See Supplemental Materials). We used the special quasi-random structure\n(SQS) technique [13{15] to generate the corresponding supercells. For the estimation of\nphase stability of ternary compounds the following equation was used:\n\u0001H=HFe3SnxM1\u0000x\u0000xHFe3Sn\u0000(1\u0000x)HFe3M (1)\nwhereHFe3SnxM1\u0000xis the enthalpy of a ternary compound and 1 \u0000xis the concentration of\nM, an impurity element on the Sn sublattice.\nThe 1 \u00021\u00022 supercell of Fe 3Sn comprising of 12 iron and 4 tin atoms, one of which\nis further substituted by an impurity (see Fig. 1), was used for the calculation of the\nmagnetic properties with help of the full-potential linear muf\fn-tin orbital (FP-LMTO)\nmethod implemented in the RSPt code [16, 17]. We performed integration over the Brillouin\nzone, using the tetrahedron method, with Bl ochl's correction [18]. The k-point convergence\nof the MAE for the chosen supercell size was found when increasing the Monkhorst-Pack\nmesh [19] to 24 \u000224\u000224, that was further used in all calculations.\nThe following impurities were considered in Fe 3Sn0:75M0:25compound: M=Si, P, Ga, Ge,\nAs, Se, In, Sb, Te, and Bi. With one substitutional impurity atom in the considered supercell,\nthe dopant concentration is \fxed to 6.25 at.%. The equilibrium lattice parameter and\nhexagonal c/a ratio were calculated for every structure using the VASP code. The e\u000bective\nexchange interaction parameters (J ij) were obtained using Lichtenstein et al. method [20,\n21], as implemented in RSPt [22]. In this technique the energy of the system is mapped\n5FIG. 2. (Color online) Enthalpy of formation (\u0001H) of the ternary Fe 3Sn0:75M0:25compound as\na function of the concentration of dopant M, calculated in comparison with the Fe 3Sn binary on\nthe left-hand side and Fe 3M (M=Ga, Ge and Hf) binary compound or a mixture of Fe and the M\nelement on the right-hand side of the dashed vertical line.\nonto a classical Heisenberg model with the following Hamiltonian:\n^H=\u00001\n2X\ni6=jJij~ ei\u0001~ ej; (2)\nwhere~ eidenotes the unit vector along the magnetic moment at the site i. The exchange\nparameter between sites iandjis de\fned in the following way:\nJij=T\n4X\nnTr\"\n^\u0001i(i!n)^G\"\nij(i!n)^\u0001j(i!n)^G#\nij(i!n)#\n; (3)\nwhere T is the temperature, \u0001 is the on-site exchange potential, Gijis an inter-site Greens\nfunction and i!nis then-th fermonic Matsubara frequency [22].\nB. Experiment\nScreening through a large variety of material compositions and possible temperature\nstability ranges using traditional equilibrium alloy approach, which is a one-alloy-at-a-time\npractice, is a laborious task. Combinatorial materials science and high-throughput screening\nmethods enable the synthesis of large number of samples at once and therefore speed up\nthe discovery of the new materials. Furthermore, using high-throughput characterization\n6methods, their properties can be e\u000eciently determined. Several phase diagrams of material\nsystems have been constructed using di\u000berent combinatorial approaches [23, 24], e.g. thin\n\flm deposition [25] and bulk high-throughput techniques, such as reactive di\u000busion method\n[26], and reactive crucible melting [7, 27{31].\nThe RCM method is \frstly introduced in Ref. [30] as a tool to search for hard magnetic\nphases. The method is based on di\u000busion processes driven by the formation of concentration\ngradient between the crucible material and other elements which are \flled into it. Production\nprocedure and working principle of the RCM method is fully described in Ref. [7], where the\nmethod is applied to the Fe-Sn binary system and all \fve known intermetallic compounds of\nthe system were synthesized in the reactive crucibles. The Fe 3Sn (3:1) phase was obtained in\nRCM by quenching of the crucibles after annealing at high temperatures (1023 K-1098 K).\nTo explore the hexagonal Fe 3Sn phase in the quest for uniaxial anisotropy, Fe-Sn xM1\u0000x\ncrucibles with M = Sb, Si, Ga, Ge, Pb, In, Bi and 0 :5< x < 0:75 were synthesized. To\nfurther extend our search, quaternary Fe-Mn, Sn, Sb crucibles were additionally produced.\nThe crucibles were made of 99.95 % pure Fe and they were \flled with about 1 g of the rest\nelements of the examined system. The \flling elements with the purity >99:9 were added\nin the form of crushed pieces. The samples were annealed at three selected temperatures of\n1013 K, 1043 K, and 1073 K for one week and subsequently quenched. For details we refer\nto [7].\nFor high-throughput characterization, the microstructure of the formed phases was stud-\nied by Philips XL30 FEG scanning electron microscopy (SEM) in back-scattered electron\n(BSE) contrast mode and their chemical compositions were determined using energy disper-\nsive x-ray (EDX) spectroscopy. In addition, a Zeiss Axio Imager.D2m magneto-optical Kerr\ne\u000bect (MOKE) microscopy was used to display the magnetic domain structure of the formed\nphases which may give a clear hint for identi\fcation of the phases with uniaxial anisotropy.\nSeveral Fe yMn3\u0000ySnxSb1\u0000xsamples (y=3 and x=1; y=2.5, 2.25, 2, 1.5 and x=0.75; and\ny=1.5 and x = 0.9) were prepared by solid state reaction (SSR) [32] through the following\nprocedure. First, stoichiometric amounts of powders of the starting elements (99.9+% purity,\nparticles of less than 50 microns in size) were handmilled with an agate mortar and pestle\nand then compacted into pellets using pressures up to 0.5 GPa at room temperature (RT).\nPellets were then encapsulated in a quartz ampoule, heated to 1073 K for 48 hours in vacuum\nand then quenched into ice water.\n7This process was repeated twice in order to homogenize the composition in the sample.\nX-Ray Di\u000braction (XRD) measurements were performed on a Philips X'Pert Pro di\u000brac-\ntometer, in the Bragg-Brentano geometry, using Cu K \u000bradiation (\u0015= 1:5418 \u0017A). The\nsamples were placed on a spinner to avoid a possible preferential crystalline orientation.\nXRD patterns were analyzed by Rietveld re\fnements, through the FullProf Suite [33], us-\ning a Thompson-Cox-Hastings pseudo-Voigt function to describe the pro\fle of the peaks.\nTemperature dependent magnetization was measured up to 823 K, using H=0.01 T, in a\nvibrating sample magnetometer EZ7-VSM from Microsense.\nIII. RESULTS\nA. Phase stability\nThe enthalpy of formation was calculated for ternary Fe 3Sn0:75M0:25alloys, where M=Sb,\nGa, Ge and Hf for di\u000berent concentrations of dopants from 6.25, 12.5, 18.75 and up to 25 at.\n% (see Fig. 2). The energies of these systems were examined and compared with the energies\nof the binary Fe 3Sn compound on the left-hand side and binary Fe 3M (or the mixture of pure\nFe and M if the binary phase was not energetically favorable) on the right-hand side of Fig.\n2 (see also Eq. 1). For the sake of simplicity, whether this particular phase was compared\nwith the binary phase or with a mixture of pure elements, is speci\fed on the right hand side\nof Fig. 2 for each considered structure. A positive slope corresponds to an unstable ternary\ncompound, while the negative one corresponds to a stable structure (See Fig.2).\nInvestigating the phase diagrams of the Fe-Ga and Fe-Ge binaries [34, 35] one can see\nthat in the region up to 10-20 at. % of dopant elements at temperatures up to 1500 K\nthe Fe 3M phase decomposes into a mixture of \u000b-Fe and pure Ga or Ge phases [34, 35]. At\nhigher concentration of dopants the phase is, however, stable. For this reason, enthalpy of\nFe3Sn0:75M0:25(M=Ga, Ge) was calculated relative to both the mixture of pure elements as\nwell as the Fe 3M (M=Ga, Ge) phase, see Fig. 2. The dashed vertical lines show the area\naround 15 at. % of dopant concentration where the mixture of these phases might exist\ndepending on temperature and the dopant. At a concentration of dopants lower than \u001815\nat. %, when compared to the mixture of pure elements on the right-hand side, the enthalpy\nis negative indicating stability of the structure (see lower red and blue lines in Fig. 2).\n8However, when compared with the binary phase Fe 3M, the stability of the ternary structure\nshould be estimated from the upper positive curves for both Ga and Ge dopants (see upper\nred and blue lines in Fig. 2).\nIn the considered Fe 3Sn0:75M0:25alloy the concentration of dopants is equal to 6.25 at %,\nwhich means the structures could be considered as stable based on this theoretical estima-\ntion. In contrast, checking the phase diagram of Fe-Sb one can see that the binary Fe 3Sb\nphase cannot exist. Furthermore, when the formation enthalpy of Fe 3Sn0:75Sb0:25phase was\ncalculated in comparison with the mixture of pure elements on the right hand side (see the\ngreen curve in Fig. 2) it appeared that the phase is unstable. However, as it is shown in\nsection III B, this phase has shown promising magnetic properties, so in order to stabilize it\nexperimentally Mn was added to the Fe sublattice, see sections III C and III D.\nSimilarly, for the Hf dopant even comparing with the mixture of elements (see the black\nline in Fig. 2), the enthalpy curvature is positive indicating thermodynamic instability of\nthis phase. For that reason Hf was further excluded from consideration and calculation of\nthe magnetic properties.\nB. Magnetocrystalline anisotropy\nWe calculated the saturation magnetic moment of Fe 3Sn0:75M0:25system. For the pristine\nFe3Sn it is equal to \u00160Ms= 1.49 T, or Ms=1.19 MA/m, in very good agreement with\nprevious theoretical and experimental estimations [6]. Due to the low concentration of the\nimpurities, 6.25 at. %, in the doped system, the obtained value of saturation magnetization\nis close to the one of the undoped system. For instance, for the case of Sb impurity it is\nequal to 1.51 T, or 1.2 MA/m.\nMagnetocrystalline anisotropy of the Fe 3Sn0:75M0:25system, where M=Si, P, Ga, Ge,\nAs, Se, In, Sb, Te and Bi was calculated from \frst principles electronic structure theory.\nThe resulting MAE values are shown in Fig. 3 for the dopants grouped according to their\npositions in the Periodic Table. For the pristine Fe 3Sn the resulting anisotropy was found to\nbe equal to -1.5 MJ/m3(see Fig. 3). In magnitude, this is one of the largest values among all\nthe considered structures, however, the minus sign indicates the easy plane of magnetization.\nThese results are in good agreement with other theoretical and experimental estimations (-\n1.59 MJ/m3and -1.8 MJ/m3respectively [6]).\n9FIG. 3. (Color online) Calculated values of magnetocrystalline anisotropy energy (MAE), K 1,\nand easy magnetization direction for di\u000berent impurities for Fe 3Sn alloy grouped according to the\npositions of dopants in the Periodic Table. A negative value of K 1indicates the in-plane easy\nmagnetization axis, while the positive one shown with the \flled blue rectangulars corresponds to\nthe desirable uniaxial magnetocrystalline anisotropy.\nAs one can see in Fig. 3, the largest magnitude of the MAE (albeit favoring in-plane\neasy axis), of all the considered structures with dopants, was obtained for the Fe 3Sn0:75In0:25\ncompound. An uniaxial anisotropy was found for M=As, Sb and Te. However, the MAE\nfor the Te doping is very small. The anisotropy of Fe 3Sn0:75Sb0:25is in agreement with the\nexisting theoretical data of 0.5 MJ/m3[6]. As it was mentioned in Sec. III A, in order to\nstabilize the Fe 3Sn0:75Sb0:25system Mn was added on the Fe subblatice. The anisotropy of\nFe1:5Mn1:5Sn0:75Sb0:25system is equal to -1.49 MJ/m3, which is very close to the value for the\nundoped Fe 3Sn system. Adding Sb to the pristine Fe 3Sn allows for the change of anisotropy\nfrom planar to uniaxial, while adding Mn for stabilization reverts it back to planar.\nThe obtained MAE data were plotted as a function of the number of valence electrons per\nformula unit in the Fe 3Sn0:75M0:25system for doping atoms from three rows of the Periodic\nTable, see Fig. 4. Positive numbers correspond to a uniaxial MAE. As one can see, a\nsimilar tendency is shown for the anisotropy in all considered rows. Starting from the lowest\nvalues, corresponding to high but planar anisotropy, for the dopants from the Group IIIA\n(Ga and In) the MAE moves towards positive values, with an increase of the number of\nvalence electrons in the system. The maximal value of MAE, which is positive for As and Sb\ndopants, is obtained for Group VA elements. This tendency is shown for all the considered\n10FIG. 4. (Color online) The magnetocrystalline anisotropy energy (MAE) in Fe 3Sn0:75M0:25com-\npound as function of the number of valence electrons, N, of the system for three rows: Si-P (shown\nwith the green line), Ga-Ge-As (shown with the red line) and In-Sn-Sb-Te (shown with the blue\nline). Doping from Group IIIA, Group IVA, Group VA and Group VIA, correspond to N=27.75,\n28.25 and 28.5, respectively.The area of interest, where the easy magnetization axis is uniaxial,\nis shown with the gradient background. The horizontal dashed line shows the magnitude of the\nFe3Sn0:75In0:25anisotropy, as its number of valence electrons is substantially lower.\nrows of elements, notifying that Group VA dopants seem to be the most promising for the\nconsideration, speci\fcally, As and Sb. Two other elements of this group, P and Bi, do not\n\rip anisotropy to uniaxial (see Fig. 3). Anisotropy of Fe 1:5Mn1:5Sn0:75Sb0:25is shown with\nthe horizontal dashed line. The number of valence electrons in the unit cell of this system\nis 26.75, which is out of the scale of Fig. 3. The above described tendency together with\nthe result of the Mn addition indicates that alloying with other elements around Sn in the\nPeriodic Table, not considered in this search, can hardly bring new candidates able to \rip\nthe easy magnetization axis to uniaxial. However, more studies can be of interest.\nThe dependence of the magnetic properties on the number of valence electrons in dopants\nwas further illustrated by the calculation of the Heisenberg exchange parameters J ijs between\nthe atoms (see Fig. 5). The Fe 3Sn system, as well as the Fe 3Sn0:75M0:25alloys, where M=Sb,\nAs and In, were shown to be strongly ferromagnetic as indicated by large and positive J 0\n(de\fned as a sum of all Jij's). In Fig. 5 the systems with planar anisotropy, the pristine\nFe3Sn (a) and Fe 3Sn0:75In0:25(d), which has the largest planar anisotropy, are shown in\n11FIG. 5. (Color online) Inter-site exchange parameters J ijbetween the iron atoms iandjseparated\nby the distance R ijof pristine Fe 3Sn compound (a), as well as of Fe 3Sn0:75M0:25(M=Sb (b), As (c)\nand In (d)). Systems with planar MAE are shown with the blue lines, while systems with uniaxial\nMAE are shown with red lines. astands for the lattice constant.\ncomparison with the systems with signi\fcant uniaxial MAE, namely, Fe 3Sn0:75Sb0:25(b) and\nFe3Sn0:75As0:25(c).\nNotice that the value of the nearest neighbor interaction, corresponding to the shortest\ninter-atomic distance, di\u000bers slightly among the considered systems. The largest nearest-\nneighbor \frst exchange interaction value was obtained for the Fe 3Sn0:75Sb0:25system ( \u0018\n1.98 mRy) with the highest positive value of MAE. A plot of the nearest neighbor exchange\n12FIG. 6. (Color online) Dependence of the nearest neighbor exchange parameter on the magne-\ntocrystalline anisotropy of Fe 3Sn and Fe 3Sn0:75M0:25(M=Sb, As and In). The area below zero\ncorresponds to the planar anisotropy while the area above zero represents the uniaxial anisotropy.\ninteraction as function of the MAE is shown in Fig. 6. The second highest nearest-neighbor\nexchange was obtained in Fe 3Sn0:75As0:25(\u00181.88 mRy). The J NNvalue for the system with\nlargest negative anisotropy, Fe 3Sn0:75In0:25is the smallest of all ( \u00181.57 mRy). The J NN\nvalue for pristine binary system Fe 3Sn has a value of \u00181.75 mRy, which is higher than that\nof the system with In, but lower than the one out the systems with uniaxial anisotropy.\nThis clearly indicates that in the presently investigated alloys, there is a certain correlation\nbetween the strength of the nearest neighbor couplings J ijs and MAE of the system.\nTaking into account that atomic relaxations substantially in\ruence the MAE, we also\nstudied the e\u000bect of the hexagonal lattice ratio, c/a, on the value of MAE. Fig. 7 shows the\nchange of the MAE when the c/a ratio increases from 1.4 to 1.9 for Fe 3Sn0:75M0:25(M= Sn,\nGe, As and Sb) systems at a \fxed equilibrium volume. The averaged equilibrium c/a ratio\n(\u00181.58) is shown with the vertical dashed line. As the dopant concentration is low (6.25 at\n%) the deviations from 1.58 of the c/a ratio for di\u000berent dopants are rather minor.\nFor all considered dopants the MAE behavior with the c/a increase is basically linear,\nillustrating that when the crystal is stretched, the easy axis can be switched from planar\nto uniaxial for all the structures. For instance, for Ge-doped Fe 3Sn a change of the c/a\nratio at the \fxed volume leads to the change of the anisotropy value from rather high with\n13FIG. 7. (Color online) Dependence of the magnetocrystalline anisotropy on the hexagonal c/a ratio\nfor M=Sn, Ge, As and Sb impurities. The averaged equilibrium c/a ratio about 1.58 is shown with\nthe dashed line.\nplanar easy magnetization direction, \u0018-2.7 MJ/m3, to almost \u00180.7 MJ/m3with uniaxial\neasy magnetization direction. This strain induced change is largest for Ge doping, out of all\nconsidered dopants. It is worth noting that changing MAE to uniaxial requires a variation\nof c/a from \u00181.58 to 1.8, corresponding to 14 %. In the experiment it is hardly possible\nto change c/a of such systems by more than few percents without structural changes, i.e.\nwithout altering the hexagonal structure.\nIt is important to mention that for all the structures there is a peak of the MAE vs. c/a\ncurve (Fig. 7) in the region near the equilibrium c/a. In some cases, like for M=Sb and\nAs it can lead to the switching of the easy magnetization direction, however, this region\nis rather narrow and a small change in the c/a ratio (like 0.5% from the equilibrium) can\nswitch the axis from planar to uniaxial and back. The highest peak is for Sb- doped Fe 3Sn;\nit corresponds to the highest anisotropy value and uniaxial easy magnetization direction.\nFor Ge-doped structure, as well as for the binary Fe 3Sn slight modi\fcations of the c/a\nnear the equilibrium are not resulting in the MAE peak su\u000eciently high to switch the easy\nmagnetization direction. It is interesting to notice, that for these two structures the peak is\nnot exactly at the equilibrium c/a ratio but shifted, meaning that the small change of the\nc/a ratio due to, for instance, alloying or heating, can slightly increase anisotropy for some\nstructures. Therefore we underline that the MAE of Fe 3Sn with any of the dopants turns\nout to be very sensitive to the change of c/a.\n14C. Reactive Crucible Melting technique\nThe e\u000bect of substituting Sn by its neighboring elements was experimentally studied by\nhigh-throughput RCM method and the production of Fe-Sn xM1\u0000xcrucibles, M= Sb, Si, Ga,\nGe, Pb, In, Bi and 0 :5< x < 0:75. The di\u000busion zone of the crucibles annealed at the\nselected temperatures in the temperature interval from 1013 K to 1073 K were carefully\nscreened by the EDX analysis. None of the Fe 3(Sn,M) structures was stable and therefore\nnot formed in any of the crucibles (for details on the results see Supplemental Material).\nOur theoretical calculations predict the change of anisotropy from planar to uniaxial\nin Fe 3Sn system by partial substitution of Sn by Sb. However, the formation enthalpy\nplot shows that Fe-Sn xSb1\u0000xis unstable in the addressed concentration range (see Fig. 2).\nBesides that, it was experimentally found that Sb destabilized the formation of 3:1 structure\nand instead the Fe 3(Sn,Sb) 2phase was formed. This phase is not desirable for permanent\nmagnet applications due to the small concentration of Fe and therefore low magnetization\nas well as its negligible MAE [6, 7]. For stabilization of Fe 3(Sn,Sb) phase, substitution of Fe\nby Mn (with one less electron) was suggested to be e\u000bective to compensate an extra electron\nof Sb in comparison to Sn. Accordingly, the Fe-Mn 0:75Sn0:5Sb0:5and Fe-Mn 1:5Sn0:75Sb0:25\ncrucibles were synthesized and annealed in the temperature range of 1013 K to 1073 K.\nIt was found that Mn and Sb substitution preserved the hexagonal Ni 3Sn structure of the\nparent Fe 3Sn phase and (Fe,Mn) 3(Sn,Sb) phase formed in the quaternary reactive crucibles.\nFig. 8 shows the microstructure forming in the di\u000busion zone of the Fe-Mn 0:75Sn0:75Sb0:25\nreactive crucible annealed at 1073 K for 5 days and subsequently quenched. A thin layer\nof (Fe 0:6Mn0:4)3(Sn0:75Sb0:25) phase is formed and bordered by Fe crucible. In addition,\n(Fe0:5Mn0:5)3(Sn0:75Sb0:25)2phase is formed in large quantity on top of the 3:1 phase. The\nphase stability of the Sb-Mn doped Fe 3Sn compound is con\frmed by RCM. However, to\nmeasure magnetic properties, such as M s, TC, and MAE; a single-phase material has to be\nsynthesized. That was done by the solid state reaction (SSR).\nD. Solid State Reaction\nSeveral Fe yMn3\u0000ySnxSb1\u0000xsamples (y=3 and x=1; y= 2.25, 2, 1.5 and x=0.75; and\ny=1.5 and x = 0.9) with di\u000berent concentrations of Mn and Sb were prepared by solid state\n15FIG. 8. BSE image of the Fe- Mn 0:75Sn0:75Sb0:25crucible annealed at 1013 K. Narrow layer of\n(Fe0:6Mn0:4)3(Sn0:75Sb0:25) is formed. However 3:2 phase is the dominant forming phase in the\ncrucible.\nFIG. 9. (Color online) XRD patterns of the Fe yMn3\u0000ySnxSb1\u0000xsamples.\nreaction. XRD patterns of the produced samples by two subsequent SSRs are shown in Fig.\n9. For all the samples, the 3:1 phase was found though only for the parent Fe 3Sn alloy we\nfound a single phase. Rietveld re\fnements were performed on all the XRD patterns and the\nresults are shown in Table I.\nThe lattice parameters of the alloys (see Table I and Figure 10) follow an increasing trend\nfor higher values of the Mn concentration. The dependence of the lattice parameters on the\nconcentration of Sb was not studied as only two points were available.\nTemperature dependent magnetization curves M(T) were measured for two selected sam-\nples with the highest amount of the 3:1 phase, namely, Fe 3Sn, and with equal concentration\nof Fe and Mn, Fe 1:5Mn1:5S0:75Sb0:25. The M(T) curves are shown in Fig. 11. The calculation\n16Sample a( \u0017A) c( \u0017A)\nFe3Sn 5.4621(5) 4.3490(6)\nFe2:25Mn0:75Sn0:75Sb0:255.4858(5) 4.3721(6)\nFe2Mn1Sn0:75Sb0:25 5.5000(4) 4.3829(6)\nFe1:5Mn1:5Sn0:75Sb0:255.5338(1) 4.4270(2)\nFe1:5Mn1:5Sn0:9Sb0:15.5545(3) 4.4453(4)\nTABLE I. Lattice parameters of the Fe yMn3\u0000ySnxSb1\u0000xalloys, obtained by Rietveld re\fnements\nof the XRD patterns\nFIG. 10. (Color online) Evolution of the lattice parameters and volume, obtained by Rietveld\nre\fnements of the XRD patterns, of the Fe yMn3\u0000ySnxSb1\u0000xalloys. These values are also shown\nin Table 1.\nof the Curie temperatures were performed by the derivative of the M(T). The obtained Curie\ntemperature of Fe 3Sn system, T C=748 K, is rather high and in very good agreement with\nthe existing experimental data (T C=743 K [5] and T C=725 K [6]), while that of the diluted\nsample, 393 K, is much lower than that of the parent composition Fe 3Sn. Similar values\nwere found previously in literature for Fe 1:5Mn1:5Sn0:9Sb0:1and Fe 1:5Mn1:5Sn0:85Sb0:15, with\nTC= 405 K [6].\nIn order to examine how the addition of Mn to the Fe 3(SnSb) compound a\u000bects the\n17FIG. 11. (Color online) M(T) curves for selected compositions within the Fe yMn3\u0000ySnxSb1\u0000xal-\nloys. The abrupt drop corresponds to the Curie temperature, determined precisely by the derivative\nof the curve.\nanisotropy, the domain structure of the formed 3:1 phase was investigated. For the fer-\nromagnets with uniaxial anisotropy observation of characteristic domain patterns, namely\nstripe or branched domains, is expected. However, our Kerr analysis on the formed Mn/Sb\nsubstituted 3:1 phase shows non-uniaxial domain structure. This observation agrees with\nthe negative theoretical value of anisotropy for the Fe 1:5Mn1:5Sn0:75Sb0:25system.\nE. Micromagnetic simulations\nA micromagnetic model was developed in order to estimate the energy density product\n(BH)maxand coercive \feld \u00160Hcof a potential magnet made of Fe 3Sn0:75Sb0:25. Using the\nsoftware tool Neper [36] we created a synthetic microstructure based on Voronoi tessellation\nas shown in Figure 12.\nThe granular grain structure consisted of 27 grains with an average grain diameter of\n50 nm. Each grains spontaneous magnetization, anisotropy energy density were taken from\nab-initio calculations ( \u00160Ms;grain= 1:52 T,Ku;grain= 0:33 MJ/m3). The exchange sti\u000bness\nconstant was assumed to be Aex;grain= 10 pJ/m. The grains easy axes were randomized\nwithin a cone angle of 5 with respect to positive z-axis.\nBetween the grains we assumed a 4 nm thick iron-rich ferromagnetic grain boundary phase\n18FIG. 12. (Color online) Synthetic microstructure used for the simulation of the demagnetization\ncurve of a nanocrystalline Fe 3Sn0:75Sb0:25magnet.\n(Ku;gb= 0). Its spontaneos magnetization was assumed to be M s;gb=0.81 T. Accordingly,\nthe exchange sti\u000bness constant was reduced to A ex;gb= 3.7 pJ/m.\nThe magnetization value of the grain boundary phase was obtained by numerical opti-\nmization. We used the numerical optimization framework Dakota [37] and maximized the\nenergy density product using \u00160Ms;gbas a free parameter. The volume fraction of the grain\nboundary phase was 10 percent. The model was discretized into a uniform mesh with an\nedge length of 2 nm. A \fnite element energy minimization code [38] was used to compute\nthe static hysteresis properties of the proposed model.\nIn order to compute the demagnetization curve we calculated the equilibrium states for\na subsequently decreasing external \feld Hext. The \feld step \u00160\u0001Hext=\u00001 mT. In order\nto compute the B(H) loop and the expected energy density product we corrected the loop\nwhich was obtained for a magnet with cubic shape with the macroscopic demagnetization\nfactorN= 1=3.\nFigure 13 shows the computed B(H) curve and Fig. 14 illustrates the equilibrium mag-\nnetic states before and after the \frst switching event, respectively. The computed coercive\n\feld is\u00160Hc=\u00000:49 T and the computed energy density product is 290 kJ/m3. This\nis about 3/4 of the maximum energy density product reported for commercially available\nNd-Fe-B magnets [2, 39, 40].\n19FIG. 13. (Color online) Computed B(H) curve (magnetic induction as function of the internal\n\feld) for Fe 3Sn0:75Sb0:25. Remanence, energy density product, and coercive \feld are marked with\na rectangle, diamond, and circle respectively.\nFIG. 14. (Color online) During magnetization reversal domain walls become pinned near the grain\nboundaries.\nIV. DISCUSSION AND CONCLUSIONS\nElectronic structure and magnetic properties of the hexagonal Fe 3Sn compound doped\nwith 6.25 at % of Si, P, Ga, Ge, As, Se, In, Sb, Te, and Bi were studied theoretically from\n\frst principles and experimentally. Our calculations show that at low concentrations of\nsome dopants, such as Ga and Ge, the considered phases are stable. In contrast to that, in\nthe case of the Sb dopant the phase was shown to be unstable against decomposition into\nthe mixture of pure elements. Our experimental study using RCM method supports this\ntheoretical prediction that the addition of Sb into Fe-Sn system destabilizes the formation\n3:1, however further doping of Mn into the Fe sublattice stabilizes the structure. The SSR\ntechnique con\frmed stabilization of the 3:1 phases with di\u000berent concentrations of Mn.\n20Theoretical simulations predict that doping with As, Sb and Te can change the easy\nmagnetization direction to uniaxial. However, the change of the c/a ratio also substantially\nin\ruences the MAE. In the case of Sb and As the peak in the MAE vs c/a dependence leads\nto the change of the easy magnetization axis from planar to uniaxial in the region close to\nthe equilibrium c/a. For the Sb and As dopants, the region where anisotropy is uniaxial,\nis rather narrow and even a small change of c/a, for instance due to alloying with small\namounts of Mn, can lead to the switch of the easy magnetization direction.\nAs follows from our calculations, the change of anisotropy due to addition of Mn to\nFe3Sn0:75Sb0:25system turns uniaxial anisotropy back to planar. This estimation indicates\nthat the predicted uniaxial anisotropy can hardly be observed experimentally. On one\nside the presence of Mn stabilizes the Sb-doped alloy, but on the other side it basically\n\rips the anisotropy back to the planar value of an undoped system. The experimentally\nstabilized (FeMn) 3SnSb phase shows non-uniaxial domain structure in nice agreement with\nthe theoretical prediction.\nWe studied all the dopants around Sn in the Periodic Table that can occupy the Sn\nsublattice and can likely be mixed with Fe 3Sn. From the MAE data grouped in the way\nas these dopants are placed in the Periodic Table of elements one can see the tendency to\nincrease the value of MAE from the III to the V group with the increase of the number\nof valence electrons. However, in the VI group there is a slight decrease for Te and Se,\nrespectively. Thus the Group V seems to be the most promising out of all the considered\ngroups and the Sb addition gives the largest uniaxial anisotropy. Looking at the vertical\ndistribution in the Periodic Table, it appears that MAE increases from P to As and to\nSb, but then for Bi there is a clear drop. We conclude that the most preferable choice of\ndopants should be the column with As and Sb, and we do not expect that other dopants\ncan substantially improve the desired properties of the Fe 3Sn compound.\nFurther, the micromagnetic simulations allowed us to estimate the magnetic induction of\nthe most promising system, Fe 3Sn0:75Sb0:25, as a function of the internal \feld. The computed\ncoercive \feld is equal to -0.49 T. The calculated density energy product, 290 kJ/m3, is at\nthe level of best known to date magnets.\nWe investigated the Fe 3Sn0:75M0:25system by di\u000berent theoretical approaches as well\nas experimentally, in order to get a wider view on the magnetic properties of the system.\nCertain dopants, like Sb, can turn MAE of the hexagonal Fe 3Sn uniaxial. Furthermore, other\n21magnetic properties of this system, such as saturation magnetization of 1.51 T and energy\ndensity product compatible with the values of the best known magnets are impressively\nhigh. However, such a turn of MAE is very sensitive to lattice deformations and the value\nof the magnetocrystalline anisotropy is reduced compared to the one of the parent phase.\nFurther, the hexagonal phase becomes unstable with respect to decomposition. Addition of\nMn allowed us to stabilize the system with dopants experimentally, however, the anisotropy\nturned back to planar. Therefore further search for better stabilizers or their combinations\n(the so-called co-doping) might be considered in order to \fnd the compromise between the\nstability and uniaxial MAE.\nACKNOWLEDGMENTS\nAuthors acknowledge support from NOVAMAG project, under Grant Agreement No.\n686056, EU Horizon 2020 Framework Programme. The computations were performed on\nresources provided by the Swedish National Infrastructure for Computing (SNIC) at PDC\nand NSC centers. O.E. acknowledges support from STandUPP, eSSENCE, the Swedish\nResearch Council and the KAW foundation (grants 2012.0031 and 2013.0020). 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Coey, Scripta Materialia 67, 524 (2012), viewpoint Set No. 51: Magnetic Materials for\nEnergy.\n24SUPPLEMENTAL MATERIAL FOR \\TUNING MAGNETOCRYSTALLINE\nANISOTROPY OF Fe3SnBY ALLOYING\"\nTheoretical calculations\nFigure S1 shows the calculated density of states for the Fe 3Sn and Fe 3Sn0:75Sb0:25com-\npounds, respectively. Figure S2 shows the three tested distributions of M impurity atoms\non the Sn sublattice in the Fe 3Sn0:75M0:25compound. For the sake of simplicity only the\nresults for 50 at. % of impurities is shown in all the cases. As one can see in the case of the\nordered distribution, (a), impurity atoms are placed in layers mixed with the layers of the\nSn atoms. In the case of the random alloy, (b) atoms are distributed in an SQS-like manner\nwith the short-range order close to 0 for a few \frst coordination shells. For the case of the\nseparated distribution of atoms, (c), the cell is split into two parts and one is fully occupied\nwith the M atoms, while the other one is occupied exclusively with the Sn atoms. According\nto our estimation, for the low concentration of dopants, the ordered distribution is the most\nenergetically preferable.\nFIG. S1. Density of states of the Fe 3Sn and Fe 3Sn0:75Sb0:25compounds without spin-orbit coupling\ntaken into account.\n25RCM measurements\nUsing the RCM method, Fe-SnM crucibles with M= Sb, Ga, Ge, Si, In, Bi and Pb were\nsynthesized. The crucibles were annealed in the temperature range of 1013 K to 1073 K for 5\ndays and subsequently quenched. Theoretical calculations predict that doping of Sb changes\nthe planar anisotropy in the Fe 3Sn compound to uniaxial. Our experimental study using\nRCM shows a destabilization of the 3:1 phase in Fe-Sn 1\u0000xSbx, 0:15<x< 0:5 crucibles. The\n3:1 phase did not form in any of the synthesized samples, however, instead the 3:2 phase was\nstabilized. Figure S3 shows the BSE image of the di\u000busion zone in Fe-Sn 0:85Sb0:15reactive\ncrucible annealed at 1073 K. The only intermetallic compound formed in the crucible is the\n3:2 phase.\nIn order to check whether substituting Ge and Ga in the Fe 3Sn compound preserves the\nparent structure, Fe-SnGe(Ga) crucibles were synthesized. The di\u000busion zone of the crucibles\nannealed at a speci\fc temperature of 1013 K are shown in \fgure S4 for Fe-Sn 0:5Ga0:5and\nin \fgure S5 for Fe-Sn 0:5Ge0:5. The observed microstructure in Fe-SnGa crucible contains a\nsolid solution Fe 1\u0000xGaxregion with x up to 20 at. %. The remaining Ga has formed the\nFe62Ga35Sn3phase (composition found by EDX) in the di\u000busion zone and the Sn content is\nleft mainly unreacted. The crucibles annealed at the higher temperatures (not shown here)\ncontained no intermetallic compounds and only solid solution Fe 1\u0000xGaxwas found. In the\nGe containing crucible, Sn remained totally unreacted. On the Fe-rich side of the crucible, a\nsolid solution Fe 1\u0000xGexregion with x up to 17 atomic percent was formed. Moreover, distinct\nlayers of Fe 3Ge and Fe 2Ge intermetallic compounds were formed on top of the solid solution\nregion. A very similar microstructure was observed for the crucibles annealed at higher\ntemperatures up to 1073 K. Considering the above mentioned conditions, the stabilization\nof 3:1 compound was not possible.\nFurther experiments have been performed in order to stabilize the Fe 3(SnM) compound\nwith M= Si, In, Bi, Pb and Sn:M varied from 2:1 to 1:1. Our RCM study shows that\nno reaction has been occurred for the 3 latter substitutions after annealing at the selected\ntemperature range. Addition of Si to the Fe-Sn crucible mimics the microstructure forming\nin the Fe-Sn binary crucibles [S1] whereas Si totally dissolves in Fe and does not di\u000buse to\nany of the formed Fe-Sn binary intermetallic compounds.\nFigures S7 and S6 show the di\u000busion zone of the Fe-Sn 0:5Si0:5crucibles annealed at 1073 K\n26FIG. S2. Structure of the Fe 3Sn0:75M0:25system. Green spheres (impurity atoms) are distributed\non the Sn sublattice, shown with the grey spheres (host atoms) in (a) ordered, (b) random or (c)\nseparated ways.\nand 1043 K respectively. In the crucible annealed at 1073 K, the high temperature phase\nFe5Sn3was formed whereas at lower temperature (1043 K), FeSn, Fe 3Sn2and Fe 3Sn phases\nwere formed. Although the 3:1 structure is formed in the crucibles, however, according to\nthe EDX quanti\fcation, Si did not di\u000buse into 3:1 structure.\n27FIG. S3. BSE image of the Fe-Sn 0:85Sb0:25crucible annealed at 1073 K. Substitution of Sb led to\nformation of Fe 3(SnSb) 2and deformation of the desirable Fe 3Sn phase.\nFIG. S4. BSE image of the di\u000busion zone in the Fe-Sn 0:5Ga0:5crucible annealed at 1073 K. Sn\nremained almost unreacted and the only ternary intermetallic compound Fe 62Ga35Sn3is formed.\n28FIG. S5. BSE image of the di\u000busion zone in the Fe-Sn 0:5Ge0:5crucible annealed at 1073 K. Sn\nremained almost unreacted and no ternary intermetallic compounds are formed.\nFIG. S6. BSE image of the di\u000busion zone in the Fe-Sn 0:5Si0:5crucibles annealed at 1073 K. The\nbinary Fe 3Sn phase is not formed in the crucible annealed at 1073 K.\n29FIG. S7. BSE image of the di\u000busion zone in the Fe-Sn 0:5Si0:5crucibles annealed at 1043 K. The\nbinary Fe 3Sn phase is formed in the crucible annealed at 1043 K. However, Si totally dissolved in\nFe in a separate region and did not di\u000buse into the 3:1 structure.\n30[S1] B. Fayyazi, K.P. Skokov, T. Faske, D.Yu. Karpenkov, W. Donner, and O. Gut\reisch, Acta\nMaterialia 141, 434 (2017)\n31"    },    {        "title": "1803.10428v5.Perpendicular_magnetic_anisotropy_at_the_Fe_MgAl__2_O__4__interface__Comparative_first_principles_study_with_Fe_MgO.pdf",        "content": "Perpendicular magnetic anisotropy at the Fe/MgAl 2O4interface:\nComparative \frst-principles study with Fe/MgO\nKeisuke Masuda1and Yoshio Miura1, 2, 3, 4\n1Research Center for Magnetic and Spintronic Materials,\nNational Institute for Materials Science (NIMS),\n1-2-1 Sengen, Tsukuba 305-0047, Japan\n2Electrical Engineering and Electronics,\nKyoto Institute of Technology, Kyoto 606-8585, Japan\n3Center for Materials Research by Information Integration,\nNational Institute for Materials Science (NIMS),\n1-2-1 Sengen, Tsukuba 305-0047, Japan\n4Center for Spintronics Research Network (CSRN),\nGraduate School of Engineering Science, Osaka University,\nMachikaneyama 1-3, Toyonaka, Osaka 560-8531, Japan\n(Dated: June 17, 2021)\nAbstract\nWe present a theoretical study on interfacial magnetocrystalline anisotropy for Fe/MgAl 2O4.\nThis system has a very small lattice mismatch at the interface and therefore is suitable for real-\nizing a fully coherent ferromagnet/oxide interface for magnetic tunnel junctions. On the basis of\ndensity functional theory, we calculate the interfacial anisotropy constant Kiand show that this\nsystem has interfacial perpendicular magnetic anisotropy (PMA) with Ki\u00191:2 mJ=m2, which is a\nlittle bit smaller than that of Fe/MgO ( Ki\u00191.5{1.7 mJ=m2). Second-order perturbation analysis\nwith respect to the spin-orbit interaction clari\fes that the di\u000berence in Kibetween Fe/MgAl 2O4\nand Fe/MgO originates from the di\u000berence in contributions from spin-\rip scattering terms at the\ninterface. We propose that the insertion of tungsten layers into the interface of Fe/MgAl 2O4is a\npromising way to obtain huge interfacial PMA with Ki&3 mJ=m2.\n1arXiv:1803.10428v5  [cond-mat.mtrl-sci]  27 Dec 2018I. INTRODUCTION\nPerpendicular magnetic anisotropy (PMA) is an essential property for ferromagnets\n(FMs) in magnetic tunnel junctions (MTJs) to realize nonvolatile magnetic random ac-\ncess memories (MRAMs) [1]. The PMA is bene\fcial for obtaining su\u000eciently high thermal\nstability and low critical current in spin-transfer-torque MRAMs (STT-MRAMs), in which\ncurrent-induced spin-transfer torque is used for magnetization switching [1]. Although large\nPMA has been observed in several FMs such as D022Mn3Ga [2, 3],D022Mn3Ge [4, 5],L10\nMnGa [3], and L10FePt [6], MTJs with these FMs did not show su\u000eciently high tunnel\nmagnetoresistance (TMR) ratios, which is another important requirement for MRAM ap-\nplications. Therefore, interfacial PMA at interfaces between FMs and insulator barriers has\nattracted much attention mainly in MTJs consisting of Fe-based FMs and MgO barriers.\nIn addition to high TMR ratios [7{10], interfacial PMA has also been obtained in the\nMgO-based MTJs. By using thin CoFeB layers ( \u00181:3 nm), Ikeda et al. observed relatively\nlarge PMA at the interface of CoFeB/MgO/CoFeB MTJ [11]. In subsequent studies [12,\n13], Koo et al. demonstrated that Fe/MgO has a larger interfacial PMA than that of\nCoFe(B)/MgO, in agreement with theoretical predictions [14, 15]. Furthermore, interfacial\nPMA has also been observed in the heterostructure composed of the Heusler alloy Co 2FeAl\nand MgO [16, 17]. The interfacial PMA is also advantageous for voltage-torque MRAMs\n[18], because high interfacial PMA gives low write error rates in voltage-driven magnetization\nswitching.\nThe underlying mechanism of such interfacial PMA in Fe-based FM/MgO heterostruc-\ntures has been discussed in several theoretical studies. By analyzing the local density of\nstates (LDOS) and band structure in Fe/MgO, Nakamura et al. [19] clari\fed that the Fe\n3d3z2\u0000r2state is distributed away from the Fermi level owing to its hybridization with the O\n2pzstate, leading to interfacial PMA. Other studies [14, 20] also indicated the importance of\nthis hybridization using di\u000berent theoretical approaches. From a di\u000berent point of view, the\nrelation between PMA and orbital magnetic moment is another signi\fcant issue. A second-\norder perturbation theory by Bruno [21] revealed a proportional relation between magnetic\nanisotropy and anisotropy of orbital magnetic moment, which is the so-called Bruno rela-\ntion. Several theoretical studies have discussed the applicability of the Bruno relation to\nvarious Fe-based heterostructures [15, 22, 23]. Moreover, by means of x-ray magnetic circular\n2dichroism (XMCD) measurements, Okabayashi et al. [24] showed that the interfacial PMA\nin Fe/MgO can be explained qualitatively by the Bruno relation. This relation gives valuable\ninformation for understanding the interfacial PMA in Fe-based FM/MgO heterostructures.\nAlthough large interfacial PMA has been observed in Fe/MgO heterostructures, the lat-\ntice mismatch between Fe and MgO is rather large ( \u00184%), which is a drawback for practical\napplications. On the other hand, spinel oxide MgAl 2O4has a small lattice mismatch ( <1%)\nwith typical FMs such as Fe, Co 0:5Fe0:5, and Co 2FeAl 0:5Si0:5[25]. Moreover, since the lattice\nconstant of MgAl 2O4can be tuned by changing the Mg/Al composition rate, one can achieve\ngood lattice matching with various FMs. Up to now, relatively high MR ratios have been\nobserved in the MgAl 2O4-based MTJs [25{28]. The interfacial PMA has also been obtained\nin some FM/MgAl 2O4heterostructures [29{31]. In particular, Koo et al. [29] reported that\nthe Fe(0:7 nm)/MgAl 2O4heterostructure has an interfacial PMA with interfacial anisotropy\nconstantKiof 0.9{1.6 mJ =m2, which is smaller than that of the Fe/MgO heterostructure\n(Ki\u00181.5{2.0 mJ=m2) with the same Fe thickness [12]. Possible reasons for such a di\u000berence\ninKishould be clari\fed; however, no theoretical study has addressed interfacial magne-\ntocrystalline anisotropy in Fe/MgAl 2O4.\nIn this work, we study interfacial magnetocrystalline anisotropy in Fe/MgAl 2O4by means\nof \frst-principles calculations based on density functional theory. We \fnd that this system\nhas interfacial PMA with Ki\u00191:2 mJ=m2. This value of Kiis smaller than that calculated\nin Fe/MgO with a similar barrier thickness ( Ki\u00191.5{1.7 mJ=m2), in agreement with the\nabove-mentioned experimental results. To clarify the origin of such a di\u000berence in Ki, second-\norder perturbation analyses are carried out, which \fnd that the smaller Kiin Fe/MgAl 2O4\nis due to a smaller positive contribution in Kifrom spin-\rip electron scattering. We show\nthat these results can be naturally understood from the features of the LDOSs and band\nstructures in these systems. We \fnally propose an interfacial insertion of tungsten (W)\nlayers into Fe/MgAl 2O4as a possible way to achieve a larger Ki. It is shown that such\nFe/W/MgAl 2O4systems with 4{5 layers of W have a large Kiof&3 mJ=m2.\nII. CALCULATION METHOD\nWe analyzed Fe/MgAl 2O4(001) and Fe/MgO(001) by means of density functional theory\n(DFT) including the e\u000bect of spin-orbit interactions, which is implemented in the Vienna\n3ab initio simulation program (VASP) [32]. We adopted the spin-polarized generalized gra-\ndient approximation (GGA) [33] for the exchange-correlation energy and used the projector\naugmented wave (PAW) potential [34, 35] to treat the e\u000bect of core electrons properly.\nFigures 1(a) and 1(b) show the supercells of Fe(5)/MgAl 2O4(9) and Fe(5)/MgO(5) used\nin this study, where each number in parentheses represents each layer number. Note that\nMgAl 2O4(9) and MgO(5) have similar barrier thicknesses, which are suitable for comparison\nof interfacial magnetic anisotropy. As mentioned in Sec. I, the most striking feature of\nFe/MgAl 2O4is the signi\fcantly small lattice mismatch between the electrode and the barrier;\nat the interface, two unit cells of bcc Fe with 2 aFe= 5:732\u0017A can be well \ftted to MgAl 2O4\nwithaMgAl 2O4=p\n2 = 5:72\u0017A. Thus, we \fxed the in-plane lattice constant aof the Fe/MgAl 2O4\nsupercell to a= 2aFe= 5:732\u0017A. On the other hand, the lattice mismatch is relatively large\nin Fe/MgO, for which we used two supercells with di\u000berent in-plane lattice constants a: one\nisa=aFe= 2:866\u0017A, and the other is a=aMgO=p\n2 = 2:98\u0017A. In all of these supercells, we\ncarried out structure relaxation, through which optimum atomic positions and the interfacial\ndistance between the electrode and the barrier were determined. Here, we used the known\nfact that an interfacial atomic con\fguration where O atoms are on top of Fe atoms [see Figs.\n1(a) and 1(b)] is energetically favored in both Fe/MgAl 2O4and Fe/MgO [36]. The details\nof our structure relaxation are given in our previous paper [40].\nIn each optimized supercell, we calculated interfacial magnetocrystalline anisotropy Ki\nusing the well-known force theorem [41]\nKi= (E[100]\u0000E[001])=2S; (1)\nwhereE[100](E[001]) is the sum of the eigenenergies of the supercell with the magnetization\nparallel to the [100] ([001]) direction, and Sis the cross-sectional area of the supercell. Note\nthat the factor 2 in the denominator re\rects the fact that each supercell has two interfaces.\nIn order to con\frm whether the force theorem gives reliable results for the present systems,\nwe also calculated Kiin the self-consistent-\feld (SCF) manner using the total energies\ninstead of the sum of eigenenergies in Eq. (1) [41]. In this paper, we represent a set of\nk-point numbers used for the calculations as Nx\u0002Ny\u0002Nz, whereNx,Ny, andNzare the\nk-point numbers used for the x,y, andzdirections of supercells, respectively. Figure 2(a)\nshows the values of Kiin Fe/MgAl 2O4as a function of the number of in-plane kpoints\nN\u0011Nx=Nyobtained from the force theorem and the SCF total-energy calculation,\n4whereNzis \fxed to 1 or 3. We see that the value of Kiis saturated for N&19 in all\nfour of the cases shown in the \fgure and that the saturated values are almost the same\n(Ki\u00181:2 mJ=m2). Similar saturations of Kiwere also obtained in two Fe/MgO systems\nwitha=aMgO=p\n2 anda=aFe, as shown in Figs. 2(b) and 2(c), respectively. In both\nof these systems, Kiis saturated to\u00181:6 mJ=m2forN&37. All these results indicate\nthat the calculation using the force theorem and 19 \u000219\u00021 (37\u000237\u00021)kpoints is\nsu\u000ecient to accurately estimate Kiin Fe/MgAl 2O4(Fe/MgO) [42]; in the following, we\nuse such calculation conditions. From Eq. (1), we can easily see that positive (negative)\nKiindicates the tendency toward perpendicular (in-plane) magnetic anisotropy. However,\nactual magnetic anisotropy is estimated by Ke\u000bt=Ki+Edemagt, wheretis the e\u000bective\nthickness of the Fe electrode and Edemagtrepresents magnetic shape anisotropy. The second\ntermEdemagtalways has a negative value, and therefore favors in-plane magnetic anisotropy.\nIn the present work, we calculated Edemagtby summing up the magnetostatic dipole-dipole\ninteraction between atomic magnetic moments [41] with the use of the Ewald-summation\ntechnique [43].\nIn addition to these calculations, we further carried out a detailed second-order pertur-\nbation analysis to understand magnetocrystalline anisotropy in Fe/MgAl 2O4and Fe/MgO\nmore deeply. By treating the spin-orbit interaction HSOas a perturbation term, the second-\norder perturbation energy is expressed as\nE(2)=X\nkunocc:X\nn0\u001b0occ:X\nn\u001bjhkn0\u001b0jHSOjkn\u001bij2\n\u000f(0)\nkn\u001b\u0000\u000f(0)\nkn0\u001b0; (2)\nHSO=X\ni\u0018iLi\u0001Si; (3)\nwhere\u000f(0)\nkn\u001bis the energy of an unperturbed state jkn\u001biwith wave vector k, band index\nn, and spin \u001b. The index occ. (unocc.) on the summation means that the sum is over\noccupied (unoccupied) states of all atoms in the supercell [44, 45]. Note here that the state\njkn\u001bican be expanded as jkn\u001bi=P\ni\u0016ckn\ni\u0016\u001bji\u0016\u001bi, where\u0016is an atomic orbital at site iand\nckn\ni\u0016\u001b=hi\u0016\u001bjkn\u001bi[44]. In the spin-orbit interaction HSO,\u0018iis its coupling constant at site i,\nandLi(Si) is the single-electron angular (spin) momentum operator. As the values of \u0018i, we\nused\u0018Fe= 54:3 meV,\u0018Mg= 47:5 meV,\u0018Al= 10:8 meV, and \u0018O= 24:3 meV for Fe, Mg, Al,\nand O atoms, respectively. The Wigner-Seitz radius of each atom was set to rFe= 1:302\u0017A,\nrMg= 1:524\u0017A,rAl= 1:402\u0017A,rO= 0:820\u0017A. All these values of spin-orbit coupling constants\n5and Wigner-Seitz radii are those listed in the pseudopotential \fles in VASP. We used wave\nfunctions and eigenenergies obtained in our DFT calculations as unperturbed states and\nenergies in Eq. (2). The magnetocrystalline anisotropy energy within the second-order\nperturbation E(2)\nMCA (/Ki) was calculated as E(2)\nMCA =E(2)\n[100]\u0000E(2)\n[001], whereE(2)\n[100](E(2)\n[001])\nis the energy for the magnetization along the [100] ([001]) direction obtained by Eq. (2).\nIn the process of such an analysis, we can decompose E(2)\nMCA =P\niEi\nMCA into four types of\nterms coming from di\u000berent electron scattering around the Fermi level:\nE(2)\nMCA=X\ni\u0000\n\u0001Ei\n\")\"+ \u0001Ei\n#)#+ \u0001Ei\n\")#+ \u0001Ei\n#)\"\u0001\n: (4)\nHere, \u0001Ei\n\")\"(\u0001Ei\n#)#) originates from spin-conserving electron scattering between occu-\npied and unoccupied majority-spin (minority-spin) states. On the other hand, \u0001 Ei\n\")#\n(\u0001Ei\n#)\") corresponds to spin-\rip electron scattering from occupied majority-spin (minority-\nspin) states to unoccupied minority-spin (majority-spin) states. The details of these cal-\nculations are given in a previous paper [44]. As we show in the next section, di\u000berences\nin magnetocrystalline anisotropy between di\u000berent systems can be explained naturally by\nthese second-order perturbation analyses.\nIII. RESULTS AND DISCUSSION\nTable I shows the values of Ki,Edemagt,Ke\u000bt, \u0001Morb;i, andMspin;ifor Fe/MgAl 2O4\nand Fe/MgO obtained in this study. Here, \u0001 Morb;iis the anisotropy of the interfacial Fe\norbital magnetic moment and Mspin;iis the spin magnetic moment at interfacial Fe atoms.\nWe see that Fe/MgAl 2O4has a positive Kiof 1:192 mJ=m2. Since this value exceeds the\nnegative shape anisotropy ( Edemagt=\u00000:895 mJ=m2), this system has interfacial PMA\n(Ke\u000bt= 0:296 mJ=m2>0). Note that O layer is the termination layer of MgAl 2O4, as\nmentioned in Sec. II. Thus, the interfacial hybridization between Fe 3 d3z2\u0000r2and O 2pz\nstates plays a key role for the interfacial PMA of Fe/MgAl 2O4in the same way as Fe/MgO.\nThe values of KiandKe\u000btin Fe/MgAl 2O4are smaller than those in Fe/MgO. As men-\ntioned in Sec. I, the relationship between magnetocrystalline anisotropy and anisotropy of\nthe orbital magnetic moment provides important information on PMA in these systems.\nFrom Table I, we \fnd that both Kiand \u0001Morb;iof Fe/MgO with a=aMgO=p\n2 are larger\nthan those of Fe/MgAl 2O4, which indicates that the Bruno relation ( Ki/\u0001Morb;i) holds\n6for these two systems. On the other hand, it seems that this relation is not applicable to\nFe/MgO with a=aFe, because this system has a larger Kibut a smaller \u0001 Morb;ithan\nFe/MgAl 2O4. Therefore, the following second-order perturbation analysis is required to\ndeeply understand interfacial PMA in all of these systems.\nIn Fig. 3(a), we show the results of the second-order perturbation analysis for the magne-\ntocrystalline anisotropy in Fe/MgAl 2O4. We see that the interfacial Fe layer has the largest\npositiveEi\nMCA, which provides the dominant contribution to the positive Kiin this system.\nThis indicates that Fe/MgAl 2O4hasinterfacial PMA. At the interfacial Fe layer (Fe1), the\nanisotropy due to minority-spin scattering (\u0001 Ei\n#)#) provides the largest contribution. In\norder to understand this feature, we utilize the following simpli\fed expressions for the local\nmagnetocrystalline anisotropy [46]:\nEi\nMCA\u0019\u0001Ei\n#)#+ \u0001Ei\n\")#; (5)\n\u0001Ei\n#)#=\u00182\niX\nu#;o#jhu#jLi\nzjo#ij2\u0000jhu#jLi\nxjo#ij2\n\u000fu#\u0000\u000fo#; (6)\n\u0001Ei\n\")#=\u00182\niX\nu#;o\"jhu#jLi\nxjo\"ij2\u0000jhu#jLi\nzjo\"ij2\n\u000fu#\u0000\u000fo\"; (7)\nwhere the meanings of \u0001 Ei\n#)#and \u0001Ei\n\")#are the same as those in Eq. (4). Here, \u0018iis the\nspin-orbit coupling constant, Li\n\u000b(\u000b=x;z) is the local angular momentum operator at site\ni, andjo\u001bi(ju\u001bi) is a local occupied (unoccupied) state with spin \u001band energy \u000fo\u001b(\u000fu\u001b).\nTo derive these expressions, it is assumed that the occupied dstates in the majority-spin\nchannel are located deep below the Fermi level. Therefore, we neglected \u0001 Ei\n\")\"and \u0001Ei\n#)\"\nin Eq. (5) [compare with Eq. (4)]. Let us now focus on the local density of states (LDOS)\nof interfacial Fe atoms in Fe/MgAl 2O4shown in Fig. 3(b). From the inset of the \fgure, we\n\fnd that the majority-spin states have quite small LDOS around the Fermi level. Thus, we\ncan expect that the term \u0001 Ei\n#)#provides the dominant contribution in Eq. (5) in the case\nof Fe/MgAl 2O4, which is consistent with our results shown in Fig. 3(a). In the previous\nparagraph, we mentioned that the Bruno relation holds in this system, which is reasonable\nbecause only \u0001 Ei\n#)#is taken into account in the derivation of the Bruno relation [47].\nIn order to obtain further information on the PMA in Fe/MgAl 2O4, we analyzed wave-\nvector-resolved magnetocrystalline anisotropy and band structures of the supercell. Previous\nstudies using this type of analysis on other ferromagnetic systems have shown that localized d\nstates around the Fermi level provide the dominant contribution to the magnetocrystalline\n7anisotropy [19, 48{50]. Figure 4(a) shows the in-plane wave vector ( kk) dependence of\n\u0001E(kk)\u0011E[100](kk)\u0000E[001](kk) [51]. Here, we plotted the case of kz= 0 because kz\ndependence is very weak owing to the long c-axis constant of the supercell. Note that Kiis\nproportional to the sum of \u0001 E(kk) over all kkin the two-dimensional Brillouin zone. We see\nthat large positive anisotropy is obtained around the \u0000 point, which provides the dominant\ncontribution to the interfacial PMA in this system. We can naturally understand this\nbehavior from the band structures of the supercell shown in Figs. 4(b) and 4(c). Actually,\nas seen from Fig. 4(c), the minority-spin bands around the \u0000 point have dzx- anddyz-\norbital components near the Fermi level, leading to \fnite values of hdyz;#jLi\nzjdzx;#iand\nhdzx;# jLi\nzjdyz;#iincluded in the \frst term of the numerator of Eq. (6). Such a band\nstructure is consistent with sharp peaks in the dyz- anddzx-orbital LDOSs in the minority-\nspin states around the Fermi level [see Fig. 3(b)].\nWe next discuss PMA in Fe/MgO to understand the di\u000berence from the case of Fe/MgAl 2O4.\nFigure 5(a) shows the results of the second-order perturbation calculations in Fe/MgO\nwitha=aMgO=p\n2 = 2:98\u0017A. In this case, we obtained similar results with Fe/MgAl 2O4;\nanisotropy energy due to minority-spin scattering (\u0001 Ei\n#)#) at the interface provides the\ndominant contribution to the PMA. The structure of the LDOS at the interfacial Fe atoms\nis also similar to that of Fe/MgAl 2O4[see Fig. 5(b)]; the majority-spin state has quite small\nLDOS around the Fermi level. In Fig. 5(a), a non-negligible di\u000berence with Fe/MgAl 2O4\nis that the Fe/MgO has a larger positive spin-\rip component \u0001 Ei\n\")#at the interfacial Fe\natoms. To clarify the reason of this behavior, we show the kkdependence of \u0001 E(kk) in Fig.\n6(a). We \fnd that positive anisotropy occurs mainly around the X point. Similarly to the\ncase of Fe/MgAl 2O4,dzxanddyzstates in the minority-spin bands give \fnite values of hLi\nzi\nas shown in Fig. 6(c), leading to positive \u0001 Ei\n#)#. In addition, the majority-spin occupied\ndxyband and minority-spin unoccupied dzxband yield \fnite values of hdzx;#jLi\nxjdxy;\"i, as\nseen from Figs. 6(b) and 6(c). This gives positive \u0001 Ei\n\")#following Eq. (7), which is the\nreason why this system has the non-negligible contribution from spin-\rip scattering in the\ninterfacial PMA.\nWe also carried out the same perturbation analysis on the other Fe/MgO with a=\naFe= 2:866\u0017A, which provides valuable insight as explained below. Figure 7(a) shows the\nresults of the calculations, in which we \fnd a clear di\u000berence from those of Fe/MgAl 2O4\nand also from those of Fe/MgO with a=aMgO=p\n2. Namely, at the interfacial Fe atoms,\n8the spin-\rip component \u0001 Ei\n\")#is quite large and has a similar value to that of the spin-\npreserving component \u0001 Ei\n#)#. This is the reason why the Bruno relation does not hold in\nthis system as mentioned above. We can naturally understand the origin of this behavior\nby analyzing the LDOSs of interfacial Fe atoms shown in Fig. 7(b). As seen from the\ninset of the \fgure, the majority-spin state has a \fnite d3z2\u0000r2LDOS around the Fermi\nlevel. Since these d3z2\u0000r2states give \fnite values of hdyz;#jLi\nxjd3z2\u0000r2;\"i, the large positive\n\u0001Ei\n\")#can occur following Eq. (7) [52]. This is the reason for the magnitude relation\n\u0001Ei\n#)#\u0019\u0001Ei\n\")#in this system. This feature can also be con\frmed by the kkdependence\nof \u0001E(kk) and band structures of the supercell shown in Figs. 8(a){8(c). The major\ndi\u000berence from the case of a=aMgO=p\n2 is that the majority-spin d3z2\u0000r2band crosses\nthe Fermi level around the \u0000 point as shown in Fig. 8(b). Thus, the occupied d3z2\u0000r2\nstates in this band and the unoccupied dyzstates in the minority-spin bands give \fnite\nvalues ofhdyz;#jLi\nxjd3z2\u0000r2;\"i[see Figs. 8(b) and 8(c)], by which \u0001 Ei\n\")#becomes larger\ncompared to the case of a=aMgO=p\n2. An experimentally realized Fe/MgO heterostructure\nis expected to have an intermediate in-plane lattice constant between aFe= 2:866\u0017A and\naMgO=p\n2 = 2:98\u0017A. Although the anisotropy energy from spin-\rip scattering \u0001 Ei\n\")#is\nsensitive to the in-plane lattice constant, we can conclude from our results that Fe/MgO has\na larger positive \u0001 Ei\n\")#than Fe/MgAl 2O4. This is a possible explanation for the fact that\nthe experimentally observed Kiin Fe/MgO is larger than that in Fe/MgAl 2O4.\nIV. A WAY TO OBTAIN LARGER PMA\nIn order to obtain larger interfacial PMA for MTJs with the MgAl 2O4barrier, we pro-\npose an insertion of thin W layers between MgAl 2O4and the Fe electrode as shown in Fig.\n9(a). The insertion of W layers at the interface of Fe/MgAl 2O4is based on the theoretical\nprediction of huge PMA in the Fe/W(001) multilayer with the in-plane lattice constant of\nbulk bcc Fe [53]. Experimentally, Matsumoto and co-workers con\frmed the large change\nof magnetic anisotropy from negative to positive by reducing the W-layer thickness in the\nFe/W(001) multilayer, where the in-plane lattice constant of W approaches from that of bulk\nW to that of bulk Fe [53, 54]. Motivated by these theoretical and experimental results, we\nexamined the possibility that the insertion of thin W layers into the interface of Fe/MgAl 2O4\nwith the in-plane lattice constant of bcc Fe can enhance the interfacial PMA in this junc-\n9tion. In Fig. 9(b), we show the calculated values of Kiin Fe/W(n)/MgAl 2O4(001) and\nFe/W(n)/MgO(001). As in-plane lattice constants, we adopted a= 2aFefor the MgAl 2O4-\nbased junction and a=aFeanda=aMgO=p\n2 for the MgO-based junction. As can be seen in\nFig. 9(b), Fe/W(3{5)/MgAl 2O4and Fe/W(4{5)/MgO with a=aFehave large positive Ki,\nindicating that the insertion of W layers signi\fcantly enhances the PMA in these junctions.\nNote that such a large enhancement was not obtained in very thin W cases ( n= 1{2). Thus,\nat least 3 layers of W are required for enhancing PMA. On the other hand, Fe/W( n)/MgO\nwitha=aMgO=p\n2 shows a negative or small Kifor any layer number of W up to n= 5,\nwhich means that the insertion of W layers degrades the interfacial PMA in this junction\nbecause of the lattice mismatch between Fe and MgO. These results are consistent with\nthose of Fe/W multilayers in Ref. [53]. We can conclude that the insertion of W layers into\nthe interface of Fe/MgAl 2O4is a promising way to obtain huge PMA owing to the good lat-\ntice matching between Fe and MgAl 2O4, indicating the advantage of MgAl 2O4as compared\nwith MgO. Perhaps it might be even better to use W layers as underlayers of Fe/MgAl 2O4,\nbecause the interfacial insertion of nonmagnetic metals between FMs and insulator barriers\ntends to decrease TMR ratios of MTJs. However, further analysis for the PMA in such a\nsystem is beyond the scope of this study and will be addressed in our future work.\nFrom the second-order perturbation analysis, we found that the large PMA in the\nFe/W/MgAl 2O4multilayer is mainly attributed to perturbation processes through unoccu-\npied majority-spin states (\u0001 Ei\n#)\"and \u0001Ei\n\")\") of the middle-layer W atoms. Figures 10(a){\n10(d) show the projected LDOSs of middle-layer W atoms in Fe(5)/W(3)/ MgAl 2O4(9) and\nFe(5)/W(3)/MgO(5) ( a=aFeanda=aMgO=p\n2). As can be seen in Figs. 10(a) and 10(c),\nthere are large unoccupied dxyanddyz(dzx) states just above the Fermi level, when the\nin-plane lattice constant corresponds to that of bcc Fe ( a=aFe). Because W is a transition\nmetal element with less than half delectrons, it has unoccupied majority-spin dstates.\nThese unoccupied majority-spin states provide a considerable contribution to the PMA\nthrough the second-order perturbation of the spin-orbit interaction between unoccupied\nmajority-spin states and occupied states in both the spin channels. In the present case,\nthe matrix elements hdxy;\"jLi\nzjdx2\u0000y2;\"iandhdyz;\"jLi\nxjd3z2\u0000r2;#ishow positive contribu-\ntions to the PMA of Fe(5)/W(3)/MgAl 2O4(9) and Fe(5)/W(3)/MgO(5) ( a=aFe) in the\nperturbation processes.\n10V. SUMMARY\nWe theoretically investigated interfacial magnetocrystalline anisotropy in Fe/MgAl 2O4,\nwhich has a potential applicability to spintronic devices because of its quite small lattice\nmismatch at the interface. By means of density functional theory, we calculated interfacial\nanisotropy constant Kiof this system and compared it with those of two Fe/MgO systems\nwith di\u000berent in-plane lattice constants. We found that Fe/MgAl 2O4has interfacial perpen-\ndicular magnetic anisotropy (PMA) with Ki\u00191:2 mJ=m2, which is slightly smaller than\nthat of Fe/MgO systems. By carrying out second-order perturbation calculations on the\nPMA in combination with detailed analyses of the LDOSs and band structures, we clari\fed\nthat the smaller Kiin Fe/MgAl 2O4is due to the smaller positive anisotropy energy from\nspin-\rip electron scattering. We \fnally proposed insertion of tungsten (W) into the interface\nof Fe/MgAl 2O4as a possible way to obtain larger interfacial PMA. We showed that such\ninsertion enhances Kiof Fe/MgAl 2O4to&3 mJ=m2.\nNote added in proof. Recently, Xiang et al. [55] reported detailed experimental results\non the interfacial PMA in Fe/MgAl2O4, which are also consistent with our present results.\nACKNOWLEDGMENTS\nThe authors are grateful to K. Hono, S. Mitani, J. Okabayashi, H. Sukegawa, M. Tsu-\njikawa, and K. Nawa for useful discussions and critical comments. This work was partly\nsupported by Grants-in-Aid for Scienti\fc Research (S) (Grant No. 16H06332) and (B)\n(Grant No. 16H03852) from the Ministry of Education, Culture, Sports, Science and Tech-\nnology, Japan, by NIMS MI2I, and also by the ImPACT Program of the Council for Science,\nTechnology and Innovation, Japan. The crystal structures of the supercells were visualized\nusing VESTA [56].\n[1] B. Dieny, R. B. Goldfarb, and K. J. Lee, Introduction to Magnetic Random-access Memory\n(Wiley, Hoboken, NJ, 2016).\n[2] F. Wu, S. Mizukami, D. Watanabe, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki,\nAppl. Phys. Lett. 94, 122503 (2009).\n11[3] S. Mizukami, F. Wu, A. Sakuma, J. Walowski, D. 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Magn. 51, 1 (2015).\n[55] Q. Xiang, R. Mandal, H. Sukegawa, Y. K. Takahashi, and S. Mitani, Appl. Phys. Express 11,\n063008 (2018).\n[56] K. Momma and F. Izumi, J. Appl. Cryst. 44, 1272 (2011).\n14FIG. 1. Supercells of (a) Fe(5)/MgAl 2O4(9) and (b) Fe(5)/MgO(5).\nTABLE I. List of Ki,Edemagt,Ke\u000bt, \u0001Morb;i, andMspin;iobtained in this study.\nSystem Ki(mJ=m2)Edemagt(mJ=m2)Ke\u000bt(mJ=m2) \u0001Morb;i(\u0016B/atom)Mspin;i(\u0016B/atom)\nFe/MgAl 2O4 1.192 -0.895 0.296 0.026 2.81\nFe/MgO (a=aMgO=p\n2) 1.617 -0.828 0.788 0.030 2.73\nFe/MgO (a=aFe) 1.552 -0.908 0.643 0.020 2.78\n15 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4\n 5  10  15  20  25  30\n 0 0.5 1.0 1.5 2.0\n 10  20  30  40  50  60\n 0 0.5 1.0 1.5 2.0\n 10  20  30  40  50  60Number of in-plane k point N (= Nx = Ny )\nNumber of in-plane k point N (= Nx = Ny )\nNumber of in-plane k point N (= Nx = Ny )Ki [mJ/m2] Ki [mJ/m2] Ki [mJ/m2](a)\n(b)\n(c)Nz=1, force theorem\nNz=1, SCF\nNz=3, force theorem\nNz=3, SCF\nNz=1, force theorem\nNz=1, SCF\nNz=3, force theorem\nNz=3, SCF\nNz=1, force theorem\nNz=1, SCF\nNz=3, force theorem\nNz=3, SCFFIG. 2. The interfacial anisotropy constant Kias a function of the number of in-plane k-point N\n(=Nx=Ny) in (a) Fe/MgAl 2O4, (b) Fe/MgO ( a=aMgO=p\n2), and (c) Fe/MgO ( a=aFe).\n16FIG. 3. (a) Results of second-order perturbation analysis on the interfacial PMA in Fe/MgAl 2O4.\nThe vertical green, yellow, blue, red, and pink bars show the values of \u0001 Ei\n\")\", \u0001Ei\n#)#, \u0001Ei\n\")#,\n\u0001Ei\n#)\", and the local anisotropy energy Ei\nMCA, respectively, at each Fe layer. [See Eq. (4) and\nthe corresponding text for details.] (b) Projected LDOSs for Fe 3 dstates at the interface of\nFe/MgAl 2O4. In panel (b), positive and negative values indicate the majority- and minority-spin\nprojected LDOSs, respectively. The inset of panel (b) shows a magni\fed view near the Fermi level.\n17FIG. 4. The wave-vector-resolved information on the interfacial PMA in Fe/MgAl 2O4. (a) The\nin-plane wave-vector ( kk) dependence of \u0001 E(kk)\u0011E[100](kk)\u0000E[001](kk). (b) and (c) The band\nstructure of the supercell in the majority- and minority-spin states, respectively. In panels (b) and\n(c), orbital components of each band are indicated by colors.\n18FIG. 5. The same as Fig. 3, but for Fe/MgO with a=aMgO=p\n2.\n19FIG. 6. The same as Fig. 4, but for Fe/MgO with a=aMgO=p\n2.\n20FIG. 7. The same as Fig. 3, but for Fe/MgO with a=aFe.\n21FIG. 8. The same as Fig. 4, but for Fe/MgO with a=aFe.\n22FIG. 9. (a) The supercell of Fe(5)/W(3)/MgAl 2O4(9) used in our calculation. (b) Values of Ki\nobtained for Fe(5)/W(3{5)/MgAl 2O4(9) and two types of Fe(5)/W(3{5)/MgO(5) with di\u000berent\nin-plane lattice constants. The data indicated by W(0) are those when W layers are not inserted\n(the values of Kiwere already shown in Table I).\n23(a) (b)\n(c) (d)-0.20-0.15-0.10-0.05 0 0.05 0.10 0.15 0.20 0.25Projected LDOS [states/eV/atom]\n-0.20-0.15-0.10-0.05 0 0.05 0.10 0.15 0.20 0.25Projected LDOS [states/eV/atom]\n-2.0 -1.5 -1.0 -0.5  0 0.5  1.0  1.5  2.0\nE-EF [eV]-2.0-1.5 -1.0 -0.5  0 0.5  1.0  1.5  2.0\nE-EF [eV]\nFIG. 10. (a){(d) Projected LDOSs for 3 dstates at the middle layer of the W atom\nin Fe(5)/W(3)/MgAl 2O4(9), Fe(5)/W(3)/MgO(5) ( a=aFe), and Fe(5)/W(3)/MgO(5) ( a=\naMgO=p\n2), where positive and negative values indicate the majority- and minority-spin projected\nLDOSs, respectively.\n24"    },    {        "title": "1804.05465v1.Strong_orientation_dependent_spin_orbit_torque_in_antiferromagnet_Mn2Au.pdf",        "content": "1 \n Strong orientation dependent spin-o rbit torque in antiferromagnet Mn 2Au \nX. F. Zhou,1,2 J. Zhang,3 F. Li,1,2 X. Z. Chen,1,2 G. Y. Shi,1 Y.  Z .  Tan ,1 Y.  D .  G u ,1 M. S. \nSaleem,1 H. Q. Wu,2 F. Pan,1,2 and C. Song1,2,∗ \n1Key Laboratory of Advanced Materials (MOE), School of Materials Science and \nEngineering, Tsinghua University, Beijing 100084, China \n2Beijing Innovation Center for Future Chip,  Tsinghua University, Beijing 100084, China. \n3School of Physics and Wuhan National High Magnetic Field Center, Huazhong University of \nScience and Technology, Wuhan 430074, China \n \nAntiferromagnets with zero net magnetic mo ment, strong anti-interference and ultrafast \nswitching speed have potential competitiveness in high-density information storage. Body \ncentered tetragonal antiferromagnet Mn 2Au with opposite spin sub-lattices is a unique \nmetallic material for Néel-order spin-orbit to rque (SOT) switching. Here we investigate the \nSOT switching in quasi-epitaxial (103), (101) and (204) Mn 2Au films prepared by a simple \nmagnetron sputtering method. We demonstrate current induced antiferromagnetic moment switching in all the prepared Mn\n2Au films by a short current pulse at room temperature, \nwhereas different orientated films exhibit di stinguished switching characters. A direction-\nindependent reversible switching is attained in Mn 2Au (103) films due to negligible \nmagnetocrystalline anisot ropy energy, while for Mn 2Au (101) and (204) films, the switching \nis invertible with the current applied along the in-plane easy axis and its vertical axis, but \nbecomes attenuated seriously during initially switching circles when the current is applied \nalong hard axis, because of the existence of magnetocrystalline anisotropy energy. Besides the \nfundamental significance, the strong orientation dependent SOT switching, which was not realized irrespective of ferromagnet and antife rromagnet, provides versatility for spintronics. \n                                                \n \n∗ songcheng@mail.tsinghua.edu.cn 2 \n I. INTRODUCTION \nAlternate arrangement of magnetic moments on adjacent atoms makes antiferromagnets \n(AFM) show zero net magnetization without stra y field and resultant immunity to external \nperturbations [1]. After more than half century of passive role of AFM for the exchange bias \neffect at the interface coupled to ferromagnet s (FM) [2], antiferromagnet spintronics have \nemerged as a fascinating research area and stimulated intense interest due to their potential for ultrafast and ultrahigh-density spintronics [1,3–7]. Much beyond the static behavior of AFM as supporting layer in spin valves and magnetic  tunnel junctions [8], different methods have \nbeen demonstrated to manipulate the AFM spin s in the AFM-based memory resistors [9] and \ntunneling anisotropic magnetoresistance [6,10], taking advantage of FM switching [11], field \ncooling [12], strain [13], electr ic field [14–18], and electric curre nt which is realized recently \n[5,19,20]. \nControlling magnetism by current, namely spin-transfer torque (STT) [21] and spin-orbit \ntorque (SOT) [22–24],\n manifests great superiority for low-power spintronics, and favorable \nprogress has been made in ferromagnetic (FM)  systems and relevant magnetic random access \nmemory (MRAM). The SOT has recently been used for the switching of both synthetic antiferromagnets [25,26] and antiferromagnetic CuMnAs [5,20] mainly attributed to the role of field-like torque  dM\nA,B/dt ~ MA,B × pA,B, where the effective field proportional to  pA = –pB \nacting on the spin-sublattice magnetizations MA,B [4,5].  The SOT in collinear antiferromagnets \nhave been studied in bulk Mn 2Au tight-binding models. The Mn 2Au crystal two sublattices \nconnected by inversion and microscopic calcula tions based on the Kubo formula show that \nthe Néel SOT in Mn 2Au is predominantly of field-like character [4,24,27]. And current-driven \nSOT was also theoretically predicted in the easy-prepared metallic antiferromagnet Mn 2Au \nwith similar opposite spin sub-lattices with CuMnAs [1,4,24,27], which shows high Néel \ntemperature above 1000 K [27], higher conductiv ity, and resultant lower energy dissipation \nfor application, compared to the arsenic- based AFM, demonstrating pressing demand and 3 \n great significance for the study of SOT in Mn 2Au. \nThe orientation dependent spin-orbit interaction and corresponding SOT have been \ncommonly ignored mainly because the perpendicular magnetic anisotropy of the ferromagnetic stacks is persistently limited by a certain preferred orientation, such as (111)-\noriented Pt/Co and (100)-oriented Ta/CoFeB. Recently, facet-dependent spin Hall angle was \nobserved in triangular antiferromagnet IrMn\n3, which provides efficient charge-to-spin current \nconversion and concomitant spin current propagating from the antiferromagnetic into an adjacent ferromagnetic material [28].\n Considering that the field-like torque in AFM with \nopposite spin sub-lattices is not restricted by a certain crystalline orientation, the orientation \ndependent magnetic switching is expected in this scenario, which has not been realized both \nin ferromagnets and antiferromagnets, providing a versatile candidate for spintronics. Compared to polycrystalline Mn\n2Au films grown by molecular beam epitaxy [29,30] and \npartially (001)-oriented Mn 2Au films deposited by sputtering  [31], experiments below attain \nquasi-epitaxial (103), (101) and (204) Mn 2Au films, and demonstrate the orientation \ndependent Néel-order spin-orbit torque switching in single layer Mn 2Au films with different \nswitching features for different orientations. \n \nII. METHODS \n40 nm-thick (103)-, (101)-, (204)-oriented Mn 2Au films, were grown on single crystal \nMgO (111), SrTiO 3 (100) and MgO (110) substrates, respectively, by magnetron sputtering at \n300 ºC. The base pressure is 2 × 10–5 Pa, and the growth rate is 0.12 nm/s using a Mn 2Au \nalloy target (atomic ratio: 2:1). High resolutio n transmission electron microscopy (HRTEM) \nwere carried out in JEM 2100. Magnetic properties were measured by a superconducting \nquantum interference device  (SQUID) magnetometry.  XAS/XMLD measurements in total \nelectron yield mode were carried out at th e Beamline BL08U1A in Shanghai Synchrotron \nRadiation Facility (at 300 K). The XAS spectra normalization was made by dividing the 4 \n spectra by a factor such that the L3 pre-edge and L2 post-edge have identical intensities for the \ntwo polarizations. After that, the pre-edge spectr al region was set to zero and the peak at the \nL3 edge was set to one. Mn L-edge XMLD curves are characterized by the difference between \nlinearly horizontal (//) and vertical ( ⊥) polarized x-ray absorption spectroscopy. The XMLD \nwas measured at different in-plane rotation angles ( θ), where θ is the angle between a \ncrystalline axis of Mn 2Au and horizontal polarization direction.  Star devices for the SOT \nswitching were fabricated using standard photo lithography and argon ion milling procedures. \nThe writing current pulses of ~107 A cm–2 were generated by a Keysight B2961A source \nmeter. After each writing pulse, a delay of 10 s was used for thermal relaxation, and then a \nmeasurement of the transversal voltage across the central part of the patterned structure was \nrecorded when applying reading current with a density of ~105 A cm–2 generated by Agilent \nB2901A source meter.  \nThe magnetic anisotropy energy (MAE) for different orientated Mn 2Au was calculated by \nusing standard Force Theory (FT). The lattice constants of Mn 2Au on different substrates used \nin the calculations were obtained by XRD experiments. The atomic position were fully \nrelaxed. First a self-consiste nce of electron charge density (electronic potential) was \nperformed in the absence of spin-orbit coupling, followed by a one-step calculation with the presence of spin-orbit coupling when the magnetization was aligned along different axes. The band energy difference was evaluated as the MAE.   III. RESULTS  \nWe first show the microstructure characterizations of 40 nm-thick Mn\n2Au films in Fig. 1  \nwith different epitaxial orientation grown by magnetron sputtering. X-ray diffraction (XRD) spectra show the (103), (101) and (204) textures for the Mn\n2Au films grown on MgO (111), \nSrTiO 3 (100) and MgO (110) substrates, respectively , in left panel of Figs. 1(a)–1(c). All the \nfilms are single phase with no evidence of diffraction peak from second phases or other 5 \n crystalline faces within the sensitivity of XRD measurements. Corresponding high resolution \ntransmission electron microsc opy (HRTEM) images of the Mn 2Au/substrate cross-sections \nare depicted in right panel of Fig. 1. Ther e is no observation for grain boundaries in the \noverall of films, though some point defects still exist in the crystalline lattice, indicating the \nquasi-epitaxial growth mode for the present Mn 2Au films by a simple sputtering method. The \nepitaxial growth mode provides a pre-condition for the observation of orientation dependent spin-orbit torque of Mn\n2Au films as discussed below. \nWe use a superconducting quantum interference device (SQUID) magnetometry to check \nantiferromagnetism of the Mn 2Au films. Magnetic field was applied in plane, i.e. parallel to \nMgO substrate (111) plane, [110] edge. Figure 2( a) presents the magnetization curves of the \nMn 2Au (103) films at 300 K. Typical diamagnetic signals up to 50 kOe are observed, which \nreflects the diamagnetic feature of the substrate, indicating the antiferromagnetism of the Mn\n2Au (103) films without any ferromagnetic signal. Mn 2Au (101) and (204) films show \nsimilar behaviors (see Figs. S1 and S2 of Ref. [32] for magnetization measurements). \nMoreover, 5 nm-thick permalloy is deposited on top of the 10 nm Mn 2Au to reaffirm the \nantiferromagnetism of Mn 2Au (103) films. Corresponding magnetization data of Mn 2Au (10 \nnm) / permalloy (5 nm) at 20 K, after 10 kOe field-cooling from 300 K, are depicted in Fig. 2(b). The hysteresis loop exhibits an apparent shift of 150 Oe compared to its coercivity of \n400 Oe, indicating the antiferromagnetic order of the Mn\n2Au (103) films. We then display a \nschematic of 3 × 3 unit cells of Mn 2Au (Mn 2Si-type tetragonal structure with a = b = 0.333 \nnm and c = 0.854 nm) [33] in Fig. 2(c), where the Mn 2Au (103) plane is highlighted with the \nhorizontal direction of [100] ( a-axis) and its vertical direction [0 31]. Also visible is the \nneighboring sublattice with opposite magnetic moments, which is the prerequisite for the current driven magnetic switching in Mn\n2Au through spin-orbit torque.  \nTo detect the switching of Mn 2Au magnetic axis, the planar Hall resistance is obtained \nperpendicular to the reading current after two orthogonal writing current injecting to the 6 \n device alternatively. Figure 3(a) depicts a schematic of the star device and current switching \nmeasurement. For this experiment, five succ essive 1 ms-width writing current pulses of Jwrite \n= 2.10 × 107 A cm−2 are applied along [010] axis (red arrow, referred to as Jwrite1) and then \nalong its vertical direction, [ 013] axis (blue arrow, Jwrite2) [Figs. 3(a) and 3(b)]. The current \ndensity is calculated with the following parameters: 42 mA applied writing current, 40 nm-\nthick Mn 2Au films and 5 μm-width of the writing current arm of the star device. Ten seconds \nare recorded after each writing current pulse to release possible tiny heat effect caused by the short current pulse of 1 ms. Then a reading current of J\nread = 4.17 × 105 A cm−2 is applied 45° \naway from the writing current, while the planer Hall resistance is recorded. \nThe [010]-directed writing pulses are expected to align a preference of domains with AFM \nspin axis perpendicular to [010]-axis, then the writing current along [ 013] direction drives the \nmagnetic switching back to [010]-axis. Following this concept, Figure 3(c) shows the Hall \nresistance ( RHall) variation as a function of the number of current pulses, five successive \ncurrent pulses along [010] and [ 013] axes alternatively [Fig. 3(b)]. The five current pulses \nalong [010] axis ( Jwrite1) set the AFM spins axis perpendicular to [010] axis, which gradually \nincreases RHall approximately 2.8 m Ω (red squared line). Remarkably, a current pulse as short \nas 1 ms along [ 013] (Jwrite2) drives the AFM spins rotating back to [010] direction, resulting \nin the RHall decreased 4.4 m Ω to about –1.6 m Ω in the opposite sign. The following four \npulses make the variation of RHall tend to be saturated (blue squa red line). After that, the five \ncurrent pulses along [010] axis increase RHall, which make RHall reverse back to 2.6 m Ω, in the \nvicinity of initial 2.8 m Ω. Note that a constant background is subtracted. Such a switching \nexperiment is conducted 10 circles. Apparently, RHall increase and decrease without clear \ndeterioration, indicating the present magnetic swit ching is reversible, repeatable, and reliable.  \nThe abrupt change of Hall resistance with th e first current pulse, followed by gradually \nenhanced Hall resistance in the following four current pulses, reflects the multi-domain 7 \n switching feature of the present Néel-order SOT,  which is quite characteristic for SOT in both \nferromagnetic system [34]  and AFM CuMnAs [5]. A temperature calibration was used to \nmonitor the stability of the test temperature at 300 K, excluding the variation of resistance caused by temperature. It is worth to mention that the Hall resistance is independent of the \ndirection of writing current, coinciding with the theory of current-driven switching of AFM \n[4]. Interestingly, the present switching is completed by a short current pulse of 1 ms, \ncompared to the 50 ms pulse used in CuMnAs system [5], which reflects strong switching capability of metallic AFM system.  \nWe then investigate the writing current density and current duration time dependent \nswitching of the Mn\n2Au films. Corresponding data are displayed in Figs. 3(d) and 3(e), \nrespectively. For these experiments, different writing current densities (2.25 × 107, 2.40 × 107, \n2.55 × 107 A cm−2) and current pulses width (3, 5, and 15 ms) are applied to the star device. \nThe measurement sequence is identical to that of Fig. 3(c). With increasing writing current \ndensity and current duration time, the magnitude of RHall becomes larger, most likely due to \nstronger SOT switching by the applied current and the resultant 90° switching of AFM moment. According to the current induced switc hing results, the first and second pulses are \ndecisive which result in sharp Hall resistance change and major magnetic moments switching, while following writing pulses causes relatively small magnetic moments switching and Hall \nresistance variety. And the switching is repr oducible during the five cycles. It should be \nmentioned that a much higher writing current and pulses width would induce non-recyclable switching or device rupture by the thermal or  electromigration effect. The current induced \nswitching results of different thicknesses point out no thickness-dependent switching behavior, reflecting the calculated inherent  switching ability of bulk Mn\n2Au (see Fig. S3 of Ref. [32] for \nthickness-dependent switching of Mn 2Au) [4,24,27]. \nIn general, the antiferromagnets show magnetic anisotropy, which might affect the SOT \nswitching. Thus the switching capability of the Mn 2Au (103) films should be dependent on 8 \n the current applied in different crystalline axes, besides two typical axes, [010] and [ 013]. We \nthen explore the SOT effect when both of the writing current and reading current directions are rotated 45° from their original counterparts. In this case, the direction of the writing current here is along the direction of reading current in Fig. 3(a). It is interesting to find in Fig. \n3(f) that R\nHall also apparently increase and decrease wi thout clear deterioration, especially the \nrecycling behavior for this switching is reliable. Also, the magnitude of RHall for the SOT \nswitching is comparable to its counterpart in Fig. 3(c). The comparable SOT switching behaviors in Figs. 3(a) and 3(c) suggest that the Mn\n2Au (103) films have no apparent \nantiferromagnetic easy axis. \nThe SOT switching experiments have revealed that the AFM spins tend to arrange \nrandomly and are comparable in each direction of (103) orientation Mn 2Au. We now address \nthe question whether an analogous spin structure can also be detected by x-ray magnetic linear dichroism (XMLD), which stands out as a unique method to identify the spin \norientations in AFM despite the absence of net moment [35,36]. Figure 4 presents Mn L-edge \nXMLD curves, characterized by the difference be tween linearly horizontal (//) and vertical ( ⊥) \npolarized x-ray absorption spectroscopy (XAS; see Fig. S4 of Ref. [32] for the original XAS). \nThe intensity of XMLD is determined by the difference between angles of AFM spin/horizontal polarization and AFM spin/vertical polarization: the larger the angle difference is, the stronger the XMLD signal is. \nTo understand the alignment of AFM spins, the XMLD at several typical angles ( θ) are \nmeasured by changing in-plane angle θ: the angle between [010] axis of Mn\n2Au and \nhorizontal polarization direction, as displayed in the inset of Fig. 4. For θ = 0°, the horizontal \npolarized x-ray is set to [010] axis of the Mn 2Au (103) films, while the vertical polarized x-\nray is along its vertical direction. The obtained XMLD signal is rather weak without characteristic “positive to negative” XMLD pattern at L\n3-edge (about 635–645 eV) [35]. \nSimilar XMLD signals are observed for the spectra measured with θ = 30°, 45°, and 60°, three 9 \n typical directions between [010] and [ 013] axes (see Fig. S5 of Ref. [32] for the pole figure \nof Mn 2Au (103) films). This finding demonstrates the absence of dominant orientation of the \nordered spin textures (Néel vectors) at Mn 2Au (103) plane. This feature guarantees that the \nmagnetic switching of Mn 2Au (103) is mainly determined by the vector of field-like torque, \nwhich realigns the spin texture perpendicular to the current channel on the basis of the theory \n[4]. The current applied alternatively at [010] and [ 013] axes lead to the reversible \nmodulation of AFM spins and the resultant planar Hall effect. \nWe then turn towards the SOT switching of Mn 2Au (101) films. The star devices for the \nMn 2Au (101) films and measurement setup are identical to those of Mn 2Au (103) in Fig. 3(a), \nbut the crystal axis at θ = 0° is [ 113] and its vertical axis is [ 311] (see Fig. S6 of Ref. [32] \nfor the pole figure of Mn 2Au (101) films). Figure 5(a) shows the Hall resistance variation \nwhen four successive writing current pulses are applied alternatively along [ 113] (red squares) \nand [311] (blue squares) of the Mn 2Au (101) films. Note that RHall changes with the direction \nof 1 ms-width writing current pulse ( Jwrite = 2.00 ×107 A cm−2) for the first circle. However, \nthis variation attenuates seriously after the first circle, and almost vanishes after five circles. \nThese features reflect that the current can not reversibly switch the AFM spin texture between \n[113] and [311] axes. The situation turns out to be dramatically different when the probe \nconfiguration rotates 45°. That is, both of th e writing current and reading current are rotated \n45° from its origin case in Fig. 5(a). Corresponding data are presented in Fig. 5(b). RHall can \nbe reversibly modulated by the writing current pulse of 1 ms-width and Jwrite = 2.00 × 107 A \ncm−2. This characterization indicates that the AFM spin texture is switched between [010] axis \n(θ = 45°) and its vertical direction. \nThe transport measurements suggest that the spin texture of the Mn 2Au (101) films should \nbe different from that of the Mn 2Au (103) films. Figure 5(c) depicts the XMLD spectra of \nthree typical angles θ = 0°, 30°, and 45°, where the angle strongly affects the XMLD signal. 10 \n For θ = 0°, the horizontal and the vertical polarized x-ray is set to [ 113] and [311] axes of \nthe Mn 2Au (101) films, respectively. It is found that the XMLD signal at Mn L3-edge is quite \nweak, indicating no obvious difference between [ 113] and [ 311]. An analogous XMLD \nspectrum is obtained for the θ = 30° but the magnitude of XMLD signals is somehow \nenhanced. The case differs completely as θ = 45°. A characteristic positive to negative XMLD \npattern at L3-edge indicates that the distinguished spin  textures between these two orientations, \n[010] axis ( θ = 45°) and [ 011] axis (θ = 135°). Because of the broken symmetry in the \nstrained Mn 2Au films, the Néel vector orientation is most likely aligned along [010] ( θ = 45°) \n[7], making [010] an easy axis for the as-grown Mn 2Au (101) films. With the emergence of \nSOT by the writing current pulse along [010] direction, the AFM spin texture can be switched to its vertical direction, and then switch reversibly and flexibly between these two directions \nby applying alternative current pulses. However, for the hard axis of [\n113] (θ = 0°) or [ 311] \n(θ = 90°), the Néel vector orientation is difficult to be arranged in these energy disfavored \ndirections. Even a part of spin texture is reluctant to be switched to these two directions by \nSOT, whereas the switching is hard to maintain, because the AFM spins are pinned gradually by the easy axis [010] at θ = 45°. \nWe then show the SOT switching and spin texture measurements of Mn\n2Au (204) films in \nFig. 6. The experiments are carried out using the same procedure as that of Mn 2Au (103) \nfilms in Fig. 3. One can see in Fig. 6(a) that the variation of Hall resistance is apparent when applying four successive 1 ms-width writing current pulses of J\nwrite = 2.00 ×107 A cm−2 along \n[010] (θ = 0°, red squares) and [ 012] axes (θ = 90°, blue squares) alternatively (see Fig. S7 of \nRef. [32] for the pole figure of Mn 2Au (204) films). Applied current along [010] initially \naligned the AFM moments perpendicular to current direction by the SOT effect and RHall turns \nout to increase. Then RHall turns out to decrease with the current along [ 012], corresponding \nto the AFM switching back to [010] axis. Two successive current pulse s would saturate the 11 \n variation of RHall to some extent. Remarkably, RHall increase and decrease with ten series of \ncurrent pulses along [010] and [ 012], demonstrating the AFM moments are switched \nreversibly between two perpendicular directions in ten circles. The switching feature is completely different when the measurement configuration rotates 45°, i.e., the writing current pulses are applied at θ = 45°. The variation of R\nHall is greatly reduced with increasing the \ncycling and keeps unchanged after three switching circles. This feature reveals that the \nmagnetic moments could not be switched effectively in these two directions at θ = 45° and  \n135°. \nXMLD signals in Fig. 6(c) support the transport measurements. For θ = 0°, the horizontal \nand the vertical polarized x-ray is set to [010] and [ 012] axes of the Mn 2Au (204) films, \nrespectively. Interestingly, the XMLD signal at θ = 0° features as a characteristic positive to \nnegative XMLD pattern at L3-edge, reflecting the easy axis along [010] for the as-grown \nMn 2Au (204) films with strain induced by the MgO (110) substrates [37]. Accordingly, the \nXMLD signals at θ = 30° are greatly reduced, followed by the absence of XMLD L3-edge \nsignal at θ = 45°, reflecting that the spin textures strongly prefer to be aligned along the easy \naxis [010]. The Néel vectors are switched between [010] and its vertical direction. Such a \nstrong AFM anisotropy is in a good agreement wi th the efficient switching between these two \ndirections, followed by the comparatively easy SOT switching and speedy saturation of RHall \nin Fig. 6(a). In contrast, the SOT switching in non-easy axis becomes more difficult with fast attenuated variation of R\nHall in Fig. 6(b).  \nIn addition, the Joule heat produced by a pulse current electric current is simulated using \nfinite element modelling by COMSOL software. The instantaneous maximum temperature of Mn\n2Au device can increase from 300 K to 424 K after a pulse of 2 × 107 A cm−2 amplitude \nand 1 ms length, which is far below the Néel temperature of Mn 2Au (>1000 K). Nevertheless, \nheat assisted switching could play a role on reducing the current density, from theoretical \n∼108–109 A cm–2 [4] to ∼107 A cm–2. The electric current induced Oster field distribution is 12 \n also simulated using finite element modell ing by COMSOL software. The maximum Oster \nfield is only ~100 Oe. Such a disturbance of the magnetic field can be ignored for antiferromagnetic Mn\n2Au (see Figs. S8 and S9 of Ref. [32] for the Joule heat and Oster field \nsimulation).  \n \n  \nIV. DISCUSSION \nIn order to understand the switching be haviour in different orientated Mn\n2Au, the \nmagnetic anisotropy energy (M AE) of different orientated Mn 2Au is calculated by using \nstandard Force Theory (FT). Acco rding to the symmetry of bulk Mn 2Au, the MAE can be \nexpressed by [7],  \n           ϕ ω ω ω ϕω 4cossin sin sin),(4\n∥44\n42\n2 K K K E + + =⊥ ⊥                        (1) \nhere ω and φ are the angles between magnetization  (M) and [001], as well as the projection \ndirection of M on Mn 2Au (001) plane and [100] axis, respectively; K 2⊥, K 4⊥, and K 4∥ \nparameterize the uniaxial MAE constant, the fourth-order out-of-plane and in-plane MAE \nconstants, respectively. Based on Eq. (1), the specific axes for the MAE are listed as follows: \nE([001]) = E(0, φ) = 0, E([100]) = E(π/2, 0) = K2⊥ + K 4⊥ + K 4// = E([010]) = E(π/2, π/2),  and \nE([110]) = E(π/2, π/4) = K2⊥ + K4⊥ – K4// = E([110]) = E(π/2, 3 π/4). Apparently, the easy \naxis is determined by the sign of K4//. The easy axis is along [110] when K 4// > 0 and [100] \nwhen K 4// < 0. The lattice constants of different orientated-Mn 2Au used in the calculations are \nobtained by XRD experiments. Through first- principles calculations, the MAE constants K2⊥, \nK4⊥, and K 4// can be obtained according to the lattice c onstant and orientation (see Table S1 of \nRef. [32] for the MAE constants), giving rise to the angular dependence of MAE in Fig. 7(a).  13 \n Note that MAE decreases with increasing angle ω, and becomes the lowest level for ω  = \n90°, corresponding to Mn 2Au (001) plane, which is the easy magnetized plane. Concerning \nthe details in (001) easy plane, (103) orientated Mn 2Au exhibits negligible difference of MAE, \naccompanied by comparable MAE value at φ = 45° ([110]) and φ = 90° ([010] direction), thus \nthe magnetic moments of  Mn 2Au (103) can be switched by current either between [100] and \n[010] or between [110] and [ 110], as illustrated in Fig. 3. Th e situation differs abruptly for \n(101) and (204) orientated Mn 2Au, MAE is different when the magnetization rotates within \nthe (001) plane, associated with lower MAE at φ = 0° ([100]) and 90° ([010]) than that at φ = \n45° ([110]) and 135° ([ 110]). Consequently, the magnetic moments of (101) and (204) \norientated Mn 2Au can be switched by current between [100] and [010] rather than [110] and \n[110], as displayed in Figs. 5 and 6. An inspection of the MAE of Mn 2Au (103), (101) and \n(204) planes in Fig. 7(b) shows that [010] direction ( θ = 0°) is indeed the energy minimum, \nwhereas its vertical directions, [ 013], [011] and [012] for (103), (101) and (204) planes are \nenergy maximum, respectively. In this senari o, it is difficult for the magnetic moments \nswitches between [010] and these directions. Instead, the magnetic moments are switched by two orthogonal currents between two easy axes, one is [010] lying within (103), (101) and (204) planes, while  the other easy axis [100] lying out of these planes, as presented in Fig. \n7(c). In fact, [\n013], [011] and [012] can be considered at the respective projection of [100] \ndirection in (103), (101) and (204) planes of Mn 2Au, which are parallel to their substrate \nsurface. The variation of Hall resistances are read out by the planar Hall effect in (103), (101) and (204) planes, respectively.   \nV. CONCLUSION \nWe have demonstrated the spin-orbit torque in body centered tetragonal antiferromagnet \nMn\n2Au with opposite spin sub-lattices. There are five main features for the present findings: (i) 14 \n The SOT switching behavior shows strong orientation dependence. The AFM moments are \nswitched reversibly when the crystalline orientation plane has no strong magnetic anisotropy or the writing current is applied along the easy ax is [010] and its vertical  direction (the (101) \nand (204) orientation in-plane projection of an other easy axis [100]), which is not only a \ntypical feature for the SOT in AFM from the fundamental viewpoint, but also provides a \nversatile candidate for antiferromagnet spintronics; (ii), Mn\n2Au was theoretically to realize \nsizable reorientations at current densities of ∼108–109 A cm–2 [4]. We demonstrate the \nswitching of Mn 2Au magnetic moments at current densities of ∼107 A cm–2, ascribed to multi-\ndomain wall and heat assisted switching; (iii ) The growth parameter of quasi-epitaxial Mn 2Au \nfilms by an easy access magnetron sputtering method has been optimized, where the buffer layer and molecular beam epitaxy are not necessary; (iv) The SOT switching has been realized in CuMnAs (arsenic-based semimetal), but only magnetic metals and alloys are generally considered as the electrode of magnetic tunnel junctions and relevant spintronics. \nOur work might advance the use of AFM alloys as a functional layer in spintronics; (v) The \nSOT switching of antiferromagnet occurs at room temperature, which is critical for the application of AFM spintronics. Our findings not only add a new dimension to spin-orbit torque, but also represent a promising step towards antiferromagnet spintronics.  ACKNOWLEDGEMENTS \nWe are grateful to Dr. Luqiao Liu of MIT for fr uitful discussions and criticial reading of the \nmanuscript. We acknowledge the Beamline BL08U1A in Shanghai Synchrotron Radiation \nFacility (SSRF) for XAS/XMLD measurements and Center for Testing and Analyzing of \nMaterials for technical support.  C.S. acknowledges the support of Young Chang Jiang \nScholars Program. This work was supported by the National Natural Science Foundation of \nChina (Grant Nos. 51671110, 51571128) and the National Key R&D Program of China (Grant Nos. 2017YFB0405704).  15 \n Note added.  \nDuring the preparation of this manuscript, we  found two relevant works reporting current \ninduced switching in Mn 2Au [38,39]. They reported the SOT switching of  Mn 2Au (001). \n \n[1] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Antiferromagnetic spintronics, Nat. \nNanotechnol. 11, 231-241 (2016). \n[2] W. Zhang and K. M. 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Givord, Revealing the \nproperties of Mn 2Au for antiferromagnetic spintronics, Nat. Commun. 4, 2892 (2013). \n[28] W. Zhang, W. Han, S. H. Yang, Y . Sun, Y . Zhang, B. Yan, and S. S. P. Parkin, Giant facet \ndependent spin-orbit torque and spin Hall conductivity in the triangular antiferromagnet IrMn\n3, Sci. Adv. 2, e1600759 (2016). \n[29] H .  C .  W u ,  Z .  M .  L i a o ,  R .  G .  S u m e s h  S o f i n ,  G .  F e n g ,  X .  M .  M a ,  A .  B .  S h i c k ,  O .  N .  \nMryasov, and I. V . Shvets, Mn 2Au: Body-Centered-Tetragonal Bimetallic \nAntiferromagnets Grown by Molecular Beam Epitaxy, Adv. Mat. 24, 6374 (2012). \n[30] H. C. Wu, M. Abid, A. Kalitsov, P. Zarzhitsky, M. Abid, Z. M. Liao, C. Ó Coileáin, H. Xu, \nJ. J. Wang, H. Liu, O. N. Mryasov, C. R. Chang, and I. V . Shvets, Anomalous Anisotropic Magnetoresistance of Antiferromagnetic Epitaxial Bimetallic Films: Mn\n2Au and \nMn 2Au/Fe Bilayers, Adv. Funct. Mater. 26, 5884-5892. (2016). \n[31] M. Jourdan, H. Bräuning, A. Sapozhnik, H. J.  Elmers, H. Zabel, and M. Kläui, Epitaxial \nMn 2Au thin films for antiferromagnetic spintronics, J. Phys. D: Appl. Phys. 48, 385001 \n(2015). \n[32] See the Supplemental Material for experime ntal details, magnetic properties of Mn 2Au \n(101) and (204) films, different thickness (103) orientated Mn 2Au switching behavior, raw \ndata for XAS, pole figure of Mn 2Au (103), (101) and (204) films, Joule heat and Oster \nfield simulation and magnetocrystalline anisotropy energy parameters. \n[33] R. Masrourab, E. K. Hlilc, M. Hamedound, A. Benyoussefbde, A. Boutaharf, and H. \nLassrif, Antiferromagnetic spintronics of Mn 2Au: An experiment, first principle, mean \nfield and series expansions calculations study, J. Magn. Magn. Mater. 393, 600-603 (2015). \n[34] O. J. Lee, L. Q. Liu, C. F. Pai, Y . Li, H. W. Tseng, P . G . Gowtham, J. P . Park, D. C. Ralph, \nand R. A. Buhrman, Central role of domain wall depinning for perpendicular 19 \n magnetization switching driven by spin torque from the spin Hall effect, Phys. Rev. B 89, \n024418 (2013). \n[35] B. Cui, F . Li., C. Song, J. J. Peng, M. S. Saleem, Y . D. Gu, S. N. Li, K. L. W ang, and F . \nPan. Insight into the antiferromagnetic struct ure manipulated by elect ronic reconstruction, \nPhys. Rev. B 94, 134403 (2016). \n[36] L. J. Zhang, Z. J. Xu, X. Z. Zhang, H. N. Y u, Y . Zou, Z. Guo, X. J. Zhen, J. F. Cao, X. Y . \nMeng, J. Q. Li, Z. H. Chen, Y . Wang, and R. Z. Tai, Latest Advances in Soft X-ray Spectromicroscopy at SSRF, Nucl. Sci. Tech. 26, 040101 (2015). \n[37] A. A. Sapozhnik, R. Abrudan, Yu. Skourski, M. Jourdan, H. Zabel, M. Kläui, and H. J. \nElmers, Manipulation of antiferromagnetic domain distribution in Mn\n2Au by ultrahigh \nmagnetic fields and by strain,  Phys. Status Solidi RRL. 11, 1–4 (2017). \n[38] S. Yu. Bodnar, L. Šmejkal, I. Turek, T. Jungwirth, O. Gomonay, J. Sinova, A. A. \nSapozhnik, H. J. Elmers, M. Kläui, and M. Jourdan, Writing and Reading \nantiferromagnetic Mn 2Au: Néel spin-orbit torques and large anisotropic \nmagnetoresistance, Nat. Commun. 9, 348 (2018). \n[39] M. Meinert, D. Graulich, and T. Matalla-Wa gner, Key role of thermal activation in the \nelectrical switching of antiferromagnetic Mn 2Au, arXiv:1706.06983. \n \n \n    \n \n  20 \n Figure Captions \nFIG. 1. Crystalline characte rization of quasi-epitaxial Mn 2Au films. XRD spectra (left panel) \nand concomitant HRTEM images (right panel) for Mn 2Au (a) (103), (b) (101), (c) (204) films \ndeposited on MgO (111), SrTiO 3 (100), and MgO (110) substrates, respectively. \n \nFIG. 2. Magnetic properties of Mn 2Au (103) films. Magnetic field was applied in plane, i.e. \nparallel to MgO substrate (111) plane, [110]  edge. (a) Magnetization curve of 40 nm Mn 2Au \n(103) films grown on MgO (111) substrates at 300 K. (b) Magnetic hysteresis loop of Mn 2Au \n(10 nm) / permalloy (5 nm) bilayer at 20 K. (c)  Schematic of 3 × 3 unit cells of Mn 2Au, where \nthe Mn 2Au (103) plane is highlighted. Magnetic moments of neighboring sublattices are \nopposite.   FIG. 3. Schematic of the switching measurement and current-driven switching of Mn\n2Au (103) \nfilms.  (a) Schematic of the star device and switching measurement geometry. Writing current \npulses are applied along red ( Jwrite1) and blue arrows ( Jwrite2) alternatively, corresponding to \ntwo orthogonal directions, [010] and [ 013], for Mn 2Au (103) films. Reading current is \nmarked as the green arrow, while the concomitant Hall resistance is detected in its transverse direction. (b) Sketch of alternative writing  current pulses along two orthogonal directions, \nJ\nwrite1 (red) and Jwrite2 (blue) of ~107 A cm−2, and the reading current of ~105 A cm−2 (green) \nafter each writing current pulse. (c) Hall resistance ( RHall) change as a function of the number \nof writing current pulses. For Mn 2Au (103), five successive writing current pulses are applied \nalong [010] and [ 013] (blue) axes alternatively. After each writing current pulse, Hall \nresistance is recorded during applying a reading current. The variation of RHall are shown by \nred and blue squares for the writing current pulse along [010] and [ 013], respectively. (d) \nComparison of the variation of RHall with different magnitude of writing current pulses. (e) 21 \n Comparison of the variation of RHall with variable writing pulses´ width. (f) The \nmeasurements of  RHall are identical as  (c), but both of the writing current and reading current \ndirections are rotated 45° from their counterparts in (c). All of the measurements are done at \nsample temperature of 300 K and a constant background of Hall resistance is subtracted. \n \nFIG. 4. Schematic of XMLD measurement and XMLD signals at different in-plane rotation angles (θ). The left panel shows the XMLD signals at  four typical angles  between horizontal \npolarized x-ray direction and [010] axis of Mn\n2Au (θ = 0°, 30°, 45°, and 60°). The right panel \ndisplays the sketch for the rotation of the horizontal ( ∥, grey solid arrows) polarized x-ray, \nand vertical ( ⊥, blue solid arrows) polarized x-ray, while the Mn 2Au [010] axis is set at θ = \n0°. The XMLD signals are measured at 300 K. \n FIG. 5. SOT switching and spin texture measurements of Mn\n2Au (101) films.  (a) Dependence \nof RHall variation on the number of writing current pulses. Three successive current pulses of \nJwrite = 2.00 × 107 A cm−2 are applied along [ 113] and [311] axes alternatively. After each \nwriting current pulse, RHall is recorded during applying the reading current of Jread = 4.17 × \n105 A cm−2, which is denoted by red and blue squares corresponding to the writing current \nalong [ 113] and [311] axes, respectively. (b) The measurement procedure is same as (a), but \nthe measurement configuration (both the writing current pulse and reading current) is rotated 45° from (a). (c) Schematic of XMLD measur ement and XMLD signals under different in-\nplane rotation angles ( θ). The left panel shows the XMLD signals at three typical angles \nbetween horizontal polarized x-ray direction and [\n113] axis of Mn 2Au (θ = 0°, 30°, and 45°). \nThe right panel displays the sketch for the rotation of the horizontal ( ∥, grey solid arrows) \npolarized x-ray, and vertical ( ⊥, blue solid arrows) polarized x-ray, while the Mn 2Au [113] \naxis is set at θ = 0°. 22 \n  \nFIG. 6. SOT switching and spin texture measurements of Mn 2Au (204) films.  (a) Dependence \nof RHall on the number of writing current pulses. Four successive current pulses of Jwrite = 2.00 \n× 107 A cm−2 are applied along [010] and [ 012] axes alternatively. After each writing current \npulse, RHall is recorded during applying the reading current of Jread = 4.17 × 105 A cm−2, which \nis denoted by red and blue squares correspondi ng to the writing current along [010] and [ 012] \naxes, respectively. (b) The measurement procedure is same as (a), but the measurement \nconfiguration (both the writing current pulse and reading current) is rotated 45° from (a). (c) \nSchematic of XMLD measurement and XMLD signals under different in-plane rotation angles (θ). The left panel shows the XMLD signals at three typical angles between horizontal \npolarization direction and [010] axis of Mn\n2Au (θ = 0°, 30°, and 45°). The right panel \ndisplays the sketch for the rotation of the horizontal ( ∥, grey solid arrows) polarized x-ray, \nand vertical ( ⊥, blue solid arrows) polarized x-ray, while the Mn 2Au [010] axis is set at θ = \n0°. \n \nFIG. 7. MAE as a function of  magnetization direction ( ω, φ) in Mn 2Au (103), (101) and (204) \norientation films.  (a) MAE of all space direction in Mn 2Au (103), (101) and (204) orientation \nfilms. (b) MAE distribution as a function of in-plane angle θ with magnetization lying in \n(103), (101) and (204) planes, respectively. (c) Schematic of relationship between easy axes \nand current directions. Hollow double arrows re present the antiferromagnetic easy axes and \nsolid thick arrows represent the current directions. The gold plane represents (001) plane and green arrow represents the projection of [100] axis in (103), (101) and (204) planes, \nrespectively.    23 \n  \n \nZhou et al.  FIG. 1. \n 24 \n \n \n \nZhou et al.  FIG. 2. \n \n 25 \n \n \n \nZhou et al.  FIG. 3.  26 \n \n \n \nZhou et al.  FIG. 4. \n \n    27 \n \n \n \nZhou et al.  FIG. 5. \n \n \n 28 \n \n \n \nZhou et al.  FIG. 6. \n \n 29 \n \n \n \nZhou et al.  FIG. 7. \n "    },    {        "title": "1804.08832v1.Antiferromagnetism_and_phase_transitions_in_non_centrosymmetric_UIrSi__3_.pdf",        "content": "1 \n Antiferromagnetism and phase transitions  \nin non -centrosymmetric UIrSi 3 \n \nJ. Valenta1, F. Honda2, M. Vališka1, P. Opletal1, J. Kaštil3, M. Míšek3, M. Diviš1,  \nL. Sandratski i4, J. Prchal1, and V. Sechovský1 \n \n1Charles University, Faculty of Mathematics and Ph ysics, Department of Condensed Matter Physics, \nKe Karlovu 5, Prague 2, Czech Republic  \n2Tohoku University, Institute for Materials Research, Narita -cho 2145 -2, Oarai, Ibaraki, Japan  \n3Institute of Physics AS CR, Na Slovance 1999/2, Prague 8, Czech Republic  \n4Max-Planck -Institute of Microstructure Physics, Weinberg 2, 06120 Halle, Germany  \n \nAbstract  \nMagnetization and specific heat measurements on a UIrSi 3 single crystal reveal Ising -like \nantiferromagnetism below TN = 41.7 K with easy magneti zation direction along the c-axis of \ntetrago nal structure. The antiferromagentic  ordering is suppressed by magnetic fields > Hc \n(µ0Hc = 7.3 T at 2 K) applied along the c-axis. The first -order metamagnetic transition at Hc \nexhibits asymmetric hysteresis reflecting a slow reentr y of the complex ground -state \nantiferromagnetic  structure with decreasing field. The hysteresis narrows with increasing \ntemperature and vanishes at 28 K. A second -order metamagnetic transition is observed at \nhigher temperatures. The point of change of the order of transition in the established H-T \nmagnetic phase diagram is considered as the tricritical point (at Ttc = 28 K and µ 0Htc = 5.8 T). \nThe modified -Curie -Weiss -law fits of temperature dependence  of the a- and c-axis \nsusceptibility provide opposite sig ns of Weiss temperatures, pa ~ -51 K and pc ~ +38 K, \nrespectively. This result and the small value of µ0Hc contrasting to the high TN indicate \ncompeting ferromagnetic  and antiferromagnetic  interactions responsible for the complex \nantiferromagnetic  ground  state. The simultaneous electronic -structure calculations focused on \nthe total energy of ferromagentic  and various antiferromagnetic  states, the U magnetic \nmoment  and magnetocrystalline anisotropy provide results consistent with experimental \nfindings and the suggested physical picture of the system.  \n \n  2 \n Introduction  \nThe growing  interest in materials adopting  crystal structure s which have no center of \nsymmetry  was boosted by the discovery of unconventional superconductivity in  CePt 3Si1. The \nabsence of a cent er of inversion  in the crystal structure along with a Rashba -type \nantisymmetric spin -orbit (s-o) coupling2, 3 leads to the possibility of a superconducting state \nwith an admixture  of spin -triplet and spin -singlet pairs4. The Rashba  s-o coupling in materia ls \ncrystallizing  in a noncentrosymmetric  crystal structure also causes spin-splitting of the Fermi \nsurface into two Fermi surfaces , which  has many intriguing implicatio ns in various branches \nof physi cs including magnetism5.  \nThe BaNiSn 3-type structure (I4mm ) illustrated in Fig. 1 is one of the ternary variants o f the \nBaAl 4 tetragonal structure . It is adopted by  several RTX 3 compound s (R: rare earth, T: \ntransition metal, X: p-electron element) . The R atoms occupy the corner s and the body center \nof the tetr agonal structure  whereas  the T-X sublattice is non -centrosymmetric . The lack of  an \ninversion center in the crystal structure bring s about a nonuniform lattice potential V(r) along \nthe c-axis, whereas the non uniform lattice potential  within the a-b plane is canceled out6.  \nThe RTX 3 compounds with  Ce are of high research interest because they exhibit diverse \ninteresting phenomena  like superconductivity with a high c ritical field , pressure induced \nsuperconductivity near a quantum critical point, coexistence of antifer romagnetism and \nsuperconductivity, vibron states , etc.7-12. The magnetic ordering in these materials is usually  \nantiferromagnetic  (AF)  with complex propagation vectors13-16 which  indicate competing \nferromagnetic and antiferromagnetic exchange inte ractions.  \n \n \nFIG. 1. Crystal structure of UIrSi 3. \n \nContrary to rare-earth compounds where the magnetic moment is usually born in  the localized  \n4f-electrons , the 5f-electron wave  functions  in U intermetallics lose, to a considerable extent , \ntheir atomic cha racter  due to  the mutual overlap  between neighboring U ions and due to the \nhybridization of 5 f-states with valence electron states of ligands  (5f-ligand hybridization) . The \nlarge direct overlap of 5 f-wave function s by rule prevents  formation  of a rigid ato mic 5f-\nelectron magnetic moment in materials in which the distance of nearest -neighbor U atoms is \nsmaller than  the Hill limit (340-360 pm)17. On the other hand , the layout of U-U nearest \nneighbors carrying 5f-electron orbital moments in the crystal lattice  usually determines the \n3 \n huge magnetocrystalline anisotropy  with the easy magnetization axis perpendicular to the \nstrong U -U bonding planes or chains18. The 5f-ligand hybridization  has similar but more \nsubtle effects on U magnetism. Its role strengthens  in compounds with a lower U content  in \nwhich the U  ion surrounding ligands prevent the direct U -U bonds19, 20. As concerns the \nmagnetic coupling the direct overlap of 5 f-wave functions of U neighbors is responsible for \nthe direct exchange interaction  between U nearest -neighbor magnetic moments whereas the \n5f-ligand hybridization mediates the indirect exchange  interaction between moments of U \nions neighboring the involved ligand .   \nOnly two uranium compounds adopting the tetragonal BaNiSn 3-type structure  are kn own, \nnamely , UIrSi 3 and UNiGa 3. Both have been  studied in the form of polycrystals only and  \nreported to order antifer romagnetically below 42  K (UIrSi 3)21 and 39  K (UNiGa 3)22, \nrespectively .  \nThis paper is dedicated to the result of our effort to advance unde rstanding of the physics \nof one of these two compounds . We have prepared  a UIrSi 3 single crystal , characterized its \ncomposition and crystal structure  and measured  the magnetization and specific heat in a wide \nrange of temperature s and external magnetic fie lds.   \nThe results confirm  that UIrSi 3 becomes antiferromagnetic below  the Néel temperature TN \n= 41.7 K with strongly anisotropic response to an external magnetic field . In the  magnetic \nfield along  the c-axis it  undergoes a metamagnetic transition (MT) at a critical fi eld Hc (µ0Hc \n= 7.3 T at 2 K ) into a field -polarized  state with a magnetic moment of  ~ 0.66 µB/f.u.  The \nobserved Hc value is much lower than expected for a simple antiferromagnet consisting of \nmagnetic moments of the order of 1 µ B with TN > 40 K.  No MT shows up in t he a-axis field \nup to 14 T.  \nAt low temperatures, MT is a first order  magnetic phase transition (FOMPT) and shows an \nasymmetric hysteresis. Hc decreases with increasing temperature while the hysteresis shrinks \nwith increasing temperatur e and eventually vanishes at 28 K . The character of MT \ndramatically changes at this temperature from FOMPT to  a second orde r magnetic phase \ntransition  (SOMPT) which is observed for 28 K  < T < TN) as manifested by the change of \ncharacter of magnetization an d specific -heat anomalies .      \nTo understand the observed phenomena further we have performed first -principles \nelectronic structure calculations focused on magnetism in UIrSi 3. The corresponding results of \nexperiments  and c alculations  as concerns the type  of anisotropy and the magnitude of \nanisotropy energy  show reasonable agreement whereas t he agreement on the size of the U \nmagnetic moment  depends  on the calculation  method . Both the experiment and theory suggest \nthat the magnetically compensated ground -state of the system is not a simple two -sublattice \nantiferromagnet of up -down - up-down type but h as a more complex nature.  \n \nExperimental and Computation al details  \nThe process of preparation of a UIrSi 3 single crystal  was started  by synthesis of \na stoichiometric poly crystal from pure element s and casting a rod ( 6.5 mm, length  85 mm). \nHigh purity elements: U  (99.9%), Ir  (99.99 %) and Si  (99.999%)  were  used. The obtained  \ningot was mounted in  a four-mirror optical furnace (by Crystal Systems Corporation ) \noptimized for t he floating zone melting method . The middle part of the final produc t was \nannealed at 700°C for 10 days . Energy d ispersive x-ray analysis  confirmed the presence of the \nsingle UIrSi 3 phase  in the annealed product.  The lattice parameters a = 417.22 pm and \nc = 996.04 pm of the tetragonal BaNiSn 3-type structure determined  by the x-ray powder 4 \n diffraction  on a pulverized pie ce of the single crystal are in  reasonable  agreement with the \nliterature 21. The U ions are coordinated solely within uranium basal plane layer s. Each U ion \nhas fo ur U nearest neighbors  located within the same basal plane and separated by dU-U = \n417.22 pm ( = a).    \nThe magnetization  and specific -heat measur ement s were carried  out with a PPMS  \napparatus (Quantum Design Inc.) in magnetic fields applied along the c-axis up to 14 T . For \ndetermination of TN from the temperature dependence of specific heat , the point of the \nbalance of the entropy released at the phase transition was taken . The field dependence  of the \nspecific heat was measured point by poin t in a stable magnetic field . At each point the \nmeasurement was repeated four times.   \nThe magnetic moments, easy magnetization axis, magnetocrystalline anisotropy energy , \nequilibrium volume and stability of antiferromagnetic structures were calculated using t he \nmethods based on density functional theory (DFT). We used the computer codes full p otential \nlocal orbitals (FPLO)23, full potential augmented plane wave s plus local orbitals (APW+lo)24 \nand in-house  augmented spherical waves (ASW)25 to solve single particle Kohn -Sham \nequations. We treated the 5f states as itinerant Bloch states in all three methods and since no \ninformation about gr ound state magnetic structure is available the simple ferromagnetic and \nantiferromagnetic arrangements of moments were applied. T he fully relativistic Dirac four-\ncomponent mode was used in all FPLO calculations. For calculations with FPLO code we \nused the division 24 ×24×24 for both the a- and c-axes corresponding to 1764 and 3756 \nirreducible k -points in the Brillouin zone, respectiv ely, to ensure the convergence of results. \nSince the total magnetic moment obtained from relativistic calculations was too small in \ncomparison with the experimental one the o rbital polarization correction26 was applied in the \nFPLO code. In  the APW+lo method we  applied spin orbit coupling with  the local spin density  \n(LSDA ) + Hubbard U approach24 to resolve the problem of the small calculated total \nmagnetic moment which points to additional electron correlations beyond the local and \nsemilocal exchange correlation potentials.  \nThe APW+ lo method was used to determine equilibrium lattice parameters.   We used \nmore than 800 augmented plane waves (more than 160 APW s per atom) and 2000 k -points in \nthe Brillouin zone to obtaine d converged results. The calculations of equilibrium volume with  \nthe APW+lo method were scalar relativistic to use forces when calculated  with local spin \ndensity (LSDA)27 and the general gradient approximations  (GGA)28-30.   \nThe ASW -LSDA method including the spin -orbit coupling (SOC) was applied to calculate \nthe total energy difference of the ferromagnetic and the two types of the antiferromagnetic \nstructures. The ASW method is well suited for calculation of complex magnetic structures in \nUranium compounds (see, e.g . in Ref.31)  \nThe performance s of LSDA and GGA have been compared with respect to the equilibrium \nvolume of UIrSi 3. The experimental c/a ratio and the symmetry free structure parameters \nobtained from minimization of the forces were used in all  the calculations.  We have \ncalculated the variation of the total energy with t he relative volume V/V 0 (V0 is the \nexperimental  equilibrium volume). The LSDA27 value of the equilibrium volume is about 3.3 \n% smaller than the experimental value. This is a typical deviation usually obtained in LSDA \ncalculations. The GGA from  Ref.28, on t he other hand, leads to a volume that exceeds the \nexperimental V0 by 1.7 % and the volume obtained with the GGA from Ref .29 is 1.5 % \nsmaller. The best results are ob tained using the GGA from Ref.30, which underestimates V0 by \nonly 0.8 %. In all forms GGA28-30 provide s a better equilibrium volume than LSDA.  \n  5 \n  \n \nResults  and Discussion  \nThe temperature dependence of the specific heat Cp(T) of UIrSi 3 displayed in Fig. 2 \nexhibits an anomaly at 41.7  K of a lambda shape , characteristic for a second -order phase \ntransition.  The anomaly is progressively shifted to lower temperatures when the crystal is \nsubjected to a gradually increasing magnetic field applie d along the c-axis (see Fig. 3). This is \na typical behavior of antiferromagnets at magnetic ordering transition.  In analogy we \nconclude that  UIrSi 3 in zero magnetic field undergoes  a magnetic phase transition between a \nparamagnetic and an AF state at Néel temperature TN = 41.7 K  which confirms  the only \npublished report on this compound21. When  the magnetic field increases above 4. 1 T the \nheight of the anomaly increases and simultaneously the anom aly becomes sharper.  This \nevolution terminates at 5 T. In fields increasing beyond  5 T the peak becomes gradually lower  \nand broader , disappearing around  7.1 T. \n \n \nFIG. 2.  Temperature dependence of the specific heat of UIrSi 3 in the temperature range \n2-80 K. The inset shows the low temperature specific heat in the Cp/T vs. T2 plot. The arrow  \nmarks TN = 41.7 K  \n \nWe have also measured temperature response s to the applied thermal p ulse in both , \nincreasing and decreasing temperature regime s. In fields higher than 5.8 T the transition \nexhibits a  temperature hysteresis which indi cates  emerging latent heat which is accompanying \na first-order magnetic phase transition.  The hysteresis of the transition vanishes in fields < 5.8 \nT.  \nWhen closely inspecting the specific heat (see Fig. 3 ) one can observe that in higher fields \nthe peak in the Cp/T vs. T plot has no lambda shape seen in fields above  5.6 T anymore .  \n \n6 \n  \nFIG. 3. Temperature depende nce of specific heat ( Cp/T vs. T plot) of single crystalline UIrSi 3 \nin various magnetic fields applied along the c-axis. \n \nIn contrast to the pronounced c-axis field influence  on the specific heat , application of \nmagnetic fields up to 14 T applied along the a-axis leaves the entire Cp(T) dependence intact.  \nThis indicates  uniaxial magnetic anisotropy with the a-axis as a hard axis.     \nThe estimated magnetic entropy associated with the AF transition in zero field is very low, \nnamely 0.14 Rln2. Such small valu e is usually attributed to  the itinerant character of \nmagnetism.  However,  the nearest U neighbor ions in UIrSi 3 are 417 pm apart which prevents \nconsiderable delocalization of 5f-electron states due to  the overlap of 5 f-electron wave -\nfunctions . The Cp/T vs. T2 plot of low temperature  (T < 4K ) specific -heat data shown in the \ninset of Fig. 2 is almost linear and points to a value of the Sommerfeld  coefficient \nγ = 31 mJ·mol-1·K-2 which is one of the lowest values among  U interme tallics .   \nAnother  evidence of the AF transition of UIrSi 3 at TN is provided by  measurements of the \ntemperature dependence of the magnetic susceptibility  (= M/H; M: magnetization, H: \nmagnetic field)  displayed in a n M/H vs. T plot in Fig. 4 . One can see that TN determined from \nspecific heat data falls to a somewhat lower temperature than the maximum of  vs. T curve. \nThis is because  TN is to coincide  with the maximum of the (T)/T derived from the \ntemperature dependence of the genuine thermodynamic variable T 32, 33. The susceptibility \nmeasured in the magnetic field applied along the c-axis is much larger than that in  the a-axis \nfield. Neither  shift of the transition temperature nor change of character of the (T) curve are \nobserved for the magnetic field applied along a-axis. On the other hand  the (T) peak near TN \nis gradually shifted to lower temperatures and broadens if the field applied along the c-axis is \nincreasing  and disappear s in a field above 7 T . This behavior correlates with the properties  of \nthe specific -heat anomaly  related to the AF transition.  \n \n7 \n  \n \nFIG. 4. Temperature dependence of susceptibility  (M/H vs. T plot) of single crystalline UIrSi 3 \nin the temperature range 2 -80 K in the magnetic field applied along the c-axis (µ 0H = 0.1 T) – \nfull red circles, and along the a-axis (µ 0H = 0.5 T) – empty blue circles.  The broken line \nmarks TN (= 41.7  K) determined from specific heat data whereas the maximum of the M/H is \nat higher  temperature (=  42.3 K). Inset: Detail of the c-axis M/H vs. T plot between 35 and \n45 K. The full black line represents the function  (T)/T vs. T dependence. The N éel \ntemperature is supposed to be at the maximum of (T)/T (= 41.4 K). \n \nThe zero-field cooled (ZFC) and field cooled (FC) thermomagnetic curves in the c-axis \nfield up to 7 T are entirely merging . In fields above 7 T the  ZFC and FC curves separate at \nlow temperatures which indicate destabilization of the AF state by the field .  \nThe temperature dependence s of the inverse magnetic susceptibility  in the paramagnetic \nregion  plotted in Fig . 5 demonstrate ubiquity  of the strong magnetocrystalline anisotropy  \nwhich is a common feature of most of magnetic uranium compounds . The magnetocrystalline \nanisotr opy in the paramagnetic range is usually manifested by the difference of Weiss \ntemperatures (paramagnetic Curie temperatures) p as parameters of fits of measured \nsusceptibility vs. temperature in the magnetic field applied along the main crystallographic \naxes by a modified Cu rie-Weiss (MCW) law: \n \n         𝜒(𝑇)=𝐶\n𝑇−𝜃𝑃+𝜒0 ,     (1) \n  \nwhere T is temperature, C is the  Curie constant  from which the value of effective moment µeff \ncan be derived and 0 is a temperature independent term representing the susceptibility of \ncond uction electrons.       \n \n8 \n  \n \nFIG. 5. Temperature dependence of the inverse susceptibility ( H/M vs. T plot) of single \ncrystalline UIrSi 3 in the temperature range 60 -600 K in the magnetic field (µ 0H = 8 T) applied \nalong the c-axis (full red circles) and along the a-axis (empty blue circles). The lines \nrepresenting  fits according to formula in the text  are hidden by experimental points . The inset \nshows the 60 -120 K detail.  The vertical broken line marks the temperature of the crossing of \nthe a-axis and c-axis H/M vs. T curves .  \n \nWe have measured the susceptibility in the temperature range from 2 to 600 K but fitted \ndata only above 60 K which is sufficiently  higher than TN to avoid influence of correlations in \nthe proximity of the magnetic ordering transition . Presuming  negligible contribution s to µeff \nfrom Ir and Si the fitted effective moment  is related to one U ion.  \n \nTABLE I. Results of the modified Curie -Weiss law fits of susceptibility data measured on a \nUIrSi 3 single crystal in the magnetic field along the a- and c-axis. \n \nH // fit T range (K)  µeff (µB/U) p (K) 0 (10-9 m3/mol)  \na 60 – 600 2.7 -51 1.3 \nc 200 – 600 2.0 -24 6.5 \nc 60 – 200 1.6 38 8.3 \n \n \nThe a-axis susceptibility data can be well  fitted over the entire temperature range 60 – 600 \nK. The fitted p value of –51 K compares with  TN (41.7  K) as expected in simple \nantiferromagnets entirely governed by an AF interaction. The fitted µeff value s are much \nlower than  the expectation values for the U3+ and U4+ free ion  (3.62  µB and 3.58  µB, \nrespectively) . \nOn th e other hand, t he c-axis (T) data cannot be fitted by a MCW law over the entire \ntemperature interval.  To get some qualitative insight into the complex situation we attempted \nto fit the data sets in the 200 – 600 K  and 60 – 200 K  sections  separat ely. The obt ained fitting \nparameters p, µeff, and 0 are displayed in TABLE I.   \nFurther on we refer to data obtained below 200 K only.  The temperature dependencies of \nthe a- and c-axis susceptibility, respectively, cross at ~ 93 K. The Weiss temperatures pa ~ - \n9 \n 51 K and  pc ~ + 38 K obtained from modified -Curie -Weiss -law fits in the  low-temperature \nregion reflect competing ferromagnetic and AF interactions . It is worth mention ing that a \nqualitatively similar situation (strong concave curvature of the 1/ c vs. T, crossing  of the 1/a \nvs. T and 1/c vs. T dependences, positive pc vs. negative pa value  at low temperatures ) is \nobserved in the case of UIr 2Si234 crystallizing in the tetragonal CaBe 2Ge2-type structure in \nwhich  the U ions appear in a noncentrosymmetric surroun ding of ligand s similar to that in \nUIrSi 3. At this stage of research we have no explanation for the origin of the anisotrop y of \ntemperature independent parameter χ0.  \nThe 2  K magnetization curves  measured in the field applied along the a- and c-axis, \nrespectively, which are displayed in Fig. 6, clearly demonstrate  the strong uniaxial anisotropy \nof UIrSi 3 in the AF state with the c-axis as the easy magnetization direction. The weak linear \nM(H) dependence of the magnetization  reach ing only 0. 2 µB/f.u. in 14 T is characteristic for \nthe field applied along the a-axis, the magnetically hard direction, indicated already by \nspecific -heat measurements . The c-axis magnetization follows almost the same M(H) \ndependence in the field up to 7 T.  When the field is increas ed above 7 T a sharp metamagnetic \ntransition  (MT) emerges resulting in a magnetization step up to 0.66 µB.  \n \n \nFIG. 6. The hysteresis loop of the magnetization measured at 2 K in the magnetic field \napplied along the c-axis (full red circles) and the M vs. H dependence of the magnetization \nalong the a-axis (empty blue circles). The arrows show the polarity of the field sweep.  \nBottom right i nset: The µ 0H = 4-8 T detail in the magnetic field applied along the c-axis. Top \nleft inset: Temperature evolution of th e hysteresis of the metamagnetic transition. The line \nrepresents the fit of experimental points with formula ( 2) in the text.  \n \nThe values of spin and orbital magnetic moments of uranium were calculated for a \nferromagnetic configuration from Dirac -Kohn -Sham  equations and the values Ms = -1.42 and \nML = 1.87 µ B providing the value of total moment M = 0.45 µ B. The total moment is much \nsmaller than experimental ly determined  moment of 0.66 µB in a field above MT . The orbital \npolarization correction as implemented  in FPLO code26 has been applied providing a total \nmagnetic moment 1.7 8 µB which overestimates the experimental value. Therefore the \nrelativistic LSDA +U method with adjustable Hubbard parameter U was applied24. The spin Ms \n= -1.20 µB and orbital ML = 1.86 µB magnetic moments were calculated with the parameter U \n= 0.28 eV that gives very good  agreement with experimental saturated moment. This finding \nshowed the 5 f electrons in UIrSi 3 are moderately correlated.  The moments calculated by the \nvarious methods an d the experimentally determined moment are compared in TABLE II.  \n10 \n    \nTABLE II. The spin MS, orbital ML and total magnetic moments M calculated by different \nmethods. Experimental value M means saturated moment.  \n \nMethod  MS [µB] ML [µB] M [µB] \nDirac  -1.42 1.87 0.45 \nDirac + OPC  -1.99 3.77 1.78 \nLSDA + U + SOC  -1.20 1.86 0.66 \nExperiment    0.66 \n \nThe total energies of the ferromagnetic and simple antiferromagnetic (up -down -up-\ndown...) structures calculated by the ASW -LSDA method including SOC were compared. \nThe AFM structure was formed by the ferromagnetic U layers in the a -b planes. The \nsubsequent layers had opposite directions of the magnetic moments. The magnetic moments \nwere collinear to the easy c-axis. Such an AFM structure appeared to be higher in energy  than \nthe FM structure by 16 meV/f.u. We increased the supercell along the c axis and tried an up-\nup-down -down antiferromagnetic structure. Interestingly, in this case there are two types of \ninequivalent U ions but the structure remains magnetically compen sated. The energy of this \nstructure appeared to be much closer to the energy of the FM one although still higher than \nthe FM energy by about 5 meV/f.u. These results are in line with our expectations of a \ncomplex antiferromagnetic ordering in UIrSi 3 due to  competition of ferromagnetic and \nantiferromagnetic exchange interactions.  The theoretical study of complex magnetically  \ncompensated structures in UIrSi 3 will be  continued.    \nWhen we extrapolate the a- and c-axis magnetization curves beyond the maximum ap plied \nmagnetic field we find them crossing at ~59  T which serves as a rough estimate of a \nmagnet ocrystalline  anisotropy field. This value is about an order of magnitude smaller than \nthe anisotropy fields of the majority of U TX and U T2X2 compounds which typically exhibit \nanisot ropy fields of several hundred teslas18. On the other hand  it compares to the anisotropy \nfields of UIr 2Si2 and UPt 2Si2 which both crystallize in the CaBe 2Ge2-type structure34, 35.  \nTo estimate magnetocrystalline anisotropy energy (MAE) the total energy with magnetic \nmoment along a- and c-axes, respectively,  have been calculated. The c-axis was found to be \nthe easy axis and the MAE is 4.57 meV  which is in fair agreement with the experimental one \n~1.1 meV.  \nThe critical field Hc of FOMPT is taken as  the midpoint of the magnetization step when \nsweeping the field u p. Hysteresis  is an intrinsic characteristic of first order transitions 36. We \nindeed have observe d a hysteresis although rather unusual . The reverse ( field-sweep -down ) \ntransition is considerably br oader and non symmetric  due to a tail in low fields. We tentatively  \nattribute  this behavior to gradual re entry of a complex ground -state AF spin arrangement with \nintermediate uncompensated phases . This picture should be verified by a relevant  microscopic \nexperiment (neutron scattering).  To characterize this transition , we have introduced the \ncharacteristic fields Hc↓ and Hc1↓ as indicated in the inset of Fig. 6.  \nThe temperature evolution of MT is demonstrated in Fig.  7 where the magnetization \nisotherms M(H)  measured at  selected  temperatures are shown.  Hc decrease s with increasing \ntemperature whereas the magnetizatio n step at MT  and the hysteresis  get reduced . The 11 \n temperature dependence of hysteresis µ0(Hc – Hc1↓) is shown in Fig. 6 and can be well fitted \nby the formula  \n \nµ0(𝐻𝑐 – 𝐻𝑐1↓)=𝑐 (1−√𝑇\n𝑇1) ,     (2),  \n which  has been taken ad hoc  from the paper on the temperature dependen ce of the \ncoercive field in single -domain particle systems applied to the Cu 97Co3 and Cu 90Co10 granular \nalloys37. The fit ting parameters c = 3.3 and T1= 24 K.  \n \n \nFIG. 7. The magnetization curves measured at various temperatures in the magnetic field \napplie d along the c-axis. \n \nTwo types of  M(H)  curves  can be seen in Figs. 7 – 9: \ni) at T  25 K they are characterized by a magnetization step M at Hc and a field \nhysteresis  H = (Hc – Hc1↓) and are practically linear for H < 0.9 Hc and saturated for H > 1.1 \nHc and Hc, M and H decrease with increasing temperature. These attributes are \ncharacteristics of a FOMPT ,     \nii) at temperatures  TN   T  30 K the M(H)  curves show an upturn  in fields  Hc \nterminated by  a cusp at Hc and followed by gradual saturation in higher fields. Hc decreases \nwith increasing temperature to become 0 T at TN. The upturn and the cusp become  \nsimultaneously less pronounced  and no hysteresis is observed.  In our scen ario th ese features  \nare characteristic of a SOMPT . \nThe magnetic -field dependence s of specific heat are shown in Figs. 8 and 9.  The plots \npresented  in Figs. 8, and 9 manifest the dramatic difference between the effect s in the specific \nheat accompanying  the FOMP T (Fig. 8) and the SOMPT  (Fig. 9), respectively , and \ncorresponding magnetization behavior . The FOMPT is manifested by a step of Cp/T at Hc \n(and hysteresis which qualitatively resemble s magnetization behavior . The positive step of \nCp/T at Hc is understood as  a result of an increased density of conduction electron states due \nto reconstruction of the Fermi surface  at MT.  \n \n12 \n  \nFIG. 8. Magnetic field dependence of specific heat ( Cp/T vs. µ0H plot) and magnetization of \nsingle crystalline UIrSi 3 at 2 K (left panel ) and 10  K (right panel ) in the magnetic field applied \nalong the c-axis (field sweep up - red, field sweep down - blue). \n \nOn the oth er hand, the SOMPT is manifested by  a -shape anomaly at Hc. The enhanced \nCp/T values in low er fields reflect  spin-flip fluctu ations from the AF state . The enhancement \nis more pronounced with temperature approaching  TN.     \n \n \nFIG. 9 . Magnetic field dependence of specific heat ( Cp/T vs. µ0H plot) and magnetization of \nsingle crystalline UIrSi 3 at 35  K (left panel) and 40  K (right panel) in the magnetic field \napplied along the c-axis (field sweep up - red, field sweep down - blue). \n \nThe magnetization and specific -heat data  allowed us to establish  the H-T magnetic phase \ndiagram shown  Fig. 10. The critical field of the MT decreases wi th increasing temperature \n13 \n towards zero  at TN. UIrSi 3 undergoes  a FOMPT at temperatures T < 28 K contrary to  a \nSOMPT observed for  T > 28 K . The strikingly different magnetization response to fields \nbelow Hc in the low and elevated temperature  areas of the magnetic phase diagram  evokes a \nquestion whether also the corresponding AF phases are different. We tentatively consider the \npoint separating the FOMPT and SOMPT regimes in the magnetic phase diagram as a \ntricritical point with coordinates Ttc = 28 K, µ0Htc = 5.8 T.  Recently two papers report ed \nsimilar to us a tricritical point in uranium intermetallic antiferromagnets without closer \nspecifi cation of the three  involved phases38, 39. We are ful ly aware of the weakness of the \nsuggested scenario until the difference  of the two antiferromagnetic phases is proven by \nrelevant experimental methods, e.g. neutron scattering, µSR, etc.    \n \n \nFIG. 1 0. Magnetic phase diagram of UIrSi 3 in the magnetic field applied along the c-axis. The \nlabels Hc, Hc↓ and Hc1↓ are defined in th e inset of Fig. 6. Colors of d ata points are representing \ndata from measurements of: M(H) … Hc - dark green, Hc↓ - red, Hc1↓ - blue; Cp(H) … Hc - \nlight blue, Hc↓ - yellow; Cp(T)… Hc - light green followed by a red hexagon indicating the \ntricritical point ( Ttc = 28 K, Htc = 5.8 T).  \n  \n \nConclusions   \nWe have gr own a single crystal of the non centrosymmetric tetragonal UIrSi 3 compound . \nThe UIrSi 3 stoichiometry and the BaNiSn 3-type structure have been confirmed by EDX and \nx-ray diffraction analysis, respectivel y. The cr ystal was subjected  to detailed measurements of \nmagnetization and specific heat with respect to  temperature and external magnetic field . To \nunderstand  the experimental results , more -detailed first-principles electronic structure \ncalculations for th is compound  have been performed .  \nThe results point to  Ising -like antiferromagnetism below TN = 41.7 K exhibiting strong \nuniaxial anisotropy  with the easy magnetization direction along the c-axis. The competition of \nantiferro - and ferromagnetic exchange in teractions plays an important role  in UIrSi 3 \nmagnetism as manifested by the  a) different sign of paramagnetic Curie temperatures of the a- \nand c-axis susceptibility  (pa ~ -51 K and pc ~ +38 K) , b) low critical field of MT (µ0Hc = 7.3 \nT at 2 K) contrasti ng with high TN (= 41.7 K) , c) asymmetric hysteresis of MT , d) existence of \nregions characterized by the first and second order phase transition , respectively,  separated by \na tricritical point  (at Ttc= 28 K, µ 0Htc = 5.8 T) , e) a higher calculated total ene rgy of a simple \nAF ground state than  a complex magnetically compensated state leading to the prediction of a \n14 \n complex AF ground -state magnetic structure. The existence of possible different \nantiferromagnetic phases remains to be proven by relevant microscop ic methods.  \n \n \nAcknowledgments  \nThis research is supported by Grant Agency of Charles University (GAUK) by project no. \n188115 , the Czech Science Foundation, grant No. P204/15 -03777S  and the Japan Society for \nthe Promotion of Science (JSPS) KAKENHI with the g rant nos. JP15K05156 and \nJP15KK0149. Experiments were performed in the Materials Growth and \nMeasurement  Laboratory MGML ( http://mgml.eu ). The authors are indebted to Dr. Ross \nColman for critical reading and correcting of the ma nuscript.  \n \nReferences  \n1   E. Bauer, G. Hilscher, H. Michor, Ch. Paul, E. W. Scheidt, A. Gribanov, Yu. Seropegin, H. \nNoël, M. Sigrist, and P. Rogl, Phys. Rev. Lett . 92, 027003 (2004).  \n2   E. Rashba, Sov. Phys . Solid State 2, 1109 –1122 (1960).  \n3   G. Dresse lhaust, Phys. Rev.  100, 580 -586 (1955).  \n4   L. P. Gor’kov, and E. I. Rashba, Phys. Rev. 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Harmon2, 5\n1Department of Physics, Kurukshetra University, Kurukshetra 136119 (Haryana), INDIA\n2Ames Laboratory, U.S. Department of Energy, Iowa State University, Ames, Iowa 50011-3020, USA\n3Department of Biochemistry, University of Nebraska, Lincoln, Nebraska 68588-0664, USA\n4Inter University Accelerator Centre, Aruna Asaf Ali Marg, New Delhi 110067, INDIA\n5Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011-3160, USA\nBy understanding the small easy axis magnetocrystalline anisotropy energy (MAE) of hexagonal\nCe2Co17, an attempt has been made to improve anisotropy and consequently to obtain better char-\nacteristics for a high energy permanent magnet via site selective substitutional doping of Ce/Co with\nsuitable elements. The present calculations of the electronic and magnetic properties of Ce 2Co17and\nrelated substituted compounds have been performed using the full potential linear augmented plane\nwave (FPLAPW) method within the generalized gradient approximation (GGA). Sm-substituted\ncompounds were simulated using Coulomb corrected GGA (GGA+ U) to provide a better represen-\ntation of energy bands due to the strongly correlated Sm- felectrons. The formation energies for\nall substituted compounds are found to be negative which indicate their structural stability. Of\nthe substitutions, Zr substitution at the Co-dumbbell site enhances uniaxial anisotropy of Ce 2Co17.\nFurthermore, Sm-substitution at Ce-2 csites favors incremental MAE whereas a La-substitution at\nboth 2b- and 2c-sites depletes the tiny MAE in Ce 2Co17. These observed trends in the MAE have\nbeen examined in terms of contributions from various electronic states. Finally, Ce 2Zr2Co15and\nSmCeCo 17are foreseen as suitable materials for designing permanent magnets derived from the\ncrystal lattice structure of hexagonal Ce 2Co17.\nPACS numbers: 71.20.Eh, 75.50.Ww, 71.15.Mb\nI. INTRODUCTION\nPermanent magnetic materials play an important role\nin improving the e\u000eciency, sustainability, and perfor-\nmance of commercial products in electric power gener-\nation, transportation, and other energy-use sectors of\nthe global economy. Besides a high Curie Temperature\n(Tc), desirable permanent-magnets (P-Ms) must have a\nhigh energy product BH maxand should be cost e\u000bective.\nBH maxis related to remnant magnetization ( Mr) and\ncoercivity ( Hc) which originate from the magnetocrys-\ntalline anisotropy energy (MAE). The MAE is one key\nfactor for determining the easiness of a particular mag-\nnetic material to become magnetized in one direction,\nwhile being resistant to magnetization in other direc-\ntions [1, 2]. The magnetic materials with high Hc, uni-\naxial MAE, and BH maxhave extraordinary applications\nin high-tech media and ferromagnetic electrodes of spin-\ntronic devices with high magnetic-noise immunity and\nthermal stability. Due to economic and geopolitical is-\nsues, there is an incentive to produce P-Ms free from\ncritical (expensive) rare earth elements. Ce-based mag-\nnets may be a prime choice in this direction due to their\nnon-criticality, relatively low price, and the abundance of\nCe [1].\nNumerous experimental and theoretical e\u000borts have\nbeen made to enhance the MAE of Ce 2Co17by vari-\nous means [3{7]. Streever [4] used their NMR results\nto evaluate the spin-orbit contribution to the magnetic\nanisotropy of Co atoms in RCo5compounds. They found\nthe easy c-axis Co anisotropy arises from the 2 csites. Keet al. [3] investigated the origin of MAE in doped Ce 2Co17\nand con\frmed that the dumbbell sites have a very nega-\ntive contribution to the MAE in Ce 2Co17, while the MAE\nis enhanced by replacing Co dumbbell sites with a pair of\nFe or Mn atoms. X-ray di\u000braction (XRD) measurements\n[5] on magnetically aligned Ce 2Co17\u0000xAlxpowders with\nx= 0\u00003 exhibit an easy-axis of magnetic anisotropy\nat room temperature. Substitution of Al for Co leads to\na change of the magnetocrystalline anisotropy of the Co\nsublattice from the basal plane to c axis. The anisotropy\nincreases with Al concentration (x), goes through a max-\nimum value of 17.7 kOe at x= 2 and then decreases.\nSun et al. [6] prepared Ce 2Co17\u0000xMnx(x= 0-4) by arc\nmelting in argon atmosphere and found by XRD mea-\nsurements that all these magnetically aligned powered\nsamples exhibit an easy axis magnetic anisotropy at room\ntemperature, attaining a maximum value of 30.1 kOe at\nx= 2.\nSm-Co magnets are strong permanent magnets de-\nveloped in the early 1960s and mainly exist in three\nhard magnetic phases: SmCo 5, SmCo 7, and Sm 2Co17.\nSm2Co17can be derived from the SmCo 5structure and\nin this process, Co enrichment with respect to SmCo 5\nyields an increase in magnetization and BH maxbut at a\ncost of the MAE [8, 9]. In a similar way, the hexagonal\nCe2Co17system (2-17 series) containing four Co sublat-\ntices de\fned by Wycko\u000b sites 12 k, 12j, 6g, and 4f, can\nalso be obtained from CeCo 5(hexagonal CaCu 5struc-\nture with 191:P6/mmm space group) by replacing ev-\nery third Ce atom by a pair of Co-atoms [3]. It is clear\nfrom the example of Sm magnets, 1-5 systems have betterarXiv:1804.10310v2  [cond-mat.mtrl-sci]  1 May 20182\nanisotropy but low magnetization due to a small number\nof Co atoms. Therefore, it is preferable to work with 2-17\nsystems if control over decreasing MAE can be achieved.\nKe et al. [10] studied R(Fe1\u0000xCox)11TiZ(R= Y\nand Ce;Z= H, C, and N) systems for identifying im-\nproved magnetic properties via saturation magnetiza-\ntionM, Curie temperature TC, and magnetocrystalline\nanisotropy energy coincident with reduction of critical\nmaterials in permanent magnets. They found the in-\nterstitial C doping signi\fcantly increases the uniaxial\nanisotropy in Ce(Fe 1\u0000xCox)11Ti for 0:7< x < 0:9 ,\nwhich may provide the best combination of all three in-\ntrinsic magnetic properties for permanent applications.\nNd2Fe14B-based sintered magnets were investigated ex-\ntensively and used widely since being discovered in\n1984 [11, 12]. Tian et al. [13] recently discovered the\npractical value of R2Fe14B (R= Pr, Nd) magnets can\nbe improved by proper doping of nitrogen into some in-\nterstitial sites. Ke and van Schilfgaarde [14] explained\nwhen Li is substituted by Fe in lithium nitride, the re-\nsultant compound; Li 2(Li1\u0000xFex)N behaves in many as-\npects similar to that of rare earth element. In addition,\nthey unveiled an analytical model to describe the magne-\ntocrystalline anisotropy energy (MAE) in solids as a func-\ntion of band \flling. The MAE decomposed into a sum of\ntransitions between occupied and unoccupied pairs. This\nmodel also yielded the MAE in good qualitative agree-\nment with its value from \frst-principles calculations for\nLi2(Li1\u0000xTx)N, withT=Mn, Fe, Co, and Ni systems.\nThese examples indicate doping substitutional replace-\nment of non-critical atoms is pivotal for enhancing the\nMAE in particular compounds.\nPure Ce 2Co17exhibits poor anisotropy characteris-\ntics and has a theoretical energy product of only\n31 MGOe [15], therefore it is not a suitable P-M.\nSun et al. [16] also established poor performance of\nCeCo 5-based compounds; Ce(Co 0:73Cu0:135Fe0:135)5:35\nand Ce(Co 0:73Cu0:135Fe0:135)5:55for permanent magnets\ndue to the formation of Ce 2Co17phase with TC<20\u000eC.\nImprovement in the anisotropy of Ce 2Co17is expected\nby suitable substitution of Co as observed in the litera-\nture, perhaps to a point beyond which it is auspicious for\nPM fabrication. Due to the presence of inequivalent sites\nof the Co and Ce atoms in Ce 2Co17, it may be possible\nto substitute these atoms by foreign atoms and design\nnew compounds with better magnetic properties and en-\nhanced MAE. The main aim of this study is to theoreti-\ncally investigate the change in the MAE via this type of\nsubstitution. For this purpose, the full potential density\nfunctional approach has been selected for studying Co\nsubstitution by Zr/Ti, and Ce substitution by Sm/La.\nII. COMPUTATIONAL APPROACH\nThe base compound; Ce 2Co17crystallizes in the\nTh2Ni17-type (space group: 194; P63/mmc) structure\n[17]. For spin polarized calculations of pristine Ce 2Co17and its 4fsite substitution, we considered the lattice pa-\nrameters of Ce 2Co17[8] whereas for 2 b/2csubstitution,\nwe have modeled the lattice parameters as an average of\nlattice parameters of Ce 2Co17and Sm 2Co17[18]/La 2Co17\n[19]. The calculations were performed using density func-\ntional theory (DFT) [20] based upon the full-potential\nlinearized augmented plane wave (FPLAPW) method\n[21] as implemented in the WIEN2k crystal program [22].\nThe generalized gradient approximation (GGA) under\nthe parameterization of Perdew-Burke-Ernzehof (PBE)\n[23] was used to construct the exchange-correlation (XC)\nfunctionals for all compounds except Sm-related ones.\nFor Sm substitutions, XC functional were considered\nwithin GGA+ Uto correct on-site f-electron correlation\nby takingUeff=U\u0000J= 6.0 eV [24], which includes\nthe Coulomb parameter, U= 6:7 eV and exchange pa-\nrameterJ= 0:7 eV for the Sm-f electrons. We used\nRmtkmax= 8.5 for basis functions, and the cut-o\u000b en-\nergy and charge convergence criteria were set to 10\u00006Ry\nand 10\u00006e, respectively, due to the sophisticated MAE\ncalculations involved. The k-space integration was per-\nformed using the modi\fed tetrahedron method [25]. Self-\nconsistency was obtained using 168 k-points in the irre-\nducible Brillouin zone (IBZ). A scalar relativistic calcu-\nlation, with GGA or GGA+ Uformalisms, was \frst per-\nformed to obtain a self-consistent potential. After \fxing\nthis potential, relativistic e\u000bects were included with the\nsecond variational treatment of spin-orbit coupling [26].\nThe MAE stems from the orbital contribution to the\nmagnetic moment mainly. We calculated the MAE by us-\ning the magnetic force theorem [27]. The MAE constant\n(K), which is MAE/volume of the cell, is then calculated\nfrom the expression:\nK=occuX\nj;k\u000fj(^n2;k)\u0000occuX\nj;k\u000fj(^n1;k) (1)\nHere\u000fjis the Kohn-Sham eigenvalue evaluated for each\nmagnetization orientation. In Eq. 1, jlabels occupied\nstates and the kis a particular k-point in the Brillouin\nzone, ^n1is [001], i.e., c-axis and ^n2is taken along the\n[100] direction. Here, the positive/negative value of E\nindicates the uniaxial/easy-plane anisotropy. The cal-\nculations of MAE demand a \fne energy resolution and\ndense mesh of in k-space, hence, we set up a 17 \u000217\u000215\nk-point mesh for all the anisotropy calculations in the\nspin orbit coupling environment.\nIII. RESULTS AND DISCUSSION\nThe identi\fcation of e\u000bective sites for substitution is\nneeded to enhance the MAE of Ce 2Co17. The litera-\nture reveals that previous researchers substituted Co 4f\n(dumbbell sites) as shown in Fig.1 by foreign elements\nand observed a signi\fcant change in the MAE [3, 5{7].\nAlong the same line, we started with similar substitution3\nby Ti and Zr. Further, Ce atoms in Ce 2Co17, presented\nat 2band 2csites, as highlighted by di\u000berent colors in\nFig. 1, are also expected to contribute signi\fcantly, thus\nwe also identi\fed these sites and investigated the substi-\ntutions at these by Sm and La. The symbolic represen-\ntation used for resultant alloys obtained by 4 f, 2band\n2csite substitutions is depicted in Fig. 2. The orbital\nmoments as obtained by proper substitutions are also\nmentioned to show the importance of the alloy within\nthis \fgure exclusively.\nFIG. 1. Schematics for Ce 2Co17showing Co dumbbell at 4 f\nsites and Co 2 c/2bsites, available for substitution.\nIn order to estimate the stability of the computed\nstructure with respect to Ce 2Co17, the formation ener-\ngies;Efor(de\fned by the di\u000berence of the accumulated\nsum of equilibrium energies of all constituent atoms from\nthe equilibrium energy of the resultant compound) was\nestimated which are depicted in the contour plot (Fig.\n3). We observe that out of the studied compounds, only\nCe2Ti2Co15and SmCeCo 17are more stable and easy to\nsynthesize as compared to Ce 2Co17. However, all com-\npounds are stable as veri\fed from the negative values\nof formation energies which is a good indication of the\nfeasibility of these compounds experimentally as well.\nThe total density of states (DOS) and atom resolved\ndensity of states at 4 f, 2band 2csites in Ce 2Co17,\nCe2Ti2Co15and Ce 2Zr2Co15are depicted in Fig.4. All\nthree total DOS are highly spin polarized, indicating the\nstrong magnetism in the alloys. With Ti/Zr substitution,\nthe states for majority spin above the Fermi level ( EF)\nincreases due to fewer d-electrons i.e. in 3 d2/4d2con\fg-\nurations as compared to Co with 3 d7con\fguration. As\nexpected, Ce- fstates contribute mainly in bands above\nEF. The states in the vicinity of EFare required for elec-\ntrical conductivity. The Co- dstates at various sites for\nTi/Zr substitution in Ce 2Co17are analyzed in Fig.5. It\nis evident that d-DOS of Co at the 12k and 6g sites is not\nmuch a\u000bected by Ti/Zr substitution. A drastic change\nin d-DOS of Co at 12j site can be identi\fed above EF.\nHowever, these states are located at energy greater than\n2 eV above EF, so these are insigni\fcant in the vicinity of\nEFfor easy excitations. Thus any change in total DOS\nis mainly due to the presence of Ti/Zr-d states in the\nFIG. 2. Symbolic site speci\fcation and nomenclature for Ce\nand Co substitution in Ce 2Co17at 4f, 2band 2c.\nFIG. 3. The contour plot for Eforof the pristine Ce 2Co17\nand 4f/2b/2csubstituted alloys (Here \u0003represents the actual\nmagnitude of Eforwith a particular substitution).\nsubstituted alloys. To emphasize the signi\fcant changes\nin total DOS, the atom resolved DOS at the substituted\nCo-site is plotted with separate colors.\nAfter identifying the e\u000bect of substitution of Co-\natoms (at dumbbell sites) on the electronic properties\nof Ce 2Co17, we have extended this study to Ce-site sub-\nstitutions by Sm and La atoms. Fig.6 depicts DOS of\nCe/Sm/La-f states at 2 band/or 2csites. On analysis, it\nis found that for the DOS at the dumbbell site; Co 4f-d\nremains una\u000bected with these substitutions. Further for\nsubstitution at 2 bor 2cby La, the unoccupied La-f are4\nFIG. 4. Total and atom resolved DOS at 2 b, 2cand 4fsites in Ce 2Co17, Ce 2Ti2Co15and Ce 2Zr2Co15\nunavailable in the vicinity of EFwhich reduces the spin\npolarization in the resultant system. Signi\fcant change\nin total DOS is governed by Sm substitution at 2 band 2c\nsites. The two Sm-sites substitutions behave quite di\u000ber-\nently. While 2 csubstitution gives rise to more occupied\nSm-f states, the 2 b-substitution produces very few Sm-f\nDOS in the occupied region. The remaining Ce-atom in\nthe unit cell on the other site, has similar type of DOS\nfor both 2b/2csubstitution. Further, these are found\nto be little perturbed compared to the case of pristine\nCe2Co17. To con\frm any change on Co-sites after sub-\nstitution, we also analyzed all the Co sites (not shown for\nbrevity). But surprisingly, all states are found to possess\nsimilar DOS as that of pristine case and do not have sig-\nni\fcant in\ruence in deciding the qualitative features of\ntotal DOS. However, the strong spin polarization at each\nsite results in better exchange spitting in all resultant\nalloys.\nTotal valence electronic charge densities, n(r) in the\n(110) and Co 4f- dumbbell planes for (i) pristine Ce 2Co17\n(ii) Ce 2Ti2Co15(iii) CeSmCo 17and (iv) SmCeCo 17com-\npounds were analyzed using the xcrysden program [28] to\nexamine the e\u000bect on the neighboring atoms after sub-\nstitution at 4 f, 2band 2csites in Ce 2Co17. Focusing on\npristine Ce 2Co17(Fig.7) the e\u000bect of symmetrical Co 4f\natoms aligned in the dumbbell form is clearly visible,\nwith maximum charge density along the dumbbell axis\n(see green color inside hexagonal contour in Fig.7 as a\nFIG. 5. Partial DOS at all types of Co-sites in Ce 2Co17,\nCe2Ti2Co15, and Ce 2ZrCo 15.5\nFIG. 6. Total and atom resolved DOS at 2 b, 2cand 4fsites in CeLaCo 17, CeSmCo 17, LaCeCo 17and SmCeCo 17\nprojection of charge density from two Co-atoms aligned\nin the dumbbell form on both sides of the (110) plane\nin perpendicular direction). Further charge density con-\ntours in the Co 4fdumbbell plane for Ce 2Co17indicate\nthat two Co atoms at 4 fsites are tightly bonded with\neach other. But the substitution of Ti in place of Co\natoms at 4 fsites decreases the charge densities along\nthe Co 4fdumbbell axis due to the presence of a lesser\nnumber of d-electrons. The dumbbell Ti-atoms are not\ntightly bonded and are expected to decrease the mag-\nnetism in the resultant compound.\nThe charge densities along the Co 4fdumbbell axis for\nSm-2b/2csubstitution are again enhanced. However, in\nthese substitutions, the e\u000bect on charge densities along\nCo4fdumbbell plane is modest and dumbbell Co-atoms\nare again tightly bonded (as in case of Ce 2Co17). The\nsubstituted Sm atom behaves di\u000berently in the resultant\ncompound which is clearly visible from the shape of Ce\nand Sm atoms as depicted in Fig.7(iii) and (iv), respec-\ntively. The Ce atom tends to polarize the neighboring\nCo-atoms but the Sm atom preserves its spherical shape.\nThe two factors, strong bonding between dumbbell Co-\natoms and spherical shape of Sm atoms seem to increase\nthe overall magnetism. These density plots demonstrate\nthe importance of Co-dumbbell sites in the present com-\npounds not only for 4 fbut also for 2 b/2csubstitutions.\nThe atom resolved spin and orbital moments at each\nsite are listed in Table-I. It is clear that 4 fsite substi-\ntution decreases the total moment of Ce 2Co17whereas\n2b/2csubstitution increases the same. The magnetiza-\ntion is found to be in excellent agreement with the pre-\nvious experiment [29] and calculation [3]. The Co spin\nmoments on each site are similar \u00191.5-1.8\u0016B. However\non 4fsubstitution, the Ti/Zr moment aligns antiparallel\nto Co moment at other sites which causes a signi\fcant\nFIG. 7. Total valence electronic charge density di\u000berence,\nn(r) in units of e=\u0017A3in Ce 2Co17, Ce 2Ti2Co15, CeSmCo 17and\nSmCeCo 17.\ndecrease in total spin moment of the resultant alloy.\nRegarding 2 b/2csubstitution by La, the La orbital mo-\nment not only drops down to zero but also its spin mo-6\nTABLE I. Calculated total orbital ( \u0016l), spin (\u0016s) and total moments ( \u0016t) in\u0016Bfor Ce 2Co17and all related substituted\ncompounds\nMaterial Magnetic moment( \u0016)\nCe(2c)Ce(2b)Co(12k)Co(12j)Co(6g)Co(4f)Inter. Total\nCe2Co17 This\nwork\nKe\net al. [3]\nExpt\n[29]\u0016s\n\u0016l\n\u0016t\n\u0016s\n\u0016l\n\u0016t\n\u0016t-0.58\n0.22\n-\n-0.84\n0.38\n-\n--0.63\n0.21\n-\n-0.90\n0.42\n-\n-1.60\n0.08\n-\n1.51\n0.10\n-\n-1.78\n0.09\n-\n1.56\n0.11\n-\n-1.55\n0.07\n-\n1.51\n0.08\n-\n-1.74\n0.06\n-\n1.65\n0.07\n-\n--4.04\n-\n-\n-1.1\n-\n-\n-23.58\n1.78\n25.36\n23.40\n2.43\n25.8\n25.4\nCe2Zr2Co15\nZr (4f)This\nwork\u0016s\n\u0016l\n\u0016t-0.47\n0.20\n--0.52\n0.47\n-1.22\n0.07\n-1.34\n0.10\n-1.07\n0.08\n--0.19\n0.01\n--3.36\n-\n-15.52\n1.95\n16.37\nCe2Ti2Co15\nTi (4f)This\nwork\u0016s\n\u0016l\n\u0016t-0.58\n0.15\n--0.66\n0.22\n-1.27\n0.07\n-1.92\n0.10\n-1.16\n0.07\n--0.52\n0.01\n--4.22\n-\n-16.07\n1.62\n17.69\nCeLaCo 17\nLa (2b)This\nwork\u0016s\n\u0016l\n\u0016t-0.62\n0.22\n--0.12\n0.00\n-1.67\n0.09\n-1.56\n0.12\n-1.63\n0.08\n-1.81\n0.07\n--4.12\n-\n-25.18\n1.86\n27.04\nCeSmCo 17\nSm (2b)This\nwork\u0016s\n\u0016l\n\u0016t-0.60\n0.22\n-0.484.82\n-1.40\n3.421.65\n0.09\n1.741.55\n0.11\n1.661.60\n0.08\n1.681.77\n0.06\n1.83-3.44\n-\n-3.4430.08\n0.38\n30.46\nLaCeCo 17\nLa (2c)This\nwork\u0016s\n\u0016l\n\u0016t-0.11\n0.00\n--0.67\n0.20\n-1.67\n0.08\n-1.56\n0.10\n-1.63\n0.08\n-1.81\n0.07\n--4.18\n-\n-25.09\n1.66\n26.75\nSmCeCo 17\nSm (2c)This\nwork\u0016s\n\u0016l\n\u0016t5.02\n-1.76\n3.26-0.66\n0.20\n-1.66\n0.08\n-1.56\n0.11\n-1.63\n0.08\n-1.79\n0.06\n--3.32\n-\n-30.56\n-0.06\n30.50\nment gets reduced. There is not much change either in\norbital or in spin moment observed for Ce as well as Co\natoms. However, this La-substitution slightly increases\nthe total moment due to dilution of antiparallel align-\nment of La and Co spin moments. On the other hand, a\nsigni\fcant change is observed for Ce substitution by Sm\nin Ce 2Co17. The Sm atom with 4 f5con\fguration has the\nability to enhance the total moment. The large increase\nin total magnetic moment is the consequence of the en-\nhanced net Sm moment of 3.42/3.26 \u0016Bat 2b/2csites\nin CeSmCo 17/SmCeCo 17. Furthermore, the Sm moment\naligns antiparallel/parallel to the Ce/Co spin moment\nwhich also favors the increase in total moment. Focusing\non orbital moment only, we have observed that all atoms\nhave very small orbital moments except for Sm-atoms\nin CeSmCo 17and SmCeCo 17which is expected due to\nSm-4f5states.\nFinally, to establish the potential of the studied alloys\nin permanent magnets, the MAE ( K) for the studiedsystems were examined in Fig.8. The value of Kfor\nCe2Co17is poor ( \u00190.03 MJ/m3) as con\frmed from our\ncalculations and there is no scope to fabricate permanent\nmagnets based on pristine Ce 2Co17alone. Further the\nprevious calculations by Ke et al. [3] and experimental\nwork by Hu et al. [29] also reported a tiny value (0.09\nMJ/m3and 0.32 MJ/ m3, respectively) of the same. The\n4f- site substitution by Zr increases the MAE of Ce 2Co17\nsigni\fcantly. We found the MAE for this substitution\n(Ce2Zr2Co15) as 1.15 MJ/ m3which is within reasonable\nagreement with the corresponding experimental value of\n2.57 MJ/m3[7] and close to that predicted by Ke et.\nal. [3]. The substituted Zr-atom contributes positively\nin the value of MAE which strains alignment of the mag-\nnetization from all atoms along any other magnetic \feld\ndirection except for the easy c-axis direction. In one ex-\nperiment with a similar type of substitution, Al atoms in\nCe2Al2Co15[5] were found to prefer the dumbbell sites\nand are responsible in increasing uniaxial anisotropy|7\n/s45/s50/s45/s49/s48/s49/s50/s51\n/s67/s101/s83/s109/s67/s111\n/s49/s55/s67/s101/s76/s97/s67/s111\n/s49/s55\n/s76/s97/s67/s101/s67/s111\n/s49/s55/s67/s101\n/s50/s67/s111\n/s49/s55/s67/s101\n/s50/s90/s114\n/s50/s67/s111\n/s49/s53\n/s67/s101\n/s50/s84/s105\n/s50/s67/s111\n/s49/s53/s75/s32/s40/s77 /s74/s47/s109 /s51\n/s41\n/s83/s109/s67/s101/s67/s111\n/s49/s55/s32/s84/s104/s105/s115/s32/s87/s111/s114/s107\n/s32/s75/s101/s32/s101/s116/s32/s97/s108/s46/s32/s91/s32/s51/s32/s93\n/s32/s69/s120/s112/s116/s46/s32/s91/s32/s55/s32/s93\nFIG. 8. Calculated MAE (MJ/ m3) for Ce 2Co17and all\nother substituted alloys. The corresponding experimental [7]\nand predicted [3] values are also presented for comparison.\nwhich also agrees with our \fndings. On the other hand,\nthe increase in MAE via similar substitution by Ti is not\nas e\u000bective and leads to an increase in Kto 0.23 MJ/ m3.\nThe di\u000berence in MAE obtained is a consequence of the\natomic size of the substituted atom (Zr/Ti). Whereas\nZr with a larger radius at dumbbell sites leads to strong\nhybridization; Ti-atom with smaller size than Co results\nin an ine\u000bective coupling.\nThe site substitution at 2 b/2csites by La has an op-\nposite e\u000bect on the MAE. Rather than increasing the\nuniaxial anisotropy, La-substitution brings out in-plane\nanisotropy of the material due to the unavailability of La-\n4felectrons. In contrast to La, site selective Sm substitu-\ntion which is rich in f-electrons, can have a considerable\ne\u000bect on the overall MAE of the resultant compound,\ndepending upon the crystal environment in the vicinity\nof substitution. In 2 bsubstitution, Sm atoms are found\naway from Co-dumbbell sites and have impact along xy\nplane, therefore, this substitution also brings down the\nanisotropy of Ce 2Co17in planner mode. However, for\n2csubstitution, Sm atoms align coaxially with the Co-\ndumbbells and thus the spin alignment along the c-axis\n(which is also an easy axis) is favourable. As a result, theinvolvement of 4 f5states gives rise to increased magnetic\nanisotropy in this case and the \fnal value of Kreaches\nto 1.4 MJ/m3.\nIV. CONCLUSION\nIn the regime of density functional theory, pristine\nCe2Co17and compounds with site substitution of Ce/Co\natom by La,Sm/Ti,Zr have been investigated. The Co-\ndumbbell sites are very crucial for the overall MAE in\nthe resultant compound. The calculated MAE is found\nmeaningful for Zr/Sm substitution at the 4 f/2csite.\nThe enhancement of MAE in these calculations for Zr-\nsubstitution is in accordance with the available exper-\nimental data. The high convergence was cross-checked\nso that these results can be used as a benchmark in fu-\nture experiments/calculations. Sm substitution at the 2 c\nsite is vital on the basis of magnetization as well as MAE\nobtained. Further experimental analysis is required to es-\ntablish the candidature of Ce 2Zr2Co15and SmCeCo 17as\npermanent magnets. Keeping in mind the cost of Sm, the\npresent analysis is important as most of the prominent\npermanent magnets are made from SmCo 5and Sm 2Co17.\nTherefore reducing signi\fcant content of Sm by substi-\ntuting cheaper Ce in the material and preserving a large\nmagnetic moment may open up new avenues for the de-\nvelopment of permanent magnets at a signi\fcant reduced\ncost.\nV. ACKNOWLEDGMENTS\nM.K.K. would like to acknowledge UGC, New Delhi\nfor providing Raman Post-Doctoral Fellowship vide grant\nNo. 5-147/2016 (IC). This research is supported by the\nCritical Materials Institute, an Energy Innovation Hub\nfunded by the U.S. Department of Energy (US-DoE),\nO\u000ece of Energy E\u000eciency and Renewable Energy, Ad-\nvanced Manufacturing O\u000ece. The Ames Laboratory is\noperated for the US-DoE by Iowa State University of\nScience and Technology under Contract No. DE-AC02-\n07CH11358. T.A.H. is grateful to the US-DoE for the\nassistantship and opportunity to participate in the SULI\nprogram for undergraduate research.\n[1] J. F. Herbst, J. J. Croat, and W. B. Yelon, Journal of\nApplied Physics 57, 4086 (1985).\n[2] J. F. Herbst, J. J. Croat, F. E. Pinkerton, and W. B.\nYelon, Phys. Rev. B 29, 4176 (1984).\n[3] L. Ke, D. A. Kukusta, and D. D. Johnson, Phys. Rev.\nB94, 144429 (2016); ArXiv: 1610.02767 , v2 (2018).\n[4] R. L. Streever, Phys. Rev. B 19, 2704 (1979).\n[5] B. gen Shen, J. yun Wang, H. wei Zhang, S. ying Zhang,\nZ. a Cheng, B. Liang, W. shan Zhan, and C. Lin, Journal\nof Applied Physics 85, 4666 (1999).[6] Z. gang Sun, S. ying Zhang, H. wei Zhang, J. yun Wang,\nand B. gen Shen, Journal of Physics: Condensed Matter\n12, 2495 (2000).\n[7] H. Fujii, M. V. Satyanarayana, and W. E. Wallace, Jour-\nnal of Applied Physics 53, 2371 (1982).\n[8] O. Gut\reisch, Journal of Physics D: Applied Physics 33,\nR157 (2000).\n[9] P. Larson and I. I. Mazin, Phys. Rev. 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Rao, The European\nPhysical Journal B - Condensed Matter and Complex\nSystems 33, 55 (2003).\n[20] M. Weinert, E. Wimmer, and A. J. Freeman, Phys. Rev.\nB26, 4571 (1982).[21] G. K. H. Madsen, P. Blaha, K. Schwarz, E. Sj ostedt, and\nL. Nordstr om, Phys. Rev. B 64, 195134 (2001).\n[22] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka,\nand J. Luitz, WIEN2K, An Augmented Plane Wave +\nLocal Orbitals Program for Calculating Crystal Properties\n(Karlheinz Schwarz, Techn. Universit at Wien, Austria,\n2001).\n[23] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett. 77, 3865 (1996).\n[24] V. I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein,\nJournal of Physics: Condensed Matter 9, 767 (1997).\n[25] P. E. Bl ochl, O. Jepsen, and O. K. Andersen, Phys. Rev.\nB49, 16223 (1994).\n[26] D. D. Koelling and B. N. Harmon, Journal of Physics C:\nSolid State Physics 10, 3107 (1977).\n[27] A. Mackintosh and O. Andersen, \\Electrons at the fermi\nsurface,\" (Cambridge University Press, Cambridge, Eng-\nland, 1980).\n[28] A. Kokalj, Computational Materials Science 28, 155\n(2003), proceedings of the Symposium on Software De-\nvelopment for Process and Materials Design.\n[29] S. Hu, X. Wei, D. Zeng, X. Kou, Z. Liu, E. Brck,\nJ. Klaasse, F. de Boer, and K. Buschow, Journal of Al-\nloys and Compounds 283, 83 (1999)."    },    {        "title": "1805.02394v1.Revealing_Controllable_Anisotropic_Magnetoresistance_in_Spin_Orbit_Coupled_Antiferromagnet_Sr2IrO4.pdf",        "content": "Adv. Funct. Mater. 28, 1706589 (2018)   \n \nRevealing Controllable Anisotropic Magn etoresistance in Spin-orbit Coupled \nAntiferromagnet Sr 2IrO 4  \n \nChengliang Lu*, Bin Gao, Haowen Wang, Wei Wa ng, Songliu Yuan, Shuai Dong*, and Jun-Ming \nLiu   Dr. C. L. Lu,  H. W. Wang, W. Wang, Prof. S. L. Yuan \nSchool of Physics & Wuhan National High Magnetic Field Center,  Huazhong University of Science and Technology, Wuhan 430074, China Email: cllu@hust.edu.cn   \nB. Gao, Prof. S. Dong \nSchool of Physics, Southeast University, Nanjing 211189, China Email: sdong@seu.edu.cn   \nProf. J. –M. Liu \nLaboratory of Solid State Micros tructures and Innovation Center of  Advanced Microstructures, \nNanjing University, Nanjing 210093, China  \nInstitute for Advanced Materials, Hube i Normal University, Huangshi 435001, China \n  Keywords: antiferromagnetic spintronics, anis otropic magnetoresistan ce, spin-orbit coupling \n  [Abstract] \n \nAntiferromagnetic spintronics activel y introduces new principles of magnetic memory, in which the \nmost fundamental spin-dependent phenomena, i.e.  anisotropic magnetoresistance effects, are \ngoverned by an antiferromagnet instead of a ferromagnet. A general scenario of the \nantiferromagnetic anisotropic magnetoresistance ef fects mainly stems from the magnetocrystalline \nanisotropy related to sp in-orbit coupling. Here we demonstr ate magnetic field driven contour \nrotation of the fourfold anisotropic magnetoresistance in bare antiferromagnetic Sr 2IrO 4/SrTiO 3 \n(001) thin films hosting a str ong spin-orbit coupling induced Jeff=1/2 Mott state. Concurrently, an \nintriguing minimal in the magnetoresistance emer ges. Through first princi ples calculations, the \nband-gap engineering due to rotati on of the Ir isospins is reveal ed to be responsible for these \nemergent phenomena, different from the traditiona l scenario where relativ ely more conductive state \nwas obtained usually when magnetic  field was applied along the magnetic easy axis. Our findings \ndemonstrate a new efficient route, i.e. via the novel Jeff=1/2 state, to realize controllable anisotropic \nmagnetoresistance in antiferromagnetic materials.   \n \n    1. Introduction \nAntiferromagnets have been garner ing increasing interest in the spintronics community, due to \ntheir intrinsic properties such as zero stray magnetic field, ultrafast spin dynami cs, and rigidity \nagainst external fields [1-4]. The observation of anisotropic magnetoresistance (AMR) and even \nmemory effect in antiferromagnetic (AFM) material s represent a major step in the field of AFM \nspintronics in which the antife rromagnet governs the tran sport instead of just  playing a passive \nsupporting role in the traditional ferromagnetic (FM) spintronics [5-8]. Here the fundamentals rely \nmainly on the magnetocrystalline an isotropy in the antiferromagnets, arising from the relativistic \nspin-orbit coupling (SOC ). Since this anisotropy energy is an even function of ordered spins [9, 10], it \nis feasible to electrically read out the spin axis  of staggered spins in an antiferromagnet through \nAMR effects which is the magnetotransport counterp art of the anisotropy energy. For instance, as \nconfirmed in several AFM materials, rotating the AF M spin-axis with respect to crystal axes leads \nto variations in electri c conductivity, and as a cons equence AMR is evidenced [5-8, 11-13]. Although \nprevious studies have successfully reali zed AFM-based AMR (AFM-AMR) through various \napproaches, it remains a great challenge to tailo r the AMR in AFM materials, which hinders the \nrecognition of the full meri ts of antiferromagnets.  \nRecently, the strong SOC in some AFM iridates is found to be essentia lly involved in Mottness \nof 5d electrons, i.e. opening a band-gap by the colla borative effect of strong SOC and moderate \nHubbard repulsion, forming a novel Jeff=1/2 state [14, 15]. Such a Jeff=1/2 state, with the SOC as its \ningredient, provides nontrivial playground to engineer the AFM-AM R, which may lead to unusual \nAMR phenomena. In addition, the strong SOC in ir idates essentially entangle both the spin and \norbital momenta, thus giving rise to a Jeff =1/2 character of the Ir magnetic moments [16]. This is \nfundamentally distinct from the situation in mo st previously studied AFM alloys where the SOC \nand magnetic moments often have different sources, i.e. the SOC arises from heavy elements while \nthe magnetic moments come from 3 d transition metals [2, 3]. Therefore, the iridates with novel \nJeff=1/2 state are leading candidates for comprehensively unde rstanding the physic al link between \nthe AMR and magnetocrystalline anisotropy rela ted to strong SOC, and exploring possibly \ncontrollable AFM-AMR.  \nHere we demonstrate a cont our rotation of the fourfold  AMR in the prototypical Jeff=1/2 AFM \nsemiconductor Sr 2IrO 4 (SIO) thin films grown on (001) SrTiO 3 (STO) substrate. Concurrently, an abnormal magnetoresistance (MR) minimal is evid enced. With the help of first-principles \ncalculations, the band-gap engineering due to the ro tation of Ir’s isospins is revealed to be \nresponsible for these phenomena, as a new route to manipulate the AFM-AMR.  \n \n2. Results and Discussion \n2.1. Sample preparation and characterization \nBecause of IrO 6 octahedra rotation ( α~11.8°) with respect to the c axis, Sr 2IrO 4 has an \nexpanded tetragonal cell. The Ir is ospins, containing remarkable or bital contribution, prefer the \nAFM order within the ab plane, and show a collective deviation of the isospins from the a axis, \ndefining the isospin canting angle ϕ~13° [17, 18]. Jackeli et al.  theoretically revealed that the angle ϕ \nrigidly tracks the lattice distortion α, i.e.,  α~ϕ [19]. A net moment was found to exist in each IrO 2 \nlayer, arising from the isospin canting [17, 20-23]. By applying small magne tic field within the ab plane, \na spin-flip transition can be triggered, resulting in a weak FM phase in Sr 2IrO 4 [20, 24, 25]. This \nprovides a natural handle to dr ive the planar AFM orders by external magnetic field in Sr 2IrO 4. The \nmagnetic structure is schematically shown in Figure 1 (a).  \nTo preserve the sp in-orbit coupled AFM orders, epitaxial Sr 2IrO 4 thin films with thickness of \n~40 nm were grown on (001) STO substrates, cons idering the good lattice f it between the film and \nthe substrate. The Sr 2IrO 4 thin films exhibit a layer-by-layer growth mode, evidenced by the \n2-dimensional reflection high energy electron diffr action (RHEED) image and intensity oscillations \n(Figure S1 ). The thin films are of pure phase and high quality, identified by detailed structural \ncharacterizations  (Figure S2 ) [26]. The x-ray reciprocal space mappi ngs (RSM) shown in Figure 1(b) \nreveals the coherent growth of th e films on the substrates, and the in-plane and out-of-plane lattice \nparameters are found to be a/√2= b/√2=3.905 Å and c/2=12.848 Å. A small tensile strain ε of only ~ \n0.46% is detected as expected. Fo r a convenience, here a pseudo-te tragonal lattice expression that \nhas a 45° in-plane rotation with respect to the tetragonal lattice is employed for the films, as \nschematically shown in Figure 1(c).  \nFigure 2 (a) presents resistivity ( ρ) as a function of T for the film. A semiconducting behavior is \nseen in the entire T range, resulting from the Jeff=1/2 Mott state [14]. In the M(T)  curves of both field \ncooling (FC) and zero field c ooling (ZFC) sequences shown in  Figure 2(b), clear FM phase \ntransition can be identified at T=TC ~230 K, arising from the Ir isospi n canting as sketched in Figure 1(a). Note that here the FM  transition is sharp and the TC is close to the value seen in Sr 2IrO 4 bulk \ncrystals [20, 24], confirming the high quality of our thin f ilms. Further evidence to the weak FM phase \nis provided by M(H)  measurements at various temperatures, and a saturated magnetization is \nestimated to be Ms ~0.03 μB/Ir at T=10 K ( Figure S3 ), which is slightly smaller than the value in \nbulk crystals [24]. Accompanying with the field indu ced isospin-flip transition at H~0.2 T, ρ shows a \nquick decrease with H, evidencing a significant suppressi on of the spin-dependent transport \nscattering effect (Figure S3).  \nBy applying H along different directions, tw o interesting features can be seen in the MR data \nmeasured at various temperatures, shown in Fi gure 2(c) and (d). First, the MR curves of H//[100] \nand H//[110] are different from each other, and present an intriguing intercross at high field region, \nevidencing an unconventional AMR phenomenon in th e thin films. Second, accompanying with the \nintercross, the MR curve of H//[110] shows a break-i n-slope upon sweeping-up H continuously. For \ninstance, with H//[110] at T=100 K, a minimal in the MR is seen at H~5.5 T. This unusual  \nH-induced MR minimal is followed by evident hysteresis, indicati ng that the system undergoes a \nmetastable state when sweeping H. Similar MR minimal was recently seen in heavy fermion metal \nCeIn 3 with large SOC, while the critical field (~60 T) was found to be one order of magnitude \nhigher than the present case [27].  \n \n2.2. AMR contour rotation \nTo gain further insight into the field induced anisotropic transport, comprehensive AMR \ncharacterizations have been carried out, as shown in Figure 3 . The device geometry for the AMR \nmeasurements is shown in Figure 3( a), in which the exciting current I is applied always along the \n[100] direction, H is rotated in-plane, and Φ is defined as the angle between H and I. Fourfold AMR \n=[R(Φ)-R(0)]/R(0) effect is generally seen below TC in the thin films, shown in Figure 3(b)-(c). The \nfourfold AMR symmetry excludes a normal AMR or igin which simply depends on the relative \nangle between the exciting current and magnetic field with a relationship of R(Φ)~sinΦ, but should \nbe mainly attributed to the magnetocrystalline anisotropy which depends on the relative angle \nbetween spin-axis and crystal-axis. This is furt her confirmed by a direct  correspondence between \nthe AMR symmetry and the tetragonal structure of Sr 2IrO 4.  \nAn obvious effect seen in Figure 3(b) is  that increasing the magnetic field up to H=7 T causes an in-plane AMR contour rotation by about 45°. For instance, the pristine fourfold AMR minima are \nchanged to be maximal positions in the newly de veloped fourfold AMR. While the AMR contour \nrotation can be seen clearly, the AMR at 9 T ma y still be mixed with little low-field AMR \ncomponent, indicating a much highe r field that is desired to co mpletely suppress the low field \ncharacteristic. This AMR contour rotation can still be seen at T=150 K and at a bit lower field H=5 \nT, shown in Figure 3(c). Such AMR contour rotation is consistent with  the intercross behavior seen \nin the MR curves (Figure 2(c) and (d)). To dis tinguish the two types of fourfold AMR appearing \nbefore and after rotation, they are denoted as ARM-I and AMR-II, respectively.   Figure 3(d) presents complete AMR map at T=50 K, which were coll ected under external \nmagnetic field ranging from H=0.1 T to 9 T.  In prior to the onset of the weak FM phase at H<0.2 T, \na twofold AMR is observed, whic h roughly follows a harmonic sin Φ dependence and can be \nascribed to a standard non-crystalline AMR.  As H>0.2 T, the weak FM phase emerges, which can \nbe utilized to drive the AFM-ax is travelling basal plane of Sr\n2IrO 4. As a consequence, the four-fold \nAMR-I arises. A closer checking id entifies minima of the AMR-I at Φ ~50°+ nπ/2 (n=0, 1, 2, 3).  \nNote that the magnetic easy axis of Sr 2IrO 4 is along the [110] direction ( Φ~45°, not exactly because \nof the isospin canting) [17, 18]. Therefore, the fourfold AMR-I can  be understood by the scheme of \nmagnetocrystalline anisotropy, while a spot of contribution from normal AMR effect cannot be \nexcluded considering the planar device geometry.  \n Further increasing H to ~5 T causes significant suppression of the AMR-I, and instead activates \nanother fourfold AMR symmetry (i.e., the AMR-II, in which the peaks are indicated by arrows) as \nH >5 T, which has a ~45° shift relative to the AMR-I. A critical field Hc of the AMR-I to AMR-II \ntransition is estimated to be H=Hc~5 T, by plotting the peak positions of the AMR as a function of H, \nshown in Figure 3(e). The peak positions can be mo re precisely determined  but don’t change the \nsymmetry of the AMR. Similar AMR transition can be seen in various temperatures below TC \n(Figure S4 ), and the critical field Hc shows gradual decrea se with increasing T, shown in Figure \n3(f). By extrapolating the Hc-T curve, a critical field of Hc ~10 T can be obtained at T=0 K. \nRegarding the AMR-II, it still has a repetition durat ion of ~90°, while its minima appear along the \npseudo-tetragonal axes of Sr 2IrO 4 (i.e., the [100] direction) which have a 45° shift relative to the \nbasal magnetic easy axes (i.e., the [110] directi on). In this sense, the emergence of the AMR-II \ncould not be explained simply by the magnetocrysta lline anisotropy as for the AMR-I. This will be discussed in details by combining with first principles calculati ons in the discussion section.  \n Stable directions of the antiferromagnetically ordered spins are separated by the magnetocrystalline anisotropy energy, and switch ing in-between these st ates by overcoming the \nenergy barrier may lead to hysteresis. On one si de, studying the switching dynamics would be of \nbenefit to deeply understand the observed AMR effect s. On the other side, for memory applications, \nhysteresis allows for non-volatile recording, while non-hysteretic may provide low-loss in magnetic \nsensors \n[6, 28]. To address this, Figure 4 (a) presents AMR traces recorded by rotating H from Φ=0° \nto Φ=360°, and then back to Φ=0° at T=50 K. Note that there is no hysteresis at the rotation starting \nposition, which can be used to disregard buckling problems in our experiments. This is further \nconfirmed by repeating the measurements with di fferent initial angles. Evident hysteresis can be \nseen in the AMR traces especially  at low field range. For instance,  the hysteresis is as large as \nΔΦ~14° at H=0.3 T. Increasing H causes notable suppression in ΔΦ, and finally a rather small \nΔΦ~1° is seen at H=9 T. Figure 4(b) shows ΔΦ as a function of H, in which three regions can be \nidentified. For H<0.5 T, ΔΦ decreases rapidly from ~14° to ~5° upon increasing H. For 0.5 T< H< 5 \nT, ΔΦ enters a plateau without evid ent change. Furt her increasing H causes a step-like sudden \ndecrease in ΔΦ(H) at H~5 T, and then ΔΦ again evolves with H steadily at a level of ΔΦ~1° till \nH=9 T.  \nBy comparing the AMR transition characteristics Φ(H) (Fig. 3(e)) and the hysteresis evolution \nΔΦ(H) (Figure 4(d)), we can see that the two show intimate correlation with each other. At low field \nregion with H<0.5 T, the fourfold AMR-I is gra dually stabilized, and the hysteresis ΔΦ decreases \nfrom 14° to 5° with increasing H. The relatively large ΔΦ may be due to additi onal contributions of \nstabilizing the weak FM phase. At 0.5 T< H< 5 T, the AMR-I possesses uniform hysteresis with a \ncertain ΔΦ~5°, arising mainly from the magneto crystalline anisot ropy energy in Sr 2IrO 4 as \ndiscussed above. At 5 T< H< 9 T, however, the AMR-II shows a nearly non-hysteretic behavior with \nminor ΔΦ~1°, suggesting that the isospins  could go freely through the crysta l axes in the thin films.  \n \n2.3. Theory of AMR of Sr 2IrO 4 \nWe now have shown the magnetic field induced fourfold AMR contour ro tation and concurrent \nMR minimal in the spin-orbit AFM SIO/STO thin  films. These phenomena have not been reported \nyet, to the best of our knowledge.  To understand the microscopic physics of the unusua l anisotropic magneto-transport in the thin \nfilms, we performed calculations based on de nsity functional theory (DFT) using Vienna ab initio  \nSimulation Package (V ASP). First, the bulk Sr 2IrO 4 is checked. The ma gnetic ground state of \nSr2IrO 4 is confirmed to be the ba sal AFM arrangement with Ir isospins pointing along the [110] \ndirection (magnetic easy axis).  The local moment of Ir is μ~0.498 μB with orbital moment μL~0.341 \nμB and spin moment μS~0.157 μB, giving rise to a ratio of μL/μS~2.17. Moreover, the calculated \ncanting angle of Ir moment is ϕ~±10.7° with respect to the [110]  direction. These results are well \nagreeing with previous experime ntal and theoretical results [17, 21, 22, 29].  \nWe then perform the calculations for strained Sr 2IrO 4 to simulate thin fi lms on the top of (001) \nSTO substrate. The calculated results are listed in Table I . After including the strain effect, the \nmagnetic easy axis remains the [110] direction.  The ratio of μL/μS is found to be ~2.58 for SIO/STO, \nalthough the total magnetic mome nt of Ir is almost unchange d (<5%) upon strain. The isospin \ncanting angle is meanwhile derived to be ϕ~5.3° with respect to the [110] direction for SIO/STO. \nPhysically, strain can modify  Ir-O-Ir bond angles in the ab plane, which tune  the single-ion \nanisotropy, Dzyaloshinskii-Moriya interaction, as well as exchanges. Thus, the canting angles are \nnaturally changed a li ttle bit upon strain. The evolutions of both μL/μS ratio and ϕ with strain are in \nagreement with previous dynamical mean  field theory calculations (DMFT) [29].  \nAs shown in Table I, the calculated magnetocrystalline anisotropy energy is ΔE ~ 1.01 meV/u.c. \n(or ΔE ~0.13 meV/Ir). This is quite close to the calculated value ( ΔE ~0.19 meV/Ir) for Ba 2IrO 4 [30]. \nSuch magnetocrystalline anisotropy can explain th e AMR-I observed under re latively low magnetic \nfield. As in many other magnetic systems, relativel y stronger suppression of magnetic scattering (a \nconsequence larger MR) would be expected when magnetic field is applie d along the magnetic easy \naxis, which is confirmed by the results shown in  Fig. 2(c) and (d). Then the AMR-I can be \nstraightforwardly understood using this conventional mechanism of axis-dependent suppression of \nmagnetic scattering.   \nRegarding the AMR-II, the relatively more conduc tive channel is shifted by ~45° to along the \n[100] direction, although the [110] dir ection is the magnetic easy axis in Sr 2IrO 4. As aforementioned, \nan AMR-I to AMR-II transition field is derived to be Hc~10 T at T=0 K, which corresponds to a \nZeeman energy difference 0.12-0.2 meV/Ir if taking the Ir moment as 0.2-0.4 μB/Ir in Sr 2IrO 4 [17, 22]. \nTherefore, Hc represents a critical fiel d to overcome the magnetocr ystalline anisotropy energy barrier. Thus, when H>Hc, all the isospins should fully rotate accompanying the external magnetic \nfield.  \nWe further calculated band structures for cases w ith the Ir isospins pointing along the [110] and \nthe [100] directions, as shown in Figure S5 . Similar to previous theoretical works [14, 29, 31], we \nfound that taking the strong SOC an d effective Hubbard repulsion Ueff into account is vital for the \ngap-opening in Sr 2IrO 4. A smaller band-gap ( Eg[100]~25.9 meV) is revealed as the Ir isospins are \naligned along the [100] di rection, as compared with  the [110] direction ( Eg[110]~35.0 meV). It should \nbe noted that the gap calculated using the GGA- PBE in V ASP is typically smaller than the \nexperimental one (or other theoreti cal values calculated using DMFT)  [14, 29]. Even though, these \nvalues are qualitatively comparable, i.e.  Eg[110]>Eg[100], which is in agreement with the previous \ncalculated results  [11]. This difference can well explain the emergence of the AMR-II in the \nSIO/STO thin films. The band gap Eg[100] (~25.9 meV) is always smaller than Eg[110] (~35.0 meV). \nThe isospin's direction can rotate with increasing ma gnetic field, not Eg's. Under small magnetic \nfields, the energy is lower when isospins pointi ng along the [110] direction and the MR is larger \nalong the [110], considering the magnetic easy axis . Under high magnetic fields, the weak FM \nmagnetization are nearly saturate d no matter the magnetic field is applied along the [100] or [110] \ndirection. Thus the AMR-I, which is contributed  by the suppression of magnetic scattering, should \nbe negligible in this situation. Instead, since all isospins follow th e magnetic field, the intrinsic band \ngaps tuned by isospins’ direction will dete rmine the transport, leading to the AMR-II.   \nAccording to the calculated band stru ctures, the minimal in the MR of H//[110] can be \nexplained as following. Magnetic twin domains have been demonstrated in bulk SIO, owing to the \ntetragonal symmetry [22]. In the present SIO/STO thin films, similar twinned magnetic domains are \nto be expected, since the tetragonal structure is preserved and the strain  in the thin films is tiny, as \nevidenced by the structural characterizations. Upon increasing H, the domains are eventually \naligned to let the FM moment approach H, leading to suppression of spin-scattering  such as quick \nenhancement in MR at low fields (Fig. 2(c) and (d)). As H ≥ Hc, the field is sufficient to overcome \nthe anisotropy energy barrier, and thus the Ir isospins can pass th e [100] direction (with smaller \nband-gap) and reach the final stable direction such  as the [110] direction (with larger band-gap). \nTherefore, a MR minimal appears at Hc in the MR with H//[110]. A sketch is shown in Figure 5 . \nThis picture is supported by the fact that the fiel ds of MR minima at various temperatures well follows the relationship of Hc(T) shown in Figure 3(f).  \n \n3. Conclusion \nIn conclusion, well controllable AMR effect a nd concurrent MR minima l are evidenced in the \nSr2IrO 4 thin films hosting a novel Jeff=1/2 Mott state, which can be observed in a broad temperature \nrange up to the AFM transition. Our first-principles  calculations reveal that these two phenomena \nshould be mainly attributed to ba nd structure engineering when rota ting the AFM-axis laterally in \nthe thin films. Our results unequivocally link the Jeff=1/2 Mott state to the AFM-based AMR, which \nwould be scientifically interesting and important for AFM spintronics, since the realization of AMR \nrepresents an important step towards the manipula tion and detection of AFM orders. In addition, our \nwork evidences a direct correlation between the inherent AFM-lattice a nd the band structure in \nSr2IrO 4, which provides useful  information to understand the natu re of magnetic interactions in \niridates.   4. Experimental section  \nThe Sr 2IrO 4 thin films were epitaxially grown on STO (001) substrate using pulsed laser \ndeposition system equipped with in-situ  RHEED. The growth parameters  were carefully optimized, \nand the details can be found in our previous work [26]. The film thickness of 40-nm was monitored \nby the RHEED intensity oscillations.  \nX-ray diffraction characterizati ons, including regular theta-2t heta scan, reciprocal space \nmapping, and rocking curves were carried out us ing a Philips X’Pert diffractometer. Magnetic \ncharacterizations were performed using a su perconducting quantum interference device by \nQuantum Design. The M(T)  curves were measured for both fi eld cooling and zero field cooling \nsequences, and the cooling and measuring field was set at H=0.1 T. Magnetization as a function of \nH were measured at various temperatures after a ZFC sequence. During the magnetization \nmeasurements, H was applied along the [100] direction. Electric transport measurements with \nexciting current I //[100] direction were performed usin g a standard four-probe method in a \nQuantum Design physical property measurement sy stem equipped with a rotator module. With \nregard to the anisotropic ma gnetoresistance measurements, I is applied always along the [100] \ndirection, and the magnetic field H is rotated within the basal plane of the films. Maximum \nmagnetic field allowed by the transport measurement set-up is 9 T.  \nIn DFT calculations, the Hubbard repulsion Ueff=U-J=3 eV is concluded and the SOC effect is \nconsidered with noncollinear spins. The pl ane-wave cutoff is 550 eV and the 6×6×2 \nMonkhorst-Pack k-points mesh is centered at Γ points. Starting from the experimental tetragonal \nstructure of bulk SIO, the lattice parameters and inner atomic posit ions are fully optimized till the \nHellman-Feyman forces are all less than 0.01 eV/A [32]. To simulate the strain in thin films, the \nin-plane lattice constants of SIO are fixed to be th e same as the substrates. Our calculations were \ndone without considering surface and interface, noting that the films are already 40 nm in thickness, \nsufficiently thick that the bulk properties are be lieved to be dominant. The isospin moments are \ninitialized along particular axes  (without canting), but not artifi cially constrained. The magnetic \ncaning is obtained via self -consistent calculations.   \n \n  Supporting Information:  Supporting Information is available from thee Wiley Online Library or \nfrom the authors.  \n \nAcknowledgements:  We acknowledge fruitful discussions with  Prof. Marin Alexe, Dr. Xavi Marti, \nDr. Ignasi Fina, and Prof. Dietrich Hesse. This work was supported by the National Nature Science \nFoundation of China (Grant Nos.  11774106, 11674055, 51332006, 51721001, and 51431006), the \nNational Key Research Projects of China (Grant No. 2016YFA0300101).  \n  \nReceived: ((will be filled in by the editorial staff)) \nRevised: ((will be filled in by the editorial staff)) \nPublished online: ((will be fille d in by the editorial staff)) \n   References:  \n[1] T. Jungwirth, X. Marti, P. Wadley, J. Wunderlich, Nat. Nanotechnol.  2016 , 11, 231. \n[2] H.-C. Wu, M. Abid, A. Kalitsov,  P. Zarzhitsky, M. Abid, Z.-M. Li ao, C. Ó. Coileáin, H. Xu, J.-J. \nWang, H. Liu, O. N. Mryasov, C.-R. 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W. Cheong, Nat. \nCommun.  2014 , 5, 3201. \n[29] H. Zhang, K. Haule, D. Vanderbilt, Phys. Rev. Lett.  2013 , 111, 246402. \n[30] Y . S. Hou, H. J. Xiang, X. G. Gong, New J Phys.  2016 , 18, 043007. \n[31] J. Matsuno, K. Ihara, S. Yamamura, H. Wadati, K.  Ishii, V . V . Shankar, H. Y . Kee, H. Takagi, \nPhys. Rev. Lett.  2015 , 114, 247209. \n[32] M. K. Crawford, M. A. Subramanian, R. L. Harlow, J. A. Fernandez-Baca, Z. R. Wang, D. C. \nJohnston, Phys. Rev. B  1994 , 49, 9198.   \nFigure captions:  \n \n \n \nFigure 1: (a) The top panel shows the lock ing effect between octahedral rotation α and isospin (red \narrow) canting ϕ. The bottom panel shows the planar antiferromagnetic structure in Sr 2IrO 4. The \nolive arrows denote the net moment in each IrO 2 layer, arising from the isospin canting. The \napplication of magnetic field in-p lane would induce an isospin-flip  transition, leading to a weak \nferromagnetic phase. (b) Reciprocal space mapping around the (109) plane of the Sr 2IrO 4 thin films \ngrown on (001) SrTiO 3 substrates. (c) Schematic of the basal planes of tetragonal (blue square) and \npsudotetragonal (black square) Sr 2IrO 4.  \n   \n \n \nFigure 2:  (a) Resistivity and (b) magnetization as a function of temperature. The TC~230 K is \nindicated by an arrow in (b). (c) and (d) show magnetoresistance of the SIO thin films measured \nwith field applied along different directions at T=50 K, and T=100 K, respectively. The inset shows \na sketchy of the measurements. Solid (dashed) curves were obtained by sweeping up (down) H. The \nminima in MR curves with H//[110] are highlighted by green lines.   \n   \n \n \n \nFigure 3: (a) Schematic of the anisotropic magnetor esistance measurement, in which the current is \napplied along the [100] direction, a nd magnetic field is rotated in-plane. Φ is the angle between \nmagnetic field H and the current I. (b) and (c) show the anisotro pic magnetoresistance measured \nunder various magnetic field at T=50 K and T=150 K for the thin films, respectively. (d) \nAnisotropic magnetoresistance map collected under a series of magnetic fields at T=50 K for the \nthin films. The high-field AMR peaks are indicat ed by arrows. (e) Peak positions of the AMR \ncurves in (d) as a function of H, in which the contour rotation is clearly seen at Hc~5 T. AMR-I and \nAMR-II represent the AMR effect that appearing be fore and after the rotation. The relative shift \nbetween AMR-I and AMR-II is ~4 5°. (f) The critical field Hc (squares) of the AMR-I to AMR-II \ntransition as a function of temperature. The dots represent fields of the MR minima at various \ntemperatures.     \n \n \nFigure 4: Hysteretic AMR traces measur ed under various magnetic fields (a) H=0.3 T, (b) H=1 T, \nand (c) H=9 T at T=50 K in SIO/STO thin films. The magnetic field is rotated from Φ=0° to 360° \n(black curves) and then backwards from 360° to 0° (red curves). (d) The derived hysteresis size ΔΦ \nas a function of H, in which step-like features are seen corresponding to the AMR-I and AMR-II.  \n   \n \n \nFigure 5: (a) and (b) schematically show the band structures as Ir isospin is aligned along the [100] \nand [110] directions, resp ectively. (c) Sketch of H driven rotation of the Ir isospins mIr (red arrows) \nas H>Hc. Olive arrows represent the FM moments due to isospin-canting. Solid (dashed) arrows \nrepresent the initial (final) positions of the moments.      \nTable I.  The energy difference for a minimal unit cell (e ight formula units), local spin moment ( μS) \nwithin the default Wigner-Seitz sphere, and orbital moment ( μL) for each Ir of Sr 2IrO 4. The item of \n“isospin angle” means a canting angle relative to the initialized direction of the isospin.  \n \nInitial moments direction [110] [100] \nBulk SIO Energy (meV) 0 1.24 \nμ=μS+μL   (μB/Ir) \nisospin angle (°)    0.498 \n±10.7 0.496 \n+9.6/-11.6 \nμS (μB) 0.157 0.155 \nμL (μB) 0.341 0.341 \nSIO/STO  (001) Energy (meV) 0 1.01 \nμ=μS+μL   (μB/Ir) \nisospin angle (°) 0.476 \n±5.3 0.473 \n+1.1/-8.7 \nμS (μB) 0.133 0.131 \nμL (μB) \nEg  ( m e V )  0.343 \n35.0 0.342 \n25.9 \n \n "    },    {        "title": "1805.06373v1.Magnetic_properties_of_single_crystalline_itinerant_ferromagnet_AlFe2B2.pdf",        "content": "Magnetic properties of single crystalline itinerant ferromagnet AlFe 2B2\nTej N. Lamichhane,1, 2Li Xiang,1, 2Qisheng Lin,1Tribhuwan Pandey,3David S. Parker,3\nTae-Hoon Kim,1Lin Zhou,1Matthew J. Kramer,1Sergey L. Bud'ko,1, 2and Paul C. Can\feld1, 2\n1Ames Laboratory, U.S. DOE, Ames, Iowa 50011, USA\n2Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA\n3Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA\nSingle crystals of AlFe 2B2have been grown using the self \rux growth method and then measured\nthe structural properties, temperature and \feld dependent magnetization, and temperature depen-\ndent electrical resistivity at ambient as well as high pressure. The Curie temperature of AlFe 2B2\nis determined to be 274 K. The measured saturation magnetization and the e\u000bective moment for\nparamagnetic Fe-ion indicate the itinerant nature of the magnetism with a Rhode-Wohlfarth ratio\nMC\nMsat\u00191:14. Temperature dependent resistivity measurements under hydrostatic pressure shows\nthat transition temperature TCis suppressed down to 255 K for p= 2:24 GPa pressure with a sup-\npression rate of \u0018 \u00008:9 K/GPa. The anisotropy \felds and magnetocrystalline anisotropy constants\nare in reasonable agreement with density functional theory calculations.\nINTRODUCTION\nIn recent years, AlFe 2B2has attracted a growing re-\nsearch interest as a rare-earth free ferromagnet that\nmight have potential as a magneto-caloric material [1, 2].\nIt is a layered material that has been identi\fed as an\nitinerant ferromagnet [3]. AlFe 2B2was \frst reported\nby Jeitschko[4] and independently by Kuz'ma and Cha-\nban [5] in 1969. AlFe 2B2crystallizes in an orthorhombic\nstructure with space group Cmmm (Mn 2AlB 2structure\ntype). The Al atoms located in 2 acrystallographic po-\nsition (0,0,0) form a plane which alternately stacks with\nFe-B slabs formed by Fe atoms; located at 4 j(0, 0.3554,\n0.5) and B atoms located at 4 i(0,0.1987,0) positions [6].\nA picture of unit cell for AlFe 2B2is shown in FIG. 1(a).\nAlMn 2B2and AlCr 2B2are the other two known iso-\nstructural transition metal compounds. Among these 3\nmembers only AlFe 2B2is ferromagnetic; however the re-\nported magnetic parameters for AlFe 2B2show a lot of\nvariation [1{3, 6, 7]. A good summary of all these vari-\nations is presented tabular form in a very recent litera-\nture [8].\nFor example, the Curie temperature of this material\nis reported to fall within a window of 274 - 320 K de-\npending up on the synthesis route. Initial work indicates\nthat, the Curie temperature of AlFe 2B2was 320 K [3].\nThe Curie temperature of Ga-\rux grown AlFe 2B2was\nreported to be 307 K and for arcmelted, polycrystalline\nsamples it was reported to be 282 K [1]. The Curie tem-\nperature for annealed, melt-spun ribbons was reported\nto be 312 K [6]. A M ossbauer study on arc-melted and\nannealed sample has reported the Curie temperature of\n300 K [2]. At the lower limit, the Curie temperature\nof spark plasma sintered AlFe 2B2was reported to be\n274 K [7]. The reported saturation magnetic moment\nalso manifests up to 25% variation from the theoretically\npredicted saturation moment of 1.25 \u0016B/Fe. The \frst\nreported saturation magnetization and e\u000bective moment\nvalues for AlFe 2B2were 1.9(2) \u0016B/f.u. at 4:2 K and4.8\u0016B/Fe respectively [3]. Recently, Tan et al. has re-\nported the saturation magnetization of 1.15 \u0016B/Fe and\n1.03\u0016B/Fe for before and after the HCl etching of an ar-\ncmelted sample [1]. The lower saturation moment, after\nthe acid etching, suggested either the inclusion of Fe-rich\nmagnetic impurities in the sample or degradation of the\nsample with acid etching. Recently, a study pointed out\nthat the content of impurity phases decreases with ex-\ncess of Al in the as cast alloy and by annealing [9]. The\nmain reason for the variation in reported magnetic pa-\nrameters is the di\u000eculty in preparing pure single crystal,\nsingle phase AlFe 2B2samples. To this end, detailed mea-\nsurements on single phase, single crystalline samples will\nprovide unambiguous magnetic parameters and general\ninsight into AlFe 2B2.\nIn this work, we investigated the magnetic and transport\nproperties of self-\rux grown single crystalline AlFe 2B2.\nWe report single crystalline structural, magnetic and\ntransport properties of AlFe 2B2. We \fnd that AlFe 2B2\nis an itinerant ferromagnet withMC\nMsat\u00191:14 and the\nCurie temperature is initially linearly suppressed with\nhydrostatic pressure at rate ofdTC\ndp\u0018\u00008:9 K/GPa. The\nmagnetic anisotropy \felds of AlFe 2B2are\u00181 T along\n[010] and\u00185 T along [001] direction. The \frst magneto-\ncrystalline anisotropic constants ( K1s) at base temper-\nature are determined to be K010\u00190:23MJ=m3and\nK001\u00191:8MJ=m3along [010] and [001] directions re-\nspectively (The subscript 1 is dropped for simplicity.).\nEXPERIMENTAL DETAILS\nCrystal growth\nSingle crystalline samples were prepared using a self-\n\rux growth technique[10]. First we con\frmed that our\ninitial stochiometry Al 50Fe30B20was a single phase liquid\nat 1200\u000eC. Starting composition Al 50Fe30B20with ele-\nmental Al (Alfa Aesar, 99.999%), Fe (Alfa Aesar, 99.99%)arXiv:1805.06373v1  [cond-mat.mtrl-sci]  16 May 20182\nand B (Alfa Aesar, 99.99%) was arcmelted under an Ar\natmosphere at least 4 times. The ingot was then crushed\nwith a metal cutter and put in a fritted alumina cru-\ncible set [11] under the partial pressure of Ar inside an\namorphous SiO 2jacket for the \rux growth purpose. The\ngrowth ampoule was heated to 1200\u000eC over 2-4 h and al-\nlowed to homogenize for 2 hours. The ampoule was then\nplaced in a centrifuge and all liquid was forced to the\ncatch side of crucible. Given that all of the melt was col-\nlected in catch crucible, this con\frms that Al 50Fe30B20\nis liquid at 1200\u000eC.\nKnowing that the arcmelted Al 50Fe30B20composition\nexists as a homogeneous melt at 1200\u000eC , the cooling pro-\n\fle was optimised as following. The homogeneous melt at\n1200\u000eC was cooled down to 1180\u000eC over 1 h and slowly\ncooled down to 1080\u000eC over 30 h at which point the\ncrucible limited, plate-like crystals were separated from\nthe remaining \rux using a centrifuge. The large plate-\nlike crystals had some Al 13Fe4impurity phase on their\nsurfaces which was removed with dilute HCl etching [7].\nThe as-grown single crystals are shown in the insets of\nFIG. 2(a).\nFigure 1. (a) AlFe 2B2unit cell (b) HAADF STEM image\nshows uniform chemistry of the AlFe 2B2crystal. The inset\nis a corresponding selected-area electron di\u000braction pattern.\n(c) High resolution HAADF STEM image of AlFe 2B2taken\nalong [101] zone axis along with projection of a unit cell rep-\nresented with Fe (red), Al(green) and B (yellow) spheres. The\nstructural pattern of Al and FeB slab layers are also visible\nin unit cell shown in pannel (a). (d) EDS elemental mapping\nwithout account of B scattering e\u000bect where green stripes are\nAl and red stripes are Fe distributions.Characterization and physical properties\nmeasurements\nThe crystal structure of AlFe 2B2was characterized\nwith both single crystal X-ray di\u000braction (XRD) and\npowder XRD. The single crystal XRD data were collected\nwithin a 4\u000e-29\u000eangle value of 2 \u0012using Bruker Smart\nAPEX II di\u000bractometer with graphite-monochromatized\nMo-K\u000bradiation source ( \u0015= 0.71073 \u0017A). The powder\ndi\u000braction data were collected using a Rigaku MiniFlex\nII di\u000bractometer with Cu-K \u000bradiation. The acid etched\nAlFe 2B2crystals were ground to \fne powder and spread\nover a zero background, Si-wafer sample holder with help\nof a thin \flm of Dow Corning high vacuum grease. The\ndi\u000braction intensity data were collected within a 2 \u0012in-\nterval of 5\u000e-100\u000ewith a \fxed dwelling time of 3 sec and\na step size of 0 :01\u000e.\nThe as-grown single crystalline sample was examined\nwith a transmission electron microscopy to obtain High-\nangle-annular-dark-\feld (HAADF) scanning transmis-\nsion electron microscopy (STEM) images, corresponding\nselected-area electron di\u000braction pattern and high resolu-\ntion HAADF STEM image of AlFe 2B2taken under [101]\nzone axis.\nThe anisotropic magnetic measurements were carried\nout in a Quantum Design Magnetic Property Measure-\nment System (MPMS) for 2 K \u0014T\u0014300 K and a\nVersalab Vibrating Sample Magnetometer (VSM) for 50\nK\u0014T\u0014700 K.\nThe temperature dependent resistivity of AlFe 2B2was\nmeasured in a standard four-contact con\fguration, with\ncontacts prepared using silver epoxy. The excitation cur-\nrent was along the crystallographic a-axis. AC resistiv-\nity measurement were performed in a Quantum Design\nPhysical Property Measurement System (PPMS) using 1\nmA; 17 Hz excitation, with a cooling at a rate of 0.25\nK/min. A Be-Cu/Ni-Cr-Al hybrid piston-cylinder cell\nsimilar to the one described in Ref. 12 was used to ap-\nply pressure. Pressure values at the transition tempera-\ntureTCwere estimated by linear interpolation between\nthe room temperature pressure p300K and low temper-\nature pressure pT\u001490Kvalues[13, 14]. p300K values were\ninferred from the 300 K resistivity ratio \u001a(p)=\u001a(0 GPa) of\nlead[15] and pT\u001490Kvalues were inferred from the Tc(p)\nof lead[16]. Good hydrostatic conditions were achieved\nby using a 4:6 mixture of light mineral oil:n-pentane as\na pressure medium; this mixture solidi\fes at room tem-\nperature in the range 3 \u00004 GPa, i.e., well above our\nmaximum pressure[12, 14, 17].3\nEXPERIMENTAL RESULTS\nStructural characterization\nThe HAADF STEM image along with selected area\ndi\u000braction pattern in the inset and high resolution\nHAADF STEM image of AlFe 2B2taken under [101] zone\naxis and EDS Al-Fe elmental mapping are presented in\npannels (b), (c) and (d) of FIG. 1. Taken together they\nstrongly suggest the uniform chemical composition of\nAlFe 2B2through out the sample.\nThe crystallographic solution and parameters re\fne-\nment on the single crystalline XRD data was performed\nusing SHELXTL program package [18]. The Rietveld re-\n\fned single crystalline data are presented in TABLES I,\nand II. Using the atomic coordinates from the crystallo-\ngraphic information \fle obtained from single crystal XRD\ndata, powder XRD data were Rietveld re\fned with RP=\n0.1 using General Structure Analysis System [19] (FIG. 2\n(a)). The lattice parameters from the powder XRD are: a\n= 2.920(4) \u0017A, b = 11.026(4) \u0017Aand c = 2.866(7) \u0017Awhich\nare in reasonable agreement with the single crystal data\nanalysis values.\nTo con\frm the crystallogrpahic orientation of the\nAlFe 2B2crystals, monochromatic Cu- K\u000bXRD data were\ncollected from the \rat surface of the crystals and found\nto bef020gfamily as shown in FIG. 2 (b), i.e. the [010]\ndirection is perpendicular to the plate. However \fnding a\nthick enough, \rat, as grown facet with [100] and [001] di-\nrection was made di\u000ecult by the thin, sheet-like morphol-\nogy of the sample and its crucible limited growth nature.\nA [001] facet was cut out of large crucible limited crystal\nas shown in the inset of FIG. 2(c). The monochromatic\nCu-K\u000bXRD pattern scattered from the cut surface con-\n\frms the [001] direction displaying the [001] and [002]\npeaks (FIG. 2 (c)). To better illustrate the crystallo-\ngraphic orientations, powder XRD, and monochromatic\nsurface XRD patterns from the plate surface and cut edge\nare plotted together in FIG. 2 (d). This plot clearly iden-\nti\fes that direction perpendicular to the plate is [010] and\ncut edge surface is (001). Slight displacement of the sur-\nface XRD peaks is the result of the sample height in the\nBragg Brentano geometry. The splitting of [080] peak is\nobserved by distinction of Cu- K\u000bsatellite XRD patterns\nusually observed at high di\u000braction angles.\nMAGNETIC PROPERTIES\nThe anisotropic magnetization data were measured us-\ning a sample with known crystallographic orientation.\nThe temperature dependent magnetization M(T) data\nalong [100] axis is presented in FIG. 3(a). Both the Zero\nField Cooled Warming (ZFCW) and Field Cooled(FC)\nM(T) data are almost overlapping for 0.01 T appliedTable I. Crystal data and structure re\fnement for AlFe 2B2.\nEmpirical formula AlFe 2B2\nFormula weight 160.3\nTemperature 293(2) K\nWavelength 0:71073 \u0017A\nCrystal system, space group Orthorhombic, Cmmm\nUnit cell dimensions a=2.9168(6) \u0017A\nb = 11.033(2) \u0017A\nc = 2.8660(6) \u0017A\nVolume 92.23(3) 103\u0017A3\nZ, Calculated density 2, 5.75 g/cm3\nAbsorption coe\u000ecient 31.321 mm\u00001\nF(000) 300\n\u0012range (\u000e) 3.693 to 29.003\nLimiting indices \u00003\u0014h\u00143\n\u000014\u0014k\u001414\n\u00003\u0014l\u00143\nRe\rections collected 402\nIndependent re\rections 7 [R(int) = 0.0329]\nAbsorption correction multi-scan, empirical\nRe\fnement method Full-matrix least-squares\non F2\nData / restraints / parameters 74 / 0 / 12\nGoodness-of-\ft on F21.193\nFinal R indices [I >2\u001b(I)]R1 = 0:0181,wR2 = 0:0467\nR indices (all data) R1 = 0:0180,wR2 = 0:0467\nLargest di\u000berence peak and hole 0.679 and -0.880 e. \u0017A\u00003\nTable II. Atomic coordinates and equivalent isotropic dis-\nplacement parameters (A2) for AlFe 2B2. U(eq) is de\fned as\none third of the trace of the orthogonalized U ijtensor.\natom Wycko\u000b\nsitex y z Ueq\nFe 4(j) 0.0000 0.3539(1) 0.5000 0.006(6)\nAl 2(a) 0.0000 0.0000 0.0000 0.006(7)\nB 4(i) 0.0000 0.2066(5) 0.0000 0.009(7)\n\feld. The M(T) data suggest a Curie temperature ( TC)\nof\u0018275 K using an in\rection point of M(T) data as a\ncriterion. This value will be determined more precisely\nbelow to be TC= 274 K using easy axis M(H) isotherms\naround Curie temperature.\nFIG. 3(b) shows the anisotropic, \feld-dependent mag-\nnetization at 2 K. The saturation magnetization ( Msat)\nat 2 K is determined to be 2 :40\u0016B/f.u., i.e. roughly half\nof bulk BCC Fe moment. The anisotropic M(H) data at\n2 K show [100] is the easy axis, [010] axis is a harder axis\nwith an anisotropy \feld of \u00191 T and [001] is the hardest\naxis of magnetization with an anisotropy \feld of \u00195 T. A4\nFigure 2. (a) Powder XRD for AlFe 2B2.I(Obs), I(Cal) and I(Bkg) stands for experimental powder di\u000braction, Rietveld\nre\fned and instrumental background data. The green vertical lines represent the Bragg re\rection peaks and the I(Obs-Cal)\nis the di\u000berential intensity between I(Obs) and I(Cal). The upper inset picture shows the crucible limited growth nature of\nAlFe 2B2. The lower inset picture is the pieces of as grown plate-like crystals. (b) Monochromatic XRD pattern from the plate\nsurface of AlFe 2B2. (c) Monochromatic XRD pattern from cut surface [001] collected using Bragg-Brentano geometry. The\nleft inset photo shows the as grown AlFe 2B2crystal. The right inset picture is the photograph of the cut section of the crystal\nparallel to (001) plane. The middle unidenti\fed peak might be due to a di\u000berently oriented shard of cut AlFe 2B2crystal. (d)\nComparison of the monochromatic surface XRD patterns from (b) and (c) with powder XRD pattern from (a) within extended\n2\u0012range of 60 - 70\u000eto illustrate the identi\fcation scheme of the crystallographic orientation.\nSucksmith-Thompson plot [20], using M(H ) data along\n[001], is shown in FIG. 3(c). The inset to FIG. 3(c)\nshows data for Halong [010]. In a Sucksmith-Thompson\nplot, the Y-intercept of the linear \ft of hard axisHint\nMvs\nM2isotherm provides the magneto-crystalline anisotropy\nconstant (intercept =2K1\nMS2,MSbeing saturation magne-\ntization at 2 K) of the material. From these plots we de-\nterminedK010= 0:23MJm\u00003andK001= 1:78MJm\u00003\nrespectively.\nGiven that AlFe 2B2hasTC\u0018room temperature,\nand is formed from earth abundant elements, it is log-\nical to examine it as a possible magnetocaloric material.\nThe easy axis, [100], M(H) isotherms around the Curie\ntemperature (shown for the Arrott plot in FIG. 3(d))\nwere used to estimate the magnetocaloric property for\nAlFe 2B2in-terms of entropy change using following equa-tion [1, 21]:\n\u0001S(T1+T2\n2;\u0001H)\u0019\n\u00160\nT2\u0000T1ZHf\nHiM(T2;H)\u0000M(T1;H)dH\n(1)\nwhereHi,Hfare initial and \fnal applied \felds and\nT2\u0000T1is the change in temperature. For this formula\nto be valid, T2\u0000T1should be small. Here T2\u0000T1\nis taken to be 1 K. The entropy change calculation\nscheme in one complete cycle of magnetization and de-\nmagnetization is estimated in terms of area between two\nconsecutive isotherms between the given \feld limit as5\n05 1 01 52 001x10102x10103x10104x10105x10106x10107x1010 \n M2(A/m)2H\nint/MAlFe2B2T = 265 KT\n = 285 KT = 276 K(d)\n1x10112x10113x101100246810129\n.0x10121.8x101300.40.81.20\n \n Hint/M (Tm/A)M\n2 (A/m)2(c)H\n || [001] \n Hint/M (Tm/A)M\n2 (A/m)2H || [010]\n01 2 3 4 5 0.51.01.52.02.53.00\n H || [100] \nH || [010] \nH || [001] \n M (µB/f.u.)µ\n0H (T )AlFe2B2T\n = 2 K(b)\n05 01 001502002503000.51.00\n0.01 T (ZFCW) \n0.01 T (FC) \n M (µB/f.u.)T\n (K)AlFe2B2H\n || [100](a)\nFigure 3. (a) Temperature dependent magnetization with 0.01 T applied \feld along [100] direction (b)) Field dependent\nmagnetization along principle directions at 2 K. [100] is the easy axis with smallest saturating \feld , [010] is the intermediate\naxis with 1 T anisotropy and [001] is the hardest axis with \u00185 T anisotropy \feld. (c) Sucksmith-Thompson plot for M(H)\ndata along [001] direction (and along [010] in the inset) to estimate the magneto-crystalline anisotropy constants. The red\ndash-dotted line is the linear \ft to the hard axes isotherms at high \feld region ( >3 T) whose Y-intercept is used to estimate\nthe anisotropy constant K. (d) Arrott plot obtained with easy axis isotherms within the temperature range of 265-285 K at a\nstep of 1 K. The straight line through the origin is the tangent to the isotherm corresponding to the transition temperature.6\nshown in FIG. 4(a). The measured entropy change as\na function of temperature is presented in FIG. 4(b).\nThe entropy change in 2 T and 3 T applied \felds\nis maximum around 276 K being 3 :78 Jkg\u00001K\u00001and\n4:87 Jkg\u00001K\u00001respectively. The 2 T applied \feld en-\ntropy change data of this experiment agrees very well\nwith reference [8], shown as 2 T\u0003data in FIG. 4(b). The\nentropy change values for our single crystalline samples\nare in close agreement with previously reported polycrys-\ntalline sample measured values as well [1, 6].\nAlthough, AlFe 2B2is a rare-earth free material, its\nmagnetocaloric property is larger than lighter rare-earth\nRT2X2(R = rare earth T = transition metal, X = Si,Ge)\ncompounds with ThCr 2Si2-type structure (space group\nI4/mmm) namely CeMn 2Ge2(\u00181:8 Jkg\u00001K\u00001) [22],\nPrMn 2Ge0:8Si1:2(\u0018 1:0 Jkg\u00001K\u00001) [23] and\nNd(Mn 1\u0000xFex)2Ge2(\u00181:0 Jkg\u00001K\u00001) [24]. The\nentropy change of AlFe 2B2is signi\fcantly smaller than\nGd5Si2Ge2(\u001813 Jkg\u00001K\u00001), it has comparable entropy\nchange with elemental Gd( \u00185:0 Jkg\u00001K\u00001) [25]. These\nresults shows that AlFe 2B2has the potential to be used\nfor magnetocaloric material considering the abundance\nof its constituents.\nTo precisely determine the Curie temperature, an Ar-\nrott plot was constructed using a wider range of M(H)\nisotherms along the [100] direction (FIG. 3(d)). In an\nArrott plot M2is plotted as a function ofHint\nM.Hint=\nHapp-N*Mis internal \feld inside the sample after the\ndemagnetization \feld is subtracted. In this case the ex-\nperimental demagnetization factor along the easy axis of\nthe sample was found to be almost negligible because of\nits thin, plate-like shape with the easy axis lying along\nthe longest dimension of the sample. The detail of de-\ntermination of the experimental demagnetization factors\nand their comparison with theoretical data is explained\nin the references [21] and [26]. The Arrott plots have\na positive slope indicating the transition is second or-\nder [27]. In the mean \feld approximation, in the limit\nof low \felds, the Arrott isotherm corresponding to the\nCurie temperature is a straight line and passes through\nthe origin. In FIG. 3(d), the isotherm corresponding to\n276 K passes through the origin but it is not a perfectly\nstraight line. This suggests that the magnetic interaction\nin AlFe 2B2does not obey the mean-\feld theory. In the\nmean-\feld theory, electron correlation and spin \ructua-\ntions are neglected, but these can be signi\fcant around\nthe transition temperature of an itinerant ferromagnet.\nSince the Arrott plot data are not straight lines, a gen-\neralized Arrott plot is an alternative way to better con-\n\frm the Curie temperature. The generalized Arrott plot\nderived from the equation of the state [28]\n(Hint\nM)1=\r=aT\u0000TC\nT+bM1=\f(2)\nis shown in FIG. 5. The critical exponents \fand\rused\n2202 402 602 803 003 200.00.51.01.52.02.53.03.54.04.55.0 \n3 T \n2 T \n2 T* \n -/s61508S (Jkg-1K-1)T\n (K)H || [100]A\nlFe2B2(b)\n5.0x1051.0x1061.5x1062.0x1062.5x10605.0x1041.0x1051.5x1052.0x1050\n(a)M  (A/m)H\n (A/m) 276 K \n275 KFigure 4. Magnetocaloric e\u000bect in AlFe 2B2obtained using\nM(H) isotherms along [100]. (a) showing the change in en-\ntropy (\u0001S) evaluation scheme at its highest value (b) change\nin entropy with 2 T and 3 T applied \felds using easy axis\n[100] isotherms. For the sake of comparison, the 2 T\u0003\feld\ndata are taken from the reference [8].\nin equation of state are derived from the Kouvel-Fisher\nanalysis[29, 30]. To determine \f, the equation used was:\nMS[d\ndT(MS)]\u00001=T\u0000TC\n\f(3)\nwhere the slope is1\n\f. The value of the spontaneous\nmagnetization around the transition temperature was ex-\ntracted from the Y-intercept of the MS4vsH\nM[31] ex-\nploiting their straight line nature with clear Y-intercept.\nThe experimental value of \fwas determined to be\n0:30\u00060:04 as shown in FIG. 6. The uncertainty in \f\nwas determined with \ftting error as \u0001 \f=\u000eslope\nslope2.\nSimilarly, the value of critical exponent \rwas deter-7\n02468101214162.0x10174.0x10176.0x10178.0x10171.0x10181.2x10181.4x10180\n \n M1//s61538 (A/m)1//s61538 \n(Hint/M)1/γTC = 274 KT = 250 KT\n = 290 K\nFigure 5. Generalized Arrott plot of AlFe 2B2with magne-\ntization data along [100] direction within temperature range\nof 250 - 290 K at a step of 1 K. The \f= 0:30\u00060:04 and\n\r= 1:180\u00060:005 were determined from the Kouvel-Fisher\nmethod. The two dash-dot straight lines are drawn to visual-\nize the intersection of the isotherms with the axes.\n260270280290300310320-30-25-20-15-10-50T\n (K)Ms(dMs/dT)-1 ( K)β = 0.30 /s61617 0.04γ\n = 1.180 /s61617 0.005 0\n51015202530/s61539\n/s61485/s61489(d/s61539/s61485/s61489/dT)-1 (K)\nFigure 6. Determination of the critical exponents ( \fand\n\r)using Kouvel-Fisher plots. See text for details.\nmined with the equation:\n\u001f\u00001[d\ndT(\u001f\u00001)]\u00001=T\u0000TC\n\r(4)\nwhere the slope is1\n\rand\u001f\u00001(T) is the initial high tem-\nperature inverse susceptibility near the transition tem-\nperature. The experimental value of the \rwas deter-\nmined to be 1 :180\u00060:005 as shown in FIG. 6.\nFinally the third critical exponent \u000ewas determined\nusing the equation:\nM/H1=\u000e(5)\nby plotting ln( M) vs ln( H) (FIG. 7) corresponding to\nCurie temperature 274 K. The experimental value of the\n\u000ewas determined by \ftting ln( M) vs ln( H) over di\u000berent\nranges of applied \feld H. Taking the average of the range\nof the\u000evalue as shown in FIG. 7 we determine \u000eto be\n101 11 21 31 41 511.512.012.5T\nC = 274 Kδ\n = 4.78 (0.7 − 3 Τ)δ\n = 4.84 (0.2 − 3 Τ)δ\n = 4.99 (0.0 − 3 Τ)  \n lnMl\nnHδave = 4.9 /s61617 0.1 Figure 7. Determination of the critical exponent \u000eusing\nKouvel-Fisher plots using M(H) isotherm at TCto check the\nconsistency of \fand\rvia Widom scaling. The data used for\ndetermining the exponent \u000eare highlighted with the red curve\nin corresponding M(H) isotherm. The data in the low \feld\nregion slightly deviate from the linear behaviour in the loga-\nrithmic scale as shown in the inset. The range dependency of\nthe value of \u000eis illustrated with di\u000berent colors tangents. The\n\feld range for the \ftted data is indicated in the parenthesis\nalong with the value of \u000e. See text for details.\n36 9 1 20 24681012140\n0.011 0 0.010.11100\n  270 K \n 271K \n 272 K \n 273 K \n 275 K \n 276 K \n 277 K \n 278 K \n279 K \n 280 K \n m = M |(T-TC)/TC|-β(105 A/m)h\n = H |(T-TC)/TC|-(β+γ)(109 A/m)T<TCT\n>TC \n m h\n T<TCT\n>TC\nFigure 8. Normalized isotherms to check the validity of scal-\ning hypothesis. The isotherms in between 270-273 K are\nconverged to higher value (T <TC) and isothems inbetween\n275-280 K are converged to lower value(T >TC). The inset\nshows the corresponding log-log plot clearly bifurcated in two\nbranches in low \feld region.\n4:9\u00060:1 which was closely reproduced (4 :93\u00060:03) with\nWidom scaling theory \u000e= 1 +\r\n\f.\nAdditionally, the validity of Widom scaling theory de-\nmands that the magnetization data should follow the\nscaling equation of the state. The scaling laws for a\nsecond order magnetic phase transition relate the sponta-\nneous magnetization MS(T)below TC, the inverse initial\nsusceptibility \u001f\u00001(T) above TC, and the magnetization8\n01002003004005006007000.20.40.60.81.01.20\n Initial Ms  f\nrom M4 vs H/M p\nlot   \nMs f\nrom Generalized \nArrottT\n (K)Ms (µB/Fe)(a)0\n200400600 I\nnitial /s61539/s61485/s61489 for  H = 1T || [100] \n/s61539\n/s61485/s61489/s61472from generalized Arrott \n/s61539/s61485/s61489 (Τ)\n2652 702 752 802 850.20.40.60\n Ms T\n (K)Ms (µB/Fe)(b)β\n = 0.295 /s61617 0.002γ\n = 1.210 /s61617 0.0030\n10 /s61539/s61485/s61489 \n/s61539/s61485/s61489 (Τ)\nFigure 9. Illustration of the consistency of the critical expo-\nnents\fand\rused for generalizes Arrott plot (a) by repro-\nducing the initial spontaneous magnetization MSand\u001f\u00001(T)\nvia Y and X-intercept of the generalized Arrott plot. (b) Fit-\nting of the extracted data (squares) from generalized Arrott\nplot with corresponding power laws (red lines) in equation 6\nand 7.\natTCwith corresponding critical amplitudes by the fol-\nlowing power laws:\nMS(T) =M0(\u0000\u000f)\f;\u000f< 0 (6)\n\u001f\u00001(T) = \u0000(\u000f)\r;\u000f> 0 (7)\nM=XH1\n\u000e (8)\nWhere M 0, \u0000 and X are the critical amplitudes and\n\u000f=T\u0000TC\nTCis the reduced temperature [32]. The scal-\ning hypothesis assumes the homogeneous order parame-\nter which with scaling hypothesis can be expressed as\nM(H;\u000f) =\u000f\ff\u0006(H\n\u000f\f+\r) (9)\nWheref+(T> TC) andf\u0000(T< TC) are the regular\nfunctions. With new renormalised parameters, m =\u000f\u0000\fM(H;\u000f) and h =\u000f\u0000(\f+\r)M(H;\u000f) equation 9 can be\nwritten as\nm=f\u0006(h) (10)\nUp to the linear order, the scaled m vs h graph is plot-\nted as shown in FIG 8 along with an inset in log-log scale\nwhich clearly shows that all isotherms converge to the two\ncurves one for T >TCand other for T <TC. This graph-\nically shows that all the critical exponents were properly\nrenormalized.\nFinally the consistency of the critical exponents \fand\n\ris demonstrated (shown in FIG. 9(a)) by reproducing\nthe initial spontaneous magnetization MSand\u001f\u00001(T)\nnear the transition temperature using the Y and X-\nintercept of generalized Arott plots as shown in FIG. 5\nwhich overlaps with MSobtained by MS4vsH\nM[31] and\nand initial inverse susceptibility \u001f\u00001(T) with 1 T applied\n\feld. The extracted data well \ft [32] with corresponding\npower laws in equation (6) and (7) as shown in FIG. 9(b)\ngiving\f= 0:295\u00060:002 and\r= 1:210\u00060:003 which\nclosely agree with previously obtained K-F values.\nTo measure the e\u000bective moment ( Meff) of the Fe\nabove the Curie temperature, a Curie-Weiss plot was pre-\npared as shown in FIG. 9. The e\u000bective moment of the\nFe-ion above the Curie temperature was determined to\nbe 2:15\u0016B. Since the e\u000bective moment above the Curie\ntemperature is almost equal to BCC Fe (2 :2\u0016B) and the\nordered moment at 2 K is signi\fcantly smaller than Fe-\nion (1:2\u0016B/Fe) giving the Rhode-Wohlfarth ratio (MC\nMsat)\nnearly equal to 1.14, where MC(MC+ 2) =Meff2, this\ncompound shows signs of an itinerant nature in its mag-\nnetization [33].\nItinerant magnetism, in general, can be tuned (mean-\ning the size of magnetic moment and Curie tempera-\nture can be altered signi\fcantly and sometime even sup-\npressed completely) with an external parameter like pres-\nsure or chemical doping. As a case study, we investigated\nthe in\ruence of the external pressure on the ferromag-\nnetism of AlFe 2B2.\nFigure 10 shows the pressure dependent resistivity of\nsingle crystalline AlFe 2B2with current applied along the\ncrystallographic a-axis. It shows metallic behaviour with\na residual resistivity of 60 \u0016\n cm. The ambient pressure\ntemperature dependent resistivity of AlFe 2B2shows a\nkink around 275 K, indicating a loss of spin disorder scat-\ntering associated with the onset of ferromagnetic order.\nAs pressure is increased to 2.24 GPa the temperature of\nthis kink is steadily reduced. To determine the transi-\ntion temperature TC, the maximum in the temperature\nderivatived\u001a/dTis used, as shown in the inset of FIG. 10.\nThe pressure dependence of TC, the temperature - pres-\nsure phase diagram of AlFe 2B2is presented in FIG. 11.\nThe transition temperature, TC, is suppressed from 275\nK to 255 K when pressure is increased from 0 to 2 :249\n0501001502002503000501001502002502\n002 503 0020\nTC/s61554/s61472( 10-6Ω cm)T\n ( K ) 0 GPa0\n.460\n.771\n.491\n.882\n.24J//aT ( K )d/s61554/dT ( 10-3Ω cm K-1 )T\nC\nFigure 10. Evolution of the single crystal AlFe 2B2resistivity\nwith hydrostatic pressure up to 2 :24 GPa. Pressure values\natTCwere estimated from linear interpolation between the\nP300K andPT\u001490Kvalues (see text). Current was applied\nalong the crystallographic a-axis. Inset shows the evolution of\ntemperature derivative d\u001a/dTwith hydrostatic pressure. The\npeak positions in the derivative were identi\fed as transition\ntemperature TC. Examples of TCare indicated by arrows in\nthe \fgure.\nGPa, giving a suppression rate of -8.9 K/GPa. Interest-\ningly, Curie temperature suppression rate of AlFe 2B2is\nfound to be comparable to the model itinerant magnetic\nmaterials like helimagnetic MnSi ( \u0018\u000015 K/GPa) [34],\nand weak ferromagnets ZrZn 2(\u0018\u000013 K/GPa) [35] and\nNi3Al (\u0018\u00004 K/GPa) [36]. A linear \ftting of the data\nas shown in FIG. 11 indicates that to completely sup-\npress theTCaround 31 GPa would be required. Usually\nsuch linear extrapolation provide an upper estimate of\nthe critical pressure.\nFIRST PRINCIPLES CALCULATIONS\nTheoretical calculations for AlFe 2B2were performed\nusing the all electron density functional theory code\nWIEN2K [37{39]. The generalized gradient approx-\nimation according to Perdew, Burke, and Ernzerhof\n(PBE) [40] was used in our calculations. The sphere radii\n(RMT) were set to 2.21, 2.17, and 1.53 Bohr for Fe, Al,\nand B, respectively. RK maxwhich de\fnes the product\nof the smallest sphere radius and the largest plane wave\nvector was set to 7.0. All calculations were performed\nwith the experimental lattice parameters as reported in\nreference [41] (which are consistent with our results) and\nall internal coordinates were relaxed until internal forces\non atoms were less than 1 mRyd/Bohr-radius. All the\ncalculations were performed in the collinear spin align-\nment. The magnetic anisotropy energy (MAE) was ob-\ntained by calculating the total energies of the system with\n0.51.01.52.02.50 250260270280 \nTcTC ( K )P\n ( GPa )Figure 11. Temperature - pressure phase diagram of AlFe 2B2\nas determined from resistivity measurement. Pressure values\nwere estimated as being described in Fig. 10 and in the text.\nError bars indicate the room temperature pressure P300K and\nlow temperature pressure PT\u001490K. As shown in the \fgure, in\nthe pressure region of 0 \u00002:24 GPa the ferromagnetic transi-\ntion temperature TCis suppressed upon increasing pressure,\nwith suppressing rate around \u00008:9 K/GPa.\n-8 -6 -4 -2 0 2 4 6 8\nE-EF (eV)-6-4-20246N(E) (eV-u.c.)-1total\nFe\nB\nAlspin-up\nspin-down\nFigure 12. The calculated density-of-states of AlFe 2B2.\nspin-orbit coupling (SOC) with the magnetic moment\nalong the three principal crystallographic axes. For these\nMAE calculations the k-point convergence was carefully\nchecked, and the calculations reported here were per-\nformed with 120,000 k-points in the full Brillouin zone.\nSimilar to the experimental observation, AlFe 2B2is\ncalculated to have ferromagnetic behaviour, with a sat-\nuration magnetic moment (we do not include the small\nFe orbital moment) of 1.36 \u0016B/Fe. This is in reason-\nable agreement with the experimentally measured value\nof 1.21\u0016B/Fe. Interestingly this calculated magnetic mo-\nment on Fe, is signi\fcantly lower than the moment on10\nFe in BCC Fe (2.2 \u0016B/Fe ) further suggesting a degree\nof itinerant behaviour. The calculated density of states\nis shown in FIG. 12. As expected for a Fe-based fer-\nromagnet, the electronic structure in the vicinity of the\nFermi level is dominated by Fe dorbitals and we observe\na substantial exchange splitting of 2-3 eV.\nFor an orthorhombic crystal structure, the magnetic\nanisotropy energy is described by total energy calcula-\ntions for the magnetic moments along each of the three\nprincipal axis [42]. For AlFe 2B2we \fnd, the [100] and\n[010] axes to be the \\easy\" directions, separated by just\n0.016 meV per Fe, with [100] being the easiest axis. The\n[001] direction is the \\hard\" direction, which lies 0.213\nmeV per Fe above the [100] axis. As in our previous\nwork on HfMnP [21], this value is much larger than the\n0.06 meV value for hcp Co and likely results from a\ncombination of the orthorhombic crystal structure and\nthe structural complexity associated with a ternary com-\npound. The 0.213 meV energy di\u000berence on a volumetric\nbasis corresponds to a anisotropy constant K 1as 1.48\nMJ/m3. (Note that we use the convention of the pre-\nvious work and simply de\fne K1for an orthorhombic\nsystem as the energy di\u000berence between the hardest and\neasiest directions.) This magnetic anisotropy constant\ndescribes the energy cost associated with the changing\nthe orientation of the magnetic moments under the ap-\nplication of a magnetic \feld, and is an essential compo-\nnent for permanent magnets. It is noteworthy that this\nanisotropy is comparable to the value of 2 MJ/m3pro-\nposed by Coey for an e\u000ecient permanent magnet [43],\ndespite containing no heavy elements, using the approx-\nimation that Ha\u00192\u00160K1=Ms, with K as 1.48 MJ/m3\nand Msas 0.68 T, yields an anisotropy \feld of 5.4 T,\nwhich is in excellent agreement with the experimentally\nmeasured value of 5 T and K001\u00191:8MJ=m3.\nCONCLUSIONS\nSingle crystalline AlFe 2B2was grown by self-\rux-\ngrowth technique and structural, magnetic and trans-\nport properties were studied. AlFe 2B2is an orthorhom-\nbic, metallic ferromagnet with promising magnetocaloric\nbehaviour. The Curie temperature of AlFe 2B2was de-\ntermined to be 274 K using the generalized Arrott plot\nmethod along with estimation of critical exponents us-\ning Kouvel-Fisher analysis. The ordered magnetic mo-\nment (Msat) at 2 K is 1 :20\u0016B/Fe at 2 K which is much\nless than paramagnetic Fe-ion moment at high temper-\nature (2:15\u0016B/Fe) indicating itinerant magnetism. The\nmagnetization in AlFe 2B2responds to the hydrostatic\npressure withdTC\ndP\u0018 \u00008:9 K/GPa. A linear extrapo-\nlation of this TC(P) trend leads to an upper estimate\nof\u001830 GPa required to fully supress the transition.\nThe saturation magnetization and anisotropic magnetic\n\feld predicted by \frst principle calculations are in closeagreement with the experimental results. The magneto-\ncrystalline anisotropy \felds were determined to be 1 T\nalong [010] and 5 T along [001] direction w. r. to easy\naxis [100]. The magneto-crystalline anisotropy constants\nat 2 K are determined to be K010\u00190:23MJ=m3and\nK001\u00191:8MJ=m3.\nACKNOWLEDGEMENT\nWe would like to thank Drs. W. R. McCallum and L.\nH. Lewis for drawing attention to this compound and\nDr. A. Palasyuk for useful discussions. This research\nwas supported by the Critical Materials Institute, an\nEnergy Innovation Hub funded by the U.S. Department\nof Energy, O\u000ece of Energy E\u000eciency and Renewable\nEnergy, Advanced Manufacturing O\u000ece. This work was\nalso supported by the o\u000ece of Basic Energy Sciences,\nMaterials Sciences Division, U.S. DOE. Li Xiang was\nsupported by W. M. Keck Foundation. This work\nwas performed at the Ames Laboratory, operated for\nDOE by Iowa State University under Contract No.\nDE-AC02-07CH11358.\n[1] Xiaoyan Tan, Ping Chai, Corey M. 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Magn. 47, 4671{4681 (2011)."    },    {        "title": "1805.11339v3.Electronic_structure_and_magnetic_anisotropies_of_antiferromagnetic_transition_metal_difluorides.pdf",        "content": "Electronic structure and magnetic anisotropies of antiferromagnetic transition-metal\ndi\ruorides\nCinthia Antunes Corr^ ea1, 2and Karel V\u0013 yborn\u0013 y1\n1Institute of Physics, Academy of Science of the Czech Republic, Cukrovarnick\u0013 a 10, Praha 6, Czech Republic\n2Department of Physics of Materials, Charles University in Prague, Ke Karlovu 5, 121 16 Prague, Czech Republic\nWe compare GGA+U calculations with available experimental data and analyze the origin of\nmagnetic anisotropies in MnF 2, FeF 2, CoF 2, and NiF 2. We con\frm that the magnetic anisotropy\nof MnF 2stems almost completely from the dipolar interaction, while magnetocrystalline anisotropy\nenergy plays a dominant role in the other three compounds, and discuss how it depends on the details\nof band structure. The last mentioned is critically compared to available optical measurements. The\ncase of CoF 2, where magnetocrystalline anisotropy energy (MCA) strongly depends on U, is put into\ncontrast with FeF 2where theoretical predictions of magnetic anisotropies are nearly quantitative.\nI. INTRODUCTION\nSeveral rutile-structure di\ruorides of transition met-\nals (TMs) such as MnF 2have long been known to be\nantiferromagnetically ordered at low temperatures. Al-\nbeit not the \frst antiferromagnets (AFMs) ever identi-\n\fed, they received signi\fcant attention in the late '50s\nand '60s when their magnetic anisotropy was e\u000bectively\ndetermined using measurements of spin \rop. These ma-\nterials are arguably one of the simplest AFMs one can\nimagine: their two magnetic sublattices are oriented in\nopposite directions (collinearity) and they exhibit uniax-\nial MA, which reduces the complexity of domain building.\nRenewed interest in these materials has arisen recently\nin the context of antiferromagnetic spintronics1. Very\nrecent device concepts using these traditional AFMs in-\nclude bilayers where spin pumping by AFM2or spin See-\nbeck e\u000bect3could be observed4. Spin currents can be\npassed through insulating AFMs5and devices involving\nMnF 2and FeF 2have been suggested2,6.\nMotivated initially by the lack of theoretical estimates\nof magnetocrystalline anisotropy (MCA), we soon real-\nized that not even the band structures of magnanese,\niron, cobalt and nickel di\ruorides are well established in\nthe literature. We therefore present DFT+U calculations\n(described in detail in Appendix C), compare them to op-\ntical measurements where available, identify the missing\ninformation (and propose experiments and calculations\nto be still carried out) and \fnally present the MCA calcu-\nlations and discuss their agreement with experimentally\ndetermined magnetic anisotropies of these materials.\nFor certain purposes, simple (in the sense explained\nabove), AFMs can be described by Stoner-Wohlfarth\nmodel7where the energy (per volume) divided by sub-\nlattice magnetization Mreads\nE\nMV=Be~ m1\u0001~ m2\u0000B~b\u0001(~ m1+~ m2)\u0000Ba[(~ m1\u0001^z)2+(~ m2\u0001^z)2]:\n(1)\nHere,~ m1;2and~bare the unit vectors giving the direc-\ntion of sublattice magnetizations and magnetic \feld B,\nrespectively. The two material-speci\fc parameters of this\nmodel are the exchange \feld Beand anisotropy \feld Baand, typically, Be\u001dBa;B. For~bjj^z, model Eq. (1) im-\nplies a spin \rop at B=Bsf= 2pBaBe, i.e., abrupt\nground state transition from ~ m1;2jj^zwith~ m1;2strictly\nantiparallel to, approximately ~ m1;2?^zwith~ m1;2slightly\ncanted (see Fig. 8 in Appendix B). Using this e\u000bect, Ba\ncan be determined from magnetometry provided that the\nexchange \feld is known or estimated.\nWe summarize the measured values of Bsffor the \frst\nthree compounds of the series in Tab. I and compare\nthem to theoretically calculated values. The latter are\nobtained by combining Ba, which comprises MCA and\ndipolar interaction (the former, B(1)\na, calculated by ab\ninitio methods detailed in Sec. III), and Be, based on\nan estimate of the exchange coupling Jfrom the N\u0013 eel\ntemperature35,\nkTN\nJ=1\n3S(S+ 1) (2)\nwhereS\u0016Bis the TM atom magnetic moment (rela-\ntion between the exchange coupling JandBeis given\nin Sec. III). We \fnd a satisfactory agreement between\nexperimental and theoretical values of spin-\rop \felds for\nMnF 2and FeF 2and the following conclusions can be\nmade. Magnetic anisotropy of MnF 2is primarily driven\nby dipole interactions (see Appendix A), which is not\nsurprising given the atomic con\fguration of manganese\n(L= 0 orbital singlet), which does not allow any ap-\npreciable MCA (see comments32on single-ion model in\nSec. III). On the other hand, this does not apply to FeF 2,\nwhere the TM 3 dshell is not half-\flled and sizable matrix\nelements of ~L\u0001~Sthen lead to a strong magnetocrystalline\nanisotropy, which translates into spin-\rop \felds as large\nas 42 T.\nCalculations of MCA in CoF 2yield ambiguous results\n(see Sec. III) and we, therefore, use opposite reasoning\nfor this material: using BsfandBe, we estimate Ba,\nwhich is then shown to imply B(1)\naconsistent with our\nab initio calculation. Again, this consistency check is ex-\nplained in Sec. III and in Tab. I, the values of Ba;B(1)\naare\nmarked with an asterisk to indicate that they are calcu-\nlated using experimental Bsf. Regarding NiF 2, we \fnd\nnegativeBain agreement with experimental evidence15\nof~ m1;2oriented in plane. Spin-\rop measurements arearXiv:1805.11339v3  [cond-mat.mtrl-sci]  12 Sep 20182\nTABLE I. Parameters of MnF 2, FeF 2, CoF 2, and NiF 2related to magnetism. Note that de\fnitions of BeandBavary\nthroughout literature.\nMnF 2 FeF 2 CoF 2 NiF 2\nexp calc exp calc exp calc exp calc\nmag.mom. [ \u0016B] 5.0484.4 3.938, 3.7593.6 2.2182.6 1.9681.63\nideal S 2.5 2 1.5 1\nBe[T] 46.510, 57.5385.5 43.410, 623116.7 32.41067.4 163.5\nBa[T] 0.69710, 0.830.42 14.910, 19.232.6 3.2100.73\u0003\u00000:50\nB(1)\na[T] 0.2 \u000110\u000032.3 0.52\u0003\u00000:71\ndipolar term 418 mT 317 mT 211 mT 203 mT\nBsf[T] 9.271112.0 41.91234.8 14.010***\nTN[K] 67.7875.8837.71374.18\nmore complicated in this case since there are multiple\neasy axes (such a system is prone to build multidomain\nstates) and therefore no data is given for Bsfin Tab. I.\nTheoretical calculation of MCA relies on a solid knowl-\nedge of the band-structure. It is essentially the di\u000ber-\nence between two large numbers EkandE?, the total\nenergy of the occupied electron states for ~ m1;2jj^zand\n~ m1;2?^z, so that even small inaccuracies may lead to\ncompletely wrong results unless such inaccuracies accu-\nrately cancel (i.e., any error in the band-structure deter-\nmination has the same e\u000bect on both EkandE?). It\nshould be pointed out that these calculations must in-\nclude the e\u000bect of spin-orbit interaction without which\nthe MCA vanishes ( Ek=E?). Band structures of the\nfour compounds considered in this article have been cal-\nculated previously under various approximations: LSDA\nband structures of MnF 2and NiF 2were \frst calculated\nby Dufek, Schwarz, and Blaha16and, a little later, the\nsame group also added FeF 2and CoF 2using GGA17(see\nalso Appendix C), albeit with unrealistically small gaps.\nThis improved with the advent of GGA+U18,22where,\nhowever, not much attention was paid to how large the\nvalues of the model parameters U;J actually should be.\nUnrestricted Hartree-Fock calculations23produce optical\ngap in excess of 10 eV for FeF 2, which is beyond any\ndoubt too large, realistic size being close to 3 eV (see be-\nlow). We now proceed to a discussion of band structures\ncalculated using GGA+U (based on the same package\nas in Ref. [17]) and critical comparison of these to ex-\nperimentally accessible quantities such as band gap, TM\nmagnetic moment, and lattice parameters.\nII. ELECTRONIC STRUCTURE\nGiven the identical crystal structure (Fig. 7) and the\nposition of Mn, Fe, Co, and Ni in the periodic table, it is\nnot surprising that structures of all four di\ruorides are\nmutually similar. It can explained using the sketch in\nFig. 1 (see also Fig. 8 in Appendix C). Fermi level ( EF)\nlies in the middle of TM d-state bands and other atomic\nZM\u0000fluorine 2s-states⇠5e Vfluorine 2p-states      lower five TM d-statesgroup B  states of TMgroup A states of TMEFconduction band\n⇠25 eVFIG. 1. Schematic band structure of rutile-type MnF 2, FeF 2,\nCoF 2, and NiF 2in their AFM state. Spin up and down bands\nare degenerate. Note that for MnF 2, all \fve upper d-bands\nare in group B (group A is an empty set).\norbitals (such as \ruorine p-states) are relatively far away.\nThe gap that opens within the TM d-state band is partly\ndue to electron-electron interactions (EEIs), which we\nmodel, within density functional theory, by GGA+U (see\nAppendix C) and partly (in the case of Mn and Ni) due\nto crystal \feld e\u000bects. Surprisingly, the size of band gaps\nat low temperature (i.e., in the AFM phase) is nowhere\nto be found in the literature and we can therefore use\nonly some indirect arguments to support the actual band\nstructure calculations in Fig. 2.\nThe band structures for spin up and down are the same\n| this is a consequence of the simple antiferromagnetic\norder (see Fig. 1 in Ref. [1] for explanation). Focusing on\none spin, the total of ten d-orbitals (for two TM atoms\nin the unit cell, see also Fig. 7) divides \frst into two3\nFeF2 atom  0    size 0.02\nZ ΓM A Z R X ΓE F Energy (eV)  0.0  5.0\n -5.0-10.0CoF2 atom  0    size 0.02\nZ ΓM A Z R X ΓE F Energy (eV)  0.0  5.0\n -5.0-10.0NiF2 atom  0    size 0.02\nZ ΓM A Z R X ΓE F Energy (eV)  0.0  5.0\n -5.0-10.0MnF2 atom  0    size 0.02\nZ ΓM A Z R X ΓE F Energy (eV)  0.0  5.0\n -5.0-10.0\nFIG. 2. Overview of all four \ruorides. Left to right: MnF 2withoutU, FeF 2withU= 0:2 Ry, CoF 2withU= 0:1 Ry, and\nNiF 2again without U.\nquintuplets that could be thought of as of bonding and\nantibonding orbitals. The lower quintuplet always lies\nbelowEF, the higher is either partly or completely above\nEF. Starting with MnF 2in its 3d5con\fguration, EFis\nlocated at the top of the lower \fve TM d-bands and all\nother \fve bands are high above ( \u00193 eV or more); in\nterms of the sketch in Fig. 1, there are no bands in group\nA and all \fve are in group B as the leftmost panel in\nFig. 2 shows. Another highly dispersive band, which we\ncall \\conduction band\" in Fig. 1, crosses these relatively\n\ratd-bands (we comment on this band in Appendix C).\nNow the e\u000bect of adding Hubbard Uis to push the bands\nin group B away from the \\lower \fve\" so that, since EF\nremains pinned at the top of the latter, the group B bands\nmove higher into the conduction band (compare also the\nleftmost panel of Fig. 2 to Fig. 8 in Appendix C). In other\nwords, the band structure of MnF 2does not change sub-\nstantially when Uis increased even though the gap does\nincrease slightly; the gap occurs mainly due to the split-\nting between the two quintuplets of d-bands and would\nbe present even in the absence of correlations. However,\nthe e\u000bect of Hubbard Uis much more dramatic for the\nother compounds under scrutiny. We do not discuss the\nbest choice36ofUin MnF 2any further since, by virtue\nof the argument of half-\flled d-shell, the MCA is anyway\nsmall in this material.\nThere is one more occupied band in FeF 2than in MnF 2\nand therefore one band from group B (Fig. 1) has to be\ntransferred into group A. Because all \fve d-bands are\nvery close to one another, forming some kind of local\nspaghetti in the band structure, this would render FeF 2\nmetallic (at least on the LDA level). A better treatment\nof EEIs is needed. In fact, a gap opens already by switch-\ning to GGA but its size is unrealistically small ( <\u00180:5 eV).\nFigure 3 shows that within GGA+U, the gap grows with\nUand for values used typically in literature18, it reaches\na reasonable9size of\u00193 eV. We point out that a room\ntemperature measurement of optical absorption24leads\nto a similar gap size; however, we will discuss below the\n 0 0.5 1 1.5 2 2.5 3 3.5\n 0 0.1 0.2 0.3 0.4 0.5gap [eV]\nU [Ry]FIG. 3. Nominal gap in FeF 2as a function of U. Note that the\napparent optical gap is larger because some optical transitions\nmay be suppressed.\nplausibility of using Uaround 2.5 eV which is somewhat\nsmaller than usual25. Our choice of Ucorresponds to the\nsecond panel from the left in Fig. 2 and seems to give\noptical spectra closer to another experimental work on\nFeF2. Impact of the choice of Uon MCA will be dis-\ncussed in Sec. III.\nThe experimental work in question26concerns room-\ntemperature ellipsometry of FeF 2layers. The gap in-\nferred in Fig. 4 of that paper is certainly smaller than in\nRef. [24] and, moreover, it turns out that the actual the-\noretical gap may be even smaller because of suppressed\ntransitions from the d-band directly below EFto the\nother low-lying bands of group B and the \\conduction\nband\" (as de\fned in Fig. 1). In fact, it is remarkable\nhow similar are theoretically calculated optical spectra\nforUas small as 0.1 Ry (shown in Fig. 4) to the experi-\nmental data26mentioned above. Given that optical gap\nat low temperatures will probably be larger, we opted4\n1.62.02.42.8\n1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.00.51.01.5\n0.00.10.20.30.40.50.60.7\n1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.03.603.653.703.753.803.853.903.95\nFIG. 4. On the left, optical spectra of FeF 2(relative permittivity) calculated for U= 0:1 Ry. Agreement with experimental\ndata on the right (based on Fig. 4 of Pi\u0014 stora et al. [26]) is striking, despite the fact that these measurements were taken above\nTN.\nfor showing band structure with U= 0:2 Ry in Fig. 2.\nCalculated magnetic moment, which is smaller than the\none experimentally determined (see Tab. I), also suggests\nthat this choice of Umay be better.\nBoth for FeF 2and CoF 2, we chose rather small values\nofUto have gaps around 2 eV in Fig. 2. This choice is\narbitrary and since measurements of structural parame-\nters and/or magnetic moment provide only a relatively\nbenevolent test27on the values of Uwithin usual ab ini-\ntiocalculations, not much progress can be expected here\nuntil low-temperature optical measurements in a broad\nspectral range are available. We show an example of\nsuch spectra (for CoF 2) in Fig. 5: There is an abun-\ndance of spectral features that could be tested against\nexperiments. In fact, absorption edge around 0.8 eV was\nmeasured28at low temperature in CoF 2, which would sug-\ngest very small value of U. This spectral feature is associ-\nated with a relatively narrow band whose origin could be\nin interband transitions (see the inset of Fig. 5) but also\nin electron-phonon interactions29, which would, in turn,\nindicate a much larger gap. Nevertheless, large values of\nthe Hubbard parameter (like U= 7 eV taken from cobalt\noxide25) seem in contradiction with relatively small MCAas explained in Sec. III.\nFinally, NiF 2again retains a gap even for vanishing U.\nUnder the action of the crystal \feld, the upper quintuplet\nofd-bands splits into a doublet and a triplet. The for-\nmer remains completely depopulated and is separated by\n\u00191:5 eV from the occupied three bands forming bands of\nthe group A (see the sketch in Fig. 1). Similar to MnF 2,\nthe e\u000bect of Hubbard Uis to increase the gap size by\nmoving the group A and B bands away from each other.\nThe small values of calculated TM magnetic moment in\nTab. I, however, suggest that, similar to MnF 2, using\nnon-zeroUmay be reasonable. To conclude this section,\nwe stress that in spite of uncertainty about what value of\nUmay lead to the best description of the actual system,\nthe qualitative character of the band structure of all four\ncompounds is clear: larger values of Upush the bands\nin group B higher and make the optical gap larger. We\nbelieve that combining data from several di\u000berent exper-\nimental sources provides a solid basis for band-structure\nvalidation and, to this end, we put emphasis on optical\nspectra in this work.5\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4\n 0 1 2 3 4 5 6 7 8permittivity\nEnergy [eV] 0 0.03\n 0 2\nFIG. 5. Calculated imaginary part of permittivity, which is\nproportional to absorption, for CoF 2(U= 0:1 Ry). The\ninset shows a feature below 2 eV, which could be related to\nthe observed narrow band28.\n-0.0500.050.100.150.200. .250.300.35\n0.05 0.10.15 0.20.25 0.30.35 0.40.45\nU [Ry]FeF2\nCoF2a anisotropy [meV/ ]\nf.u.\nFIG. 6. Calculated MCA for FeF 2and CoF 2as a function of\nthe Hubbard parameter U.\nIII. MAGNETOCRYSTALLINE ANISOTROPY\nBased on su\u000eciently accurately calculated electronic\nbands (with e\u000bects of spin-orbit interaction included37),\ntotal energy in the in-plane ( Ek) and out-of-plane mag-\nnetic con\fgurations ( E?) can be calculated. Sublattice\nmagnetization M=S\u0016B=V(whereVis the volume of\nunit cell which contains one TM atom of each sublat-\ntice) can then be used to obtain Ba= (Ek\u0000E?)=Mand\nalso, using Eq. (2), Be=NJS2=MV . The central ques-\ntion now is how the total energies depend on U: other\nquantities such as optimal structure parameters (lattice\nconstants) or TM magnetic moment do27, and it is not a\npriori clear how sensitive the MCA is to the variation of\nU.\nKeeping in mind that MCA must be very small for\nMnF 2(recall the argument of half-\flled d-shell), there is\nno need to investigate its dependence on U. The more-\nthan-satisfactory agreement between calculated and mea-suredBsfin Tab. I (the former one being larger by 29%)\nrelies on the dipolar term, which is not as di\u000ecult to\nevaluate. Regarding the quantitative agreement between\ncalculations and experiment, we should once again stress\nthat the limiting factor is now probably the estimate of\nBe, based on TN. The situation is di\u000berent with FeF 2:\nthe valueB(1)\na= 2:3Tin Tab. I corresponds to calcula-\ntions withU= 0:2 Ry. Now, as Fig. 6 shows, while the\nMCA does depend on U, the variation is moderate. It is\npossible to conclude that for FeF 2, magnetic anisotropies\ncan be fairly well predicted theoretically (Tab. I shows\nthat theoretical estimate of Bsfis only about 17% lower\nthan the measured value). Considering the fact that\nsometimes18theab initio calculations are extended to\ninclude another Hubbard-like parameter J, we also cal-\nculated MCA for U= 0:44 Ry and J= 0:07 Ry and\nfound it to be somewhat smaller than what would corre-\nspond toUe\u000b\u0011U\u0000J= 0:37 Ry. This further highlights\nthe limits of quantitative predictions of MCA based on\nab initio calculations.\nAs an alternative to ab initio calculations, we note that\nthe sign and, to some extent, also the order of magni-\ntude of MCA in FeF 2can be deduced from the single-ion\nmodel19. Orbital multiplet of the Fe2+ion (L= 2) is\nfully split by the crystal \feld in the rutile structure and\nthe action of spin-orbit interaction HSOon the lowest\n(non-degenerate) level can be written in terms of a spin\nS= 2 Hamiltonian, Hs=DS2\nz. Corrections to this form\nofHsare small19, derivation of this result is explained\nbelow when we discuss CoF 2; note that the argument in\nEq. (5) explains the negative sign of Das a consequence\nof level repulsion. Exchange splitting \foriented along\nthe direction of the N\u0013 eel vector ^ eL, combined with Hs,\nleads to a simple model exhibiting MCA: Hs+\f~S\u0001^eL.\nThe lowest energy (with respect to spin) for ^ eLk^zand\n^eLk^x, respectively, is thus obtained by diagonalizing\nDS2\nz+\fSzandDS2\nz+\fSx; (3)\nwhich yields 4 D\u00002\fandD\u00002\ffor the lowest eigenvalue\nin the\f\u001dDlimit. Given D < 0, the former direction\nis preferred implying uniaxial anisotropy. The values of\nD(around 1 meV) determined by various experimental\ntechniques20are consistent, yet not quite in agreement\nwith, calculated and measured magnetic anisotropy of\nFeF2.\nA very di\u000berent situation is found with CoF 2. For\nsmall values of U, MCA even changes sign (see again\nFig. 6) and if we take the experimental value of Bsfin\nTab. I as a means to estimate Baand, once the dipo-\nlar term has been subtracted, also B(1)\na, we \fnd that the\nMCA changes with Urapidly around the corresponding\nvalue (0.05 meV per formula unit). Hence the conclusion,\nat minimum, that it is not possible to rely on theoretical\ncalculations of MCA in this case, unless some additional\nguidance is provided. Moreover, values of Uthat pro-\nduce MCA of this size are rather small (below 0.1 Ry)\nwhile more commonly25larger values are used. It should\nbe noted, however, that CoF 2seems to be anomalous6\nFIG. 7. Crystal structure (rutile) applies to all four di\ruorides under study. Magnetic structure on the left corresponds to\norientation along the easy axis (except for NiF 2) and we denote its energy by Ek. Magnetic structure on the right is de\fned to\nhave energy E?.\nwithin the series of four materials considered in this pa-\nper (contrary to the other three, it has a signi\fcantly\nlowerTN) and it is possible that the estimate of Bein\nTab. I is too large. This would allow for larger B(1)\naand,\nin the spirit of Fig. 6, for larger values of Uas well. Re-\nliable experimental determination of optical gap (at low\ntemperatures) could resolve this issue.\nAnalysis based on the single-ion model21for Co2+\nleads quantitatively to an even worse estimate of MCA\nthan for FeF 2but still predicts the correct sign and also\nthe negligible in-plane anisotropies. The orbital multi-\nplet (L= 3) is now split by octahedral crystal \feld and\nthe lowest lying \u0000 4triplet is further split by30\u0001\u00190:1 eV\ninto a ground state doublet and an excited state (singlet\njLzi=j0i; in the following, we will use this notation for\nthe orbital part of wave functions). Rhombohedral crys-\ntal \feld lifts the degeneracy of the doublet, producing\nstates\njai=p\n5\n4j\u00003i+p\n3\n4j\u00001i+p\n3\n4j1i+p\n5\n4j3i(4)\njbi=p\n5\n4j\u00003i\u0000p\n3\n4j\u00001i+p\n3\n4j1i\u0000p\n5\n4j3i\nwhose energy splitting Eb\u0000Eais a fraction30of \u0001. The\nperturbative action of HSO=\u0015~L\u0001~Son the lower state\ncan now be evaluated to the second order in spin-orbit\ninteraction \u0015. Provided we neglect coupling to the jLzi=\nj0istate, we obtain\n\u00152haj~L\u0001~Sjbihbj~L\u0001~Sjai\nEa\u0000Eb=9\n4\u00152\nEa\u0000EbS2\nz\u0011DS2\nz (5)\nbecause~L\u0001~S=LzSz+1\n2(L+S\u0000+L\u0000S+) and, given\nin Eq. (4), the matrix elements of the raising (lowering)\noperatorsL+(L\u0000) vanish. This construction predicts\nD< 0 by virtue of Ea<Ebbut, quantitatively, it implies\na larger MCA than for FeF 2since both \u0015is larger (for\nCoF 2) and the energy splitting of the lowest two states\nsmaller. The absent in-plane anisotropy amounts to Hs,\ncontaining no Sx,Syoperators and this, in turn, is aconsequence ofhajL\u0006jbi= 0. Perturbative coupling to\nthejLzi=j0isinglet will introduce the Sx;yterms toHs,\nhowever, their coe\u000ecients will be small (\u0001 \u001djEa\u0000Ebj).\nConcerning the quantitative disagreement between the\nsingle-ion model for CoF 2andEk\u0000E?calculated by ab\ninitio , a more advanced approach seems necessary such as\nsome kind of cluster model, e.g., FeF 6, constructed along\nthe lines of Ref. [31] where a model of MnAs 4cluster\nwas used to explain certain magnetic anisotropy terms\nin (Ga,Mn)As dilute magnetic semiconductor. Such an\nattempt to make sense of the ab initio calculations is\nnevertheless clearly beyond the scope of this paper. On\nthe other hand, the single-ion model is successful in case\nof FeF 2and also32MnF 2.\nIV. CONCLUSION\nMagnetic anisotropies of MnF 2, FeF 2, CoF 2, and NiF 2\nhave been investigated theoretically and it was found\nthat, with the exception of CoF 2,ab initio calcula-\ntions described in Appendix C lead to reliable results.\nFor comparison to experiments, we used well-established\nspin-\rop measurements (spin-\rop \feld Bsf, see Tab. I).\nRegarding CoF 2, we conclude that while the calcula-\ntions are consistent with experimentally determined Bsf,\nthe MCA depends too sensitively on Hubbard parameter\nUso that quantitative prediction is impossible, without\nknowing in advance what the correct result is.\nWe pointed out that band structures should be vali-\ndated, for example, through optical measurements, be-\nfore using them for further calculations. It would be de-\nsirable to perform such low-temperature measurements\nfor all four compounds and determine the optical gap.\nThis would a\u000bord greater con\fdence in the values of U\nused in ab initio calculations.\nACKNOWLEDGEMENTS\nIt is our pleasure to express gratitude to Pavel Nov\u0013 ak\nfor his guidance in crystal \feld theories and litera-7\nture dating back deep into the 20th century. We ac-\nknowledge support from National Grid Infrastructure\nMetaCentrum provided under the programme \\Projects\nof Large Research, Development, and Innovations In-\nfrastructures\" (CESNET LM2015042) and also discus-\nsions with Ond\u0014 rej \u0014Sipr, Jind\u0014 rich Koloren\u0014 c, and Ger-\nhard Fecher, and also funding via Contract No. 15-\n13436S (GA \u0014CR) and EU FET Open RIA Grant no.\n766566 (ASPIN). C.A.C. acknowledges the extended \f-\nnancial support by the ERDF, Project NanoCent No.\nCZ.02.1.01/0.0/0.0/15 003/0000485.\nAppendix A: Dipolar interactions\nDipolar magnetic energy (per unit cell) of an (in\fnite)\nlattice of magnetic moments is E=\u00001\n2P\nj~Bj\u0001~ \u0016jwhere\nthe sum goes over all magnetic moments ~ \u0016jin the unit\ncell. Magnetic \feld generated, at the position of given\n~ \u0016j, by all other magnetic moments is\n~Bj=\u00160\n4\u0019X\ni3(^\u0016i\u0001^rij)~ rij\u0000~ \u0016i\nj~ rijj3(A1)\nwhere~ rijis the relative position of ~ \u0016iwith respect to\n~ \u0016j. The dipolar magnetic energy depends on the orien-\ntation of the magnetic moments; values labeled \\dipolar\nterm\" in Tab. I are E?\u0000Ekrecalculated into \feld us-\ning the same procedure as for MCA (see Sec. III). Mag-\nnetic momentsj~ \u0016jjused in Eq. (A1) were taken from\nexperiments8, as given in Tab. I.\nDipolar interactions do not contribute to MA in cu-\nbic lattices while they may even constitute its dominant\nsource if the high symmetry is broken (or completely\nabsent). To explain qualitatively the e\u000bect of the bro-\nken symmetry, we consider a \fve-atom cluster (magnetic\nsublattice A atom located at the center of coordinate\nsystem and four atoms of magnetic sublattice B located\nat (\u0006a;0) and (0;\u0006b) with strictly antiparallel magnetic\nmoments) and calculate the energy of the four B atoms\nin the dipolar \feld ~BAimplied by Eq. (A1). This en-\nergy,E(\u001e), depends in general on the magnetic moment\norientation (sin \u001e;cos\u001e). Fora=b= 1, however, E(\u001e) is\nconstant owing to sin2\u001e+cos2\u001ebeing independent on \u001e.\nOnce the symmetry is broken ( a6=b), the con\fguration\nwith moments parallel to x(\u001e=\u0019=2) ceases to have the\nsame dipolar energy as the \u001e= 0 case, the ratio of the\nrespective energies being (4 \u00002a3=b3)=(4a3=b3\u00002)6= 1.\nAppendix B: Structural parameters\nCrystal structure with two orientations of magnetic\nmoments is shown in Fig. 7. Antiferromagnetic ( J1) and\nferromagnetic ( J2) interactions between nearby magnetic\nmoments are highlighted, the coupling Jdiscussed in\nSec. III is a weighted average of them. Lattice constants\nand atom positions (taken from Ref. [33]) are given inTABLE II. Crystal structure of MnF 2, FeF 2, CoF 2, and\nNiF 233. Crystal system is tetragonal, space group P42=mnm\nNo. 136 for all four compounds.\nChemical formula MnF 2\na, c ( \u0017A) 4.8738(1), 3.3107(1)\nV (\u0017A3) 78.642(3)\natomic positions\nMn1 (0,0,0)\nMn2 (0.5,0.5,0.5)\nF1 (0.3053(12),0.3053(12),0)\nF2 (0.8053(12),0.1947(12),0.5)\nChemical formula FeF 2\na, c ( \u0017A) 4.6945(4), 3.3097(1)\nV (\u0017A3) 72.940(9)\natomic positions\nFe1 (0,0,0)\nFe2 (0.5,0.5,0.5)\nF1 (0.3010(8),0.3010(8),0)\nF2 (0.8010(8),0.1990(8),0.5)\nChemical formula CoF 2\na, c ( \u0017A) 4.6954(4), 3.1774(4)\nV (\u0017A3) 70.051(12)\natomic positions\nCo1 (0,0,0)\nCo2 (0.5,0.5,0.5)\nF1 (0.3052(8),0.3052(8),0)\nF2 (0.8052(8),0.1948(8),0.5)\nChemical formula NiF 2\nCrystal system, space group Tetragonal, P42=mnm No. 136\na, c ( \u0017A) 4.6498(3), 3.0838(1)\nV (\u0017A3) 66.674(6)\natomic positions\nNi1 (0,0,0)\nNi2 (0.5,0.5,0.5)\nF1 (0.3012(13),0.3012(13),0)\nF2 (0.8012(13),0.1988(13),0.5)\nTab. II. It is important to note that, upon introducing\nthe magnetic order, the space group of the crystal struc-\nture is modi\fed from the tetragonal P42=mnm to the\northorhombic Cmmm because the TM atoms sitting at\n(0;0;0) and (1=2;1=2;1=2) are no longer equivalent by\nsymmetry, as their spins are antiparallel (Fig. 7). In our\ncalculations, we used mu\u000en-tin radii ( RMT) as shown in\nTab. III and all data shown in this paper are based on\nRkmax= 7.8\nTABLE III. RMTin Bohr radii for individual atoms used in\nour calculations.\nMn, F 2.08, 1.88 Co, F 1.97, 1.78\nFe, F 1.97, 1.78 Ni, F 1.95, 1.77\nAppendix C: Electronic structure calculations\nFor our density functional theory (DFT) calcula-\ntions, we use the linearized augmented plane wave\nmethod38, with added orbital-dependent correction (so\ncalled DFT+U). The chosen orbitals for this Hubbard-\nlike term are the 3 d-states of the TM, scheme of Ref. [39]\nis used to avoid double counting and GGA to the den-\nsity functional is employed. Scalar relativistic approach\nto the spin-orbit interaction is taken from the very be-\nginning of our calculations. Convergence with respect to\nthe number of points in the k-space (nk) is achieved al-\nready around nk= 10000, meaning that energies Ekand\nE?are converged to several \u0016Ry while their di\u000berenceis about an order of magnitude larger. Within this pre-\ncision, we \fnd no signi\fcant di\u000berence between in-plane\ndirections, which agrees, assuming the single-ion model\nto be valid, with the perturbative argument [see Eq. (4)\nand below].\nAs described in the main text and Fig. 1, the main\ne\u000bect of increasing Uis to push the group B d-bands\naway from the lower quintuplet of the d-states and, if\npresent (as for FeF 2, CoF 2, and NiF 2), also from the\ngroup Ad-bands. This can be seen by comparing Fig. 8\n(exemplifying the e\u000bect of large U) to the leftmost panel\nof Fig. 2. As a side remark, we note that the position of\nthe low-lying \ruorine 2 sstates also depends on the TM\nion type.\nFigure 8 also clearly shows the \\conduction band\"\n(lowest lying unoccupied parabolic band). Its e\u000bective\nmass is moderately anisotropic and smaller than the free\nelectron rest mass m0; wave functions of this band are\nlargely localized in the interstitial space. Averaged over\ndirections, we \fnd me\u000b=m0about 0.22 for MnF 2, 0.25\nfor FeF 2, 0.51 for CoF 2, and 0.36 for NiF 2.\n1V. Baltz et al., Rev. Mod. Phys. 90, 015005-1 (2018); T.\nJungwirth et al., Nat. Phys. 14, 200 (2018).\n2\u001f. Johansen and A. Brataas, Phys. Rev. B 95, 220408\n(2017).\n3S. M. Rezende et al., Phys. Rev. B 93, 014425 (2016).\n4S.M. Wu et al., Phys. Rev. Lett. 116, 097204 (2016).\n5T. Moriyama et al., Appl. Phys. Lett. 106, 162406 (2015).\n6S.A. Gulbrandsen and A. Brataas, Phys. Rev. B 97, 054409\n(2018).\n7A.N. Bogdanov et al., Phys. Rev. B 75, 094425 (2007).\n8J. Strempfer et al., Phys. Rev. B 69, 014417 (2004).\n9P. Nov\u0013 ak et al., Phys. Stat. Sol. (B) 243, 563 (2006).\n10A.S. Carri\u0018 co et al., Phys. Rev. B 50, 13453 (1994).\n11I.S. Jacobs, J. Appl. Phys. 32, s61 (1961).\n12V. Jaccarino et al., J. Magn. Magn. Mater. 31-34, 1117\n(1983).\n13T. Nagamiya et al., Adv. Phys. 4, 1 (1955).\n14M. Balkanski et al., J. Chem. Phys. 40, 1897 (1964).\n15H. Shi et al., Phys. Rev. B 69, 214416 (2004).\n16P. Dufek et al., Phys. Rev. B 48, 12672 (1993).\n17P. Dufek et al., Phys. Rev. B 49, 10170 (1994).\n18S. L\u0013 opez{Moreno et al., Phys. Rev. B 85, 134110 (2012).\n19M.T. Hutchings et al., J Phys C: Sol St Phys 3, 307 (1970).\n20M.E. Lines, Phys. Rev. 156, 543 (1967).\n21A. Ishikawa and T. Moriya, J. Phys. Soc. Jpn. 30, 117\n(1970).\n22J.A. Barreda{Arg ueso et al., Phys. Rev. B 88, 214108\n(2013).\n23G. Valerio et al., Phys. Rev. B 52, 2422 (1995).\n24R. Santos-Ortiz et al., Appl. Mater. Interfaces 5, 2387\n(2013).\n25P. Thunstr om et al., Phys. Rev. Lett. 109, 186401 (2012).\n26J. Pi\u0014 stora et al., J. Phys. D:Appl. Phys. 43, 155301 (2010).\n27Z. Yang et al., Trans. Nonferrous Met. Soc. China 22, 386\n(2012).\n28V.V. Eremenko and A.I. Zvyagin, Sov. Phys.Solid State 6,\n1013 (1964), in Russian: Fiz. Tverd. Tela 6, 1013 (1964).\nZ ΓM A Z R X ΓE F Energy (eV)  0.0  5.0\n -5.0\n-10.0\n-15.0\n-20.0\n-25.0FIG. 8. Band structure of MnF 2calculated with large U\ncorresponding to Ref. [34] (0.43 Ry).\n29P.A. Young, Thin Sol. Films 4, 25 (1969).\n30H. Kamimura and Y. Tanabe, J Appl Phys 34, 1239 (1963).\n31H. Subramanian and J.E. Han, J. Phys.: Cond. Matter 25,\n206005 (2013).\n32Within the single-ion model, the ground state of a Mn2+\nion is an orbital singlet according to the Hund's rule.\nHence, the ground state cannot be split by crystal \feld\nand the only way for HSO, de\fned above Eq. (4), to come\ninto play is to mix the6Ssinglet with an excited4Pstate.\nEnergy of the latter is orders of magnitude larger than the\ncrystal \feld splittings in FeF 2, CoF 2, and NiF 2so that the\nperturbative e\u000bect of HSOon the Mn2+ground state and9\nhence MCA is small.\n33W.H. Baur and A.A. Khan, Acta Cryst. B 27, 2133 (1971).\n34E. Stavrou et al., Phys. Rev. B 93, 054101 (2016).\n35This is a standard result of Weiss theory of magnetism\n[see, for instance, Eq. (11) in Phys. Rev. 123, 425], where\nHi=\u0000Jh~Si\u0001~Siis assumed for interaction between a given\nlocalized magnetic moment ~Siand a mean \feld h~Sipro-\nduced by all other magnetic moments.\n36Note that the calculated value of magnetic moment in\nTab. I is signi\fcantly lower than the experimental value.This suggests that appreciable Ushould in fact be used.\n37Band structure does not change when direction of magnetic\nmoments is varied unless spin-orbit interaction is included\nin the DFT calculations.\n38P. Blaha et al., WIEN2k, An Augmented Plane Wave +\nLocal Orbitals Program for Calculating Crystal Proper-\nties(Karlheinz Schwarz, Techn. Universit at Wien, Austria,\n2001).\n39V.I. Anisimov et al., Phys. Rev. B 48, 16929 (1993)."    },    {        "title": "1806.01233v2.Site_Specific_Spin_Reorientation_in_Antiferromagnetic_State_of_Quantum_System_SeCuO__3_.pdf",        "content": "Site-Specific Spin Reorientation in Antiferromagnetic State of Quantum System SeCuO 3\nNikolina Novosel,1William Lafargue-Dit-Hauret,2ˇZeljko Rapljenovi ´c,1Martina Dragi ˇcevi´c,1\nHelmuth Berger,3Dominik Cin ˇci´c,4Xavier Rocquefelte,2,\u0003and Mirta Herak1, †\n1Institute of Physics, Bijeni ˇcka c. 46, HR-10000 Zagreb, Croatia\n2Institut des Sciences Chimiques de Rennes UMR 6226,\nUniversit ´e de Rennes 1, Campus de Beaulieu, 35042 Rennes, France\n3Institut de Physique de la Mati `ere Complexe, EPFL, CH-1015 Lausanne, Switzerland\n4Department of Chemistry, Faculty of Science, University of Zagreb, Horvatovac 102A, HR-10000 Zagreb, Croatia\n(Dated: June 19, 2018)\nWe report on the magnetocrystalline anisotropy energy (MAE) and spin reorientation in the antiferromagnetic\n(AFM) state of the spin S=1=2 tetramer system SeCuO 3observed by torque magnetometry measurements in\nmagnetic fields H<5 T and simulated using density functional calculations. We employ a simple phenomeno-\nlogical model of spin reorientation in finite magnetic field to describe our experimental torque data. Our results\nstrongly support a collinear magnetic structure in zero field with a possibility of only very weak canting. Torque\nmeasurements also indicate that, contrary to what is expected for an uniaxial antiferromagnet, only fraction\nof the spins exhibit a spin-flop transition in SeCuO 3, allowing us to conclude that the AFM state of this sys-\ntem is unconventional and contains two decoupled subsystems. Our results demonstrate that the AFM state in\nSeCuO 3is composed of a subsystem of AFM dimers, forming singlets, immersed in a long-range ordered AFM\nstate, both states coexisting at the atomic scale. Furthermore, we show using ab-initio approach that both sub-\nsystems contribute differently to the MAE, corroborating the existence of decoupled subnetworks in SeCuO 3.\nThe present combination of torque magnetometry, phenomenological and density functional theory approach\nto magnetic anisotropy represents a unique and original way to study site-specific reorientation phenomena in\nquantum magnets.\nI. INTRODUCTION\nLow-dimensional spin systems represent a fertile ground to\nstudy the influence of quantum effects on the formation of ex-\notic states of matter. Zero-dimensional (0D) systems, in par-\nticular, are of significance since simple finite lattices represent\na fruitful playground for theoretical investigations, while, at\nthe same time, 0D magnetic lattices can be found in real ma-\nterials allowing the theory to be tested. The simplest example\nof a 0D system is a spin dimer consisting of two spins coupled\nby exchange energy J. The two allowed states are singlet and\ntriplet separated by an energy gap Jand the ground state is\ndetermined by the sign of J(antiferromagnetic or ferromag-\nnetic coupling). In real materials small, but finite, interactions\nbetween the 0D units can lead to a long range magnetic or-\ndering. Combined with the quantum effects of underlying 0D\nmagnetic units, this can lead to exotic phase diagrams where\ndifferent phases can be obtained by tuning the relative strength\nof the exchange couplings.\nAnother example of a 0D system is a spin tetramer where\nfour spins Sa,Sb,Scand Sdinteract forming a 0D mag-\nnetic unit with slightly more complex excitation spectrum than\nfound in a spin dimer [1]. When the coupling between spins\nSaandSbis equal to the coupling between spins ScandSd, the\nspin Hamiltonian can be written as\nH=J12(Sa\u0001Sb+Sc\u0001Sd)+J11(Sb\u0001Sc): (1)\nAn interesting limit for the spin tetramer is a case when the\ncoupling J11between the two spins in the middle, SbandSc,\n\u0003xavier.rocquefelte@univ-rennes1.fr\n†mirta@ifs.hris antiferromagnetic (AFM) and much stronger than the cou-\npling J12of spins in the middle with spins on the sides of\nthe tetramer, SaandSd(see Fig. 1). In this case SbandSc\nare expected to form a singlet state which persists in the back-\nground of weakly connected paramagnetic spins SaandSd. At\nlow temperatures weak intertetramer interactions can lead to\na long-range magnetic order. When this happens, the question\narises whether singlet states are broken with SbandScspins\njoining the long-range order, or do they somehow persist as\nsinglets in the background of long-range magnetically ordered\nspins SaandSd. The latter scenario was proposed by Hase et\nal.for spin tetramer system CdCu 2(BO 3)2based on the high\nfield magnetization measurements [2], and it was recently\nconfirmed by nuclear magnetic resonance (NMR) and zero-\nfield muon spin relaxation (ZF- µSR) [3]. In the ordered AFM\nstate of CdCu 2(BO 3)2, spins S aand S dare related to Cu2 site,\nwhile S band S cto Cu1 site. In this system, the spins of Cu1\natoms form strongly coupled singlets, but at T N= 9.8 K, an-\ntiferromagnetic long-range order sets in due to much weaker\nintertetramer interactions. Neutron powder diffraction (NPD)\nmeasurements on this system reported significant magnetic\nmoments on both Cu1 and Cu2 site, although it is smaller on\nCu1 [4]. Detailed theoretical investigation of CdCu 2(BO 3)2\nshowed dominant coupling between Cu1 spins forming AFM\ndimers, while further couplings forming intratetramer and in-\ntertetramer interactions are responsible for low-temperature\nAFM LRO of Cu2 spins which polarize Cu1 singlet states [5].\nThe same study revealed that polarization of Cu1 singlets is\npossible because the field from Cu2 spins is staggered and\nthus does not commute with the exchange interaction on the\ndimer [5]. Janson et al. further showed that a significant mag-\nnetic moment can be induced on Cu1 spins by the staggered\nfield from Cu2 spins [5]. This picture, later confirmed exper-arXiv:1806.01233v2  [cond-mat.str-el]  18 Jun 20182\nimentally by NMR and ZF- µSR measurements [3], is differ-\nent from the usual long-range order which is induced by the\ninteractions between the spins. In CdCu 2(BO 3)2magnetic in-\nteractions are responsible for the magnetic order of Cu2 site\nonly, while Cu1 moments are polarized. The decoupling of\nthe Cu1 and Cu2 spins in the ordered state is witnessed by\na magnetic anomaly at T\u0003=6:5 K observed in NMR, which\nwas attributed to the reorientation of Cu2 spins, while Cu1\nremain intact [3]. To better understand how this type of or-\ndering emerges it is important to find and study new systems\nwhich might host such exotic behavior. SeCuO 3studied in\nthis work is in many aspects similar to CdCu 2(BO 3)2and thus\nrepresents an ideal candidate to study site-selective quantum\nmagnetism, especially since, unlike for the latter, high quality\nsingle crystals are available.\nThe monoclinic SeCuO 3was recently proposed to be a\nhost to a quantum linear spin tetramer system described by\nthe Hamiltonian (1) where site-selective quantum correlations\nmight play a significant role in establishing unusual magnetic\nproperties [6]. SeCuO 3crystals belong to the monoclinic\nspace group P21=nwith unit cell parameters a=7:712 ˚A,\nb=8:238 ˚A,c=8:498 ˚A, and b=99:124\u000e[7]. Two crystallo-\ngraphically inequivalent copper sites, Cu1 and Cu2 are present\nin the monoclinic phase of SeCuO 3, each surrounded by\nsix oxygen ligands forming a Jahn-Teller distorted elongated\nCuO 6octahedra. This ligand configuration suggests dx2\u0000y2\norbital state for the unpaired copper spin S=1=2. Taking\ninto account the local environment of the magnetic ion Cu2+,\nit was proposed in Ref. 6 that isolated linear spin tetramers\nCu2-Cu1-Cu1-Cu2 are present in SeCuO 3. Two magnetically\ninequivalent spin tetramers in SeCuO 3are shown in Fig. 1.\nHowever, the temperature dependence of the magnetic sus-\nceptibility cannot be explained by a simple tetramer Hamilto-\nnian Eq. (1) [8], even if the gfactor temperature dependence\nis taken into account [9].\nPrevious studies of SeCuO 3proposed that the rotation of\nmacroscopic magnetic axes and the temperature variation of\nthe electron gtensor in the paramagnetic state [6, 9] are not\na consequence of the temperature change of the crystal struc-\nture, but rather of a site-selective correlation which emerges\nfrom the large difference between J11andJ12couplings [6].\nA recent nuclear quadrupole resonance (NQR) study proposed\nthat Cu1 spins are strongly coupled forming a spin singlet\nstate at temperatures T<J11\u0019200 K [10].\nBelow TN=8 K, SeCuO 3exhibits a long-range AFM order\n[6, 9–11]. Temperature dependence of the magnetic suscep-\ntibility anisotropy in low magnetic field, below TN, is typical\nfor uniaxial antiferromagnets [6, 9]. A spin-flop transition is\nobserved around HSF\u00191:8 T at T=2 K when the magnetic\nfield is applied along the easy axis [6]. Torque magnetometry\nreveals an unusual weak rotation of the magnetic axes with\ntemperature in the acplane [6, 9], while NMR measurements\ndemonstrate a possible rotation of the magnetic moments in\nthe AFM state [11] connected to different moments devel-\noped on the Cu1 and Cu2 sites and different coupling between\nthem. The high-field magnetization measurements showed the\nexistence of a half-step magnetization plateau [11] similar to\nwhat was found in the previously mentioned CdCu 2(BO 3)2\nJ11J12S1S2S3S4Cu1Cu1Cu2Cu2SaSbSScdFIG. 1. Two magnetically inequivalent tetramers in SeCuO 3as pro-\nposed in Ref. 6. Cu1 atoms are shown in orange colour, Cu2 in blue\nand O atoms in red. Se atoms are not shown for simplicity.\ncompound where the plateau emerges from the polarization of\nweakly coupled Cu2 spins while Cu1 dimers remain in the sin-\nglet state [2]. The77Se NMR measurements revealed different\ntemperature dependence of the Cu1 and Cu2 spin-spin relax-\nation rate 1 =T2in the AFM state which could be a signature of\nthe presence of two subsystems in SeCuO 3, strongly coupled\nCu1 dimers and weakly coupled Cu2 spins [11]. All these ob-\nservations confirm that SeCuO 3represents an ideal new host\nto study exotic magnetic behavior influenced by quantum phe-\nnomena.\nThe magnetic structure of SeCuO 3recently proposed\nfrom neutron powder diffraction measurements is highly non-\ncollinear with different magnetic moment values at Cu1 and\nCu2 sites, mCu1\u00190:4µBandmCu2\u00190:7µBat 1.5 K [10]. The\nsmaller value for Cu1 site, mCu1=0:35µB, was confirmed by\nNQR measurements, but with significantly different orienta-\ntion of magnetic moments with respect to the CuO 4plaquette\nthan observed in the neutron diffraction experiment [10]. Ob-\nviously, the AFM state in SeCuO 3is more interesting than a\nsimple uniaxial N ´eel state and calls for further studies. In this\nwork we experimentally probe the magnetic anisotropy of the\nAFM state in SeCuO 3using torque magnetometry measure-\nments in magnetic fields H.5 T significantly higher than\nthe spin-flop field HSF\u00191:8 T. This allows us to determine\nthe magnetocrystalline anisotropy energy (MAE) of SeCuO 3\nas well as study the field-induced spin reorientation. We com-\nplete the description of the MAE by a theoretical investigation\nbased on first-principles calculations.\nII. METHODS\nA. Experimental\nSingle crystals of monoclinic SeCuO 3have been grown by\na standard chemical vapor phase method, as described in lit-\nerature [6].\nThe magnetic torque was measured by a home-built torque\napparatus based on the torsion of a thin quartz fibre. The mag-\nnetic field was supplied by the Cryogenic Consultants 5 T\nsplit-coil superconducting magnet with a room-temperature3\nbore. The quartz sample holder is placed in a separate cryostat\nwhich is mounted in the room-temperature bore of the mag-\nnet cryostat. The monitoring and control of the sample tem-\nperature were performed by the Lakeshore 336 temperature\ncontroller. For magnetic torque measurements a single crystal\nof mass (246\u00068)µg was used with the baxis parallel to the\nlongest crystal axis and with the two crystal planes, (101)and\n(1 01), easily distinguishable.\nB. Theory\nThe present calculations are based on the spin-polarized\ndensity functional theory as implemented in the Wien2k pack-\nage [12] using a full potential linear augmented plane wave\nmethod. The Perdew-Burke-Ernzerhof approximation (PBE)\n[13] is considered for the exchange and correlation part. A\nHubbard effective term within the Anisimov approach [14]\nwas used to describe the Cu 3 dorbitals more properly, allow-\ning us to obtain magnetic moments for the copper sites close to\n0.73 µB. We also checked our calculations considering the on-\nsite PBE0 hybrid functional [15], giving similar observations.\nThe R MTMuffin-Tin radii for Se, Cu and O atoms were set to\n1.65, 1.96 and 1.49 bohr and the RK maxto 6. The separation\nbetween valence and core states was set to -6 Ry, except for\nthe calculations including zinc atoms (set to -8 Ry, with R MT\n= 1.96 bohr). The Brillouin zone sampling was done using a\n5\u00024\u00024 k-mesh [16].\nIII. RESULTS\nA. Experimental Results\nFor a simple collinear uniaxial antiferromagnet in low mag-\nnetic field H\u001cHSF, the magnetization is linearly dependent\non the magnetic field. Consequently (see Appendix), angular\ndependence of the measured component of magnetic torque tz\nis then described by the expression\ntz=t0sin(2j\u00002j0); (2)\nwhere amplitude t0is given by\nt0=m\n2MmolDcxyH2: (3)\nmis the mass of the sample, Mmolis the molar mass and H\nis the magnitude of applied magnetic field. Dcxy=cx\u0000cyis\nthe magnetic susceptibility anisotropy in the xyplane in which\nthe magnetic field rotates. xandyare, respectively, directions\nof the maximal and minimal susceptibility components in the\nplane of measurement. jis the goniometer angle and j0is\nthe angle at which the field is parallel to x. Eqs. (2) and (3)\nshow that, in the case of a linear response, the magnetic torque\nis proportional to H2andDcand the angular dependence of\ntorque is a sine curve with a period of 180\u000e. The previously\npublished low-field ( H.0:2 T) torque data [9] in both para-\nmagnetic and antiferromagnetic states are well described by\n100150200250300-2-10121\n00150200250300-1011\n00150200250300-4-202401 2232425201[-101]* \n \n1T    2T \n2.5T 3.5T \n4.5TTorque (dyn cm)A\nngle (deg)T = 4.2K[\n101](a)A\nngle (deg)T = 4.2K \n  \n1T \n2T \n2.5T \n3.5TTorque (dyn cm)b[ -101]*T\n = 4.2K \n  \n1T \n2T \n3T \n4Tτ 2Mmol/mH2 (10-4 emu/mol)A\nngle (deg)b[ 101]*(d)( c)(b) \nExperiment \nSimulation \nτ0 (dyn cm)H\n2 (T2)FIG. 2. Angular dependence of torque tmeasured in three crystal\nplanes in different magnetic fields. (a) The torque measured in the ac\nplane. (b) Full symbols: the dependence of torque amplitude t0on\nH2in the acplane [see (a)]. Solid line is fit to the experimental data\nforH>HSF. Dashed line represents expected H2dependence of the\ntorque amplitude for antiferromagnet with no reorientation (extrapo-\nlation of H\u001cHSFdata using Eq. (3) and low-field anisotropy data\n[9]). Empty symbols represent simulation (see Sec. III B). (c) An-\ngular dependence of torque measured in the plane spanned by band\n[1 0 1]\u0003axes, and (d) the plane spanned by band[1 0 1]\u0003axes. In (d)\nthe torque is multiplied by 2 Mmol=(mH2)resulting in practically the\nsame curve for all applied fields. This is consistent with no reorien-\ntation of spins in this plane [see Eq. (2)]. The angles corresponding\nto the specific crystal directions are pointed by arrows.\nEqs. (2) and (3).\nThe magnetic torque was measured in the AFM state at\nT=4:2 K by rotating the magnetic field in three crystal\nplanes: the acplane, the plane spanned by band[1 0 1]\u0003axes\nand the plane spanned by band[1 0 1]\u0003axes. Angular de-\npendence of torque for these three planes is shown in Fig. 2.\nWith the exception of the data shown in Fig. 2(d) for plane\nspanned by band[1 0 1]\u0003axes, the measured torque cannot be\ndescribed by Eq. (2). The torque curves in Figs. 2(a) and 2(c)\nare not regular sine curves and the amplitude of torque does\nnot increase linearly with H2, as can be seen for the acplane\nin Fig. 2(b). The deviation of the torque curves from Eq. (2)\nis most pronounced in the acplane for H=2 T which is close\nto the spin-flop field HSF\u00191:8 T, see Fig. 2(a).\nFor the plane spanned by band[1 0 1]\u0003axes we show\ntorque multiplied by 2 Mmol=(mH2)in Fig. 2(d). In this way\nthe torque measured in different fields gives practically the\nsame curve, which is expected from Eqs. (2) and (3) and im-\nplies that there is no spin reorientation in this plane. In this\nwork we show that the deviation from low-field behaviour ob-\nserved in the two other crystal planes is a result of the spin axis4\nreorientation induced by magnetic field comparable in magni-\ntude to the spin-flop field.\nIn Fig. 2(b) we plot the dependence of torque amplitude\nt0onH2measured in the acplane. The dashed line repre-\nsents the slope given by m=2MmolDcxy[see Eq. (3)] with\nDcobtained from the low-field measurements [9]. The de-\nviation from the low-field behavior is observed already for\nH&1:5 T. Interestingly, the H2dependence seems to be\nrestored for H\u00152 T but with a much smaller slope which\nwould, according to Eq. (3), correspond to a weaker suscepti-\nbility anisotropy Dc. This result is crucial for our interpreta-\ntion of the spin reorientation in SeCuO 3and in the following\nsections we will demonstrate how the observed behavior is a\ndirect consequence of the site-specific spin reorientation in-\nduced by the magnetic field, comparable to or larger than the\nspin-flop field.\nB. Phenomenological model of spin reorientation in SeCuO 3\nThe spin reorientation in a collinear antiferromagnet in a fi-\nnite applied magnetic field was first proposed by N ´eel in 1936\n[17, 18] who observed that the competition between the ori-\nentation of spins defined by the magnetocrystalline anisotropy\nenergy (MAE) and the orientation preferred by the Zeeman\nenergy results in a reorientation of spins in such a way to min-\nimize the total energy. The MAE determines the spin orien-\ntation in the absence of magnetic field (easy axis direction),\nwhile Zeeman energy for an antiferromagnet is minimal when\nspins are perpendicular to the applied magnetic field. De-\npending on the magnitude and direction of the applied field,\nspins will be oriented in such a way to minimize the total en-\nergy while still maintaining an almost collinear AFM struc-\nture due to the exchange energy which is much larger than the\nanisotropy energy. When magnetic field is applied along the\neasy axis, the spins reorient perpendicularly to magnetic field\nwhen the field reaches a critical value HSFcalled the spin-\nflopfield. The magnitude of the spin-flop field depends on the\nmagnitude of magnetocrystalline anisotropy energy, as well as\nthe magnetic susceptibility anisotropy [19]. Spin-flop transi-\ntions were observed in many antiferromagnets and studied in\ndetail in literature (see e.g. Ref. 20 and references therein). A\nsimple phenomenological approach proposed already by N ´eel\ncan be used to study field-induced spin reorientation in anti-\nferromagnets with different symmetries. Specifically, torque\nmagnetometry measurements can be employed to determine\nthe MAE shape and also to study the spin axis reorientation\nin a finite magnetic field. We have successfully used this ap-\nproach previously to study the magnetocrystalline anisotropy\nin a uniaxial antiferromagnet [21], but also in antiferromag-\nnets with higher symmetries and multiple antiferromagnetic\ndomains [22, 23].\nTo employ such a phenomonological approach to study the\nmacroscopic spin reorientation in collinear antiferromagnets\nwe need to know the magnetic susceptibility tensor, as well as\nthe MAE. In accordance with Neumann’s principle both the\nsusceptibility tensor and MAE of SeCuO 3must obey point\ngroup symmetry elements of P21=nspace group.The magnetic susceptibility tensor can be determined by\ntorque measurements in low magnetic field, as shown in Ap-\npendix. The temperature dependence of the magnetic suscep-\ntibility anisotropy of SeCuO 3in AFM state [6, 9] is typical\nfor an uniaxial antiferromagnet: susceptibility goes to zero as\ntemperature decreases from TNto zero when magnetic field\nis applied along the easy axis, and is almost temperature-\nindependent along the hard and the intermediate axes. Our\nprevious torque measurements in low magnetic field [9] have\nshown that in the AFM state below T\u00196 K the easy magnetic\naxis is along\n1 0 1\u000b\u0003, while the intermediate and hard axes are\nalongh1 0 1iandh0 1 0i=\u0006baxes, respectively. This allows\nus to write the magnetic susceptibility tensor of SeCuO 3in the\nAFM state at T.6 K as\nˆccc0=2\n4c[1 0 1]\u0003 0 0\n0c[1 0 1]0\n0 0 cb3\n5: (4)\nThe coordinate system spanned by the magnetic eigenaxes\nin the AFM state below \u00196 K is ([1 0 1]\u0003;[1 0 1];b). This\nis in accordance with the symmetry requirements which dic-\ntate that the baxis must be one of the magnetic eigenaxes,\nwhile the other two eigenaxes can have any orientation in the\nacplane [24]. One might note here that the measurement of\nsusceptibility tensor allows the detection of symmetry break-\ning. A signature of symmetry lowering in monoclinic SeCuO 3\nwould be observed if the baxis would no longer represent one\nof the magnetic eigenaxes, as explained in the Appendix. We\nhave performed detailed measurements to determine if there\nis such a symmetry breaking in SeCuO 3and, within the small\nerror induced by a possible misorientation of the sample, our\nresults show that symmetry is preserved in the AFM state of\nSeCuO 3, in agreement with the recent neutron powder diffrac-\ntion and NQR measurements [10]. Small rotation of the mag-\nnetic axes observed for 6 K .T.TN=8 K [6, 9] is confined\nto the acplane and is thus symmetry-preserving.\nNext, we write down the MAE, which also must satisfy the\npoint group symmetry operations for SeCuO 3, as follows\nFa(q;f) =K1cos2q+K2sin2qcos2f; (5)\nwhere qandfare polar and azimuthal angles in the spherical\ncoordinate system, and K1andK2are the anisotropy constants\nexpressed in units of erg/mol. In principle, the anisotropy en-\nergyFacan be written to higher-order terms and can also in-\nclude extrinsic contributions to the anisotropy such as a shape\nanisotropy. However, second order terms, written in Eq. (5),\nare sufficient to study the reorientation of the spin axis in a fi-\nnite magnetic field, as long as we do not try to describe critical\nbehaviour in magnetic fields very close to the spin-flop field\n[20]. Eq. (5) describes the amount of energy needed to rotate\nthe spin axis away from the easy axis direction. Depending\non the value and sign of the anisotropy constants (both can\nbe positive or negative), the anisotropy energy (5) can have\nseveral different shapes allowed by symmetry. Torque mag-\nnetometry is a technique which can be employed to experi-\nmentally determine the actual shape of the MAE for a specific\ncrystal. As already mentioned, our previous low-field torque5\nFIG. 3. The MAE shape in SeCuO 3obtained from the torque mea-\nsurements in this work. Red lines represent the magnetic axes which\nare also extrema of the anisotropy energy. Easy axis direction (a\nglobal minimum) is along the\n1 0 1\u000b\u0003axis.\nresults along with the results shown in Fig. 2 show that the\neasy axis direction in the AFM phase for T.6 K is along\nthe\n1 0 1\u000b\u0003direction. This is then the direction of the global\nminimum of the anisotropy energy (5). So, the other two ex-\ntrema must be along the h1 0 1iand\u0006bdirections. The torque\nresults presented in this work show that the global maximum\ni.e. hard axis is along the \u0006bdirection, while the interme-\ndiate axis is found along the h1 0 1idirection. This puts the\nfollowing constraints on the anisotropy constants: K2<0 and\nK2<K1. In Fig. 3 we plot the resulting magnetocrystalline\nanisotropy energy in SeCuO 3.\nIn a finite magnetic field, H(y;x), the Zeeman energy is\ngiven by\nFZ(y;x) =\u00001\n2H(y;x)\u0001ˆccc\u0001H(y;x) (6)\nwhere ˆcccis the magnetic susceptibility tensor of the sample\nexpressed in the same coordinate system as the MAE (5), and\nyandxare polar and azimuthal angles in the spherical coor-\ndinate system representing the direction of the magnetic field\nH=H(cosxsiny;sinxsiny;cosy). In very low magnetic\nfield the susceptibility tensor ˆcccis given by (4). In finite mag-\nnetic field, the spin axis will in general start to rotate away\nfrom the direction of the easy axis. We describe this rotation\nby allowing the susceptibility tensor to rotate\nˆccc(q;f) =R(q;f)\u0001ˆccc0\u0001RT(q;f) (7)\nwhere ˆccc0is the low-field ( H\u001cHSF) susceptibility tensor\ngiven by the expression (4) and R(q;f)is the rotation matrix.\nOur torque measurements were performed at T=4:2 K where\nc[1 0 1]\u0003=4\u000110\u00004emu/mol, c[1 0 1]=3:5\u000110\u00003emu/mol and\ncb=3:8\u000110\u00003emu/mol are the eigenvalues of the susceptibil-\nity tensor obtained from our previous susceptibility and torquemeasurements [6, 9] .\nThe total phenomenological energy Ftotof the sample in\nfinite magnetic field is the sum of MAE and Zeeman energy\nFtot(q;f;y;x) =Fa(q;f)+FZ(y;x;q;f): (8)\nFor an anisotropic antiferromagnetic sample placed in finite\nmagnetic field H(y;x), the new direction of the spin axis\nis obtained numerically by minimizing the total energy Ftot\nwith respect to qandf. To simulate the experimental re-\nsults we start by finding K1andK2values which satisfy the\nabove mentioned requirements for the MAE extrema. A cor-\nrect choice of values must reproduce the experimental HSF\nvalue of\u00191:8 T at T=2 K when the magnetic field is applied\nalong the easy axis. In the present case, the spin-flop field\nis given by HSF= [2(K1\u0000K2)=(c[101]\u0000c[101]\u0003)]1=2[20, 25].\nBy simulating magnetic field dependence of magnetization to\nreproduce the experimental HSFvalue we can only pinpoint\nthe difference K1\u0000K2. The values of K1andK2will thus\nbe obtained from the simulation of the angular dependence of\ntorque data.\nWe proceed by simulating the field dependence of magne-\ntization for magnetic field applied along the easy axis direc-\ntion at T=2 K and by comparing this result to the exper-\nimental data. Setting K2\u0000K1=\u00006:35\u0001105erg/mol gives\nHSF=1:87 T for values of the susceptibility tensor measured\natT=2 K [9]. We fix our spin-flop field to this value since it\nis well within the margin of error for experimental data, and\nit also reproduces well our other results. We apply magnetic\nfield H(y0;x0)along the easy axis by setting y0=p=2,x0=0\nin the chosen coordinate system and then minimize numeri-\ncally the total energy (8) for each value of the applied field\nto obtain q0andf0. Using the values obtained in this man-\nner we calculate a rotated susceptibility tensor (7) and finally\nthe magnetization from M=ˆccc(q0;f0)\u0001H(y0;x0). In order to\nsimulate the dc magnetization measurements we only take a\ncomponent of the magnetization along the applied magnetic\nfield. The result of our calculation, shown by solid black line\nin Fig. 4 (simulation 1), is compared to the measured values\nfrom Ref. 6 represented by empty blue squares in Fig. 4. The\nspin-flop transition is clearly observed at HSF=1:87 T in the\ncalculated curve.\nOur simulation also gives a new direction of the spin axis\nwhich for H\u0015HSFrotate from the easy axis direction to the\nintermediate axis direction h101i, as shown in inset of Fig. 4.\nHowever, for H>HSF, the calculated magnetization (solid\nblack line in Fig. 4) is somewhat larger than the measured one\n(empty blue squares) and in fact falls on values obtained for\nmagnetization measured along the intermediate and the hard\naxes (empty green triangles and red circles in Fig. 4). We\nshould mention here that our result agrees with behaviour that\nis usually obtained in the experimental spin-flop. Significantly\nsmaller magnetization observed in the experiment is in fact\nanomalous and suggests that the spin reorientation in SeCuO 3\nmight be unconventional. We will return to this point later.\nExperimental value of the spin-flop field is the same at 4.2 K\nas at 2 K, within the experimental uncertainty [6]. Since sus-\nceptibility tensor components are slightly different at 4.2 K,\nwe need to take K2\u0000K1=\u00005:42\u0001105erg/mol in order to re-6\n0\n1\n2\n3\n4\n5\n0\n50\n100\n150\n200\n250\n300\neasy axis\nintermediate axis\nbaxis\nsimulation 1\nsimulation 2a\nsimulation 2b\nM(emu Oe / mol)\nH(T)\nT= 2 K\nFIG. 4. A simulation of the magnetization dependence on magnetic\nfield when His parallel to the easy axis compared to the experi-\nmental result at T=2 K published in Ref. 6 (empty blue squares).\nInset: The dependence of the spin axis direction on applied magnetic\nfield obtained from simulation. Simulation 1 allows all spins to ro-\ntate simultaneously. Simulation 2 allows only fraction of the spins\nto rotate, as described in the main text. Solid red line (2a) represents\nresult for Hkeasy axis, while dashed line (2b) simulates a misorien-\ntation of the sample by 10\u000e, to mimic probable misorientation in our\nexperiment.\nproduce HSF\u00191:87 T at T=4:2 K.\nFinally, we proceed with calculating the angular depen-\ndence of torque for all planes probed during the measure-\nments. We simulate a rotation of magnetic field by appro-\npriately changing yandxforH(y;x). By minimizing the\ntotal energy (8) we obtain the rotated susceptibility tensor\nfrom which we calculate magnetization Mexpressed in emu\nOe/mol, as described above. Then we calculate torque from\nt=m=MmolM\u0002H, and plot only the component perpendic-\nular to the plane of rotation of magnetic field, since only that\ncomponent is measured in experiment.\nThe results obtained for the acplane shown by dashed lines\nare compared to the measured torque curves in Fig. 5. For\ntheacplane, our simulations agree very well with measure-\nments for H\u00142 T. However, at higher magnetic fields the\ncalculated curves have smaller amplitude than the measured\ncurves, and the discrepancy increases as the field increases.\nIn order to understand the discrepancy between the calculated\nand the measured amplitudes t0, we compare measurements\nwith the calculated H2dependence of t0in the acplane in\nFig. 2(b). The measured amplitude increases nonlinearly with\nH2in low magnetic field, and for H>2 T the increase be-\ncomes linear with H2, as we already pointed out in Section\nIII A. In comparison, the calculated torque amplitude follows\nthe measured one below H.2 T and then it becomes con-\nstant for H\u0015HSF[empty squares in Fig. 2(b)]. Independence\nof the torque amplitude on magnetic field for H\u0015HSFob-\ntained from our calculation is also observed in torque experi-\nment for conventional spin reorientation in uniaxial collinear\nantiferromagnet [25, 26] and is in agreement with the results\n100\n150\n200\n250\n300\n-0.2\n0.0\n0.2\n100\n150\n200\n250\n300\n-1.0\n-0.5\n0.0\n0.5\n1.0\n100\n150\n200\n250\n300\n-1.0\n-0.5\n0.0\n0.5\n1.0\n100\n150\n200\n250\n-1.0\n0.0\n1.0\n100\n150\n200\n250\n-1.0\n0.0\n1.0\n1T\n(a)\n(b)\nAngle (deg)\nTorque (dyn cm)\nT= 4.2K\n[101]\n[-101]*\n2T\n(a)\n(b)\nTorque (dyn cm)\nAngle (deg)\n2.5T\n(a)\n(b)\nAngle (deg)\nTorque (dyn cm)\n3.5T\n(a)\n(b)\nAngle (deg)\nTorque (dyn cm)\n4.5T\n(a)\n(b)\nTorque (dyn cm)\nAngle (deg)\nFIG. 5. The torque measured in the acplane in different magnetic\nfields compared to two different simulations, as described in the main\ntext. (a) A simulation with reorientation of all spins. (b) A simula-\ntion allowing the site-specific reorientation. Bottom right panel: the\nangle-dependent reorientation of the spin axis obtained from the sim-\nulation shown in laboratory coordinate system. A plane of rotation\nof the magnetic field is shown as dark square in the accompanying\ncoordinate system. The angle is measured with respect to the baxis,\nwhile in other panels a goniometer angle is shown.\nof N´eel [17, 18]. Our experimental result points to a more ex-\notic spin reorientation in SeCuO 3. Thus, in the remainder of\nthis section we consider possible reasons for the deviation of\nthe measured data from the results expected for a conventional\nuniaxial collinear antiferromagnet.\nThe deviation of our experimental curves from the cal-\nculated ones could be a consequence of the lowering of the\nsymmetry. This symmetry lowering would result in the MAE\nshape different form the one shown in Fig. 3 and consequently\ncould influence the measured angular dependencies of torque.\nHowever, the lowering of the symmetry can be disregarded,\nas we explained earlier.\nAnother possibility for the observed discrepancy is a mis-\norientation of the sample. A simulation of misorientation of\nthe sample showed that the increase in torque amplitude mim-\nicking the measured one can be reproduced only for an un-\nrealistic assumption of misorientations of \u001950\u000eor so, which\nwere certainly not realized in experiment.\nHaving discarded the symmetry lowering and the misori-\nentation as possible reasons for discrepancy of the measured\nand the simulated data, we turn again to the torque ampli-\ntude dependence on H2shown with full circles in Fig. 2(b).\nAccording to Eq. (3), we expect the amplitude to increase7\nlinearly with H2when there is no spin reorientation, i.e. in\nan antiferromagnet for H\u001cHSF. If all the spins in the\nsample would remain oriented along the easy axis in ap-\nplied field, the torque amplitude would increase with H2by\nfollowing the dashed line in Fig. 2(b) which represents the\nlow-field behavior. Instead, for H\u0015HSF, the data can be\nfitted to t0=a+b\u0001H2. The result of the fit, shown by\nsolid line in Fig. 2(b), gives a= (0:47\u00060:1)dyn cm and\nb= (0:0539\u00060:0009)dyn cm /T2. Using Eq. (3), from the\nfitting constant bwe obtain Dc= (8:4\u00060:1)\u000110\u00004emu/mol.\nThis value is significantly smaller than the total susceptibility\nanisotropy measured in the acplane at 4.2 K in low magnetic\nfield, Dcac=3:1\u000110\u00003emu/mol.\nThe analysis given above can be interpreted as if a frac-\ntion of the spins do not reorient in applied magnetic field, but\npreserve the low-field behaviour described by Eq. (2). This\nallows us to attempt to obtain better agreement between the\nexperiment and the simulation by assuming reorientation of\nonly a fraction of the spins. We start by dividing the suscepti-\nbility tensor in two parts,\nˆccc=ˆccc1+ˆccc2: (9)\nIn principle, one can expect different tensors ˆccc1andˆccc2for two\nsubsystems. However, in experiment we can only measure the\nmacroscopic total tensor ˆccc. So, we try to simulate our data\nby making a simple reasonable assumption: both subsystems\nparticipate in the long range AFM order, and thus their tensors\nare described by the measured tensor (4). However, following\nthe result of Ref. [10] which claims the magnetic moments\non different Cu sites are different, we allow different weights\nforˆccc1andˆccc2. Consequently, we write for tensors of separate\nsubsystems\nˆccc1=nˆccc0;\nˆccc2= (1\u0000n)ˆccc0; (10)\nwhere ˆccc0is given by Eq. (4). The expression (10) is written\nunder assumption that both tensors share magnetic eigenaxes.\nThe parameter ndescribes a contribution of tensor ˆccc1to the\ntotal tensor ˆccc0, i.e. a contribution of induced magnetization\nof subsystem 1 to the total magnetization M=M1+M2=\n(ˆccc1+ˆccc2)\u0001H. In the following we choose ˆccc1as the suscepti-\nbility tensor of the spins that do not reorient in finite magnetic\nfield and ˆccc2as the tensor which is allowed to rotate in an ap-\nplied magnetic field. Rotation of the tensor ˆccc2is described by\nEq. (7) where instead of ˆccc0we put ˆccc2from expression (10).\nTo fix the value of nwe calculate again the dependence of\nmagnetization on magnetic field and compare it to the mea-\nsured values in Fig. 2. Since we assume that only spins of the\nsubsystem 2 reorient, now HSF= [2(K1\u0000K2)=Dc2]1=2, where\nDc2represents the susceptibility anisotropy only for subsys-\ntem 2 [see Eq. (10)]. To reproduce HSF=1:87 T at T=2 K\nwe set K2\u0000K1=\u00004:95\u0001105erg/mol and n=0:22. The result\nof simulation for these parameters is shown by red solid line\n(simulation 2a) in Fig. 4. The spin-flop transition is sharp, as\nexpected for perfect orientation of the sample. Furthermore,\nour calculation now reproduces measured magnetization val-\nues in all applied fields. We expect some misorientation in theexperiment (less than 10\u000e). Taking this into account we ob-\ntain the result shown by the dashed red curve (simulation 2b)\nin Fig. 4. The agreement between the experimental and the\ncalculated data is excellent.\nWe now proceed to simulate the angular dependence of\ntorque measured at T=4:2 K assuming reorientation of only\nfraction of the spins. To reproduce HSF=1:87 T we set\nK2\u0000K1=\u00004:23\u0001105erg/mol. We also set n=0:22 as ob-\ntained above and proceed our calculation with the susceptibil-\nity tensor values measured at T=4:2 K [6, 9]. Furthermore,\nnow it is possible to pinpoint the values of K1andK2to ob-\ntain the best agreement with the measured torque curves. The\nabove mentioned constraints on anisotropy constants leave\ntwo choices for the sign of K1:K1>0 results in an easy\naxis kind of anisotropy, while K1<0 results in an anisotropy\nshape that is closer to the easy plane kind of anisotropy. Our\ndata could be well described only by assuming K1<0. From\nour simulations we find that measured torque for all orienta-\ntions is best described by the following values of anisotropy\nconstants: K1=\u00000:6\u0001105erg/mol and K2=\u00004:83\u0001105at\n4.2 K. The magnetocrystalline anisotropy energy with those\nanisotropy constants is shown in Fig. 3. The result for the ac\nplane is shown by solid lines in Fig. 5. The agreement with\nexperiment is now quite remarkable.\nIn Fig. 6 we compare the torque measurements for field ro-\ntating in the plane spanned by band[1 0 1]\u0003axes to the simu-\nlation in the plane spanned by easy and hard axis. Sharp tran-\nsitions are expected in case of a perfect orientation (dashed\ncurves) while a simulation of small misorientation (solid\nlines) gives a better agreement with the experiment. In both\nFig. 5 and Fig. 6, bottom right panel, we plot the simulated re-\norientation of the spin axis corresponding to the torque curves\nmeasured in different fields. Finally, in Fig. 7 we see excellent\nagreement between the result of measurement for the plane\nspanned by band[1 0 1]\u0003axes and the one obtained by sim-\nulation. However, in this plane there is no reorientation in\napplied field for obtainable values of magnetic field, so results\nin this plane are not sensitive to whether we allow all spins to\nrotate or just a fraction of them.\nC. Magnetic properties estimated from Density Functional\nTheory\nIn parallel with our experimental investigations of the mag-\nnetic anisotropy of SeCuO 3, we have carried out density func-\ntional theory (DFT) calculations including spin-orbit coupling\n(SOC), using a similar strategy to the one we considered for\nthe low-dimensional magnetic compound CuO [27]. Indeed,\nwe demonstrated that the estimation of the MAE of CuO, con-\nsidering its antiferromagnetic ground state, allows to properly\npredict its easy axis of magnetization. In contrast to CuO,\nspin fluctuations seem to play a major role in SeCuO 3mag-\nnetic properties. ˇZivkovi ´cet al. have already pointed out the\npossibility to observe these quantum fluctuations in SeCuO 3,\nand evoked a possible difference of the magnetic moment val-\nues between the Cu1 and Cu2 sites [6]. More recently, Lee\net al. mentioned that the site-specific spin correlation may be8\n100\n150\n200\n250\n300\n-0.2\n-0.1\n0.0\n0.1\n0.2\n100\n150\n200\n250\n300\n-1.0\n-0.5\n0.0\n0.5\n1.0\n100\n150\n200\n250\n300\n-1.0\n0.0\n1.0\n100\n150\n200\n250\n300\n-2.0\n-1.0\n0.0\n1.0\n2.0\n100\n150\n200\n250\n300\n-2.0\n-1.0\n0.0\n1.0\n2.0\nAngle (deg)\n1T\n2a\n2b\nTorque (dyn cm)\nT= 4.2K\nAngle (deg)\n2T\n2a\n2b\nTorque (dyn cm)\nAngle (deg)\n2.5T\n2a\n2b\nTorque (dyn cm)\n3T\n2a\n2b\nTorque (dyn cm)\nAngle (deg)\n3.5T\n2a\n2b\nTorque (dyn cm)\nAngle (deg)\n[-101]*\nb\nFIG. 6. The torque measured in the plane spanned by band[1 0 1]\u0003\naxes compared to the results of simulation, as described in the main\ntext. Dotted lines represent a perfect orientation (2a), and solid lines\nsimulate a slightly misoriented sample (2b). Only site-specific simu-\nlation is shown. Bottom right panel: angle-dependent spin axis reori-\nentation for perfect orientation of the sample shown in the laboratory\ncoordinate system. The plane of rotation of the magnetic field is\nshown as a dark square in the accompanying coordinate system. The\nangle is measured with respect to the baxis, while in other panels\ngoniometer angle is shown.\nexplained by considering two subsystems based on strongly\ncoupled Cu1 dimers and weakly interacting Cu2 spins [11].\nThey conclude that such a scheme will lead to smaller or-\ndered magnetic moments for Cu1 than for Cu2 due to singlet\nfluctuations. Also, the reduced ordered magnetic moment of\nCu1 sites has been confirmed based on the neutron powder\ndiffraction and NQR measurements [10]. However, none of\nthe magnetic models proposed so far allows to explain all the\nexperimental measurements. It is thus essential to provide a\ntheoretical basis to clarify the present picture.\nOur previous investigation, based on magnetic susceptibil-\nity measurements, leads to the conclusion that the Cu2-Cu1-\nCu1-Cu2 tetramer is based on two antiferromagnetic cou-\nplings, namely J 11= 225 K and J 12= 160 K [6]. Thus, we\nhave generated an antiferromagnetic order noted AF 1shown\nin Fig. 8(a) respecting these conditions, which has been used\nto estimate the MAE of SeCuO 3with the Wien2k code.\nSOC is included as a perturbation of the antiferromagnetic\ncollinear state, leading to an energy lowering given by\nDESOC=\f\f\fD\nijbHSOCjjE\f\f\f2\n\f\fei\u0000ej\f\f(11)\n100150200250300-0.040.000.041\n00150200250300-0.2-0.10.00.10.21\n00150200250300-0.4-0.20.00.20.41\n00150200250300-0.40.00.4Angle (deg) \n  \n1T \nsim.Torque (dyn cm)T\n = 4.2Kb\n[ 101]*Angle (deg) \n  \n2T \nsim.Torque (dyn cm)A\nngle (deg) \n  \n3T \nsim.Torque (dyn cm) \n  \n4T \nsim.Torque (dyn cm)A\nngle (deg)FIG. 7. The torque measured in the plane spanned by band[1 0 1]\u0003\naxes compared to the results of the site-specific simulation (sim.), as\ndescribed in the main text.\nwhich accounts for an interaction between an occupied state\niwith an energy eiand an unoccupied state jwith an energy\nejvia the matrix elementD\nijbHSOCjjE\n. The resulting MAE is\nrepresented in Fig. 9(b), showing an uniaxial anisotropy along\nthebdirection, while hard axis is in the acplane. A similar\nresult is observed considering the on-site PBE0 hybrid func-\ntional. To be more quantitative, Table I gathered the MAE\nvalues for the magnetic eigenaxes of the AFM state (below\nT=6 K), deduced from torque measurements and highlighted\nin Fig. 3. Given values are expressed relatively to the MAE in\nthe[0 1 0]crystal direction.\nFirst of all, we have tested two different functionals, i.e.\nGGA+U and on-site PBE0 hybrid. It appears that GGA+U\nwith Ue f f= 5 eV leads to similar MAE values to the ones\nobtained with PBE0. The difference between the easy axis\n(along b) and the two others directions is about 10 µeV/f.u. If\nwe consider a larger correction ( Ue f f= 9 eV) as in Ref. 28\nfor SrCu 2(BO 3)2, the MAE values are reduced by a factor of\ntwo, but the trend is conserved, i.e. bis still predicted to be\nTABLE I. MAE values ( µeV /f.u.) for the magnetic eigenaxes of the\nAFM state, obtained with PBE0, GGA+U with Ue f f= 9 and 5 eV\n(noted 9 and 5, respectively) or by substituting zinc for copper ( noted\nZn). Directions\u0002\n1 0 1\u0003\u0003and[1 0 1]represent easy and intermediate\naxes obtained from experiment.\nUCu1\ne f f=UCu2\ne f f[1 0 1]\u0002\n1 0 1\u0003\u0003\n9 / 9 4 5\n9 / Zn -2 -1\nZn / 9 5 9\n5 / 5 9 9\n5 / Zn -4 -2\nZn / 5 11 19\nPBE0 / PBE0 11 149\nFIG. 8. (a) Antiferromagnetic order considered in our DFT calcula-\ntions. The Cu2+sites are depicted as filled and empty circles, rep-\nresenting up-spin and down-spin, respectively. (b) Orbital moment\nof one tetramer. Red and blue colors show the maximum and mini-\nmum of the orbital moment, respectively. The arrow on the right side\nshows the vector sum of orbital moments represented by brown and\nblue vectors on Cu1 and Cu2 sites, respectively.\nFIG. 9. (a) Experimental MAE, (b) Contributions MAE1 (MAE2) to\nthe total MAE (top) determined substituting Cu2 (Cu1) by Zn atoms\nand considering an Ue f f= 5 eV correction for Cu sites.\nthe easy axis in disagreement with the experimental facts. To\nunderstand this discrepancy, we first consider the Bruno re-\nlation [29]. According to this model, which is based on the\nSOC perturbation expression of Eq. (11) and ignoring spin-\nflip terms, the MAE is directly proportional to the orbital mo-\nment anisotropy\nMAE =Ehard\u0000Eeasy=x\n4\f\f\fhLzihard\u0000hLzieasy\f\f\f (12)\nwherehLziis the orbital angular momentum. The hLziterm in\nEq. (12) is shown in Fig. 8(b) for the four copper sites of one\ntetramer. For clarity, arrows highlight the direction for which\nthe orbital momentum is maximum, pointing along the normal\nof each CuO 4plaquette. Summing the orbital moment of all\ncopper sites leads to a total orbital moment which is maximal\nalong the bdirection, i.e. an easy magnetization axis is along\nb. Thus, both arguments based on orbital moments and total\nenergy lead to the same conclusion, i.e. an easy axis along\nthebdirection, in contradiction to the experimental data. It\nshould be noticed that as expected for a spin-half system, the\ndipolar contribution is negligible, less than 0.7 µeV/f.u.\nIn addition, the present DFT calculations cannot repro-\nduce another experimental fact which is the different mag-\nnetic moments at Cu1 and Cu2 sites, 0.46 and 0.73 µB, re-spectively, from NPD [10]. Indeed, DFT gives similar mag-\nnetic moments for Cu1 and Cu2, i.e. 0.84 and 0.75 µBcon-\nsidering Ue f f= 9 and 5 eV , respectively. While Ue f f= 5 eV\nallows to properly describe the magnetic moment of Cu2, it\ncannot explain reduced value obtained from experiment for\nthe Cu1. Such a feature appears to be related to the fact that,\nas in CdCu 2(BO 3)2, Cu1 ions form strongly coupled singlets,\nwhich are polarized by the staggered field of Cu2 spins, and\nCu1 and Cu2 magnetic subnetworks are decoupled.\nWe have then estimated the contribution of each inequiva-\nlent copper site to the MAE. MAE of Cu1 sites (noted MAE1)\nwas calculated by substituting zinc for copper on all Cu2 sites,\nand MAE of Cu2 sites (noted MAE2) reversely [30]. Indeed,\nZn2+and Cu2+cations share nearly the same radii, 0.74 and\n0.71 ˚A, respectively. In addition, Zn2+is non magnetic (d10\nelectronic configuration) allowing suppression of magnetic re-\nsponse of Zn-substituted sites. A representation of these par-\ntial MAE is given in Fig. 9(b). In particular, the easy magne-\ntization axis is located in the acplane and along the bdirec-\ntion for MAE1 and MAE2, respectively. However, MAE2 is\nlarger in amplitude than MAE1, leading to a total contribution\nto MAE dictated by MAE2, i.e. an easy axis along the b.\nOur calculations demonstrate the predominant role of the\nCu2 subnetwork in the magnetic anisotropy of SeCuO 3, but\nremain incomplete because we do not reproduce the magnetic\nmoment reduction of Cu1. As already mentioned above, such\na feature is a consequence of the formation of Cu1 dimers\nwhich are in a singlet state at low temperature ( T<200 K),\nthe spin fluctuations of Cu1 spins which are different from\nthe ones of Cu2, leading to the decoupling of the two sub-\nnetworks, and the staggered field of Cu2 subnetwork which\npolarizes the magnetic moments of Cu1, leading to a strong\ndecrease of its value [5].\nFrom our point of view, all these experimental data con-\nverge to one model for SeCuO 3, consisting of nearly isolated\nCu1 dimers immersed in the staggered field of the AFM long\nrange order of the Cu2 subnetwork. One simple approach is\nto reduce the Hubbard correction on Cu1 site, and indeed this\nleads to decrease of Cu1 magnetic moments from 0.84 to 0.75\nand 0.60 µB, with Ue f f= 9, 5 and 0 eV , respectively. Reduc-\ningUe f feven more to negative values will lead to entering an\nattractive electron-electron interaction regime. Such attractive\nHubbard model has been previously used as an effective de-\nscription for systems involving strong electron-phonon cou-\npling [31]. Indeed, strong spin-lattice coupling in the low-\ntemperature state of SeCuO 3has been proposed by Lee et al. ,\nbased on the measurement of the nuclear spin-lattice relax-\nation rate 1/T 1[11].\nHere, the idea is to simulate an extreme situation for which\nthe electron-phonon coupling involving the Cu1 dimers would\nbe enough to overcome the electron-electron Coulomb repul-\nsion. It will correspond to the observation of attractive and\nrepulsive regimes on low and high energy scales, respectively\n[31]. Interestingly, calculating the MAE with two sizable dif-\nferent treatments for Cu1 and Cu2 subnetworks reproduces\nthe experimental observation. More specifically, we have used\nUe f f= 0 eV for Cu1 and Ue f f= 5 eV for Cu2. This choice\nallows us to reproduce in an effective manner the magnetic10\nTABLE II. MAE values ( µeV/f.u.) for the magnetic eigenaxes of the\nAFM state obtained with different GGA+U treatments for copper\natoms. We report values obtained for Ue f f= 5 eV and U\u0000J=\n(5\u00000:5)eV . Results are given with respect to the MAE along the b\naxis direction.\nUe f f U\u0000J\nCu1 / Cu2 5 / 5 0 / 5 5-0.5 / 5-0.5 0 / 5-0.5\n[1 0 1] 9 -10 -7 -26\u0002\n1 0 1\u0003\u00039 -7 -20 -32\nmoment of Cu2 and the reduction of magnetic moments of\nCu1. Such treatment leads to a 3D shape shown in Fig. 10(b),\nfor which the bdirection is properly found as being the hard\naxis and the easy axis lying in the acplane. To be more quan-\ntitative, we compared in Table II the MAE values for the mag-\nnetic eigenaxes of the AFM state with respect to the MAE\nvalue along the baxis direction. It now appears that, among\nthese three directions, [0 1 0]is systematically the hard one,\nand[1 0 1]the easy one. In other words, by considering that\nCu1 and Cu2 subnetworks are decoupled and by taking into\naccount the reduction of the magnetic moment of Cu1, we are\nable to reproduce the experimental hard magnetization axis\nalong the bdirection. The easy axis is found to be in the ac\nplane in agreement with experimental refinements, but still\nnot in the\u0002\n101\u0003\u0003direction determined from experiment (see\nFig. 10).\nAt this stage, it should be mentioned that Bousquet\net al. [32] have demonstrated that defining explicitely the\nexchange-correction parameter J, in LSDA+U treatment,\nstrongly affects the non-collinear magnetic ground state, and\nmore specifically the spin canting and the magnetocrystalline\nanisotropy shape. This constitutes a really delicate issue be-\ncause it implies an adjustment of the amplitude of two pa-\nrameters, UandJ. Our results for U\u0000J=5\u00000:5 eV are\nsummarized in Table II. It should be noticed that a similar\ntrend of values is obtained using J=1 eV , confirming that the\nmore important aspect is to explicitly specify the Jvalue. In-\nterestingly, the experimental MAE eigenaxes are properly de-\nscribed as soon as Jis explicitly defined [see Fig. 10(c)]. More\nspecifically, when both Cu1 and Cu2 are corrected, MAE\nvalues are 0, -7 and -20 µeV .fu\u00001for the [0 1 0],[1 0 1]and\u0002\n1 0 1\u0003\u0003directions, respectively. When the correction is added\nonly on Cu2, the MAE values are 0, -26 and -32 µeV .fu\u00001for\nthe identical respective directions. Such results demonstrate a\ndrastic change of the MAE, mainly on the intermediate eige-\nnaxes. In such a case, both explicit Jdefinition on Cu2 and\nreduced Hubbard correction on Cu1 are necessary to properly\norientate the theoretical easy axis along the experimental one,\nas represented in Fig. 10(c). In order to verify how the MAE2\nbehaves with such treatment, we redo a chemical substitution\nby Zn atoms on Cu1 sites. As expected, the MAE is strongly\nmodified with respect to the one determined using an Ue f f\ntreatment [Fig. 10(b)], i.e. with an easy, intermediate and hard\naxes in really good agreement with the experimental ones, as\ncan be witnessed from Figs. 10(a) and 10(c). It should be no-\nticed that the overall shape of the MAE was unchanged when\nFIG. 10. (a) Experimental MAE, (b) MAE considering Ue f f= 0 eV\nfor Cu1 and Ue f f= 5 eV for Cu2, (c) similar as previously, except a\nU - J = 5 - 0.5 eV correction on Cu2.\nconsidering J = 0.5 and 1 eV for U = 5 eV .\nIV . DISCUSSION\nTorque magnetometry is a convenient method for studying\nmagnetocrystalline anisotropy and spin reorientation phenom-\nena in finite magnetic field since angular dependence of torque\nis very sensitive to the orientation of the spin axis. We em-\nployed this method to determine the MAE shape in antiferro-\nmagnetic state of SeCuO 3. Previous magnetic susceptibility\nresults showed that, below TN, SeCuO 3might be considered\na conventional collinear uniaxial antiferromagnet since sus-\nceptibility measured along one of the axis goes practically to\nzero as T!0 (easy axis), while along other two axes conly\nslightly increases as the temperature decreases [6]. This be-\nhaviour is typical for uniaxial collinear antiferromagnets [18],\nalthough it does not disqualify a very weak canting of spins.\nThe magnetization measurement at 2 K revealed a spin-flop\ntransition in field of HSF\u00191:8 T applied along the easy axis,\nalso a feature of antiferromagnets with weak magnetocrys-\ntalline anisotropy [17, 18].\nUsing a symmetry allowed MAE and experimentally deter-\nmined susceptibility tensor we calculated the magnetization as\na function of applied field and angular dependence of torque\nunder assumption that all spins collectively rotate in finite\nmagnetic field. This produced results which are expected for\nuniaxial antiferromagnet, [18, 25, 26], but which only qual-\nitatively agree with experiment. In H>HSFthe magnitude\nof magnetization should be equal to the value obtained when\nthe field is applied along the intermediate axis, while the ex-\nperimental value was significantly smaller (see Fig. 4). Also,\ninH>HSFthe torque amplitude is expected to become in-\ndependent on applied magnetic field [25, 26], while in exper-\niment the amplitude showed H2dependence, but with much\nsmaller slope than in low magnetic field. The H2dependence\nof the torque amplitude is characteristic for an AFM system\nin magnetic field H\u001cHSF, in which there is no reorientation\nof spins. This led us to perform simulations where we distin-\nguished two separate subsystems, each represented by its own11\nsusceptibility tensor, ˆccc1andˆccc2. The tensor of one subsystem,\nˆccc1, remains fixed in applied magnetic field, while the tensor\nof the other subsystem, ˆccc2, is allowed to rotate. In this way we\nobtained quantitative agreement of our simulations with both\nthe magnetization and the torque experiment (see Figs. 4 - 7).\nA possibility of the existence of two subsystems was al-\nready mentioned in Ref. 6 where it was suggested that the cor-\nrelations between Cu1 and Cu2 in SeCuO 3are site-selective\nand strong coupling between Cu1 spins might form a singlet\nstate at higher temperatures thus separating the Cu1 from the\nCu2 spin sublattice. The scenario of two subsystems made\nof the strongly coupled Cu1 dimers and the weakly coupled\nCu2 spins in the AFM state was recently proposed from NMR\nmeasurements which witnessed different temperature evolu-\ntion of 1 =T2assigned to Cu1 and Cu2 spins in the AFM state\n[11]. The NQR measurements showed that Cu1 dimers indeed\nform singlets already at high temperatures T<200 K, while\nCu2 spins are only weakly coupled to the central pair [10].\nThese results allow us to construct a more rigorous model\nof the spin reorientation in SeCuO 3. Previously published\nsusceptibility anisotropy in the AFM state strongly supports\na picture of collinear or very weakly canted AFM state in zero\nmagnetic field [6, 9]. From our macroscopic measurements\nwe are not able to discern the canting so we proceed with the\ncollinear picture. The magnetically ordered state which com-\nplies with our torque measurements, both previous [9] and\nfrom this work, is shown in Fig. 11(a) and 11(c). Our the-\noretical investigations based on a similar magnetic ordering\ndo not allow us to obtain a better picture of this magnetic or-\ndering. Moreover, actual magnetic structure cannot be deter-\nmined from our macroscopic experiment, only orientation of\ncollinear spins with respect to the crystal axes. To propose a\nspecific orientation of the spins shown in Fig. 11(a) and 11(c),\nwe rely on symmetry elements as well as results from liter-\nature which allow us to assume AFM coupling between Cu1\nspins [10] and AFM coupling between Cu1 and Cu2 spins on\nthe tetramer [6]. In zero field all spins are oriented along the\n1 0 1\u000b\u0003direction (Ref. [9] and this work). This result is in\ndisagreement with the neutron powder diffraction data which\npropose a very noncollinear magnetic structure [10]. We point\nout here that the structure from Ref. 10 is in disagreement\nwith the measured magnetic susceptibility anisotropy [6, 9]\nwhich only allows for collinear or very weakly canted spins,\nas we mentioned above. A large canting proposed in Ref. 10\nwould not produce a magnetic susceptibility measured along\neasy axis which goes to zero as T!0 [6], a typical feature of\ncollinear or only very slightly canted antiferromagnet.\nIn order to confirm the present picture, we have realized\nDFT+U+SOC calculations [13, 33] using the V ASP code [34–\n36] to take into account the potential non-collinearity. More\nspecifically, we have used an energy cutoff of 550 eV , a similar\nkmeshto the one in Wien2k and the convergency criterion was\nfixed at 10\u00007eV . Starting from a collinear antiferromagnetic\narrangement with all the spins oriented along the\n1 0 1\u000b\u0003,\nwe obtained a small but significant non-collinearity between\nCu1 and Cu2 spins, with a canting angle ranging from 0 :2\nto 1\u000edepending on the U(from 5 to 7 eV) and J(from 0 to\n1 eV) values. This last result confirms that SeCuO 3can be\nH=0\na a\na ab b\nbbc c\nc cH=0(a) (b)\n(c) (d)H>H  SF\nH>H  SFFIG. 11. The magnetic structure in AFM state in zero field (a) and (c)\nand in field H\u0015HSFapplied along the easy axis (b) and (d) obtained\nin this work. (a) and (b) show the acplane to which the spins are\nconfined.\nviewed as a slightly canted antiferromagnet, but the canting\nis too weak to produce an effect in our macroscopic measure-\nments. It also justifies the Wien2k calculations reported in the\npresent paper, which have been done using a collinear anti-\nferromagnetic model. These calculations have demonstrated\nthat an explicit definition of the exchange-correction term J\nin the GGA+U+SOC calculations is needed to properly de-\nscribe the MAE eigenaxes. Moreover, by taking into account,\nin an effective manner, the different correlation regime of Cu1\nand Cu2 subnetworks, we are able to reproduce the reduction\nof the Cu1 magnetic moment and the relative amplitudes of\nMAE1 and MAE2.\nA chemical interpretation of these results can be reached\nby examining the projected densities of states (pDOS). In-\ndeed, such analysis allows to determine the most important\ninteractions, which are the ones with the smallest energy gap\f\fei\u0000ej\f\f, as defined in Eq. (11). More precisely, the ob-\nserved spin orientations of such Cu2+S=1=2 system can\nbe interpreted by the inspection of the pDOS and thus the\ninteractions involving the crystal-field split d-states of each\nmagnetic Cu2+ion, under the action of the spin-orbit cou-\npling. Figs. 12(a) and 12(b) show the split of Cu1 and\nCu2 d-states using GGA+U calculations, with Ue f f= 5 eV .\nIt should be noted that the pDOS obtained with and with-\nout specifying explicitly the exchange-correction term Jare\nsimilar. Here, we defined the local coordinate system with\nxand yinside the CuO 4plane, pointing towards oxygen\natoms, and zperpendicular to the CuO 4plaquette as shown\nin Fig. 12(a). The overall features for pDOS of Cu1 and\nCu2 are the same, with an empty ( x2\u0000y2) state. In such\na situation, theD\ndxy#jbHSOCjdx2\u0000y2#E\n,D\ndxz#jbHSOCjdx2\u0000y2#E12\nandD\ndyz#jbHSOCjdx2\u0000y2#E\ninteractions will be non-zero and\nmainly active because closer to the Fermi level. Spins with\norientation inside the plaquette ( kxyspin orientation) will be\nfavored if dxz#ordyz#states are closer to the empty dx2\u0000y2#\nstates. In contrast, if dxy#states are closer, spins with ori-\nentation perpendicular to the CuO 4plaquette (?xyspin ori-\nentation) will be favored. In the present case, the interpre-\ntation is not straightforward due to the significant distortion\nof Cu1 and Cu2 sites, which are far from regular CuO 4pla-\nquettes. Only Cu1 pDOS show relevant features, which can\nbe interpreted. Indeed, the inset of Fig. 12(b) shows that the\nstates which are mainly contributing on the top of the valence\nband are dxzanddyz, which leads to favour the kxyspin ori-\nentation, as witnessed in our DFT+U+SOC calculations when\nconsidering only Cu1 subnetwork, i.e. MAE1 which shows\nan easy magnetization axis in the acplane. In contrast, Cu2\npDOS does allow us to determine which d-state is mainly con-\ntributing to the top of the valence band. Indeed, while such\nan analysis is relevant for systems exhibiting regular environ-\nments, it should be used with care for distorted environments,\nbecause the choice of the local axes for the pDOS is not any-\nmore unique and may influence the results. Fig. 12(c) shows\nthe pDOS of Cu1 when considering no Hubbard correction\n(Ue f f= 0 eV). The main consequence is a significant band\ngap reduction and an increase of the dxzanddyzcharacters on\nthe top of the valence band. Both modifications lead to an in-\ncrease of the spin-orbit coupling which mixes the dxzanddyz\noccupied states with the dx2\u0000y2unoccupied state. Such treat-\nment leads to have a larger contribution of MAE1 to the total\nMAE, which then develops an easy axis in the acplane and\nhard axis along the bcrystallographic direction.\nTo summarize, there are some similarities between the\nstructure from Ref. 10 and our proposal shown in Fig. 11(a)\nand 11(c). Magnetic moments on Cu2 almost lie within the\nCuO 4plaquette in both cases and their general direction seems\nto be similar although it is difficult to make quantitative com-\nparison since authors in Ref. 10 only plotted magnetic struc-\nture and did not explicitly provide directions of the magnetic\nmoments. The moments on Cu1 in our structure are nei-\nther inside the plaquette nor perpendicular to it, similar to\nwhat is proposed by neutron diffraction [10]. The direction\nof moments on Cu1, however, is in our case very different\nfrom the one given in Ref. 10. Our results state that Cu1\nmoments should be collinear or almost collinear to Cu2 mo-\nments, in agreement with the susceptibility, magnetization and\ntorque data, while neutron powder diffraction gives a very\nnoncollinear structure [10]. Further investigation of magnetic\nstructure by neutron diffraction on single crystal would re-\nsolve this issue.\nTorque data presented in this work suggest a reorientation\nfor only fraction of the spins in SeCuO 3in magnetic field\ncomparable to the spin-flop field. Present DFT calculations\nand results from literature [10, 11] allow us to propose that\nthis partial reorientation is in fact site-specific reorientation.\nSince NMR measurements witness two separate subsystems\nin AFM state [11], while NQR shows existence of strongly\ncoupled singlet [10], we propose that the susceptibility ten-\nsor which rotates in finite magnetic field corresponds to Cu2spins, while Cu1 spins do not reorient in field applied in our\nexperiment. This results in the magnetic structure shown in\nFig. 11(b) and 11(d) for H\u0015HSFapplied along easy axis,\nwhere spins on Cu2 rotate in the acplane and are perpendicu-\nlar to spins on Cu1. This supports the picture where Cu1 and\nCu2 subsystems are decoupled in some way. Since half of the\nspins in SeCuO 3are Cu1 spins, and the other half are Cu2\nspins, we can now write for the susceptibility of Cu1 given in\nEq. (10) ccc1=nccc0=m\u0001(0:5\u0001ccc0). In this equation, 0 :5\u0001ccc0\npresents the total contribution the Cu1 spins would give to the\nsusceptibility if they were equivalent to the Cu2 spins, and in\nour model this contribution is reduced by mdue to the de-\ncoupling of the Cu1 and Cu2 sublattices. For an obtained\nvalue n=0:22 we get m=0:44. The magnetization induced\nby magnetic field on Cu1 spins is only 44% of the value it\nwould have if these spins were equivalent to Cu2 spins. It is\ntempting to compare this to the ratio of magnetic moments\nmCu1=mCu2\u00190:4=0:7=0:57 obtained for magnetic moments\non Cu1 and Cu2 from neutron data [10]. However, this com-\nparison may not be justified if Cu1 and Cu2 belong to de-\ncoupled subsystems. The ratio c1=c2=n=(1\u0000n) =0:282\nconfirms that the magnetization induced on Cu2 spins is sig-\nnificantly larger than on Cu1 spins. This picture corroborates\nthat the dominant contribution to the total MAE comes from\nthe Cu2 site.\nThe separation of Cu1 and Cu2 subsystems is compara-\nble to the observations in Cu 2Cd(BO 3)2[3]. The question in\nSeCuO 3is, are the Cu1 spins polarized singlets in the under-\nlying AFM state formed by interaction between Cu2 spins,\nor do both Cu1 and Cu2 spins interact mutually to form the\nAFM state? If Cu1 spins indeed form singlet states even in\nAFM state, their susceptibility and susceptibility anisotropy\nshould be much smaller than that of the antiferromagnetically\nordered Cu2 spins. In fact, if a true singlet state persists in the\nAFM state, for an intradimer interaction J\u0019200 K between\nCu1 spins we should expect susceptibility and its anisotropy\nto be zero at low temperatures [37]. A finite slope of torque\namplitude for the non-flopped spins [see Fig. 2] suggests Cu1\nspins contribute with finite susceptibility anisotropy. Also,\nc1=c2=0:282 obtained from our simulations is small but fi-\nnite. From our data we cannot distinguish if Cu1 spins in the\nAFM state form polarized singlets or contribute to the AFM\norder with a much smaller magnetic moment than Cu2 spins.\nSite-specific spin reorientation we observe favors the former\npicture, but further experiments are needed to confirm this.\nIf Cu1 spins are in fact polarized singlets, the neutron\ndiffraction result for Cu2 sublattice [10] would then be\nin agreement with other experimental data as well. For\nCdCu 2(BO 3)2neutron diffraction gave sizable magnetic mo-\nment on Cu1 sites [4], while theoretical and experimental data\nshowed that the Cu1-Cu1 dimer in fact forms a singlet in the\nAFM state [3, 5]. A similar scenario appears to apply to\nSeCuO 3, and indeed our DFT calculations which mimic the\nreduction of the Cu1 magnetic moment reproduce properly\nthe experimental MAE. Magnetic order in CdCu 2(BO 3)2is\nalmost collinear with very small canting, similar to what we\npropose for SeCuO 3. One way to check our proposed mag-\nnetic structure in zero and finite field rigorously is to perform13\nFIG. 12. Projected density of states (pDOS) on (a) Cu2 and (b) Cu1\nsites considering an Ue f f= 5 eV , and on (c) Cu1 only at he GGA\nlevel. Projection axes for each Cu site is represented on the scheme\nin (a). Insets zoom the spin minority states over the range of 2 eV\nbelow the Fermi energy. Pink stars correspond to first excitations\nallowed with the excited states.\nsingle crystal neutron diffraction experiments in zero and ap-\nplied magnetic field.\nV . CONCLUSION\nThe present paper proposes a combined experimental and\ntheoretical investigation of the magnetic properties of a low-\ndimensional spin 1 =2 system, which appears to be based on\ntwo decoupled magnetic subsystems. This finding, based on\nmeasurements on high-quality single crystals and state of the\nart density functional calculations, opens a way to very ex-citing physics, with the possibility to control separately two\nmagnetic subsystems in one material. SeCuO 3was previously\nproposed as a candidate for a system with site-selective spin\ncorrelations where Cu1 copper atoms form strongly coupled\nAFM dimers, while the coupling including Cu2 spins results\nin a long range AFM order at low temperatures. Our torque\nmagnetometry results demonstrate site-specific spin reorienta-\ntion in an applied magnetic field in AFM state of SeCuO 3. Us-\ning ab-initio approach we show that Cu1 and Cu2 contribute\ndifferently to the magnetic anisotropy energy. These results\nstrongly suggest that Cu1 and Cu2 spin systems are decoupled\nin SeCuO 3. Combining our experimental and theoretical find-\nings we propose an antiferromagnetic structure of SeCuO 3in\nzero field, as well as in field H&HSF, to be verified by future\nexperiments on this system.\nACKNOWLEDGMENTS\nM. H., N. N., ˇZ. R. and M. D. acknowledge full support of\ntheir work (torque experiment and simulation of torque data)\nby the Croatian Science Foundation under Grant No. UIP-\n2014-09-9775. M. H., M. D., W. L.-d.-H. and X. R. acknowl-\nedge support by COGITO project ”Theoretical and experi-\nmental investigations of magnetic and multiferroic materials”\nfunded by Croatian Ministry of Science and Education and\nThe French Agency for the promotion of higher education, in-\nternational student services, and international mobility. W. L.-\nd.-H. and X. R. thank the CCIPL (Centre de Calcul Intensif\ndes Pays de la Loire) for computing facilities. The theoretical\nwork was also performed using HPC resources from GENCI-\n[TGCC/CINES/IDRIS] (Grant 2017-A0010907682). Crystal\nand magnetic structures in this work were drawn using 3D vi-\nsualization program VESTA [38].\nAppendix: Torque magnetometry, susceptibility tensor and\nmagnetic axes\nFor a sample with magnetization Mplaced in magnetic field\nH, magnetic torque tttacting on the sample can be written as\nttt=VM\u0002H; (A.1)\nwhere Vis the volume of the sample. For systems in which\nresponse of magnetization to magnetic field is linear (such as\nparamagnets and antiferromagnets in magnetic fields well be-\nlow the spin-flop field HSF)M=ˆccc\u0001H, where ˆcccis the mag-\nnetic susceptibility tensor. Taking into account Neumann’s\nprinciple and symmetry restrictions for SeCuO 3[24], the ten-\nsorˆcccwritten in crystal axes coordinate system (a\u0003;b;c)is\ngiven by\nˆccc=2\n64ca\u0003a\u00030ca\u0003c\n0cbb 0\nca\u0003c0ccc3\n75: (A.2)\nThe expression (A.2) states that the baxis is one of the eige-\nnaxes of the susceptibility tensor, or magnetic axes, while two14\nother magnetic axes restricted to the acplane are not neces-\nsarily along any of the the crystal axes ( ca\u0003c6=0).\nThe previously published torque magnetometry results\nshowed that in paramagnetic state, apart from the baxis, eige-\nnaxes of the susceptibility tensor in the acplane are not in\nthe direction of crystal axes [9]. Furthermore, torque results\nshowed that magnetic axes in the acplane rotate as the tem-\nperature changes [6, 9]. To understand how this rotation of\nmagnetic axes can be observed in torque measurements, we\nwrite the tensor ˆcccfor the most general orientation of the sam-\nple in laboratory coordinate system (x;y;z)\nˆccc=2\n64cxxcxycxz\ncxycyycyz\ncxzcyzczz3\n75: (A.3)\nIn experiment magnetic field is rotated in the xyplane, H=\nH(cosjsinj;0), and only the component of torque along the z\naxis is measured. When tensor cccfrom Eq. (A.3) is introduced\nin expression (A.1) we obtain for the measured component of\ntorque tz\ntz=m\n2MmolH2[(cxx\u0000cyy)sin2j\u00002cxycos2j]:(A.4)\nwhere mis the mass of the sample given in units of g,Mmolis\nthe molar mass given in g/mol and components of the tensor\n(A.3) are expressed in emu/mol . The measured torque is then\ngiven in dyn cm .jis the goniometer angle which describes\nthe direction of magnetic field with respect to the laboratory\naxes. When eigenvectors of the susceptibility tensor are along\nthexandyaxis, the off diagonal component cxyis zero and\ncxxandcyyrepresent the maximal and minimal components of\ncccin that plane. Otherwise, the maximal and minimal compo-\nnents in the xyplane will be rotated by some angle q0with re-\nspect to the (x;y)axis. If we term these new directions (x0;y0)\nwe can write for the measured torque\ntz=m\n2MmolH2\u0000\ncx0\u0000cy0\u0001\nsin(2j\u00002j0); (A.5)where correspondence of (A.4) and (A.5) is established\nthrough\ntan(2j0) =2cxy\ncxx\u0000cyy; (A.6a)\ncx0\u0000cy0=cxx\u0000cyy\ncos(2j0)=2cxy\nsin(2j0): (A.6b)\nj0is the angle that the x0axis makes with the xaxis. In exper-\niment the measured component of torque (for paramagnet and\nAFM in H\u001cHSF) can be described by Eq. (A.5). From the\namplitude of torque we obtain the susceptibility anisotropy\nin the plane of measurement xydefined as Dcxy=cx0\u0000cy0.\nUsing equations (A.6a) and (A.6b) we can determine the dif-\nference of the diagonal tensor components cxx\u0000cyy, and the\noff-diagonal tensor component cxy.This approach shows that\ntorque magnetometry, in combination with the magnetic sus-\nceptibility measurement along one of the directions x,yor\nz, allows for determination of the temperature dependence of\nthe susceptibility tensor ˆccc. By diagonalizing the tensor one\ncan then obtain the temperature dependence of the magnetic\nsusceptibility along magnetic eigenaxes (or magnetic axes for\nshort) and also the orientation of the eigenaxes with respect to\nthe crystal axes.\nDetermining magnetic eigenaxes is crucial in systems\nwhere they rotate as temperature changes, as was observed\nin SeCuO 3in both the paramagnetic and the AFM state\n[6, 9], since magnetic eigenaxes also define the MAE shape.\nNamely, Neumann’s principle dictates that the extrema of the\nMAE are expected to coincide with the magnetic eigenaxes.\nOur previous results showed that in the AFM state the mag-\nnetic axes rotate in the acplane. Another important result\nof Eq. (A.4) and Eq. (A.5) is that the torque measurements\ncan be utilized to observe a symmetry lowering. In case of\nSeCuO 3this would mean the appearance of finite tensor com-\nponents ca\u0003borcbcin tensor (A.2). In experiment this means\nthe following. 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Izumi, Journal of Applied Crystallography\n44, 1272 (2011)."    },    {        "title": "1806.01536v2.Efficient_technique_for_ab_initio_calculation_of_magnetocrystalline_anisotropy_energy.pdf",        "content": "arXiv:1806.01536v2  [cond-mat.mes-hall]  1 Nov 2018Efficient technique for ab-initio calculation of\nmagnetocrystalline anisotropy energy\nJunfeng Qiaoa,b, Weisheng Zhaoa,b,∗\naFert Beijing Institute, BDBC, Beihang University, Beijing 100191, China\nbSchool of Electronic and Information Engineering, Beihang University, Beijing 100191,\nChina\nAbstract\nAb-initio calculation of magnetocrystalline anisotropy energy (MCAE) often\nrequires a strict convergence criterion and a dense k-point mesh to sample the\nBrillouin zone, making its convergence problematic and time-consumin g. The\nforce theorem for MCAE states that MCAE can be calculated by the band\nenergy difference between two magnetization directions at a fixed p otential. The\nmaximally localized Wannier function can be utilized to construct a comp act\nHilbert space of low-lying electron states and interpolate band eigen values with\nhigh precession. We combine the force theorem and the Wannier inte rpolation\nof eigenvalues together to improve the efficiency of MCAE calculation s with no\nloss of accuracy. We use a Fe chain, a Fe monolayer and a FeNi alloy as examples\nand demonstrate that the Wannier interpolation method for MCAE is able to\nreduce the computational cost significantly and remain accurate s imultaneously,\ncompared with a direct ab-initio calculation on a very dense k-point mesh.\nThis efficient Wannier interpolation approach makes it possible for larg e-scale\nand high-throughput MCAE calculations, which could benefit the des ign of\nspintronics devices.\nKeywords: magnetocrystalline anisotropy energy, ab-initio , Wannier function\nPACS: 75.30.Gw, 71.15.-m, 85.75.-d\n1. Introduction\nIn ferromagnetic (FM) materials, the magnetic anisotropy energy (MAE) is\nthe difference between magnetizing energies along two directions. M AE can be\nmainly separated into the extrinsic part and the intrinsic part. The e xtrinsic\nshape anisotropy, in which MAE correlates with the shape of the FM m ate-\nrial, originates from the magnetic dipole-dipole interaction [1, 2]. The intrinsic\nmagnetocrystalline anisotropy energy (MCAE) arises from the colle ctive effect\n∗Corresponding author\nEmail address: weisheng.zhao@buaa.edu.cn (Weisheng Zhao)\nPreprint submitted to Computer Physics Communications Nov ember 2, 2018of the crystal structure and spin-orbit coupling (SOC). The orbit al motions of\nelectrons are restricted to specific orientations by the crystal fi eld, and the SOC\ncouples the spin degree of freedom and the orbital motion. Conseq uently, the\nenergy of the crystal depends on the magnetization orientation [ 3].\nUsually, the MCAE is on the order of meV at surfaces and interfaces [4, 5, 6],\nabout 3 orders of magnitude larger than that of bulk crystals [7, 1]. Apart\nfrom the interest in fundamental science, the practical applicatio ns have also\nstimulated lots of research into the understanding and optimization of MCAE [8,\n9, 10, 11, 12, 13]. One of the applications is the magnetic tunnel jun ction (MTJ),\nwhich is the core structure of magnetic random access memory (MR AM) [14, 15].\nMTJ stores one bit of information by the relative magnetization orien tation of\nthe two FM layers, and the read process is based on tunneling magne toresistance\n[16, 17, 18]. The large MCAE induced by the CoFe/MgO interface is cru cial for\nthe stability of the information.\nThe MCAE is hard to be captured because of its small magnitude. The\ntotal energy of a crystal unit cell is on the order of 1 ×103eV, while the MCAE\nis on the order of 1 ×10−3eV, this significant contrast poses serious challenges\nnot only to experimental instruments but also to numerical algorith ms. As a\nresult, calculation of MCAE still remains an intriguing problem in the ab-initio\ncommunity [19, 1, 20, 21, 22].\nInab-initio calculations, plane wave (PW) algorithm is widely adopted be-\ncause it is mathematically simple and in principle complete. The PW method\nprovides the same accuracy at all points in space, this advantage o n one side\nis appealing but on the other side is disappointing since it often require s large\namounts of plane waves to converge and results in bad scaling behav ior when\nsystem size increases. A more efficient approach is always desired. M aximally\nlocalized Wannier function (MLWF) is the Fourier transform of the re ciprocal\nspace PW wave function under the criterion of maximal localization of Wan-\nnier function (WF) in real space [23]. MLWF not only gives a clearer ph ysical\ninsight of chemical bondings but also manifests its necessities and ac curacies in\nthe calculations of many physical properties, such as electric polar ization [24],\nanomalous hall conductivity [25] and orbital magnetization [26], etc. [23, 27, 28].\nHere we extend the Wannier interpolation method to MCAE calculation\nfor the purpose of improving efficiency. Our steps are summarized a s follows:\nFirst, an ab-initio calculation is performed on a relatively coarse k-point mesh\n(kmesh). Second, MLWFs are extracted from the acquired ab-initio wave func-\ntions. Third, MCAE calculations are performed on a much denser kme sh in the\nWannier representation. Compared to a direct ab-initio calculation on the dense\nkmesh, this Wannier interpolation approach has equivalent accurac y but greatly\nreduced computational cost. The initial coarse kmesh is too spars e for MCAE\nto converge but the extracted WFs meet the criterion of “good” lo calization.\nOur Wannier interpolation approach could not only reduce the compu tational\ncost but also make it possible to calculate MCAE for large systems req uiring\nextremal computational resources, and calculations requiring ex ceptionally high\naccuracy.\n22. Methods\nThe Wannier interpolation method for calculating MCAE is based on two\ntheories: The force theorem (FT) states that MCAE can be calcula ted by the\ndifference between band energies of two magnetization directions; the MLWF\nprovides an efficient way for interpolating eigenvalues on arbitrary d ense kmesh\nbased on coarse kmesh ab-initio calculation. We first give a brief and self-\ncontained introduction to the Wannier interpolation of band struct ure, then\nshow our procedures for combining Wannier interpolation with the FT of MCAE.\nWe followed the theory and mathematical notations used in Yates et al. Ref.[29]\nand Marzari et al. Ref.[23]. The disentanglement of entangled bands is omitted\nfor conciseness and detailed theory on MLWFs can be found in Ref.[29 ] and\nRef.[23].\n2.1. Wannier interpolation\n2.1.1. Construction of WFs\nIn a periodic crystal, the Bloch theorem allows us to write down the wa ve\nfunction as\nψnk(r) =eikrunk(r), (1)\nwherekis thek-point vector, nis the band index, ψnk(r) is the Bloch wave\nfunction,unk(r) is periodic in real space. Inserting Bloch function into the\nKohn-Sham equation, we arrive at the equation for the periodic par t of the\nBloch function,\nˆH(k)u(k) =ǫku(k), (2)\nwhere ˆH(k) is the transformed Hamiltonian ˆH(k) =e−ikrˆHeikr,ǫkis the\neigenvalue. Usually one needs several thousand plane waves to exp and theu(k)\nin the PW method and the diagonalizations of the Hamiltonian matrices h aving\nthe rank of several thousand are performed on each k-point. This is why a direct\nab-initio calculation on a dense kmesh is rather time-consuming.\nGenerally, Fourier transform enables us to have a different repres entation of\nfunctions in reciprocal space, and this may be useful for analysis o f the problem,\ne.g. the frequency spectrum of audio signals and images. If treate d properly,\nthe reciprocal space should be equivalent to the original space, i.e. they are\nequivalent Hilbert spaces. We apply the Fourier transform to the Blo ch wave\nfunctionψnk(r) which lies in the Brillouin Zone (BZ),\n|Rn/an}bracketri}ht=V\n(2π)3/integraldisplay\nBZdke−ik·R|ψnk/an}bracketri}ht, (3)\nthus|Rn/an}bracketri}htform an orthonormal set and span the same Hilbert space as ψnk(r).\nIn principle, a smooth function in real space results in a localized func tion in its\nreciprocal space, and vice versa. It is not naturally guaranteed t hat the simply\nsummed Bloch function of Equ. (3) results in a smooth function |Rn/an}bracketri}htin real\nspace. Fortunate enough, there is a gauge freedom left in the defi nition of Bloch\nfunction,\n|˜ψnk/an}bracketri}ht=eiϕn(k)|ψnk/an}bracketri}ht, (4)\n3or equivalently,\n|˜unk/an}bracketri}ht=eiϕn(k)|unk/an}bracketri}ht. (5)\nWe can utilize this freedom to construct localized WFs in real space, t he so-\ncalled maximally localized Wannier function.\nWe define the unitary transformation which takes the original Bloch function\n|unk/an}bracketri}htto the smoothed function |˜unk/an}bracketri}ht(from now on written as |u(W)\nnq/an}bracketri}ht, we use\nqhere as to differentiate another kmesh in the following Wannier interp olation\nstep) as\n|u(W)\nnq/an}bracketri}ht=M/summationdisplay\nm=1|umq/an}bracketri}htUmn(q), (6)\nwhereMis the number of states needs to be considered for our targeted p hysical\nproperties, usually the low-lying states below Fermi energy or plus f ew empty\nstates above. We call this unitary transformation as the transfo rmation from\nBloch gauge to Wannier gauge.\nThus, the Fourier transform pair between the smoothed Bloch fun ctions and\nthe MLWFs are\n|Rn/an}bracketri}ht=1\nN/summationdisplay\nqe−iq·R|u(W)\nnq/an}bracketri}ht,\n/arrowdblbothv\n|u(W)\nnq/an}bracketri}ht=/summationdisplay\nReiq·R|Rn/an}bracketri}ht,(7)\nwhere,Nis the number of points in BZ.\n2.1.2. Interpolation on arbitrary kmesh\nFor the reciprocal space Hamiltonian operator ˆH(q), we define the M×M\nHamiltonian matrix in the Wannier gauge as\nH(W)\nnm(q) =/an}bracketle{tu(W)\nnq|ˆH(q)|u(W)\nmq/an}bracketri}ht= [U†(q)H(q)U(q)]nm, (8)\nwhereHnm(q) =Enqδnm,δnmis the Kronecker delta function. If the H(W)\nnm(q)\nis diagonalized by\nU(q)†H(W)(q)U(q) =H(H)(q), (9)\nwhereH(H)\nnm(q) =E(H)\nnqδnm, thenE(H)\nnqwill be identical to the original ab-initio\nEnq.\nTransforming the Hamiltonian operator from reciprocal space to r eal space,\nH(W)\nnm(R) =1\nN/summationdisplay\nqe−iq·RH(W)\nnm(q), (10)\nand then performing inverse Fourier transform\nH(W)\nnm(k) =/summationdisplay\nReik·RH(W)\nnm(R), (11)\n4we succeed in interpolating the Hamiltonian operator on arbitrary k-pointk.\nSince the WFs we have chosen are maximally localized, the H(W)\nnm(R) is expected\nto be exponentially localized in real space, a few Rare sufficient in the sum of\nEqu. (11).\nThe final step is diagonalizing H(W)\nnm(k),\nU(k)†H(W)(k)U(k) =H(H)(k), (12)\nthen the acquired eigenvalues on arbitrary k-pointkcan be used for later extrac-\ntions of the targeted physical properties. We comment here that sinceH(W)(k)\nare of dimensions M×M, their diagonalizations are very “cheap”, compared\nwith the diagonalizations of the Hamiltonian matrices in PW method, whic h\nare on the order of more than 1000 ×1000 dimensions.\nFigure 1 shows the interpolated band structure of our later used F e chain\ncompared with the original coarse kmesh ab-initio calculation and a dense kmesh\nab-initio calculation. The calculation details will be described in section 3.1.\nThe interpolated band structure is in excellent agreement with the d ense kmesh\nab-initio band, whether it locates at the coarse kmesh ab-initio q-points or the\ninterpolated k-points between the q-points. Apparently, the coarse kmesh ab-\ninitio calculation, from which the WFs are constructed, is far from conve rgence\nwith respect to the dense kmesh ab-initio calculation.\n2.2. MCAE\nAs mentioned in the introduction, one of the difficulties of calculating M CAE\nis its small magnitude compared with the total energy. Another diffic ulty is\nits rapid variation in k-space. As shown in Fig. 2(b), the k-space resolved\nMCAE, defined by Equ. (14), displays sharp peaks and steps along kzdirection.\nTo capture these tiny but critical features, a sufficiently dense km esh must be\nadopted, and slow convergences relative to kmesh are often the c ase in the\ncalculations of MCAE.\nIn theab-initio calculation of MCAE, the most natural and rigorous method\nis performing two self-consistent (SCF) calculations of different ma gnetization\ndirections, and the MCAE should be the difference between the tota l energies.\nWritten down by notations, the MCAE between two crystallographic directions\ncan be defined as\nMCAE =E90◦−E0◦, (13)\nwhereEαis the total energy of magnetization pointing towards the θ=α\ndirection,θis the spherical polar angle with respect to the crystallographic c\naxis. In our case, we choose α= 90◦and 0◦, and the azimuthal angle φis\nkept fixed to 0 in all the calculations. According to this definition, MCA E>0\nstands for perpendicular magnetic anisotropy (PMA) while MCAE <0 stands\nfor in-plane magnetic anisotropy (IMA).\nHowever, the cumbersome diagonalizations are rather time-consu ming, es-\npecially when SOC is included, and reaching the energy minimal becomes much\nharder in the SCF iterations. According to the FT [7, 20, 1, 30, 31, 32], the\n50.0 0 .2 0 .4 0 .6 0 .8 1 .0\nkz(2π·c−1)−6−5−4−3−2−10123E(eV)Fermi energydis froz max\nab-initio, kmesh 2600\nWannier, kmesh 2600\nab-initio, kmesh 24\nFigure 1: Comparison of Fe chain band structures obtained fr om a coarse kmesh 1 ×1×24\nab-initio calculation (red circles), a dense kmesh 1 ×1×2600ab-initio calculation (green lines),\nand the interpolated band structure on the dense kmesh 1 ×1×2600 by MLWFs (blue dashed\nlines). The MLWFs are constructed from the coarse kmesh 1 ×1×24ab-initio calculation. The\nblack dashed horizontal line corresponds to the Fermi energ y, the orange dashed horizontal\nline corresponds to the upper limit of the frozen inner windo w in the disentanglement process\nwhen constructing MLWFs. The k-points on the horizontal axis are along the crystallograph ic\ncaxis and are in fractional coordinates.\n6main contribution to the MCAE originates from the difference of band energies\nbetween two magnetization directions at fixed potential,\nMCAE(k) =/summationdisplay\nif90◦\nikǫ90◦\nik−/summationdisplay\ni′f0◦\ni′kǫ0◦\ni′k,\nMCAE =1\nN/summationdisplay\nkMCAE(k),(14)\nwherekis thek-point vector, iandi′are the band indexes of magnetization\ndirections along θ= 90◦andθ= 0◦, respectively. fikis occupation number and\nǫikis the energy of band iatk-pointk,Nis the number of k-points in the BZ.\nThus by non-self-consistent (NSCF) calculation of only a one-step diagonal-\nization, the eigenvalues are obtained and the MCAE can be calculated by Equ.\n(14). The FT is widely adopted and its validity has been proved by prac tical cal-\nculations [33, 34, 22]. Good matches between experimental and ab-initio results\nare also found in the literature [35, 36]. Another merit of FT is that b y Equ.\n(14) somek-space resolved analyses can be performed, such as the contribu tion\nof the quantum well states to the oscillation of MCAE [10].\nThe practical calculation of MCAE involves two steps. First, charge density\nis acquired self-consistently without taking into account SOC. Seco nd, reading\nthe SCF charge density, two NSCF calculations are performed includ ing SOC,\nwith magnetization pointing towards the θ= 90◦and theθ= 0◦directions,\nrespectively. Finally, MCAE is calculated via Equ. (14).\nBefore we finish the discussion of FT for MCAE, we would like to emphas ize\nthe importance of Fermi energy EF. Since MCAE is such a small quantity\nthat is on the order of meV, a small displacement of EFwill propagate into\nthe occupation numbers fik, and then the MCAE by Equ. (14). The MCAE\nwill be significantly modified, even the sign could be changed. To accur ately\ncalculate MCAE, the small difference of Fermi energy between magn etizations\nalongθ= 90◦andθ= 0◦must be considered. Since the number of electrons\nNelecare the same between two magnetization directions, the Fermi ene rgies\nare calculated separately for θ= 90◦andθ= 0◦in the Wannier interpolation\nsteps by\nNelec=/integraldisplayE90◦\nF\nn90◦(E)dE=/integraldisplayE0◦\nF\nn0◦(E)dE, (15)\nwhereEα\nFandnα(E) are the Fermi energy and density of states for magne-\ntization along θ=α,α= 90◦or 0◦. Also, the occupation numbers fikare\ncalculated separately for the two magnetization directions. Finally, MCAE is\ncalculated by Equ. (14).\nIn all, Wannier interpolation gives us the ability to efficiently interpolate\nband energies on arbitrary kmesh, the FT tells us the MCAE can be ca lculated\nby the difference of band energies. Combining these two theories to gether,\nMCAE calculations can be carried out on a much denser kmesh with no lo ss of\naccuracy but greatly reduced computational cost.\nWe implemented the code for MCAE calculations on the basis of Wan-\nnier90 package [37, 38, 39, 40]. In the next section, we choose a Fe chain\n7as an example to demonstrate our method of MCAE calculation and dis cuss\nsome convergence issues in the calculations. On one hand, due to th e lowered\ncrystal symmetry, the Fe chain system is expected to have medium magnitude\nMCAE; on the other hand, this system is small enough that an extre mely dense\nkmeshab-initio calculation can be performed so that our Wannier interpolation\nmethod can be directly compared with high accuracy ab-initio results. After\ndiscussing all the critical convergence issues in the Wannier interpo lation ap-\nproach, we further carry out MCAE calculations of Fe monolayer, w hich has\nMCAE on the order of sub meV and the results can be validated agains t lit-\nerature. Finally, the MCAE of bulk materials are on the order of µeV and\ntheirab-initio calculations are very challenging, we show that even in such a\ntough situation as FeNi alloy, our Wannier interpolation approach st ill faithfully\nrecover the converged MCAE from a coarse kmesh calculation, and the results\nwell match other ab-initio calculations and experiments. The three examples are\nrespectively 1-dimension, 2-dimensions, and 3-dimensions, with MCA E varying\nfrom meV to µeV. Their convergence criteria are increasingly tighter, and their\ndensities of kmesh increase linearly, quadratically and cubically when t esting\nthe MCAE convergence. These examples are good test grounds fo r verifying\nWannier interpolation approach of MCAE.\n3. Fe chain\n3.1. ab-initio calculation and Wannierization\nTheab-initio calculations were performed using the Quantum ESPRESSO\n(QE) package based on the projector-augmented wave (PAW) me thod and a\nplane wave basis set [41, 42]. The exchange and correlation terms we re described\nusing generalized gradient approximation (GGA) in the scheme of Per dew-\nBurke-Ernzerhof (PBE) parameterization, as implemented in the pslibrary\nversion 0.3.1 [43]. We used a wave function cutoff of 90 Ry and electron density\ncutoff of 1080Ry. A Marzari-Vanderbilt cold smearing [44] of width 0 .002 eV\nwas adopted. Convergence relative to smearing width and kmesh will be de-\ntailedly discussed in the subsequent paragraph. Since the MCAE of t he Fe chain\nis on the order of meV, the energy convergence criteria of all the c alculations\nwere set as 1 .0×10−8Ry. We kept a 15 ˚A vacuum space in the xy plane to\neliminate interactions between periodic images. Since enough vacuum space\nwas left in the xy plane, we set the Monkhorst-Pack k-point mesh to 1 ×1×24\nand confirmed that increasing it to 2 ×2×24 or more had negligible impacts\non MCAE. The unit cell contains one Fe atom [shown in Fig. 2(a)] and th e\nlattice constant along caxis (i.e. the atomic spacing along the Fe chain) was\nset as 2.2546 ˚A, which was acquired by relaxation until the force acting on the\nFe atom was less than 1 ×10−2Ry/Bohr.\nTo transform Bloch functions into MLWFs, first, the overlap matric esMk,b\nmn\nand the projection matrices Ak\nmnare extracted from the Bloch functions [23].\nThen after disentanglement and Wannierization the MLWFs are cons tructed.\nWe used 20 spinor WFs having the form of sp3d2,dxy,dxz,dyz, ands-like Gaus-\nsians. The disentanglement outer energy window was set as [ −100.0,10.0]eV,\n8Figure 2: (a) Crystal structure of the Fe chain. The chain is a long the crystallographic caxis.\nThe cuboid represents the unit cell. (b) k-resolved Fe chain MCAE along kzaxis, defined by\nEqu. (14). Only half of the k-space is shown.\nand the inner window was set as [ −100.0,−3.41]eV, which is slightly higher\nthan the Fermi energy −4.38 eV. The spreads of WFs signify the quality of\nlocalization, as defined by [23]\nΩ =/summationdisplay\nn[/an}bracketle{t0n|r2|0n/an}bracketri}ht−/an}bracketle{t0n|r|0n/an}bracketri}ht2], (16)\nwhere|0n/an}bracketri}htare the WFs in the home unit cell. The convergence threshold of\nspread for both the disentanglement and Wannierization were set a s 1×10−10˚A2.\nApart from 2 WFs having the spread Ω around 2 .0˚A2, the spreads of the re-\nmaining 18 WFs were smaller than 1 .0˚A2, mostly around 0 .2˚A2. The total\nspread of WFs was around 10 .0˚A2and the convergence issues will be discussed\nin the next section.\n3.2. Convergence issues of Wannier interpolation\nThree main parameters should be tested to reach a reliable result of MCAE:\nthe number of coarse ab-initio kmeshnab\nkfor the construction of MLWFs, the\nnumber of Wannier interpolation kmesh nwan\nkfor the interpolation of MCAE\nfrom MLWFs, and the smearing width σthat determines the fictitious smearing\ncontributions to the total energies.\nNote since the kmesh in the xy plane is always set as 1 ×1, we will use the\nnumber ofk-points along z-axis nkas the shorthand of the kmesh 1 ×1×nk.\nWe differentiate the coarse ab-initio kmesh and the dense Wannier interpolation\nkmesh by the notation nab\nkandnwan\nk.\nTo make a clearer separation among different tests, we obey the no tation\nthat for a function f(x;y), thexbefore the semicolon is the variable while\ntheyafter the semicolon is the parameter which is held fixed. For example,\nMCAE(nab\nk;nwan\nk) represents the variation of MCAE relative to the coarse ab-\ninitio kmeshnab\nkwhen the Wannier interpolation kmesh nwan\nkis held fixed.\n9Next, we test the convergence of the spread relative to the numb er of coarse\nab-initio kmesh Ω(nab\nk), the convergence of MCAE relative to the the number of\ncoarseab-initio kmesh MCAE( nab\nk;nwan\nk,σ), the convergence of MCAE relative\nto the number of Wannier interpolation kmesh MCAE( nwan\nk;nab\nk,σ), the con-\nvergence of MCAE relative to the smearing width MCAE( σ;nab\nk,nwan\nk) and the\nminimum number of Wannier interpolation kmesh needed relative to sme aring\nwidthnwan\nk(σ).\nBefore carrying out quantitative analyses, we introduce several definitions\nto avoid ambiguities.\n1. Theconvergence indicator δf(x) of a function f(x) is defined as\nδf(x) =|max\ny≥xf(y)−min\ny≥xf(y)|, (17)\nwhere|...|means taking the absolute value.\n2. Letǫbe a positive real number, we say that the function f(x) isconverged\natx0under the threshold ǫifδf(x)<ǫis satisfied for any x≥x0.\n3. Theconverged value ¯fis defined as\n¯f=/summationdisplay\n{x|δf(x)<ǫ}f(x), (18)\ni.e. iff(x) is converged at x0, then ¯f=/summationtext\nx≥x0f(x).\n4. Thedeviation of a function f(x) is defined as\n∆f(x) =f(x)−¯f (19)\n3.2.1. Spread of WFs\nThe construction of WFs relies on a uniform ab-initio kmesh, we call it as\ncoarseab-initio kmesh, since this kmesh does not need to be dense enough for\nthe MCAE to be converged but is sufficient once “good” localization of WFs\nare reached.\nHere we show the relation of WFs’ spread and the density of ab-initio kmesh\nin Fig. 3(a). In the MCAE calculations, the “good” criterion for the lo caliza-\ntion of WFs is ultimately determined by the convergence of the MCAE. We\nmention it in advance that a 1 ×1×2600 interpolation kmesh and a smearing\nwidth of 0.0012 eV are sufficient for MCAE to be converged on the order of\n1×10−6eV, and the converged MCAE is 2 .358±0.001meV with Fe magnetic\nmoment parallel to the chain. The subsequent paragraphs will deta iledly discuss\nthe convergence issues related to interpolation kmesh and smearin g width. We\nuse this conclusion here to exclude the influence of the Wannier inter polation\nkmesh and the smearing width when discussing the convergence of t he WFs’\nspread and the MCAE relative to the density of the ab-initio kmesh.\nWe use the indicator δΩ(nab\nk) to represent the convergence trend of the spread\nΩ calculated on ab-initio kmeshnab\nk\nδΩ(nab\nk) =|max\ni≥nab\nkΩ(i)−min\ni≥nab\nkΩ(i)|. (20)\n10As shown in the inset of Fig. 3(a), when nab\nk≥24, theδΩ are less than 0 .2˚A2.\nWe use ∆MCAE( nab\nk) to represent the deviation of MCAE( nab\nk) relative to the\nconverged MCAE. In this case, they are defined as\n∆MCAE(nab\nk) = MCAE( nab\nk)−MCAE,\nMCAE =1\n930/summationdisplay\nnab\nk=22MCAE(nab\nk),(21)\nwherenab\nkis theab-initio kmesh used for the construction of WFs, MCAE is\nthe mean value of MCAE calculated with ab-initio kmesh from 22 to 30 and\nthese are the converged value since the variation between these 9 values are on\nthe order of 1 ×10−6eV, as shown in the inset of Fig. 3(b).\nAn interesting anomaly appears in the inset of Fig. 3(b) at nab\nk= 21. This\nmay be caused by the insufficiencies of coarse ab-initio kmesh or the Wannier\ninterpolation kmesh. To separate the influence of these two kmesh es, we show\nthe convergence indicator δMCAE(nwan\nk;nab\nk) by the red error bar in the inset\nof Fig. 3(b). The δMCAE(nwan\nk;nab\nk) is defined as\nδMCAE(nwan\nk;nab\nk) =|max\ni≥nwan\nkMCAE(i;nab\nk)−min\ni≥nwan\nkMCAE(i;nab\nk)|,(22)\nwherenwan\nkis in the range of 2500 to 2700. At each specific nab\nk,δMCAE(nwan\nk;nab\nk)\nonly varies less than 3 ×10−6eV whennwan\nkchanges from 2500 to 2700, much\nsmaller than the anomaly. This indicates that the fixed Wannier interp olation\nkmesh ofnwan\nk= 2600 is high enough and the anomaly at nab\nk= 21 is caused\nby theab-initio kmesh alone. This means to get a very accurate MCAE, a suf-\nficiently dense ab-initio kmesh is needed for constructing high-quality MLWFs.\nHowever, compared with the ab-initio kmeshnab\nk= 2600 used for a direct ab-\ninitio computation of MCAE, this kmesh is much sparser and only use 0 .8 % of\nthe number of k-points.\nFornab\nk≥22, the total spread of WFs converges to approximately 0 .2˚A2,\nat the same time the MCAE converge on the order of 1 ×10−6eV as shown in\nthe inset of Fig. 3(b). As in many other cases, the interpolated ban d structure\nagrees very well with high-density kmesh ab-initio calculation [23]. While the\nband structure of the coarse ab-initio kmesh, which is used for the construction\nof MLWFs, deviates largely from the “true” dense kmesh band stru cture needed\nfor MCAE to converge. The comparison of band structures is show n in Fig. 1.\n3.2.2. Interpolation kmesh\nThe former paragraphs consider the convergence of MCAE with re spect to\nab-initio kmesh, which should be dense enough for constructing MLWFs. Fro m\nnow on, we fix the nab\nkto 24. Once high-quality MLWFs are constructed, the\nnext step for reaching converged MCAE is the Wannier interpolation . This is\nwhere the computational cost is significantly reduced.\n118 10 12 14 16 18 20 22 24 26 28 30\nab-initio kmeshnab\nk05101520Ω (/RingA2)(a)\n12 14 16 18 20 22 24 26 28 300.00.20.5δΩ (/RingA2)θ= 0°\nθ= 90°\n8 10 12 14 16 18 20 22 24 26 28 30\nab-initio kmeshnab\nk0123MCAE (×10−2eV)(b)\n12 14 16 18 20 22 24 26 28 30−3−2−1012∆MCAE ( ×10−5eV)\nAverage of 22 to 30: 2 .359meV\nFigure 3: For Fe chain: (a) Spread of WFs relative to the densi ty ofab-initio kmesh, the\ninset shows the convergence trend δΩ(nab\nk) of the spread defined by Equ. (20). Blue color and\norange color are the calculations of magnetization directi on perpendicular to and along the\nFe chain, respectively. (b) The MCAE relative to the density ofab-initio kmesh. The inset\nshows the deviation of MCAE when approaching convergence, d efined by Equ. (21). The red\nerror bars in the inset represent the convergence trend δMCAE(nwan\nk;nab\nk) relative to nwan\nk\nas defined by Equ. (22). The small error bars indicate that the anomaly at nab\nk= 21 is caused\nby theab-initio kmesh alone and this means ab-initio kmesh plays a critical role in Wannier\ninterpolation of MCAE.\n122500 2525 2550 2575 2600 2625 2650 2675 2700\nWannier interpolation kmesh nwan\nk−0.20.00.20.40.6∆MCAE ( ×10−5eV)\naverage of 2500to2700:2.358meV100 200 300 400 500 600 700 800−40−2002040∆MCAE ( ×10−5eV)\nFigure 4: The Fe chain MCAE deviation relative to the converg ed value when Wannier inter-\npolation kmesh nwan\nkvaries, defined by Equ. (23). The inset is the overview with nwan\nkfrom\n100 to 800.\nIn such circumstance, the ∆MCAE is defined as\n∆MCAE(nwan\nk) = MCAE( nwan\nk)−MCAE,\nMCAE =1\n2012700/summationdisplay\nnwan\nk=2500MCAE(nwan\nk),(23)\nand this is shown in Fig. 4.\nThe inset of Fig. 4 shows the overview of the ∆MCAE( nwan\nk) fornwan\nkfrom\n100 to 800. Apparently, nwan\nkon the order of 100 is far from convergence. For\nnwan\nkaround 2600, the MCAE converge under 4 ×10−6eV.\n3.2.3. Smearing contribution\nSmearing is very important for the convergence of ab-initio calculations.\nLarge smearing is beneficial for fast convergence, but this may int roduce a fic-\ntitious and un-negligible contribution to the total energy. Too small smearing\nmay slow down the convergence significantly, and often one needs a much denser\nkmesh to converge. This problem can be largely alleviated because th e Wannier\ninterpolation is very “cheap”—as we mentioned in the Sec.2 it only involv es\ndiagonalizations of M×Mmatrices. One could easily increase kmesh density\nand reduce smearing width, or even not resorting to smearing.\n13The relation between the minimum number of interpolation kmesh nwan\nkand\nthe smearing width σis shown in the inset of Fig. 5 and the specific values are\ntabulated in Table 1. When σis in the range of 0 .20 eV to 0.05 eV, only less\nthan 300nwan\nkis needed to converge the MCAE on the order of 2 ×10−6eV.\nHowever, as indicated by the blue line in the inset of Fig. 5, the smearin g\ncontribution to the MCAE, ∆MCAE( σ;nwan\nk), is on the order of 2 ×10−3eV,\nleading to unreliable results. ∆MCAE( σ;nwan\nk) is defined as\n∆MCAE(σ;nwan\nk) = MCAE( σ;nwan\nk)−MCAE,\nMCAE =1\n10/summationdisplay\nσ∈ΣMCAE(σ;nwan\nk),where Σ = {0.0001×i|0≤i≤9},(24)\nwhere thenwan\nkare fixed to 2600, and the 0 .0000 in Σ means calculation without\nsmearing.\nWe find a smearing width of less than 0 .003 eV is able to reduce the smear-\ning contribution to under 2 ×10−6eV, and at such a minimal smearing width\na kmesh of nearly 1600 should be adopted. To totally exclude the sme aring\ncontribution, i.e. calculation without smearing, a dense kmesh of 260 0 should\nbe used [data listed in Table 1].\nFinally, as the best touchstone for the validity of Wannier interpolat ion, we\nperformed an extremely accurate ab-initio calculation, with a kmesh of 1 ×1×\n2600 and a smearing width of 0 .0010 eV. As shown in Table 1, considering the\nnumerical accuracy of 0 .001 meV, the Wannier interpolation results of rows No.9\nto 11 are exactly equal to the dense kmesh ab-initio result of row No.15. The\nMLWFs used for interpolating MCAEs in rows No.1 to 11 are construct ed from\ntheab-initio calculation of row No.13. It is clear from rows No.12 to 14 that the\nab-initio calculation of row No.13 is not converged but the constructed MLWF s\nsuccessfully recover the “true” MCAE.\n3.3. Computational resources\nSo far we have demonstrated that our Wannier interpolation appro ach for\ncalculating MCAE is sufficiently accurate. To illustrate its effectivenes s in re-\nlieving the computational cost, we directly compare the resources used in the\ndifferent calculations.\nThe computational resources and time spent for each calculation a re shown\nin Table 2. All of the ab-initio results are calculated on 96 CPU cores. The ab-\ninitio calculation on kmesh 2600 took 9 .0 h, while ab-initio calculation on kmesh\n24 took several minutes. Note the most time-consuming step in the Wannier\ninterpolation method—the disentanglement and Wannierization proc esses—can\nbe boosted by selecting better initial projection orbitals. In the cu rrent work,\nwe do not further investigate the choice of initial projection orbita ls and we\nexpect a better choice can greatly reduce the time spent on disent anglement and\nWannierization. The biggest advantage of the Wannier interpolation method is\nthat once the MLWFs are constructed, interpolations on kmesh of arbitrary\ndensity can be performed with negligible time. Apart from the comput ational\n14Table 1: The Fe chain MCAE and the minimum Wannier interpolat ion kmesh needed for\nconvergence with respect to smearing width σ. The data in this table are plotted in the inset\nof Fig. 5. The MCAEs in rows of No.12 to 15 are directly calcula ted by QE, as indicated by\nQE in the Notes column. While the MCAEs in rows of No.1 to 11 are calculated by Wannier\ninterpolation, as indicated by Wan in the Notes column. The σ= 0.0000 in the row No.11\nrepresents calculation without smearing. The MLWFs used fo r interpolating MCAEs in rows\nNo.1 to 11 are constructed from the ab-initio calculation of row No.13.\nNo.σ(eV)nab\nk MCAE (meV) min nwan\nk Notes\n1 0.2000 24 3 .678 62 72 Wan\n2 0.1700 24 4 .298 54 81 Wan\n3 0.1500 24 4 .698 43 93 Wan\n4 0.1300 24 5 .039 60 113 Wan\n5 0.1000 24 5 .144 34 158 Wan\n6 0.0700 24 3 .857 64 225 Wan\n7 0.0500 24 2 .121 76 319 Wan\n8 0.0200 24 2 .115 46 725 Wan\n9 0.0030 24 2 .358 76 1575 Wan\n10 0.0012 24 2 .358 21 2670 Wan\n11 0.0000 24 2 .357 39 2598 Wan\n12 0.0272 23 3 .262 19 None QE\n13 0.0272 24 1 .766 03 None QE\n14 0.0272 25 1 .239 16 None QE\n15 0.0010 2600 2 .358 67 None QE\n150.000 0.001 0.002 0.003 0.004 0.005 0.006\nSmearing width σ(eV)0.00.20.40.60.8∆MCAE ( ×10−5eV)\naverage of 0.0009to0.0:2.358meV0123\nminnwank(×103)\n0.00 0.05 0.10 0.15 0.20\nSmearing width σ(eV)−10123∆MCAE ( ×10−3eV)\nFigure 5: Fe chain MCAE convergence relative to smearing wid thσ, defined by Equ. (24).\nThe inset is the overview for σfrom 0.20eV to 0 .00eV. The magenta line in the inset is the\nminimum number of Wannier interpolation kmesh nwan\nkneeded for each smearing width σ.\ntime being saved, the computational resources needed for the int erpolation is\nrather small, only 1 CPU core is sufficient for interpolation on arbitrar y kmesh.\nIf simply use\nCost = CPU cores ×Wall time, (25)\nas an estimation of computational cost, the direct ab-initio calculation costs\n864 hour ·core, the total cost of the Wannier interpolation method is 18 hour ·core,\ni.e. we achieved a 48 times speedup.\nWe comment here that the adaptive refinement of kmesh and the ad aptive\nsmearing algorithms frequently used in MLWF interpolation of other p hysical\nproperties [29, 23] are not necessary since it is already enough con venient to\ninterpolate MCAE on arbitrary dense kmesh. For MCAE calculations, smear-\ning should be avoided and the main concern is the quality of MLWFs. Onc e\nhigh-quality MLWFs are constructed, the following interpolation can be easily\nperformed with very high accuracy and negligible cost.\n4. Fe monolayer\nThe prototypical calculations on Fe chain have demonstrated the e ffective-\nness of Wannier interpolation and we have discussed several conve rgence issues\nwhen employing Wannier interpolation to calculate MCAE. In this sectio n, we\nperform calculations on Fe monolayer, which can be directly validated against\n16Table 2: Computational resources spent by direct dense kmes hab-initio calculation and by\nWannier interpolation method in the MCAE calculations of Fe chain. The cost is defined\nby Equ. (25). The Time in the third column is the wall time, and they are the sum of\ncalculations for two magnetization directions. The unit of Cost is hour ·core and is the sum of\nall the calculation steps. The Time = 0 in the last row means th e time spent was less than a\nsecond, negligible.\nCalculations CPU cores Time Cost\ndirectab-initio , kmesh 2600 96 9 .0 h 864\nWannier interpolation\n18a. coarse ab-initio , kmesh 24 96 5 .8 min\nb. projections & overlaps 96 3 .8 min\nc. disentanglement &\nWannierization1 2.5 h\nd. interpolation, kmesh 2600 1 0\nformer results in the literature. In the next section we will show calc ulations\non FeNi alloy, which can be compared with many ab-initio calculations and\nexperimentally measured values.\nThe MCAE of Fe monolayer was calculated as 0 .7 meV based on ultrasoft\npseudopotential (USPP) result [31]. To make a direct comparison, w e used the\nsame structure as in Ref.[31], i.e. (001) Fe monolayer with lattice cons tant 2.85˚A\nand 8 ˚A vacuum separation in the z-direction. The ab-initio calculations were\nperformed using QE based on PAW-GGA. We used a wave function cut off of\n60 Ry and an electron density cutoff of 500 Ry. A Methfessel-Paxto n smearing\nof width 0.05 eV was adopted. To accurately determine MCAE, a kmesh of\n50×50×1 was used in the SCF calculation.\nAs is shown in Fig.6(a), for a direct ab-initio calculation, a kmesh higher\nthan 62×62×1 is needed to converge MCAE under 0 .05 meV, and the converged\nvalue of 0.48±0.05meV (magnetic moment perpendicular to the plane) is similar\nto that of 0.70meV in Ref.[31]. The 0 .20 meV difference between our results and\nthe Ref.[31] may be caused by different type of pseudopotentials.\nThen we constructed MLWF from coarse kmesh ab-initio calculations and it\nwas found that a kmesh of dimension 11 ×11×1 was enough for the convergence\nof MCAE when using Wannier interpolation. The results of the conver gence\ntests are compiled in Table 3. Apparently, from Fig.6(a) we can conc lude that\nab-initio calculations of kmesh around 11 ×11×1 are far from convergence,\nbut from Table 3 rows No.4 to 6, the 11 ×11×1 kmesh ab-initio calculation\nis sufficient for the Wannier interpolation of MCAE. With the help of MLW F,\nequivalent accuracy MCAE value can be extracted from a calculation with only\n3% the number of k-points needed for the converged ab-initio calculation. The\naccuracy of Wannier interpolation is even more convincing when comp aring\nFig.6(a) and (b). Two curves are nearly identical, i.e. the MCAE is faith fully\nrecovered for each kmesh on the horizontal axis.\nAs has been shown in Fig.5 and Table 1, smearing can be used to inhibit\n1710 15 20 25 30 35 40 45 50 55 60 65 70\nab-initio kmeshnab\nk01MCAE (meV)converged MCAE: 0 .48±0.05 meV(a)\n10 15 20 25 30 35 40 45 50 55 60 65 70\nWannier kmesh nwan\nk01MCAE (meV)converged MCAE: 0 .48±0.05 meV(b)\nFigure 6: Fe monolayer MCAE convergence relative to (a) ab-initio kmeshnab\nk, and (b)\nWannier interpolation kmesh nwan\nk. The MLWFs were constructed from ab-initio kmesh\n11×11×1, and it is clear from (a) that this kmesh is far from converge nce if evaluating\nMCAE by direct ab-initio calculation. In fact, a kmesh of 62 ×62×1 is needed for the\nab-initio calculation to converge. The error bars are the convergence indicators obeying the\ndefinition of Equ.(17). To save computational time, we selec tively calculated some of the\nkmeshes when nab\nk>60 in (a).\nMCAE fluctuation and greatly reduce the density of kmesh needed f or the con-\nvergence. However, the fictitious smearing will give rise to deviation s around\nconverged MCAE. In such circumstance, the Wannier interpolation is very help-\nful because we can cheaply increase the density of kmesh without r esorting to\nsmearing. We calculated the MCAE of Fe monolayer with different smea ring\nwidth [Table 3 rows No.7 to 9] and it was found that kmesh should increa se to\nmore than 86 ×86×1 to converge the MCAE. Since we have already used a\nsmall smearing width of 0 .05 eV in the ab-initio calculation, the MCAE hardly\nchanges when reducing the smearing width. One noticeable irregular ity is that\nthe kmesh for convergence decreases when smearing width decre ases. We spec-\nulate that this is caused by small numerical noises rather than a gen eral trend.\nIn all, for the MCAE on the order of sub meV, the Wannier interpolatio n\nsuccessfully recovers high accuracy ab-initio results from coarse kmesh ab-initio\ncalculations.\n18Table 3: The convergence of Fe monolayer MCAE with respect to theab-initio kmeshnab\nk,\nWannier interpolation kmesh nwan\nkand smearing width σ. The MCAEs in rows of No.1 to\n3 are directly calculated by QE, while the MCAEs in rows of No. 4 to 9 are calculated by\nWannier interpolation. The No.10 row is the result of USPP-G GA calculation from Ref.[31].\nThe number nkin columns 2 and 3 stands for kmesh nk×nk×1.\nNo.nab\nknwan\nkσ(eV) MCAE (meV) Notes\n1 10 None 0.05 0 .78 QE\n2 11 None 0.05 0 .39 QE\n3≥62 None 0.05 0 .48±0.05 QE\n4 10 ≥74 0.05 0 .67±0.05 Wan\n5 11 ≥64 0.05 0 .48±0.05 Wan\n6 12 ≥64 0.05 0 .47±0.05 Wan\n7 11 ≥109 0.02 0 .50±0.05 Wan\n8 11 ≥99 0.01 0 .50±0.05 Wan\n9 11 ≥86 0.00 0 .50±0.05 Wan\n10 40 None 0.05 0 .70 USPP[31]\n5. FeNi alloy\nMCAE tends to be larger when symmetry is lower, and the MCAE of sur -\nfaces and interfaces are on the order of meV. Our calculations on F e chain and\nFe monolayer have validated that Wannier interpolation is capable of e fficiently\nreducing the computational costs for MCAE in the region of meV. Th e cal-\nculation of bulk MCAE is much harder since their MCAE are usually on the\norder ofµeV. What is even worse is that in 3 dimensions we need to sample\nthe BZ along 3 directions, this means the number of k-points will increase cu-\nbically when we densify the kmesh for the convergence of MCAE. To f urther\ndemonstrate the usefulness of Wannier interpolation, we perform ed calculations\non FeNi alloy since a practical comparison between our Wannier inter polation\nresults and other ab-initio calculations as well as experimental results can be\nmade. Also, as a demonstration of the versatility of Wannier interpo lation,\nwe useGPAW [45, 46] as the ab-initio calculator in this subsection, since the\nconstruction of MLWF is irrelevant with the kind of the underlying ab-initio\ncode and the Wannier interpolation approach is a post-processing t ool capable\nof cooperating with different ab-initio calculator seamlessly.\nBased on ab-initio calculations and experimental measurements, the MCAE\nof FeNi alloy is in the range of 0 .48 MJ/m3to 0.77 MJ/m3[47, 48, 49, 50]. We\nused the same structure as in Ref.[49, 48], i.e. L10FeNi with in-plane lattice\nconstanta= 3.56˚A and out-of-plane lattice constant c= 3.58˚A. The calculated\ncell is 45◦rotated relative to the conventional cell and the in-plane lattice co n-\nstant isa/√\n2. Theab-initio calculations were performed using GPAW based\non PAW-GGA with 600 eV wave function cutoff and a Fermi-Dirac smear ing of\nwidth 0.1 eV. A kmesh of 25 ×25×17 was used in the SCF calculation and the\nconvergence criterion was set as 1 ×10−6eV.\n19For theab-initio calculation, a kmesh of dimension 19 ×19×13 is needed to\nconverge MCAE [Fig.7 and Table 4 rows No.1 to 3]. The converged MCAE is\n52µeV (magnetic moment along [001] axis), i.e. 0 .37 MJ/m3, a bit smaller than\nthe value in the literature around 0 .5 MJ/m3[47, 48, 49, 50]. Next, we construct\nMLWF based on coarse kmesh ab-initio calculations. It was found that kmesh\nof dimension 12 ×12×8 is sufficient for constructing MLWFs and a kmesh of\n21×21×21 is enough for Wannier interpolation. Due to the limitations of\nGPAW-Wannier90 interface, the quality of the constructed MLWFs was not so\ngood as that of QE. Nevertheless, the Wannier interpolated MCAE o f 0.41±\n0.05MJ/m3[Table 4 row No.6], still successfully recovered high density kmesh\nab-initio result of 0.37±0.05MJ/m3[Table 4 row No.3]. In this case, MLWFs\nenable us to get equivalent accuracy MCAE result from coarse kmes h of 25 %\nthe number of k-points relative to the dense kmesh ab-initio calculation. We\nexpect that by improving the GPAW-Wannier90 interface, the dens ity of the\n12×12×8 kmesh for constructing MLWFs can be further reduced, thus hig her\nefficiency when using Wannier interpolation.\nIn theab-initio calculations we used a large smearing width of 0 .1 eV, next\nwe utilize Wannier interpolation to recover the MCAE without smearing . Upon\ndecreasing smearing width to 0, a Wannier interpolation kmesh of dime nsion\n67×67×67 is needed to converge MCAE [Table 4 row No.7] and the result\nof 0.52±0.05MJ/m3is quite similar to the literature. A 67 ×67×67 kmesh\nmeans that the number of k-points is on the order of 300 000, such a large\nnumber ofk-points poses serious challenges to ab-initio calculations. However,\nsince in MLWF space the number of WFs is on the order of 10 to 100, th e\ndiagonalization of the Hamiltonian matrix is much cheaper thus the eige nvalues\nare efficiently computed. The Wannier interpolation is very helpful wh en it\nrequires an extremely dense kmesh, and the case of FeNi alloy has p roven that\nWannier interpolation is capable of computing even bulk magnetocrys talline\nanisotropy energy.\n6. Conclusions\nWe combine the Wannier interpolation of eigenvalues and the force th eorem\nof magnetocrystalline anisotropy energy together to develop an e fficient and\naccurate method for calculating MCAE. First, coarse kmesh ab-initio calcula-\ntions are performed for different magnetization directions. Secon d, maximally\nlocalized Wannier functions are constructed based on the overlap m atrices and\nprojection matrices obtained from the first step. Third, MCAE is ca lculated\nbased on FT and Wannier interpolation of eigenvalues on a dense kmes h. This\nWannier interpolation method serves as a post-processing step fo r economically\ncalculating MCAE other than brute-force ab-initio calculation. The ultimate\naccuracy of the calculated MCAE is determined by the underlying ab-initio code\n(the choice of exchange-correlation functionals, type of basis se ts, etc.), since it\nis theab-initio code that produces the Bloch functions from which the MLWFs\nare constructed. Nevertheless, the Wannier interpolation is indep endent of the\nchoice of the underlying code, since the only quantities needed from the code are\n2010 12 14 16 18 20 22 24 26 28 30 32\nab-initio kmeshnab\nk0.000.250.500.75MCAE (MJ /m3)\nconverged MCAE: 0 .37±0.05 MJ/m3(a)\n10 12 14 16 18 20 22 24 26 28 30 32\nWannier kmesh nwan\nk0.000.250.500.75MCAE (MJ /m3)\nconverged MCAE: 0 .41±0.05 MJ/m3(b)\nFigure 7: FeNi alloy MCAE convergence relative to (a) ab-initio kmeshnab\nk, and (b) Wannier\ninterpolation kmesh nwan\nk. The MLWFs were constructed from ab-initio kmesh 12 ×12×8,\nand it is clear from (a) that this kmesh is far from convergenc e if evaluating MCAE by direct\nab-initio calculation. In fact, a kmesh of 19 ×19×13 is needed for the ab-initio calculation\nto converge. The error bars are the convergence indicators o beying the definition of Equ.(17).\n21Table 4: The convergence of FeNi alloy MCAE with respect to th eab-initio kmeshnab\nk,\nWannierinterpolation kmesh nwan\nkand smearingwidth σ. TheMCAEsinrowsofNo.1to 3are\ndirectly calculated by GPAW, while the MCAEsin rowsof No.4 t o 7 are calculated by Wannier\ninterpolation. The No.8 to 10 rows are the results of VASP[49 ], WIEN2K[48] and SPR-\nKKR[48] calculations respectively. The No.11 and 12 rows ar e the results of experiments[50,\n47]. The number nkin columns 2 and 3 stands for kmesh nk×nk×int(a\ncnk) where aand\ncare the in-plane and out-of-plane lattice constants, int(x) means truncating real number x\ninto its integer part, i.e. nk×nk×int(a\ncnk) means a uniformly distributed kmesh. The nab\nk\nin rows No.8 to 10 are the total number of k-points in the BZ.\nNo.nab\nknwan\nkσ(eV) MCAE (MJ /m3) Notes\n1 11 None 0.1 0 .34 GPAW\n2 12 None 0.1 0 .15 GPAW\n3≥19 None 0.1 0 .37±0.05 GPAW\n4 10 ≥21 0.1 0 .43±0.05 Wan\n5 11 ≥22 0.1 0 .54±0.05 Wan\n6 12 ≥21 0.1 0 .41±0.05 Wan\n7 12 ≥67 0.0 0 .52±0.05 Wan\n8 19800 0 .56 VASP[49]\n9 20000 0 .48 WIEN2K[48]\n10 160000 0 .77 SPR-KKR[48]\n11 0 .63 Exp.[50]\n12 0 .58 Exp.[47]\nthe overlap and projection matrices which can be obtained merely fr om Bloch\nfunctions.\nFirst, we take a Fe chain as an example and demonstrate that our Wa nnier\ninterpolation approach for MCAE achieves a 48 times speedup compa red to the\ndirect dense kmesh ab-initio calculation, and in this example we discuss three\ncritical factors for successfully getting high accuracy MCAE: the coarseab-initio\nkmesh for constructing MLWFs, the dense Wannier interpolation km esh and\nthe smearing width. It is found that the most important issue is the q uality of\nMLWFs, and we need a sufficiently dense kmesh to reach a reliable MCAE , but\nthis kmesh is still much sparser than direct ab-initio calculation. The Wannier\ninterpolation kmesh can be easily increased and smearing width can be reduced\nto 0. Then, the Fe monolayer and the FeNi alloy example demonstrat e that our\nWannier interpolation approach is capable of calculating MCAE on the o rder\nof meV to µeV, greatly mitigating the computational burden on calculating\ninterface and bulk MCAE. In summary, to get the most accurate MC AE, no\nsmearing should be adopted and the coarse kmesh ab-initio calculation should\nbe fully tested. The convergence relative to the Wannier interpolat ion kmesh\ncan be easily achieved.\nThe cooperations of QE-Wannier90 and GPAW-Wannier90 demonstr ate that\nthe Wannier interpolation approach can work with different ab-initio calculators\nseamlessly. We expect that the Wannier interpolation approach for MCAE can\n22be more useful when the quality of the interfaces between ab-initio calculators\nand Wannier90 are improved. This Wannier interpolation method redu ces the\ncomputational cost significantly and maintains high accuracy simulta neously.\nBesides, it makes it possible for MCAE calculations which are hard to co n-\nverge or unfeasible due to the computational cost, as well as high- throughput\ncalculations to identify material candidates for spintronics devices .\nAcknowledgments\nThis work was supported by the National Natural Science Foundat ion of\nChina [grant numbers 61627813, 61571023]; the International Co llaboration\nProject [grant number B16001]; and the National Key Technology P rogram\nof China [grant number 2017ZX01032101].\nReferences\n[1] G. H. O. Daalderop, P. J. Kelly, M. F. H. Schuurmans, First-princ iples\ncalculation of the magnetocrystalline anisotropy energy of iron, co balt, and\nnickel, Phys. Rev. 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Theoretical calculation suggests the 50% doping with Ce at 4f-site can lead to competitive magnetic anisotropy to that of the champion magnet Nd2Fe14B. Electronic structure calculations are performed using the full-potential linearized augmented plane wave method by inclusion of the spin-orbit coupling and Hubbard (U) interaction in the calculation for the rare-earth elements to get the correct influence of the localized 4f orbitals. Detailed analysis of the magnetic moment and magnetic anisotropy change has been studied by individually inserting the La and Ce atoms at the two inequivalent sites (4g and 4f sites) of the 2-14-B tetragonal structure. Accurate prediction of the total magnetic moment with the orbital contribution in the 2-14-B structure shows the maximum moment for Ce2Fe14B (3.86 µB/f.u. less) compared to Nd2Fe14B. Theoretical analysis confirms that regardless of the anti-parallel spin moment emerging in the Ce atom the complex structure of the Ce substituted compound at 4f-site gives the maximum anisotropy of 2.27 meV/cell with lowering the magnetic moment by 1.26 µB/f.u. compared to the Nd2Fe14B compound. _____________________________ *Electronic mail: rajivchouhan@gmail.com; rajiv.chouhan@usm.edu    I. INTRODUCTION Permanent magnets can be classified due to their high magnetocrystalline anisotropy energy (MAE), and strong magnetization. Because of their superior properties, permanent magnets have wide range of technological applications such as; they are a key component of wind turbines, electric vehicle motors, smartphones, magnetic resonance imaging (MRI) medical machines and much more. Rare-earth (RE) based permanent magnets have high coercivity which is an important factor to maintain the magnetic stability against the demagnetizing field.1 Usage of RE ions in the compound enhanced the coercivity through the interaction of their anisotropic localized (4f) electron clouds with the crystal electric field generated by the surrounding charges.2,3 As a result to the interactions within the compound under the influencing spin-orbit coupling, the magnetic moment of the compound is preferably aligned towards a specific crystalline direction, and therefore, the magnetocrystalline 2  anisotropy energy involved in disrupting the preferred magnetic alignment provides intrinsic stability to the rare earth based magnets. Among the rare-earth based magnets, Nd2Fe14B is still superior4 with its unique combination of high magnetic anisotropy and magnetic moment is still the best among other permanent magnets,5 it is widely known as champion magnet. Following the discovery of Nd2Fe14B,4 the understanding of the basic underlying physics about the atomic origin of the magnetic anisotropy within the structure is still incomplete. It is very clear from the previous studies3,4,5,6,7 that in the 2-14-B structure Nd is the most efficient rare earth element with the dominating axial anisotropy in the P42/mnm tetragonal structure, whereas effective magnetic moment comes from the Fe atoms. In-spite of several studies, it is not very clear how different Wyckoff positions (mainly the two non-equivalent sites 4f and 4g of rare-earth) in the complex geometry of 2-14-B are responsible in maintaining such a high magnetocrystalline anisotropy. Studies8,9 also confirmed that the rare earth sites 4f and 4g have a potential to be replaced with some abundant rare-earth elements (for example Ce, La, Pr) which can lower the cost of the permanent magnet with less or negligible usage of Nd in the compound. The experimental finding8 suggest that the Nd site (4g) strongly prefers the easy axis [001] direction anisotropy at ambient temperature, however the other 4f-site has a tendency to reduce the intrinsic stability by aligning partially towards the planar direction. Henceforth, the substitution mechanism can impact the overall criticality to 50% by usage of non-magnetic La atoms or Ce atoms which has recently been reported to perform mix valence states9,10 of Ce3+ and Ce4+ with the magnetic d-block elements. In addition to this the correct explanation to the total effective magnetic moment (6 µB/f.u.of Nd atoms and 31 µB /f.u. of Fe atoms) are missing needing to be re-defined. This is mainly due to the spin-orbit coupling contribution associated with the individual atoms in the 2-14-B structure.  In this manuscript, we report the detailed theoretical understanding of the origin of magnetocrystalline anisotropy in the Nd2Fe14B structure. The importance of sites (Wyckoff positions 4f and 4g) is discussed and its overall effect to the magnetic anisotropic performance is calculated by the inclusion of La/Ce with site substitution mechanism. We included the onsite correlation (Hubbard U and J parameter) to capture the better spin-orbit coupling effect in the presence of localized 4f-electrons in the rare-earth atoms. Comprehensive theoretical calculations predict that the Cerium substitution at 4f-sites in the tetragonal 2-14-B structure is the optimal position to maintain the MAE 3  (reduce by 3.23%) and the total magnetic moment (reduces by 4.5%) which is almost negligible compared to the Nd2Fe14B compound. The electron charge density difference analysis confirms the involvement of the geometrical arrangement underlying effect to the hybridization of the B atom in the ab-plane with the 4g-sites responsible for enhancing the MAE contribution of the Nd atoms along c-axis. This is also confirmed by the experimental work done by D. Haskelet.al.8 for Nd2Fe14B compound. The c-axis anisotropy in Nd atoms at 4g-site is maintained with the overlapped hybridization occuring due to the presence of light weight boron atoms at 4g-sites which eventually act as a catalyst for localized 4f-electrons of Nd atoms, whereas the insignificant 4f-site when occupied by Ce atoms a mixed valence state of Ce3+ and Ce4+ ions is observed and further confirmed by the magnetic moment distribution analysis.  II. METHODS           We performed the theoretical calculation using the density functional theory (DFT)11 to study the physical properties of 2-14-B type compounds by the site substitution mechanism at RE(4f) and RE(4g) Wyckoff positions, respectively. In this study our focus is to see the effect of individual sites in maintaining the high MAE in the tetragonal P42/mnm 2-14-B structure. In addition to the site effects, we performed the doping process on the two non-equivalent sites with the abundant La/Ce atoms in order to reduce the Nd (criticality) usage in the compound. We have analyzed the variation of the MAE, magnetic moments, charge densities, and density of states (DOS) to understand the change of intrinsic magnetic performance. We utilized the full-potential linearized augmented plane wave (FP-LAPW)12 method, as implemented in the WIEN2k code to study the electronic structure of RE2Fe14B. Calculations are performed using generalized gradient approximation (GGA)13 along with spin-orbit coupling (SOC) and onsite electron correlation (Hubbard U)14 . The Hubbard parameters used here were U = 6.2 eV and J = 0.7 eV. Around muffin-tin region the spherical harmonics with lmax=10 were used to expand the wavefunctions, charges and potentials. The plane wave energy cutoff parameters are set as RKmax = 7.0 and Gmax = 12, with the atomic radii for RE (Nd, La, Ce), Fe, and B as 2.5, 2.25 and 1.67a.u., respectively. The k-space integrations have been performed with 6×6×4 Brillouin zone mesh, which was sufficient for the convergence of total energies (10-6 Ryd), magnetic moments, and the 4f crystal field (CF) splitting of these 68-4  atoms per unit cell systems. We used the force method to extract the MAE by taking the difference between the c-axis and the planar total spin polarized eigenvalues energies given by the expression below: 𝑀𝐴𝐸≈%𝜀'𝑏)⃗+,-./0112+3/4'56−\t\t%𝜀'𝑏)⃗9:-;3<0112+3/4'56 Where, ϵ is the sum of eigenvalues for both spins in corresponding directions and 𝑏)⃗1:-;3< and 𝑏)⃗+,-.-=are easy and planar directions. The negative (positive) values for the corresponding MAE relate to the planar and uniaxial anisotropy. The naming convention used for doping is RE(4f)RE(4g)Fe14B throughout the paper. III. NUMERICAL RESULTS RE2Fe14B type of compounds forms in the complex tetragonal structure with a space group of P42/mnm,3 with challenging 68-atoms per unit cell. This structure contains two inequivalent rare earth’s (RE) sites namely 4f and 4g, surrounded by six inequivalent Fe sites, i.e. Fe(16k1), Fe(16k2), Fe(8j1), Fe(8j2), Fe(4e), Fe(4c), and one B(g) site, respectively as shown in figure-1. The representation of sites can be distinguished by different colors as RE (green as 4f and blue as 4g), Fe (pink with numeric values for 16k1, 16k2, 8j1, 8j2, 4e, and 4c, respectively) and B (black). The lattice parameters used for the calculations are a= 8.80/8.73/8.82 Å and c= 12.20/12.06/12.34 Å for Nd2Fe14B, Ce2Fe14B and La2Fe14B,5,15 respectively. We performed the GGA+U+SOC calculations for the Nd2Fe14B compound to investigate the origin of the c-axis intrinsic magnetocrystalline anisotropy and compared them with the previous available results. Eventually the results reveal that the highest MAE is still governed by the Nd2Fe14B (2.34 meV/unit cell) among all kinds of doping performed with the Ce/La in the compound. Previous studies16 have shown that to accurately capture the localized effect of f-orbital in the rare earth elements a proper value of Hubbard U parameter must be included in the theoretical calculations in addition to the spin-orbital contribution. Henceforth, by including all the desired parameters in the calculations the total effective magnetic moment for Nd2Fe14B is found to be 34.33 µB/f.u. which is very close to the earlier results3 (i.e. 32.5 µB/f.u.). Individual spin (orbital) moments for the Nd(4f) and Nd(4g) sites are 2.81(-1.50) µB and 2.81(-1.50) µB, respectively which are similar, whereas significant differences can be seen in the iron moments. The experimental3 average value of Nd is 2.35µB which is reported by taking the difference between Nd2Fe14B and Y2Fe14B per f.u. whereas, one theoretical study17 reports the average moment of Nd as 2.86(-3.25) µB with the 5  spin orbit coupling. Eventually both the values are inaccurate because taking reference of Y2Fe14B diminishes the effect of the 4f-orbital contribution in Nd2Fe14B, whereas the theoretical value was done with less k-points (5x5x3), which results in bad converged values. The Fe spin moments for the individual sites are tabulated in Table-1, and compared side by side with the available experimental and theoretical results.18,19,20 Our calculated spin magnetic moments for Fe atoms are much closer to the experimental values than other previous calculated values. The calculated results are more accurate because of the inclusion of onsite-correlation and spin-orbit coupling effect in the calculations.  To capture the doping effect of La and Ce atoms in 2-14-B compound, we perform the calculation with substitution at 4f and 4g sites separately. We found a competitive uniaxial magnetocrystalline anisotropy of 2.26 meV/f.u. for the CeNdFe14B compound compared to 2.34 meV/f.u. for the Nd2Fe14B compound, when we substitute the 4f site with the Ce atoms. Surprisingly all substituted structures have shown the uniaxial MAE, which is plotted as a bar graph plot in figure-2. The lowest magnetocrystalline anisotropy is noticed for La2Fe14B (0.7 meV/f.u.) which is trivial because of the presence of non-magnetic atoms, whereas the La substitution at 4f and 4g sites results in a MAE of 1.76 meV/f.u. and 1.63 meV/f.u., respectively. This indicates the importance of the 4g site over the 4f site in terms of the uniaxial MAE caused by Nd atoms in 2-14-B structure, which is also confirmed by a recent element-specific x-ray magnetic circular dichroism (XMCD) experiment.8 Close observation of density of states in figure-3 shows that there are two localized peaks below the Fermi energy for the Nd2Fe14B compound which is mostly coming from the localized 4f-orbital of Nd atoms. A small broadening can be seen in the 4f site DOS compared to the 4g site. When we see the individual DOS of Nd atoms for the 4f and 4g sites in the presence of a non-magnetic La atom doped case, we notice a sharp peak in Nd atomic DOS. In addition to this, a noticeable low energy shift (left below Fermi energy) in Nd density of state occurs when both 4f and 4g sites are occupied by Nd atoms. This is due to the interaction between the Nd atoms of 4f and 4g sites in the presence of Fe atoms in its surroundings. In the case of Ce contained 2-14-B compounds, the Ce electron’s density of states is always aligned in negative directions to the majority density of states of the compound. This opposite spin alignment was also observed in the other permanent magnetic system CeCo5,21 which is because of the formation of mixed valence states of Ce3+ and Ce4+ in the compound. The similar effect of the density of 6  states splitting and broadening in the 4f site due to presence of Nd occupied 4g sites can also be seen in CeNdFe14B, whereas these phenomena don’t occur for Nd atoms at 4g sites.  We found that the non-magnetic La atom in 2-14-B structure in the doped case compared to the pure (La2Fe14B) form has a small magnetic moment (-0.18 and -0.20 µB) in opposite direction to the total magnetic field direction. Similar trend of magnetic moment alignment of Ce atoms (opposite to total magnetic moment of the cell) are observed in Ce2Fe14B, CeNdFe14B and NdCeFe14B structures. Absolute spin moments of -1.16 and -1.18 µB and orbital moments of 0.48 and 0.51 µB are found at 4f and 4g sites. Hence, the individual total magnetic moment of Ce (4f site) and Ce (4g site) in Ce2Fe14B are the same (i.e. 0.68 µB) and are opposite to the total magnetic moment 30.47 µB/f.u.  in the unit cell. In spite of having negative moment, the total magnetic moment of the cell is only decreased by a maximum of 1.56 µB in Ce substitution at 4f site among all other doping of La and Ce atoms. This can be understood by considering the Fe moments (table-1) and its density of states (figure-4) trends in the pure compounds i.e. La2Fe14B and Ce2Fe14B, respectively. Comparing the iron magnetic moment of individual sites of pure La2Fe14B and Ce2Fe14B structures with respect to the Nd2Fe14B structure, we found that there is negligible but small increment in the moment. And because of the presence of 56 Fe-atoms, we cannot see the major effect to the change of total magnetic moments even in the presence of non-magnetic La atoms at RE sites in the 2-14-B system. We performed the electron difference charge density analysis to reveal the importance of Nd occupied 4g site in the 2-14-B structure. Figure-5 shows the two-dimensional electron charge difference plots for La2Fe14B, Ce2Fe14B, Nd2Fe14B and CeNdFe14B along the ab-plane of the unit cell structure. In all the structures, a strong boron-boron hybridization is observed which is further hybridized with the co-axially aligned Fe(16k1) and Fe(4e) atoms. This unique stable pattern of electrons charge densities results in the uniaxial MAE in all the configurations. The lowest MAE structure (La2Fe14B) has no effect on the RE(4f) and RE(4g) sites because of the presence of the non-magnetic La-atoms, however we can see a noticeable change at the 4g sites which are situated along the diagonal direction of the boron-boron hybridized pattern. For all cases, Ce2Fe14B, Nd2Fe14B and CeNdFe14B there are strong influences of a diagonal hybridized boron-boron pattern on the 4g sites, which has introduced the stability in the atomic charge alignment, henceforth making all the structures favorable uniaxial 7  MAE. Whereas the above influence is missing in 4f-sites as they lie away from unique diagonal arrangement of boron atoms.  \n FIG. 1.  (Color online) Crystal structures for RE2Fe14B having 68 atoms per unit cell and space group P42/mnm.   Table 1.  Calculated spin (orbital) magnetic moments for different RE (La/Ce/Nd) and Fe sites, and the total magnetic moment per formula unit (Mf.u.) in 2-14-B structure (in µB)    La2Fe14B LaNdFe14B NdLaFe14B Ce2Fe14B CeNdFe14B NdCeFe14B Nd2Fe14B  GGA+U+SOC [ spin (orbital) magnetic moment] in µB Expt18 others19,20 RE(4f) -0.18 (0) -0.18 (0) 2.80 (-1.34) -1.16 (0.48) -1.16 (0.48) 2.80 (-1.44) 2.81 (-1.50)  2.86 (-3.25)17  RE(4g) -0.20 (0) 2.80 (-1.49) -0.19 (0) -1.18 (0.51) 2.80 (-1.50) -1.18 (0.52) 2.80 (-1.47)  2.85 (-3.08) 17 Fe(16k1) 2.35 2.32 2.32 2.30 2.32 2.34 2.32 2.60 2.15, 2.45 Fe(16k2) 2.43 2.40 2.39 2.38 2.40 2.42 2.39 2.60 2.18, 2.43 Fe(8j1) 2.35 2.32 2.32 2.31 2.32 2.35 2.30 2.30 2.12, 2.33 Fe(8j2) 2.81 2.79 2.79 2.77 2.79 2.80 2.79 2.85 2.74, 2.72 Fe(4e) 2.22 2.20 2.20 2.16 2.20 2.21 2.21 2.10 2.13, 2.44 Fe(4c) 2.54 2.50 2.50 2.48 2.51 2.52 2.50 2.75 2.59, 2.51 B(4g) -0.16 -0.16 -0.15 -0.15 -0.15 -0.15 -0.16   Mf.u. 32.07 33.39 33.46 30.47 32.77 33.08 34.33     \n8    FIG. 2. Magnetocrystalline anisotropy energy (MAE) for various configurations of 2-14-B structures with La/Ce/Nd substitution.     \n   FIG. 3. Calculated site dependent density of states of RE(La/Ce/Nd) atoms for RE(4f) and RE(4g) sites for various configurations in 2-14-B structure.   \n9    FIG. 4. Calculated site dependent density of states of Fe atoms for La2Fe14B, Ce2Fe14B and Nd2Fe14B structures.   \n  FIG. 5. Two-dimensional electron charge density difference plots for La2Fe14B, Ce2Fe14B, Nd2Fe14B and CeNdFe14B structure along projected c-axis (ab-plane).  \n10  IV. CONCLUSIONS In conclusion, we have demonstrated the importance of the RE(4f) and RE(4g) sites in the 2-14-B structure. We theoretically confirmed that 4g site is the suitable replacement for the Nd-atoms occupancy to maintain the uniaxial magnetocrystalline anisotropy in the structure. Cerium doping confirms the maximum MAE of 2.26 meV/f.u. when substituted at 4f sites which is comparable to MAE value of 2.34 meV/f.u. for Nd2Fe14B. The Ce magnetic moment (0.68 µB) is oppositely aligned to the Nd atoms, which also confirms the presence of mixed valence state (Ce3+ and Ce4+) in the 2-14-B structure. The electron charge density difference analysis confirms the involvement of the geometrical arrangement underlying effect to the hybridization of the B atom in the ab-plane with the 4g sites responsible for enhancing the MAE contribution of the Nd atoms along c-axis. V. ACKNOWLEDGEMENT This work is supported by the Critical Materials Institute, an Energy Innovation Hub funded by the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, Advanced Manufacturing Office. The Ames Laboratory is operated for the U. S. 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Franse, D. B. de Mooij, and K. H. J. Buschow, Journal of Magnetism and Magnetic Materials 44, p333-p34 (1984). 16     I.L.M. Locht, Y.O. Kvashnin, D.C.M. Rodrigues, M. Pereiro, A. Bergman, L. Bergqvist, A.I. Lichtenstein, M.I. Katsnelson, A.   Delin, A.B. Klautau,B. Johansson, I. Di Marco, O. Eriksson, Phys. Rev. B 94 085137 (2016). 17Imran Khan, and Jisang Hong, Journal of the Korean Physical Society, 68, 1409 (2016). 18Givord, D., H. S. Li, and F. Tasset, J. Appl. Phys. 57, 4100, (1985).  19Jaswal, S. S., Phys. Rev. B 41, 9697 (1990).   20Szpunar, B., W. E. Wallace, and J. Szpunar, Phys. Rev. B 36, 3782 (1987). 21Rajiv K. Chouhan, and D. Paudyal, Journal of Alloys and Compounds 723 208 (2017). "    },    {        "title": "1806.06234v1.First_principles_prediction_of_sub_10_nm_skyrmions_in_Pd_Fe_bilayers_on_Rh_111_.pdf",        "content": "arXiv:1806.06234v1  [cond-mat.mtrl-sci]  16 Jun 2018First-principles prediction of sub-10 nm skyrmions in Pd/F e bilayers on Rh(111)\nSoumyajyoti Haldar,1,∗Stephan von Malottki,1Sebastian Meyer,1Pavel F. Bessarab,2,3and Stefan Heinze1\n1Institute of Theoretical Physics and Astrophysics,\nUniversity of Kiel, Leibnizstrasse 15, 24098 Kiel, Germany\n2School of Engineering and Natural Sciences, University of I celand, 107, Reykjavik, Iceland\n3ITMO University, 197101, St. Petersburg, Russia\n(Dated: June 19, 2018)\nWe show that stable skyrmions with diameters of a few nanomet ers can emerge in atomic Pd/Fe\nbilayers on the Rh(111) surface. Based on density functiona l theory we calculate the exchange and\nthe Dzyaloshinskii-Moriya interaction as well as the magne tocrystalline anisotropy energy. The later\ntwo terms are driven by spin-orbit coupling and significantl y reduced compared to Pd/Fe bilayers\non Ir(111) as expected since Rh and Ir are isoelectronic 4 dand 5dtransition-metals. However, there\nis still a spin spiral ground state at zero magnetic field. Ato mistic spin dynamics simulations show\nthat a skyrmion phase occurs at small magnetic fields of ∼1 T. Skyrmion diameters amount to 2\nto 8 nm and skyrmion lifetimes are up to 1 hour at temperatures of 25 to 45 K.\nMagnetic skyrmions, localized topologically protected\nspin structures, are promising candidates in the field\nof spintronics and magnetic information storage due to\ntheir unique properties1–3. Based on micromagnetic and\nphenomenological model calculations, skyrmions have\nbeen widely investigated in magnetic materials since the\nlate 1980s.4–7However, the experimental observation\nof skyrmions was achieved only very recently in MnSi\nbulk material.8Since then, researchers have shown that\nskyrmions can be stabilized experimentally in various\nsystems such as bulk crystals,9,10crystal thin films,11,12\nmagnetic multilayers,13,14or ultrathin films.15–17\nTodate, anatomicPd/FebilayerontheIr(111)surface\nisthebeststudiedultrathinfilmsystemwithaspinspiral\nground state and formation of a nanoscale skyrmion lat-\ntice in an external magnetic field.16–23Skyrmions with\nsmall diameters of about 4–6 nm at magnetic fields of\n1–3 T have been observed using low temperature spin-\npolarized scanning tunneling microscopy (STM).16,17\nThe key element of stable N´ eel-type skyrmions with\na unique rotational sense in this ultrathin film is the\nDzyaloshinskii-Moriya interaction (DMI).24,25The DMI\noccurs due to strong spin-orbit coupling (SOC) from the\n5dtransition metal (TM) with the broken structural in-\nversion symmetry at the interface. A critical value of\nDMI (Dc) is needed to stabilize noncollinear spin struc-\ntures.5,7The value of Dcdepends on spin stiffness ( A)\nand effective magnetocrystallineanisotropyconstant( K)\nand is given by Dc∝√\nAK. To stabilize spin spirals or\nskyrmionlattices, the DMIhastobe largerthan Dc. One\napproach to achieve this goal is to create interfaces of 3 d\nand 5dTM layers with large SOC. Another approach is\nto consider 3 d/4dinterfaces to reduce the value of Kand\nthus the value of Dc. However, since DMI is also reduced\nfor a 3d/4dinterface due to a lower SOC, very good con-\ntrol of magnetic interactions is needed for this approach\nto be successful. Recently Herv´ e et al.26have reported\nexperimental realization of skyrmions with radii of about\n40–50 nm in Co monolayers on Ru(0001) in the limit of\nalmost vanishing magnetic anisotropy.\nHere,westudyanatomicPd/FebilayerontheRh(111)surface using density functional theory (DFT), atom-\nistic spin dynamics and harmonic transition state the-\nory. This system is similar to Pd/Fe/Ir(111) since Rh\nis the 4dTM isoelectronic to Ir. Moreover, Fe/Rh(111)\npseudomorphic films have already been experimentally\ngrown.27We find that the exchange interactions are very\nsimilar to Pd/Fe/Ir(111) while the DMI and Kare sig-\nnificantly reduced since SOC is weaker in Rh. However,\nthe magnetocrystalline anisotropy energy (MAE) is af-\nfected much more strongly than the DMI because it is a\nsecond-order perturbation term in SOC while the DMI\narises in first order. This leads to a spin spiral ground\nstate at zero magnetic field as in Pd/Fe/Ir(111)and with\na very similar period. From DFT we parameterize an ex-\ntended Heisenberg model which we study by atomistic\nspin dynamics. The zero temperature phase diagram\nshows transitions between the spin spiral ground state,\nskyrmion lattice, and ferromagnetic state under external\nmagnetic fields on the order of 1 T. The skyrmion diam-\neters of isolated skyrmions amount to 2 to 8 nm. The\ncalculated energy barriers and lifetimes for collapse of\nisolated skyrmions demonstrate their stability at up to\n25 K to 45 K depending on Pd overlayer stacking.\nWe have used DFT as implemented in the fleur\ncode28to study the electronic and magnetic properties\nof Pd/Fe/Rh(111). We have used the fcc stacking of the\nFe ML on Rh(111). For the Pd overlayer, we have con-\nsidered both fcc and hcp stacking. In orderto investigate\nthe ground state and magnetic interactions in this sys-\ntem, wecalculatedthetotalenergyE( q)ofhomogeneous,\nflat spin spirals along the high symmetry directions Γ–\nK andΓ–M in the two-dimensional Brillouin zone (2D\nBZ) with and without SOC.29–31Spin spirals are charac-\nterized by the wave vector qin the 2D BZ with a con-\nstant angle ( φ=q·R) between two magnetic moments\nat adjacent lattice sites separated by R. We calculated\nthe MAE using the force theorem starting from a self-\nconsistent scalar-relativistic calculation and constraining\nthespinquantizationaxisalongin-planeandout-of-plane\ndirections in calculations with SOC (see Supplemental\nMaterial for details).2\n−MΓ\n2468\n0\n-0.2-0.1 0.10.2 0.0\nq-Vector( )a2πCycl oidal S pinSpiraldirection\nFe\nRh(111)fcc-Pd\ntotal-202468Righ trotating Left rota ting\n-0.2 -0.1 0.00 .1 0.2-2-1012(meV/Fe- atom)\nPd\nFe\nRh−KΓλminwithout\nSOCwith\nSOC\n(a)\n(b)∆Esoc(q) (me V/Fe-atom) E(q) - E FM\nFigure 1. (a) Energy dispersion E( q) of homogeneous cy-\ncloidal flat spin spirals for fcc-Pd/Fe/Rh(111) without (gr een\ndots) and with SOC (red dots) in Γ−K high-symmetry direc-\ntionfor bothsenses ofrotation. Theenergyis givenrelativ e to\ntheferromagnetic state. Thedispersion isfittedtotheHeis en-\nberg model (green line) and includes the DMI and MAE (red\nline). Inset is the energy dispersion along the Γ−M direction.\n(b) Total and element resolved contributions of ∆ ESOC(q)\nupon including SOC.\nFig.1(a)displaysthecalculatedenergydispersionE( q)\nof spin spirals along the Γ−K high-symmetry direc-\ntion for fcc-Pd/Fe/Rh(111). First, we discuss the results\nneglecting SOC. We observe that the energy dispersion\nhas a minimum at the Γ point, i.e. the ferromagnetic\n(FM) state, and is degenerate for right-( q >0) and left-\nrotating ( q <0) spirals. For larger values of |q|, the\nenergy rises due to the FM nearest-neighbor (NN) ex-\nchange interactions. A similar energy dispersion is ob-\ntained for the Γ−M direction (inset of Fig. 1(a)) and for\nhcp-Pd/Fe/Rh(111) (see Fig. S1).\nFor both Pd stackings we obtain an easy magnetiza-\ntion axis along the out-of-plane direction for the ferro-\nmagnetic state due to SOC. The values of the MAE are\nK=−0.17 and−0.31 meV/Fe-atom for fcc and hcp\nstacking, respectively. The inclusion of SOC for spin spi-\nrals induces DMI in this system due the broken inversion\nsymmetry at the 3 d/4dinterface.24,25DMI leads to non-\ncollinearspinstructureswithcantingofthemagneticmo-\nments. The energy contributions due to the DMI along\nwith the element decomposition are shown in Fig. 1(b)\nfor fcc-Pd/Fe/Rh(111). We observe that the major DMIcontributions stem from the Rh surface together with a\nminor contribution from the Pd overlayer. Incorporating\nthe DMI, the energy dispersion has a minimum value for\na homogeneous cycloidal flat spin spiral state with a spe-\ncific rotational sense.32As can be observed in Fig. 1(a),\nan energy minimum of −0.33 meV/Fe-atom compared\nto the FM state occurs for a right rotating spin spiral\nwith a pitch of λ= 2π/q= 4.8 nm. Note that there\nis an energy shift of K/2 for spin spirals with respect\nto the ferromagnetic state due to the MAE which favors\na collinear magnetic state. For hcp-Pd/Fe/Rh(111) we\nfind an energy minimum of −0.22 meV/Fe-atom for a\nright rotating spin spiral with a pitch λ= 6.7 nm. (see\nFig. S1).\nWe use the atomistic spin model given by\nH=−/summationdisplay\nijJij(mi·mj)−/summationdisplay\nijDij·(mi×mj)\n+/summationdisplay\niK(mz\ni)2−/summationdisplay\niµs(B·mi). (1)\nto analyze the magnetic properties of Pd/Fe/Rh(111).\nEq. (1) describes the magnetic interactions of spins Mi\nandMjbetween two Fe atoms at sites RiandRj, re-\nspectively, where mi=Mi/µs. The parameters for the\nexchange interactions ( Jij) and the DMI ( Dij) are ex-\ntracted for both fcc and hcp stacking of the Pd overlayer\nfrom the fitting of DFT energy dispersion curves calcu-\nlated without SOC and with SOC, respectively. The\nmagnetic moments ( µs) and MAE ( K) are also calcu-\nlated using DFT for both stackings. The parameters Jij,\nDij,K, andµsare given in Table I.\nThe exchange constants suggest a strong frustration\ndue to competition of ferromagnetic NN exchange and\nantiferromagnetic for second and third NN exchange in-\nteractions. The values and their variation with the Pd\noverlayer stacking are quite similar to those reported for\nPd/Fe/Ir(111)33. This is due to Rh being the isoelec-\ntronic to Ir which leads to a similar hybridization with\nthe Fe layer. Since the spin-orbit coupling constant ξ\nstrongly increases with nuclear charge the DMI and the\nMAE are much smaller for Pd/Fe bilayers on Rh(111)\nthan on Ir(111). However, the MAE is reduced much\nmore since it arises in second order perturbation the-\nory with ξwhile the DMI occurs in first order. The\nratios of the MAE and the DMI between the two sys-\ntems are quite consistent with this expectation. Taking\nthe values of table I and Ref. 33 we find for the MAE\nKPd/Fe/Rh/KPd/Fe/Ir≈0.39 and 0 .24 and for the DMI\nDPd/Fe/Rh\neff/DPd/Fe/Ir\neff≈0.73 and 0 .62 for hcp and fcc\nstacking, respectively. Deviations from the simple scal-\ning can be due to the contribution of both the Pd and\nthe Rh (Ir) interface to the total DMI.\nTo investigate the zero temperature magnetic phase\ntransitionsin Pd/Fe/Rh(111)in the presenceofan exter-\nnal magnetic field, we have used atomistic spin-dynamics\nsimulations using the model described by Eq. (1) (see\nSupplementalMaterial). Fig.2(a,b)displaythezerotem-3\nTable I. Exchange constants ( Jij), Dzyaloshinskii-Moriya interaction parameters ( Dij), magnetocrystalline anisotropy ( K) and\ntotal magnetic moment ( µs) obtained from DFT for fcc- and hcp-Pd/Fe/Rh(111). The posi tive sign of Ddenotes right rotating\nspin spirals. The negative sign of Kindicates an easy out-of-plane magnetization direction. T he parameters are given in meV.\nJ1J2J3J4J5J6J7J8J9J10J11D1D2D3D4D5Kµs\nfcc +13.35−2.69−2.84 +0.62 +0.46−0.10−0.30−0.11 +0.03 +0.17−0.06+0.62−0.04 +0.04 +0.00 +0.05−0.173.2\nhcp +12 .23−1.18−2.78 +0.27 +0.40 +0.03−0.14−0.10 +0.00 +0.12−0.06+0.87 +0.02−0.09 +0.02 +0.04−0.313.1\n-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4\nSkXSS\nFM\n-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4\n 0  0.5  1  1.5  2  2.5  3  3.5SkXSS\nFMEnergy w.r .t th e homog enious SS  (meV / Fe atom )\nMagnetic Field (T)(a)\n(b)Fe\nRh(111)fcc-Pd\nFe\nRh(111)hcp-Pd\nFigure 2. Zero temperature phase diagram for (a) fcc and (b)\nhcp Pd overlayer stacking on Fe/Rh(111) obtained based on\nDFT parameters for the magnetic interactions. The energies\nof the FM, skyrmion lattice (SkX) and spin spiral (SS) states\nare shown relative to the homogeneous spin spiral (zero line ).\nBlue, red, and green color represents the regime of the SS,\nSkX, and FM ground state, respectively.\nperature phase diagrams using the parameters extracted\nfrom DFT for fcc- and hcp-Pd/Fe/Rh(111), respectively.\nForboth stackings, the ground stateat zeroapplied mag-\nnetic field is a spin spiral consistent with the energy min-\nimum and the pitch λobserved in the DFT energy dis-\npersion curves (cf. Fig. 1 and Fig. S1). At a certain\ncritical field value of ≈1.06 T (0.82 T) for fcc (hcp)\nstacking, the skyrmion lattice is energetically favorable.\nFor a larger critical field value of ≈2.76 T (1.74 T) for\nfcc (hcp), the skyrmion lattice phase changes to the FM\nphase. These values are on the order of the fields at\nwhich skyrmions have been observed experimentally for\nPd/Fe/Ir(111)16,17.\nIn our simulation, isolated skyrmions are metastable\nin the FM phase. We obtain their profiles starting from\na theoretical profile34and then relaxing the spin struc-\nture using spin dynamics. For both fcc and hcp stacking,\nwe observe skyrmions with radii ≈3–5 nm for magnetic\nfields of 1–2 T (see Fig. 3(a)). The radii decrease rapidlyMagnetic Fiel d (T)Energy b arrier (meV)(a)\n(b)fcc-Pd/Fe/Rh\nhcp-Pd/Fe/ Rh\nhcp-Pd/Fe/Irfcc-Pd/Fe/Ir 0 1 2 3 4 5 6 7 8 Skyrmion Radius (nm)\n 0 60 120 180\n 0  1  2  3  4  5  6  7  8DKSkyrmion Radius (nm)\nK (meV)D (meV )\n 0.5 1 1.5 2 2.5 3\n 0  0.4  0.8  1.2 0  0.4  0.8  1.2\nB = 4  T\nFigure 3. (a) Radii of skyrmions in Pd/Fe/Rh(111) obtained\nfor DFT parameters as a function of the applied magnetic\nfield. The radii of the skyrmions were computed as in Ref. 34.\nInset shows the radii variations as a function of K and D\nfor a fixed B = 4.0 T. (b) The energy barriers of isolated\nskyrmion collapse based on DFT parameters are shown as a\nfunction of applied magnetic field for Pd/Fe/Rh(111). For\ncomparison the radii and energy barriers for Pd/Fe/Ir(111)\nare also shown33.\nwith increasing value of applied magnetic field and are\nlarger for fcc than for hcp stacking. These values are\nvery similar to those observed in Pd/Fe/Ir(111) from ex-\nperiment16and theory.33This may seem surprising at\nfirst glance since SOC is much smaller for Rh and both\nDMI and MAE are reduced significantly in our system.\nIn order to understand the origin of similar radii val-\nues, we have computed the radii for isolated skyrmions\nwith given DFT exchange constants and either with the\nfixed MAE value but varying the nearest-neighbor DMI\nvalueorwith fixedDMI from DFT but varyingthe MAE.\nAs shown in the inset of Fig. 3(a), the skyrmion radius\ndecreases with reduced DMI but it rises with reduced\nMAE. This observation is consistent with micromagnetic\ncalculations.35Since both, DMI and MAE, are reduced4\nFigure 4. Lifetime of skyrmion collapse in (top) fcc-\nPd/Fe/Rh(111) and (bottom) hcp-Pd/Fe/Rh(111) obtained\nin harmonic transition state theory based on DFT parame-\nters as a function of magnetic field and temperature.\nfor Pd/Fe on Rh(111) vs. on Ir(111) the skyrmion radii\nremains almost unchanged.\nWe have calculated the minimum energy path for the\ncollapse of isolated skyrmions in the FM background us-\ning the geodesic nudged elastic band (GNEB) method.36\nThe obtained energy barriers [Fig. 3(b)] decrease with\nmagnetic field and display a nonlinear behavior for both\nstackings of Pd on Fe/Rh(111). The values of the energy\nbarriers are in between those reported for fcc and hcp Pd\non Fe/Ir(111).33Note that using the effective nearest-\nneighbor model (see Supplemental Material) leads to a\nlarge reduction of the obtained energy barriers and re-\nduced skyrmion stability as previously reported.33\nThe decomposition of the energy barrier for skyrmion\ncollapse in both fcc and hcp Pd stacking with respect to\ndifferent interactions [see Fig. S4] shows that the DMI\nis mainly responsible for skyrmion stabilization. How-\never, we also find that the exchange energy leads to an\nadditional energy barrier which is caused by exchangefrustration (cf. table I) and is larger than for hcp Pd\nstacking(see Fig. S4). This automaticallyleads to higher\nenergybarrierheightsobservedforfcc thanforhcpstack-\ning in Fig. 3(b) at a given magnetic field. The effec-\ntive exchange parameters fail to describe the exchange\nenergy barrier properly (see Fig. S5). The mechanism\nof skyrmion collapse is similar to that reported ear-\nlier.33,36–43The skyrmion starts to shrink along the path\nand at the saddle point all the spins are nearly planar.\nThe topological charge vanishes at this point and the\nskyrmion gradually collapse into the FM state.\nThe skyrmion lifetime τdepends exponentially on\nthe energy barrier, ∆ E, and temperature, T, asτ=\nτ0exp(∆E/kBT).44Typically, the pre-factor τ0is as-\nsumed to be a constant estimated value, which in some\ncases could be too crude approximation.45–48We have\nexplicitly calculated τ0within harmonic transition state\ntheory44,48and thereby τas a function of temperature\nandexternalmagneticfield [Fig. 4]. We find that isolated\nskyrmions should be stable in fcc-Pd/Fe/Rh(111) and\nhcp-Pd/Fe/Rh(111)up to hoursat temperatures ofup to\n25 K and 45 K, respectively. Hence, these skyrmions can\nbe observed experimentally e.g., in STM experiments.\nHowever, STM experiments on skyrmions in ultrathin\nfilms have only been performed at low temperatures of\nabout8K16,17,21,23andthestabilityofskyrmionsinthese\nsystems remains an open experimental issue. We predict\na finite skyrmion lifetime for both fcc and hcp [cf. Fig. 4]\nstacking of the Pd overlayer on Fe/Rh(111) making ex-\nperiments on these systems promising.\nIn conclusion, we predict the formation of stable iso-\nlated skyrmions with diameters of a few nanometers at\na 3d/4dtransition-metal interface. Our approach starts\nfrom density functional theory to obtain first-principles\nparameters of an atomistic spin model which is solved by\nspin dynamics simulations. Using the geodesic nudged\nelastic band method and transition state theory enables\nus to obtain quantitative results for the stability of iso-\nlated skyrmions, i.e. energy barriers as well as lifetimes.\nThis allows us to explore novel skyrmion systems which\nmay guide future experimental work.\nThis work was supported by the European Unions\nHorizon 2020 research and innovation program under\ngrant agreement No 665095 (FET-Open project MAGic-\nSky). S.J.H. and S.H. acknowledge the DFG via SFB677\nfor financial support. P.F.B. acknowledges support from\nthe Icelandic Research Fund (Grants No. 163048-053,\n184949-051). We gratefully acknowledge the computing\ntime at the supercomputer of the North-German Super-\ncomputing Alliance (HLRN).\n∗Corresponding author: haldar@physik.uni-kiel.de\n1A. Fert, V. Cros, and J. Sampaio, Nat. Nano. 8, 152\n(2013).\n2R.Wiesendanger,Nature Reviews Materials 1, 16044 (2016).\n3N. S. Kiselev, A. N. Bogdanov, R. Sch¨ afer, and U. 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Delin, Scientific Reports 8, 3433\n(2018)."    },    {        "title": "1806.11339v1.Stability_and_magnetic_properties_of_Fe_double_layers_on_Ir__111_.pdf",        "content": "Stability and magnetic properties of Fe double-layers on Ir (111)\nMelanie Dup ´e,1Stefan Heinze,2Jairo Sinova,1, 3and Bertrand Dup ´e1, 2\n1INSPIRE Group, Institute of Physics, Johannes Gutenberg-Universit ¨at, 55128 Mainz, Germany\n2Institute of Theoretical Physics and Astrophysics, Christian-Albrechts-Universit ¨at zu Kiel, 24098 Kiel, Germany\n3Institute of Physics ASCR, v.v.i., Cukrovarnicka 10, 162 53 Praha 6 Czech Republic\n(Dated: November 7, 2019)\nWe investigate the interplay between the structural reconstruction and the magnetic properties of Fe double-\nlayers on Ir (111)-substrate using first-principles calculations based on density functional theory and mapping\nof the total energies on an atomistic spin model. We show that, if a second Fe monolayer is deposited on Fe/Ir\n(111), the stacking may change from hexagonal close-packed to bcc (110)-like accompanied by a reduction of\nsymmetry from trigonal to centered rectangular. Although the bcc-like surface has a lower coordination, we find\nthat this is the structural ground state. This reconstruction has a major impact on the magnetic structure. We\ninvestigate in detail the changes in the magnetic exchange interaction, the magnetocrystalline anisotropy, and\nthe Dzyaloshinskii Moriya interaction depending on the stacking sequence of the Fe double-layer. Based on our\nfindings, we suggest a new technique to engineer Dzyaloshinskii Moriya interactions in multilayer systems em-\nploying symmetry considerations. The resulting anisotropic Dzyaloshinskii-Moriya interactions may stabilize\nhigher-order skyrmions or antiskyrmions.\nPACS numbers: 73.20.-r, 71.15.Mb, 75.70.Ak\nKeywords: density functional theory, spintronics\nI. INTRODUCTION\nThe next generation of high-density and low-energy data\nstorage devices or neuromorphic computing based units will\nrequire novel materials and phenomena. Skyrmions in mag-\nnetic materials have high potential to meet the demands for\nthese new technologies.1,2In condensed matter, magnetic\nskyrmions were predicted and first studied based on continu-\nous micromagnetic models.3,4Their existence was confirmed\nexperimentally in bulk and thin-film semiconductors,5,6in\nmetallic multilayers7,8and ultra-thin films.9,10The presence\nand the manipulation of isolated skyrmions in magnetic thin-\nfilms and multilayers make them promising for technological\napplications such as the race-track memory.11–14\nThe presence of isolated skyrmions is attributed to\nthe Dzyaloshinskyi-Moriya interaction (DMI) which occurs\nwhere spatial inversion symmetry is broken.15,16In B20 com-\npounds such as MnSi5or Fe 0:5Co0:5Si6this symmetry is\nbroken due to the crystal lattice whereas in the multiferroic\nCu2OSeO 317the polarization breaks inversion symmetry. At\nsurfaces and interfaces, the inversion symmetry is broken due\nto the interface between different materials.\nThe DMI originates from spin-orbit coupling (SOC). At\nmetal surfaces and interfaces, the DMI can be understood via\nthe model of Fert and Levy,18which gives a general direction\nto control the DMI. In ultra-thin films, DMI can be engineered\nby combining 3dtransition metals with 4dor5dsubstrates,\nwhich provide large SOC.19–21In magnetic multilayers, two\ninterfaces can be used to control different magnetic interac-\ntions. One interface can be used to tune the magnetic ex-\nchange while the other one can generate a large DMI.21When\nboth interfaces are composed of 5dmetals, the contribution of\neach interface can be engineered to obtain a giant DMI.7,22\nIn ultra-thin films, not only the DMI can be tuned via the\ninterface but also the magnetic exchange interactions, which\nmake them an ideal playground to study magnetism.23Amongthem, the Fe monolayer on Ir (111) has attracted particular at-\ntention due to its versatility. If the Fe atoms are adsorbed on\nthe fcc surface sites, the magnetic ground state is a square\nlattice of skyrmions.9If Fe is adsorbed in the hcp stacking,\nthe ground state is a hexagonal lattice of skyrmions.24If two\nmonolayers of Fe are deposited on Ir (111), the growth is\nnot epitaxial anymore but results in a complex reconstruction\nleading to a mixture of fcc, hcp and bcc-stacking of the second\nFe-layer characterized by a certain pattern of reconstruction\nlines.25,26These reconstruction lines play a prominent role in\nthe triple layer of Fe on Ir (111) for the writing and deleting\nof skyrmions by applying an electric field.27In this system,\nthe surface reconstruction stabilizes skyrmions with an oval\nshape, which was attributed to an environment anisotropy.28\nRecently, it was found that the symmetry of the interface and\nthereby the symmetry of the DMI could also determine the\ntype of skyrmions that can be stabilized, i.e. skyrmions or\nantiskyrmions.29\nIn the case of an fcc (100) or an fcc (111) interface, the\nsymmetry of the interface imposes that the DMI has the same\nsign along each neighboring bond. This configuration fa-\nvors the presence of skyrmions and explains why Pd/Fe/Ir\n(111) exhibits isolated skyrmions.10,30,31In the case of a bcc\n(110) interface, the sign of the DMI may change depending\non the nearest neighbor bonds.29,32Therefore, antiskyrimons\nmay be more stable, as was illustrated in the case of 2Fe/W\n(110).29Independently from the symmetry argument, it was\nalso shown that skyrmions and antiskyrmions may coexist in\nthe case of frustrated exchange interaction.21,30,33–36There-\nfore, an accurate theoretical description of all magnetic in-\nteractions is required.\nHere, we study the double-layer of Fe on Ir (111) (2Fe/Ir\n(111)) via density functional theory (DFT) with a particu-\nlar focus on the different stackings of the two Fe layers of\n2Fe/Ir (111). We base our DFT calculations on the experi-\nmental observations of certain structural phases.25,26First, aarXiv:1806.11339v1  [cond-mat.mtrl-sci]  29 Jun 20182\npseudomorphically strained double-layer with fcc-stacking of\nthe surface Fe-layer was identified, which exhibits spin spi-\nrals of short periodicity of about 1.2 nm without a preferred\npropagation direction, indicated by a grainy contrast in the\nspin-polarized scanning tunneling microscopy (SP-STM) im-\nages. These areas seem to be prone to defects such as vacan-\ncies and substitutional atoms.37Second, reconstructed areas\nwere suggested with differently oriented bcc domains sepa-\nrated by reconstruction lines with a characteristic distance of\n5.2 nm. The reconstruction lines compensate for the lattice\nmismatch between the Fe layer at the interface, which has\na fcc (111)-structure, and the Fe layer at the surface which\nadopts the bcc (110)-structure. In the bcc domains spin spi-\nrals with periodicities of about 1.9 nm were observed which\npropagate only along the [100] directions of the bcc-unit cells\nand thus, in the presence of the reconstruction lines and dif-\nferent domain orientations, give rise to a characteristic zig-zag\nshaped herringbone-like magnetic contrast with pearls along\nthe spin spiral propagation directions.\nWe show that, counterintuitively, the second monolayer\ndoes not grow in the fcc or hcp absorption site but in the bcc\nabsorption site. This induces a reduction of the crystal sym-\nmetry of 2Fe/Ir (111) which loses the 3-fold rotation axis. We\ncalculate the magnetic exchange interaction for each of the\nstackings of the Fe second layer and show that the frustration\nof exchange interaction varies considerably from one stacking\nto the other. Our DFT-parameterized atomistic spin model in-\ncludes both intra- and inter-layer exchange interactions. Then,\nwe compute the DMI for each of the stackings and show that\nthe interfacial symmetry does not impose the symmetry of the\nDMI alone. Finally, we deconvolute the DMI contribution of\neach of the layers and infer a new method based on symme-\ntry considerations to obtain an anisotropic DMI which may\nstabilize antiskyrmions or higher order skyrmions.\nThe paper is organized in two parts. The first part is ded-\nicated to the methodology used to compute the magnetic ex-\nchange interaction and the DMI. The second part presents the\nresults regarding the different magnetic ground states of 2Fe/Ir\n(111) depending on the double-layer stacking.\nII. MODEL AND COMPUTATIONAL METHODS\nA. Stacking of the Fe double-layers\nWe want to study the effect of simple variations of stack-\ning sequences in the Fe double-layer deposited on an Ir (111)\nsurface and how their structural differences influence the sta-\nbility and magnetic interactions between the Fe-atoms in the\ndouble-layer. We chose double-layer structures which are de-\nrived from the metallic bulk structures fcc, hcp and bcc.\nThe fcc structure consists of hexagonal close-packed lay-\ners in the (111)-plane, where every atom has six equidistant\nnearest neighbours within the plane as shown in Fig. 1(a).\nThese close-packed layers follow an ABC-stacking sequence\nperpendicular to the plane as shown on the right, where the\nnumbers indicate the x- and y-coordinates of the atoms in the\ndifferent layers. Each layer occupies a set of hollow sites ofthe sub-layer.\nAlso the hcp structure in Fig. 1(b) is formed by hexagonal\nclose-packed layers, which correspond to the (0001)-planes\nof the hexagonal unit cell. In hcp, the layers follow an ABA-\nstacking sequence. The hollow sites C remain empty in this\ncase.\nThe third stacking-type we include in our study, is the\npseudomorphically strained bcc (110) structure. In contrast\nto the fcc and hcp structures, the bcc bulk structure usually\ndoes not possess any close-packed crystallographic planes.\nHowever, the (110)-plane of the bcc unit cell can be (consid-\nerably) strained ( \u000fxx=\u000010:7%,\u000fyy= +9:6%) in order to fit\nthe same hexagonal unit cell as the fcc and hcp structures, as\nindicated in white. The main difference is that the Fe-atoms\ndo not occupy the hollow site positions B or C, but the bridge\npositions marked D, giving rise to the stacking sequence\nADA. Fe atoms in this position have a reduced coordination\nnumber as they possess only two nearest neighbours in the\nplane below instead of three.\nTo characterize the stacking sequences in the Fe double-\nlayer with respect to the fcc-stacked Ir (111) substrate,\nthroughout this paper, we use a modified h-f stacking se-\nquence notation, borrowed from the description of close-\npacked (bulk) crystal structures. In contrast to the original h-f\n(a) fcc (111) \nB BB\nCC C\nAC2/3\nB1/3\nA 0\n(b) hcp (0001)\nBB\nB\nAB1/3\nA 0A 0\n(c) pseudomorphically strained bcc (110)\nDD\nDD\nAD1/2\nA 0A 0\nFIG. 1. (Color online) Given are the stacking sequences for the bulk\ncase for structures (a) fcc, (b) hcp and (c) pseudomorphically strained\nbcc. All layers are close-packed in this scenario.3\ncm p3m1Ir@FeFe@IrFe@vac\nglide reflectionreflection\nrotation centerfb* ff\nFIG. 2. (Color online) Top views of the atomic configurations of\nfilms with stackings fb\u0003and ff and symmetry elements of their re-\nspective plane groups c mand p 3m1. Shown are the atoms of the\nthree outmost layers, only.\nstacking sequence notation, where the symbol always refers\nto the middle layer of the sequence-triple, i.e. ”f” for the B-\nlayer in the ABC-sequence of the fcc structure or ”h” for the\nB-layer in the ABA-sequence of the hcp-structure, our sym-\nbols refer to the top layer of the sequence-triple, as we want\nto characterize surface structures. In addition to f and h, we\nintroduce the stacking sequence b\u0003to indicate a pseudomor-\npically strained bcc-like top-layer.\nIn Tab. I we give an overview of the stacking sequences\nthat we have studied. The first symbol indicates the stacking\nsequence of the Fe atom at the interface Fe@Ir and the second\nsymbol the stacking sequence of the Fe atom at the surface\nFe@vac. For Fe@Ir only stackings f and h were considered,\nwhile for Fe@vac also b\u0003was taken into account. Besides\nthe stacking sequence in ABC-notation also the coordination\nnumbers (CN) of the two Fe-atoms are given, demonstrating\nthat coordination numbers are reduced by one, if the top layer\nadopts the bcc-like structure. Two different symmetries result\nin the close-packed structures ff, fh, hf and hh, we find the\ntrigonal plane group (PG) p 3m1and in the bcc-like structures\nfb\u0003and hb\u0003the centered rectangular/rhombic plane group c m.\nFigure 2 illustrates these symmetries. In an isolated double-\nlayer, the symmetry is higher in the bcc-like stackings (c 2mm\ninstead of cm), but unchanged in the others.\nTABLE I. Overview of the double-layer structures: Given are the\nnames and stacking sequences in ABC-notation, where the stacking\nof the substrate is given in parentheses. Also indicated are the co-\nordination numbers (CN) of the Fe atoms in the two layers of the\ndouble-layer, as well as the plane groups (PG) of the isolated double\nlayer and the full film.\nStacking CN CN PG PG\nName sequence Fe@Ir Fe@vac double-layer film\nff (ABC) AB 12 9 p 3m1 p3m1\nfh (ABC) AC 12 9 p 3m1 p3m1\nfb\u0003(ABC) AD 11 8 c 2mm cm\nhf (ABC) BA 12 9 p 3m1 p3m1\nhh (ABC) BC 12 9 p 3m1 p3m1\nhb\u0003(ABC) BD 11 8 c 2mm cmTABLE II. Inter-layer distances for the outmost three layers in the\nrelaxed structures. Values are given in ˚A.\nff fh fb\u0003hf hh hb\u0003\ndFe-Fe 2.02 2.08 2.01 2.01 2.08 2.02\ndFe-Ir 2.09 2.11 2.12 2.09 2.11 2.11\ndIr-Ir 2.25 2.25 2.24 2.23 2.23 2.23\nB. Stability of the stackings\nWe study via density functional theory (DFT) calcula-\ntions the energies and magnetic interactions of the six differ-\nent structural stackings presented in Table I. We have used\nthe F LEUR ab initio package.38The F LEUR code utilizes\nthe full potential linearized augmented plane wave approach\n(FLAPW),39–41which ranks among the most accurate elec-\ntronic structure techniques. Especially, F LEUR allows us\nto study complex magnetic states at interfaces such as non-\ncollinear magnetic states,42skyrmion lattice ground states9\nand the presence of isolated topologically protected states in\nbilayers such as skyrmions or antiskyrmions.21,29,30,34\nTo optimally describe the geometry of the Fe/Ir interface, a\nmixed exchange correlation functional was employed.43This\nmixed functional applies the generalized gradient approxi-\nmation in the parameterization of Perdew et al. (GGA)44to\nthe interstitial region and to the muffin-tin spheres of the Fe\natoms, whereas in the Ir atoms’ muffin-tin (MT) spheres the\nlocal density approximation (LDA)45is applied. This method\nhas been shown to capture the magnetic and structural proper-\nties of the 3delements on 5dsubstrates.46–48\nThe muffin-tin radii were set to 2.23 bohr (1.18 ˚A) for Fe\nand 2.31 bohr (1.22 ˚A) for Ir. We chose a plane-wave cut-\noffkmaxof 4.0 bohr\u00001and a mesh of 256 k-points within the\nirreducible part of the first Brillouin zone.\nStructural relaxations were performed using a symmetric\nfilm consisting of 11 layers of Ir and two layers of ferromag-\nnetic (FM) Fe on the top and bottom of the Ir slab, slightly dif-\nferent from the slab shown in Fig. 5. The equilibrium hexag-\nonal lattice parameter of the Ir substrate of aIr= 5:10bohr\n(2:70˚A) was used. The atoms of the outmost three layers,\ni.e. Fe@vac, Fe@Ir and Ir@Fe, were allowed to relax along\nthe z-direction until forces were smaller than 0.001 Hartree\nper bohr (0.04 eV/ ˚A). Table II provides the resulting inter-\nlayer distances. dIr-Iris with 2.23–2.25 ˚A larger than the dis-\ntance in the bulk material of 2.20 ˚A, it shows little variation\nfor the different stackings but tends to be smaller for the h x-\nstackings.dFe-Irdepends weakly and dFe-Fe depends strongly\non the stacking of the Fe@vac atoms. The values for dFe-Ir\nare in the range between 2.09 and 2.12 ˚A. WhiledFe-Fe is\nwith 2.08 ˚A largest for fh- and hh-stacking, it is considerably\nsmaller in ff, hf, fb\u0003and hb\u0003with 2.01–2.02 ˚A.\nC. Symmetry aspects of the spin spirals\nA powerful way of describing and understanding the under-\nlying mechanisms and magnetic interactions leading to non-4\ncollinear magnetic structures in ultrathin films is to study the\nenergy dispersion of spin spirals.49,50\nHomogeneous spin spirals possess static periodic struc-\ntures, which can be incommensurate with the chemical unit\ncell of the crystal lattice. Therefore, we describe them by\nwave vectors qin reciprocal space so that the magnetic mo-\nmentMat site ris given by:\nM(r) =M0\n@sin (qr)\ncos (qr)\n01\nA; (1)\nThis spin spiral propagation vector qcan be chosen arbi-\ntrarily in the irreducible Brillouin zone. In Fig. 3, we present\nthe irreducible Brillouin zones of the different double-layer\nstackings. In Fig. 3(a), we show how the unit cells of the real\nspace (spanned by aandb, in grey) and the reciprocal space\n(spanned by a\u0003andb\u0003, in blue) are related to each other for a\nhexagonal lattice (the length of the vectors is arbitrary here).\nIn internal coordinates, the high-symmetry points \u0016\u0000 = (0;0),\n\u0016M = (1\n2;0)and\u0016K = (1\n3;1\n3)within the first Brillouin zone\n(BZ, shown in yellow) are indicated as well, along with the ir-\nreducible part of the BZ (dotted line). To extract the exchange\nand the DM interactions, we employ the cartesian coordinate\nsystem within the BZ, which is spanned by xandy(in yel-\nlow).\nAs we learned before (see Tab. I), the Fe double-layer does\nnot possess the six-fold rotation axis of the hexagonal symme-\ntry. Instead, depending on the stacking sequence, the symme-\ntry is reduced to trigonal (plane group p 3m1) in the hexagonal\nclose-packed stackings ff, fh, hf, and hh or to centered rectan-\ngular (plane group c 2mm) in the bcc-like stackings fb\u0003and\nhb\u0003. Therefore, some of the high-symmetry points are lost\nin the close-packed stackings or become fully obsolete in the\nbcc-like stackings, as indicated in Fig. 3(b) and (c).\nIn the case of trigonal symmetry, the \u0016K-points are not\nequivalent anymore as shown by the additional \u0016K0along~b.\nThe \u0016M-points remain unchanged. Therefore, the irreducible\nBZ has doubled in size as compared to the case shown in\nFig. 3(a). In the centered rectangular symmetry, the BZ\nchanges its shape and size. The former \u0016M-point along y\nbecomes the \u0016Y-point at (0;1p\n3)(in cartesian coordinates)\nand the \u0016K-point becomes obsolete. A new high-symmetry\n\u0016X-point results at (1;0)and a new \u0016M-point at (1p\n3;1p\n3).\nAs\u0016\u0000-\u0016Mand\u0016\u0000-\u0016Xalong xas well as \u0016\u0000-\u0016Kand\u0016\u0000-\u0016Yalong y\nare high-symmetry lines in both the trigonal and centered rect-\nangular symmetries, we utilize these throughout this study to\ncompare the magnetic interactions in the different stackings.\nThe spin spiral propagation vector qdetermines the propa-\ngation direction of the spin spiral and the periodicity length or,\nin other words, the angle between the neighbouring magnetic\nmoments. A spin spiral with q= \u0000 describes the ferromag-\nnetic state. At q=\u0016Mthe row-wise antiferromagnetic state is\ncharacterized by an angle of 180\u000ebetween neighbouring mag-\nnetic moments and a periodicity length of 2a.q=\u0016Kon the\nother hand characterizes the N ´eel state with an angle of 120\u000e\nand a periodicity length of 3a.\nreal space\nunit cell\nab\n(b) trigonal: ff, fh, hf, hh (c) centered rectangular: fb*, hb*Cartesian \ncoordinate\nsystem \nFirst Brillouin \nzone (BZ)xy\nreciprocal \nunit cellb*\na*(a) hexagonal reference\nxM\nΓyMK'\nKxX ΓyY MMK ΓFIG. 3. (Color online) Brillouin zones of the double-layer stack-\nings. In (a) the relationships between real space (grey) and reciprocal\n(blue) unit cells including the first Brillouion zone (yellow) for the\nhexagonal reference structure is given. For simplicity, the vectors\nof the real space and reciprocal space were chosen to have the same\nlength. The dashed black triangle indicates the irreducible part of the\nfirst Brillouin zone. Also shown is the cartesian coordinate system\nwithin the first Brillouin zone (yellow), which is used for the calcula-\ntion of magnetic interaction energies. The high symmetry q-points \u0016\u0000,\n\u0016Kand\u0016Mare indicated as well as the irreducible part of the Brillouin\nzone (dotted line). In the double-layers, the hexagonal symmetry is\nreduced to trigonal and centered rectangular. In (b) the trigonal set-\nting is shown where half of the points \u0016Kare lost, indicated by the\nadditional \u0016K0and a doubling of the size of the irreducible Brillouin\nzone. In the centered rectangular systems, given in (c), the Brillouin\nzone changes its shape and size (the hexagonal one is shown for com-\nparison) and new special k-points \u0016X,\u0016Yand\u0016Mresult.\nIn Fig. 4 two examples are shown corresponding to q=\n1\n62\u0019\napropagating along the two directions \u0016\u0000-\u0016Kwith the wave\nvector (q;0;0)and\u0016\u0000-\u0016M with the wave vector (0;q;0).\nD. Magnetic interactions from DFT calculations\n1. Magnetic exchange interaction\nTo study the magnetic properties of these films, we calcu-\nlate the total energy of flat spin spirals as a function of the\nangle between neighboring magnetic moments via the gener-\nalized Bloch theorem.50It allows the calculation of homoge-\nneous spin spirals which are incommensurate with the chem-\nical unit cell of the crystal lattice. Here, we only consider\nflat spin spirals which are propagating in the xy-plane and are\ndescribed by the propagation vector q. To minimize the com-\nputational cost, we have used an asymmetric slab consisting5\n(a) Γ-Κ direction of reciprocal space\n(b) Γ-Μ direction of reciprocal space\nabcabc\n6a\n6a\nFIG. 4. (Color online) Schematic pictures of flat spin spirals on the\nhexagonal lattice in real space corresponding to q=1\n62\u0019\napropagat-\ning along (a) \u0016\u0000-\u0016Kdirection with the wave vector (q;0;0)and (b)\n\u0016\u0000-\u0016M direction of reciprocal space with the wave vector (0; q;0).\nFe@vac\nFe@Ir\nIr-substratevacuum\nvacuum\nunit cell with periodic boundary conditions in x and y-directionIr@Fe\nFIG. 5. (Color online) Asymmetric film geometry as it was used in\nthe calculations of the magnetic properties. Shown is the ff-stacking\nof the Fe double-layer.\nof two iron atoms on nine layers of iridium substrate as shown\nin Fig. 5. In these calculations, the spin spiral is propagating\nin both the iron and the iridium layers unless stated otherwise.\nWe use akmax= 4:0bohr\u00001and 1936k-points in the full BZ.\n2. Spin-orbit coupling contributions\nSOC contributes to the magnetocrystalline anisotropy en-\nergy (MAE) and to the DMI. The MAE contribution is ob-\ntained from calculations of collinear spin structures. Whereas\nthe DMI energy contribution is accessed via calculations of\nspin spirals.The MAE can be obtained by evaluating the energy contri-\nbution of the SOC in the collinear Fe double-layers with all\nmagnetic moments oriented parallel along the three cartesian\naxesx,yandz. This SOC contribution is calculated by per-\nforming self-consistent scalar-relativistic calculations, which\nrequire an increased accuracy, therefore we use a kmax =\n4:3bohr\u00001and 1936k-points. To study the stability of the\nMAE with respect to the number of Ir layer, we have succes-\nsively turned off the SOC contribution in the muffin tin of the\nIr atoms. In that case, we have converged self-consistently the\ncharge density when the quantization axis was applied in the\nz-direction and applied the magnetic force theorem to evalu-\nate the energy when quantization axis was applied in the x-\nandy-direction.\nFor the non-collinear case of a flat spin spiral, the SOC can\nbe included via first-order perturbation theory.51In that case,\nthe band energies are corrected via the SOC Hamiltonian\nHSOC=X\ni\u0018i\u001b\u0001Li; (2)\nwhere\u0018iis the SOC strength at site i,\u001bis the Pauli matrix\nandLiis the orbital momentum operator at site i.HSOC\ndescribes the odd part of the magnetic exchange tensor which\ncan be interpreted as the DMI, which stabilizes a left or a right\nrotating spin-spiral. Within F LEUR , the FLAPW basis set\nprovides a natural framework for the atomic decomposition\nof the SOC. The atomically resolved SOC contribution allows\nthe determination of the layer-dependent contributions to the\nDMI. If the magnetic moments in the MT of the layer iare\nkept in a FM state perpendicular to the rotation plane of the\nspin spiral in the other layers then the SOC contribution of\nthis layer will be reduced to zero. The DMI can be attributed\nto the SOC contributions of the remaining layers. For these\ncalculations, we have used kmax = 4:3bohr\u00001and 1936\nk-points.\nE. Extended Heisenberg model\nWe use an extended Heisenberg model to analyze the en-\nergy dispersion curves E(q)of the different stacking models.\nThis model involves the magnetic exchange interactions up\nto the fifth nearest neighbors and the Dzyaloshinksii-Moriya\ninteractions.\n1. Heisenberg exchange interaction\nThe exchange interactions in the magnetic Fe double-layer\ncan be separated into two contributions:21the intra-layer inter-\nactions within the layers parallel to the film (labeled Fe@vac\nand Fe@Ir) and the inter-layer interactions between these two\nlayers perpendicular to the film. This decomposition results\nin the spin Hamiltonian\nH=Hk;Fe@vac+Hk;Fe@Ir+H?: (3)6\nBoth the intra- and the inter-layer interaction Hamiltonian\nare expressed as\nHk;?=\u0000X\nijJk;?\nij(mi\u0001mj); (4)\nwhere the sum runs over sites within both Fe-layers. This\nHamiltonian may be expressed as a series of Cosines by in-\nserting the magnetization of a homogeneous spin spiral as\nH=\u0000X\n\u000eJ\u000eX\nicos(q\u0001R\u000ei); (5)\nwhere R\u000eiis the position of the atom iin the shell\u000eandqis\nthe propagation vector of the spin spiral in units of2\u0019\na.\nIn all structures presented in this study, both Fe layers\nadopt a hexagonal close-packed structure. Therefore Hk\nremains unchanged as compared to previous works.21In\ncontrast,H?depends on the double layer stacking. In the\ncase of the close-packed stackings ff, fh, hf and hh, there\nare three next nearest neighbors in the adjacent plane at\npositions (a\n2;a\n2p\n3),(\u0000a\n2;a\n2p\n3)and(0;\u0000ap\n3). The resulting\nequations for the inter-layer interactions can be found in the\nSupplemental Material of Ref[21].\nFor the bcc-like stackings fb\u0003and hb\u0003, there are two next\nnearest neighbor atoms at positions (a\n2;a\n2)and(\u0000a\n2;a\n2). This\ngives for the first four neighbor shells in the adjacent plane the\nfollowing expressions in cartesian coordinates H?\nnn(qx;qy):\nH?\n1= 2J?\n1h\ncos\u0010aqx\n2\u0011\n\u00001i\n(6)\nH?\n2= 2J?\n2h\ncos\u0010p\n3aqy\n2\u0011\n\u00001i\n(7)\nH?\n3= 4J?\n3h\ncos(aqx)\u0010p\n3aqy\n2\u0011\n\u00001i\n(8)\nH?\n4= 2J?\n4h\ncos\u00103aqx\n2\u0011\n\u00001i\n(9)\nThus, the energy dispersions obtained for three different\nspin spirals in the Fe-layers can be fitted to the spin Hamil-\ntonian (4) in order to obtain the exchange interaction coeffi-\ncientsJk;Fe@vac\nij ,Jk;Fe@Ir\nij andJ?\nijup to the fifth and fourth\nneighbor shell, respectively. Jk;Fe@vac\nij andJk;Fe@Ir\nij are ob-\ntained when the spin spiral only propagates in the Fe@vac or\nFe@Ir layer, respectively. J?\nijis obtained when the spin spiral\npropagates in both Fe layers simultaneously.\nIn addition, we provide effective exchange coefficients Je\u000b\nresulting from the first nearest neighbor exchange interaction\nobtained in the range jqj<0:1\u0002(2\u0019\na), i.e. in the vicinity of\nthe\u0016\u0000-point.21,36\n2. Dzyaloshinskii-Moriya interaction\nThe DMI arises when SOC occurs in a system with broken\ninversion symmetry such as a surface or an interface. This in-teraction favours a perpendicular orientation between neigh-\nboring spins instead of the parallel or anti-parallel orientation\nfavoured by Heisenberg exchange. Its effect is to favour cy-\ncloidal spin spirals and a certain rotational sense, thus, it de-\ntermines whether a spin spiral rotates clockwise or counter-\nclockwise. The Hamiltonian can be written as\nHDM=\u0000X\nijDij\u0001\u0000\nmi\u0002mj\u0001\n; (10)\nwhere the sum runs over sites within both Fe-layers.\nWe consider contributions to the DMI up to the third\nneighbor shell and also provide effective DMI coefficients\nDeffresulting from the next nearest neighbor interaction as\nobtained close to the \u0016\u0000-point (q2[\u00000:1;0:1]).\nIII. RESULTS AND DISCUSSION\nThis section is organized as follows. We first investigate\nand discuss the stability of the different stackings of the Fe\ndouble-layer. We consider the low temperature case (A) as\nwell as a high temperature case (B), where the effect of ther-\nmal expansion of the substrate is taken into account. For com-\npleteness, we also study the kinetic stability of the stackings\nwith respect to the ground state structure (C). Afterwards we\nprovide an in-depth investigation of the magnetic interactions\nplaying a decisive role in this material system (D), i.e. Heisen-\nberg exchange (D 1), magnetocrystalline anisotropy (D 2) and\nthe Dzyaloshinskii Moriya interaction (D 3). In order to un-\nderstand the obtained relative stabilities and magnitudes of\nmagnetic interactions we present densities of states (E). Fi-\nnally, after having determined all energy contributions to the\nmagnetic texture resulting from the different stackings in the\nFe double-layer, we can discuss the occurrence of spin spirals\nin the studied system and compare our findings with experi-\nments.\nA. Thermodynamic stability of stackings\nWe start by presenting the stability of the different double-\nlayer stackings given in Tab. I. In Fig. 6 the total energies of\nthese different structures are given relative to the energy of\nthe ff-stacking. We find three groups of stackings in terms\nof stability, the xf-stackings with energies around zero meV\nper Fe atom (where xis f, h), the xh-stackings with higher\nenergies at about 45 meV per Fe atom and the more stable\nxb*-stackings at about \u000030meV per Fe-atom, of which fb\u0003\ngives the ground state of the Fe double-layer on Ir (111).\nThis result can be considered surprising in two ways.\nFirstly, the Fe@vac-layer uniquely determines the stability of\nthe double layer. Secondly, the low-symmetry structures xb*\nare more stable than the close-packed stackings xh orxf. It\nseems that two monolayers of Fe are sufficient to give rise\nto bulk-like properties (i.e. the FM bcc-ground state struc-\nture) in the ultra thin-film, although it is extremely strained:7\n−40−2002040\nff hf fb* hb* fh hhTotal energy (in meV/Fe)\nStacking of Fe double-layer\nGround \nstate\n0+2\n-36-26+44+47\nFIG. 6. (Color online) Total energies of the different stackings rela-\ntive to the energy of the ff-stacking. Decisive for the stability of the\ndouble layer is the top-layer configuration.\n\u000fxx=\u00004:6%and\u000fyy= +17:0%with respect to the calcu-\nlated lattice constant of bcc iron (compare a= 2:83˚A of Fe\nin cubic unit cell vs. a= 2:70˚A of Ir in hexagonal unit cell).\nThere is no explanation so far for the experimental finding that\nboth fcc and bcc-top layers can be found to coexist,25accord-\ning to our calculations only fb\u0003and hb\u0003stackings should be\nobserved in the Fe double-layer.\nB. Effect of epitaxial strain\nAs the Fe double-layers are grown at elevated temperatures\nof about 700 Kelvin, thermal expansion of the Ir substrate\nmight affect the relative stabilities of the stackings. In or-\nder to investigate the effect of an increased epitaxial strain\ndue to thermal expansion of the substrate we studied the f x-\nstackings at different in-plane lattice-constants expanded by\n0.5 and 1.0% as compared to the DFT equilibrium value of\na0= 2:70˚A. Iridium has an expanded lattice constant by\nabout 0.34% at 600 K as compared to the value at tempera-\ntures close to 0 K,52while the DFT value is 0.45% smaller\nthan the experimental 0 K-value. Thus, our range is suffi-\nciently large to cover for both the DFT-underestimation and\nthe thermal expansion effect. The total energies are shown\nin Fig. 7. All values are given relative to the energy of the\nff-stacking at the same lattice constant.\nThe energy difference between ff- and fb\u0003-stackings\nslightly decreases by 10 meV from the equilibrium lattice con-\nstanta0to +1.0% epitaxial strain, while the energy of the fh-\nstacking stays almost constant at a value of about 45 meV\nper Fe-atom higher than the ff-stacking. Therefore, epitaxial\nstrain alone cannot change the stability hierarchy of the stack-\nings we studied.\n−40−2002040\nff fh ff ff fb* fhTotal Energy (in meV/Fe)+1.0%\nfb*a0\nfb* fh\nStacking of Fe double-layer+0.5%FIG. 7. (Color online) Total energies of the different stackings rel-\native to the energy of the ff-stacking with different epitaxial strains.\nEpitaxial strain does not alter the relative stabilities of the different\ndouble-layer stackings.\n01020304050607080\n0.0 0.5 1.0 1.5 2.0\nShift along 1 _[112]6ff\nfb*fh\nDC B AEnergy relative to fb* (in meV/Fe)\nFIG. 8. (Color online) Total energies along the transformation paths\nfrom ff and fh to fb\u0003-stacking. The line is a guide to the eye.\nC. Kinetic stability of the bcc-like stacking\nSince epitaxial strain cannot explain the presence of ff-\nstacked areas in experimentally investigated Fe double-layers,\nwe turn now to the kinetic stabilization of the hexagonal-close\npacked stackings, i.e. if there are any energy barriers between\nthose and the bcc-like stackings.\nWe envision that the growth of the second layer could start\nat the hollow sites, i.e. positions B or C of the close-packed\nstructures with three next neighbors in the Fe@Ir-layer (see\nFig. 1), in contrast to the energetically unfavored D-positions\nwith only two next neighbors characteristic of the bcc-like sur-\nface structure. An energy barrier between the close-packed\nstructures and the bcc-like structure would then explain why\nlarger islands cannot transform into the ground state fb\u0003but\nstay in the metastable states ff or fh. Thus, the structure would\nbe determined by growth and kinetic considerations although\nthermodynamics favor another stacking.8\nTherefore, starting from the ff-stacking and fh-stacking, we\ncontinuously shifted the Fe@vac-layer along the [112] direc-\ntion relative to the fcc (111)-plane of the Fe@Ir-layer until\nwe reached the lattice sites of the fb\u0003-stacking. In the inset\nof Fig. 8, these movements correspond to shifts from posi-\ntion C (ff-stacking) and position B (fh-stacking) to position\nD (fb\u0003-stacking), respectively. Thus, we follow the paths\nff!fb\u0003 fh.\nThe corresponding energies can be found in Fig. 8. There\nare no energy barriers to overcome for reaching the fb\u0003ground\nstate, at least in the 0 K limit of our calculations. This means\nactually that both the ff and fh-stacking are mechanically un-\nstable. Small distortions towards fb\u0003shall lead to a phase tran-\nsition.\nAt higher temperatures this simple picture may change\nthough. As was shown previously for the martensitic phase\ntransition in bulk titanium,53structures, which are mechani-\ncally unstable at 0 Kelvin, can be stabilized at elevated tem-\nperatures by phonon contributions. Such study is beyond the\nscope of this communication though.\nMoreover, we observe that the fb\u0003-stacking ground state\nis actually degenerate. A small off-centering towards the\ndirection of the ff-stacking by 0.08 ˚A leads to a slightly lower\nenergy by about 1.4 meV per Fe-atom as compared to fb\u0003.\nThis result is confirmed by supercell calculations.26\nWe recently found that at specific growth conditions even\nmore complex bcc-like superstructures resulting in zigzag pat-\nterns may be thermodynamically stable.26\nD. Magnetic interactions\nWe investigate next the effect of the change of symmetry on\nthe magnetic interactions for three different stackings: ff, fb\u0003\nand fh. In addition, we included the hf-stacking to check for\nthe influence of the Fe@Ir-layer. We first present the energy\ndispersion curves of spin spirals without SOC, which are fitted\nto the Heisenberg model to obtain exchange constants within\nand between the two magnetic Fe-layers. We then show the\neffect of the double-layer stacking on the magnetocrystalline\nanisotropy. Finally, we investigate the Dzyaloshinskii-Moriya\ninteraction in detail.\n1. Total magnetic exchange: Mapping of energy dispersions on\nextended Heisenberg model\nThe total magnetic exchange interaction can be studied\nwhen the spin spiral is propagating within both Fe layers as\nwell as the Ir substrate. Figure 9 shows the energy disper-\nsion curves of the four investigated Fe double-layer stackings.\nTwo types of dispersion curves can be distinguished. In the\ncase of ff (turquoise) and fh (dark blue), the magnetic ex-\nchange interactions favor spin spirals. The dispersion curve\nof fh has a deep minimum at E=\u000013:1meV/Fe with\nq= 0:24\u0002(2\u0019\na)which corresponds to a spin spiral wave-\nlength of\u0015= 1:1nm. Although the overall symmetry does\n04080120160200240280Energy E(q)-EFM (in meV)\n0.4 0.2 0.0 0.2 0.4 0.6 0.8 1\nSpin spiral vector q (in 2π/a)fb*\nfh ffhfΓ M/Y K/- M/X\n−20−100102030\n−0.4−0.2 0.20.4 0.0FIG. 9. (Color online) Spin spiral energy dispersions in the stackings\nff, fb\u0003, hf and fh. Exchange stabilized spin spirals are found in ff and\nfh, while fb\u0003and hf exhibit a ferromagnetic ground state.\nnot change, when the hcp stacking of Fe@vac is replaced by\nthe fcc stacking, the energy minimum of the dispersion curve\nof ff is reduced to E=\u00001:8meV/Fe and the wavelength in-\ncreases to\u0015= 1:9nm withq= 0:14\u0002(2\u0019\na). These two\nstackings have a spin spiral ground state.\nOn the other hand, the dispersion curves of hf (yellow)\nand fb\u0003(red) exhibit an energy minimum at q= 0:0\u0002(2\u0019\na)\nwhich corresponds to the ferromagnetic state. In the bcc-like\nfb\u0003-stacking, the N ´eel state and the row-wise antiferromag-\nnetic state at the BZ edges become extremely unfavorable\n(E(\u0016X)=437 meV and E(\u0016Y)=221 meV , which are more\nthan twice as large as the corresponding energies for the\nclose-packed stackings).\nThe occurence of isolated skyrmions depends on the rise\nof the dispersion curve close to q= 0:0\u0002(2\u0019\na)which can\nbe estimated by using the effective nearest neighbor exchange\nconstantJe\u000bas explained in section II E 1.21The values of Je\u000b\n(see Tab. III) confirm the qualitative findings of Fig. 9. They\nare in the same range as the values found in the Pd/Fe/Ir(111)\nultra-thin films, where Je\u000bof\u00002:3and+4:4meV were re-\nported depending on the stacking of the Pd overlayer.34The\nstackings ff and hf possess dispersion curves with energy min-\nima along each direction \u0016\u0000-\u0016Kand\u0016\u0000-\u0016Mwhich correspond to\nJe\u000b<0. On the other hand, the hf and fb\u0003stackings possess\npositive slopes and Je\u000b>0. The disperion curve of hf rises\neven faster close to \u0016\u0000than the one of fb\u0003, although the oppo-\nsite is true further from \u0016\u0000. In both stackings, the magnetic ex-\nchange interaction favors a ferromagnetic ground state. In all\nstackings,Je\u000bare small, which facilitates the presence of iso-\nlated skyrmions at finite magnetic fields, as we could demon-\nstrate previously for Pd/Fe/Ir(111).30\nWe want to analyze in more detail how the exchange-\ninduced spin spirals in ff and fh develop by inspecting the\nexchange contributions from each Fe layer separately.\nThe layer dependent energy dispersion curves are shown9\nTABLE III. Effective next nearest neighbor exchange coefficients Jeff\ncorresponding to the dispersion curves of Fig.9 for jqj<0:1\u0002(2\u0019\na).\nAll values are given in meV/Fe.\ndouble-layer stacking \u000b fb\u0003fh hf\nJe\u000b \u00003:0 +1 :8\u00006:9 +3 :2\nin Fig. 10, where a spin spiral was imposed only in one of\nthe Fe-layers and the Ir-substrate, while spins in the other Fe-\nlayer were kept parallel to each other and perpendicular to the\nrotational plane of the spin spiral resulting in a mixed spin\nconfiguration (spin spiral + ferromagnetic alignment).\nFigure 10(a) shows the dispersion curves of the different\nstackings with a spin spiral in the layer of Fe@vac. All curves\npossess a flat energy dispersion close to the \u0016\u0000-point. This in-\ndicates that the magnetic exchange interaction in this layer\nis generally frustrated. Only the ff-stacking possesses a spin\nspiral ground state in this mixed spin configuration, while the\nremaining stackings are fully ferromagnetic.\nFigure 10(b) shows the dispersion curves for spin spirals in\nthe layer of Fe@Ir. The dispersion curves of hf and ff are al-\nmost identical, although the local structure of the investigated\nlayer is different. The curves of ff,fh and fb* rise faster close\nto the \u0016\u0000-point than in the Fe@vac case indicating less frustra-\ntion. However, the dispersion curve of the fh stacking shows\na deep minimum at \u00005meV/Fe for q= 0:2\u0002(2\u0019\na).\nThus, the spin spirals in the ff and fh double-layers have\ndifferent origins. In fh, its formation is driven by the Fe@Ir-\nlayer, while in ff, it emerges from the Fe@vac-layer.\nIt is interesting that the strong FM behavior of the fb\u0003\nstacking observed in Fig. 9 is lost for the single layer spin spi-\nrals (sinceE(\u0016K) = 80 and 110 meV/Fe) and that the 3-fold\nsymmetry of the BZ is also recovered (since E(\u0016X) =E(\u0016Y)).\nIn order to analyze in detail all magnetic exchange interac-\ntions in the different stackings, we provide in Tab. IV the fitted\nnearest neighbor interactions Jijup to the fifth and the fourth\nnearest neighbor in JkandJ?, respectively.\nAll intra-layer exchange constants Jkshow a high degree\nof frustration. They oscillate between ferromagnetic (FM)\nand antiferromagnetic (AFM) coupling depending on the shell\nnumber. In both Fe-layers, Jk\n1is positive which shows that the\nFM state is more stable than the AFM states but the Jkbeyond\nthe first nearest neighbors can become negative.\nAllJkhave values below 12 meV/Fe which is between\nJ1= 5:7meV/Fe for Fe/Ir(111)54andJ1is in the range\nof 13 to 14 meV/Fe for Pd/Fe/Ir(111).30Nevertheless, the\ninter-layer exchange constants J?can reach up to J?\n1=\n62:7meV/Fe and J?\n2= 18:1meV/Fe in the case of fb\u0003.\nThese values are much higher than J?\n1= 24:73meV/Fe for\n[Rh/Pd/2Fe/2Ir ]1.21This strong inter-layer coupling is the ori-\ngin of the marked FM character of the fb\u0003Fe double-layer ob-\nserved in Fig. 9. These large values are originating from the\nsymmetry lowering of the (110) surface where the first and the\nsecond shell contains only two atoms. They will induce a large\nanisotropy in the magnetic exchange interaction which can ex-\n−20020406080100120140\n0.4 0.2 0.0 0.2 0.4 0.6 0.8 1\nSpin spiral vector q (in 2π/a)fhhfff\nfb*Γ M/Y K/- M/X−20020406080100120140\nfhfb*hf\nffΓ M/Y K/- M/X\n0.4 0.2 0.0 0.2 0.4 0.6 0.8 1\nSpin spiral vector q (in 2π/a)Energy E(q)-EFM (in meV)(a) spin spiral in Fe@vac\n(b) spin spiral in Fe@Ir\nEnergy E(q)-EFM (in meV)Fe@vac\nFe@Ir\nFe@vac\nFe@IrFIG. 10. (Color online) Layer-dependent spin spiral energy disper-\nsions from DFT without SOC in the stackings ff, fb\u0003, hf and fh. In\n(a), the spin spiral propagates only in the Fe@vac-layer exposed to\nthe vacuum, but not in the interfacial Fe@Ir-layer. In this spin config-\nuration, the spin spiral is only stable in stacking ff, all other stackings\nfavour the ferromagnetic state. In (b), the spin spiral propagates in\nthe interfacial Fe@Ir-layer, but not in the Fe@vac-layer at the sur-\nface. In this case, we find a deep energy minimum for the spin spiral\nstate in the hf-stacking, while the others remain ferromagnetic.\nplain the occurrence of non-centrosymmetric skyrmions.25–28\nTo summarize, in ff- and fh-stackings, spin spirals are stabi-\nlized by magnetic exchange interaction. Their formation can\nbe driven by only one of the Fe-layers. In contrast, in fb\u0003- and\nhf-stacking a flat dispersion curve is observed close to the \u0016\u0000-\npoint indicating high magnetic frustration in these structures.\nIn the latter structures, spin spiral ground states might be sta-\nbilized by the DMI, as we will see later. We will now consider\nthe SOC contributions to the total energy.\n2. SOC contribution to collinear states: magnetocrystalline\nanisotropy\nThe MAE is determined by calculating the SOC contribu-\ntion to the total energy when all spins are pointing parallel10\nTABLE IV . Heisenberg exchange coefficients Jijas obtained from\nfits to the spin Hamiltonians Eq. (4). All values are given in meV/Fe.\nJi \u000b fb\u0003fh hf\nJk;Fe@vac\n1 +5:0 +12 :3 +7 :8 +6 :1\nJk;Fe@vac\n2 +1:1 +1 :3\u00001:8 +1 :4\nJk;Fe@vac\n3\u00002:3\u00005:2\u00001:0\u00001:1\nJk;Fe@vac\n4 +0:2 +0 :9 +0 :1\u00000:2\nJk;Fe@vac\n5\u00000:5\u00001:2 +0 :7\u00000:4\nJk;Fe@Ir\n1 +5:3 +9 :6 +6 :0 +4 :9\nJk;Fe@Ir\n2 +0:1 +0 :02 +0 :5 +0 :4\nJk;Fe@Ir\n3\u00000:7\u00002:3\u00002:7\u00000:4\nJk;Fe@Ir\n4\u00000:3 +0 :6\u00000:1\u00000:2\nJk;Fe@Ir\n5 +0:5\u00000:6\u00000:2 +0 :2\nJ?\n1 +24:0 +62 :7 +12 :1 +22 :1\nJ?\n2 +0:1 +18 :1 +0 :9 +2 :0\nJ?\n3\u00008:1\u00006:1\u00008:6\u00005:8\nJ?\n4 +2:9\u00002:3 +2 :5 +1 :3\nalong the x,yandzdirections. The MAE is then determined\nby the energy difference Ei\u0000Eminwithi=x;y;z . The easy\naxis or easy plane is the axis or plane along which the SOC\ncontribution to the total energy is lowest, i.e. Emin.\nSince we perform calculations for asymmetric films there\ncan be unphysical contributions from the Ir terminated side of\nthe film. It is therefore important to explore the dependence\non the MAE on the number of Ir MT spheres in which SOC is\napplied as shown in Fig. 11. The MAE is rather independent\nfrom the number of Ir MT for the ff (a), fh (c) and hf (d)\nstackings, for which the energy difference oscillates around\nthe values 1.2 meV , \u00000:5meV and\u00000:5meV , respectively.\nHowever, the MAE is much smaller for the fb\u0003stacking and its\nvariation oscillates around 0 meV . We approximate the MAE\nof fb\u0003to 0.1 meV .\nIn Fig. 12 we summarize the MAE calculated for the four\nstackings ff, fb\u0003, fh and hf. The coloured axes and planes\nindicate the easy axis or easy plane, respectively. Numbers\nindicate the anisotropy coefficients K1with their associated\ndirections. All energies are in the typical energy range of\nfew meV for ultrathin-film systems. For ff we find an easy-\naxis pointing out-of-plane. Whereas fb\u0003, hf and fh possess\nan easy-plane in the basal plane of the film. The anisotropy\ncoefficients K1in the easy-axis system is 1.0 meV (ff). The\nsystems with the easy-plane have coefficients of \u00000:1meV\n(fb\u0003) and\u00000:5meV (fh and hf).\nIt is surprising that the centered rectangular structure fb\u0003\nposesses an easy-plane, where x- andy-directions are degen-\nerate, although these directions are not equivalent by symme-\ntry.\nIn the limit of small q, the effect of the MAE on the sta-\nbility of a spin spiral is a constant energy shift of the whole\ndispersion curve by +1\n2K1, i.e. it equally disfavors any kind\nof rotation of the magnetic moments.\nEnergy ΔESOC (in meV)\n−1.0−0.50.00.51.01.5\n5 6 7 8 9\nNumber of Ir layersx−z\ny−z\nx−yhf(d)x−z(a)ff\n−1.0−0.50.00.51.01.5\ny−z\nx−yEnergy ΔESOC (in meV)\n(c)\n−1.0−0.50.00.51.01.5\nl sx−z\ny−z\nx−yfhEnergy ΔESOC (in meV)(b)\n−1.0−0.50.00.51.01.5x−z\ny−z\nx−yfb*Energy ΔESOC (in meV)FIG. 11. Energy difference between FM states when SOC is ap-\nplied along the xandzdirection (dashed green triangle), along the y\nandzdirection (dashed blue square) and along the xandzdirection\n(dashed red dot) as a function of the the number of Ir layers in which\nSOC is applied in the calculation for the ff (a), fb\u0003(b), fh (c) and hf\n(d).\n3. SOC contribution to non-collinear states:\nDzyaloshinskii-Moriya interaction\nThe DMI originates from SOC in non-centrosymmetric\nsystems. In our systems, the broken inversion symmetry\ncomes from the Ir-Fe interface, while the strong SOC origi-\nnates from the Ir 5d-states. The DMI is thus expected to be\ndominated by the atomic SOC contributions from the Ir(111)\nsubstrate.\nFigure 13 shows the total and atomic SOC contributions to\nthe energy dispersions of spin spirals in the ff and fb\u0003stack-\nings. Figure 13(a) shows the atomic SOC contributions for the\nff stacking. This stacking is also representative for the other 3-\nfold symmetric stackings (therefore, fh and hf are not shown).\nAs expected, the SOC respects the 3-fold symmetry: Its total11\n+0.1\nfb* fh ffMagnetocrystalline \nanisotropy energy (in meV)0.00.20.40.60.81.01.21.41.61.8+0.5+1.0+0.5\nhfz\nxyxyxy\nStacking of Fe-double layer\nFIG. 12. (Color online) Shown are the preferred (in color and bold)\nand disfavoured spin orientations (in grey and fine) for the stackings\nhf, fb\u0003, fh and ff. The preferred orientation of spins changes from an\neasy-plane in hf, fb\u0003and fh-stacking to an out-of-plane easy-axis in\nthe ff-stacking. The anisotropy coefficients K1increase at the same\ntime from 0.1 meV in fb\u0003to 1.2 meV in ff.\namplitude (red dots and solid lines) does not depend on the\npropagation direction of the spin spiral and is maximum for\nthe 90\u000espin spirals at q=\u00000:29\u0002(2\u0019\na)corresponding to\nqcart= (0;1\n2p\n3;0)andq= +0:33\u0002(2\u0019\na)corresponding\ntoqcart= (1\n3;0;0). From the atomic SOC contributions we\ncan see that the total SOC energy is dominated by contribu-\ntions of the Ir substrate. The contributions of the two Fe layers\n(turquoise triangles and blue squares) are of opposite sign and\ntherefore, cancel each other out.\nThe situation is different for the fb\u0003stacking as shown in\nFig. 13(b). The total SOC energy along the direction \u0016\u0000\u0000\u0016X\nhas the same amplitude as in the ff case. However, the SOC\nenergy along \u0016\u0000\u0000\u0016Yis reduced by a factor of three as compared\nto\u0016\u0000\u0000\u0016X. This reduction of SOC interaction along the ky-\ndirection may be surprising. Especially, if one assumes that\nthe symmetry of the DMI is determined solely by the sym-\nmetry of the interface. Fe@Ir occupies in the fb\u0003-stacking\nthe same sites with respect to the Ir substrate as in the ff and\nfh stackings. Therefore, only the surface Fe-layer (Fe@vac)\nreduces the symmetry of the ultrathin film, which is usually\nnot expected to modify the hybridization of the interfacial Ir\natoms.\nIn Tab. V we compare the DMI coefficients De\u000bandD1\u00003\nacting on the interfacial Fe (Fe@Ir) of the stackings ff, fh, fb\u0003\nand hf, which we obtained from fitting the total SOC ener-\ngies. For stackings ff and fh, we find De\u000bof about 2.0 meV ,\nwhereas fb\u0003and hf give about 1.2 meV . As De\u000bin all stack-\nings adopts positive values it favours clockwise-rotating spin\nspirals. These DMI coefficients are in the same range as those\nfor the Fe monolayer on Ir(111)9and those of the Fe/Ir(111)-\nsystem with Pd-overlayers.30The centered rectangular sym-\nmetry of the fb\u0003stacking is reflected in the existence of a sec-\nondDe\u000b. The DMI in the Cartesian y-direction is strongly\nsuppressed to De\u000b= 0:2meV , only one sixth of the value\nalong the Cartesian x-direction.\nIn Tab. V, also the coefficients for the DMI up to the third\nnext nearest neighbor are provided. The values of D1are\nvery similar for all stackings. For the ff and the fh stackings,\n(a)\n(b) Y Γ\nfb*\nSpin spiral vectorq(in 2π/a)Γ Κ Μ\nff\nSpin spiral vectorq(in 2π/a)Energy ΔE soc(q) (in meV) Energy ΔE soc(q) (in meV)Fe@IrFe@vac\nall atoms Ir(111)-substrate\n−8.0−6.0−4.0−2.00.02.04.0\n−0.4 −0.2 0.0 0.2 0.4 0.6\nFe@vac\nIr(111)-substrateall atoms\n−8.0−6.0−4.0−2.00.02.04.0\n−0.4 −0.2 0.0 0.2 0.4 0.6Fe@IrFIG. 13. (Color online) Spin-orbit coupling contribution to energy\ndispersion of cyloidal flat spin spirals related to the Dzyaloshinskii-\nMoriya interaction along high symmetry lines for stackings (a) ff and\n(b) fb\u0003. Given are the contributions of the different atoms as well as\nthe fit for the total energy considering DMI up to the third nearest\nneighbor shell (see Tab. V). The DMI is considerably suppressed in\nthe\u0016\u0000-\u0016Ydirection of the fb\u0003-stacking.\nTABLE V . DMI coefficients of Fe@Ir in all stackings determined\nalong \u0016\u0000-\u0016Kand\u0016\u0000-\u0016Mdirection for ff, fh, and hf and along \u0016\u0000-\u0016Xdi-\nrection for fb\u0003(along \u0016\u0000-\u0016Yin parentheses). All values are given in\nmeV .\ndouble-layer stacking ff fb\u0003fh hf\nDe\u000b 2.0 1.2 (0.2) 2.0 1.3\nD1 1.25 1.17 1.25 1.21\nD2 0.16 0.90 0.16 -0.05\nD3 0.11 -0.78 0.11 0.10\nD1\u00003are quasi identical and they are all positive, which\ncreates aDe\u000bof around 2 meV/Fe. The De\u000bis significantly\nreduced to 1.2 meV/Fe for the hf and the fb\u0003stackings due to\nthe negative contribution of the D2and theD3, respectively.\nTheDe\u000bof fh and ff differ due to small energy differences of\nthe SOC contributions close to \u0016\u0000.12\n(a) spin spiral in Fe@Ir (b) s pin spiral in Fe@vac\nall atoms\nIr(111) substrateFe@IrFe@vacfb*\n−4.0−3.0−2.0−1.00.01.0Γ K M\n2.0\n−0.4−0.2 0.0 0.2 0.4 0.6\nSpin spiral vector q (in 2π/a)all atomsIr(111) substrate\nFe@IrFe@vacfb*\n−4.0−3.0−2.0−1.00.01.02.0\n−0.4−0.2 0.0 0.2 0.4 0.6Energy ΔE soc(q) (in meV)\nSpin spiral vector q (in 2π/a)Energy ΔE soc(q) (in meV)Γ K M\nno DMI contribution \nfrom Fe-IrIr@FeFe@IrFe@vac\nIr-Fe DMI\nFe-Fe DMI(c) symmetry aspects of DMI\nΓ_M KΓ_Γ_M KΓ_spin spiral in Fe@Ir spin spiral in Fe@vac\nqsoc\nqsoc q\nsocq\nsocFe@vacFe@IrFe@vacFe@Ir\nFIG. 14. (Color online) Total energy contributions to spin spiral energy in the fb\u0003stacking dispersions due to SOC resulting from spin spirals\nin only one of the Fe-layers. Given are the atomic contributions and the total SOC energies. In (a), the spin spiral propagates in the interfacial\nFe@Ir-layer, but not in the Fe@vac-layer at the surface. The magnitudes along both propagation directions are the same. Thus, the SOC\npossesses 3-fold symmetry in this spin configuration. The corresponding DMI coefficient is D1(Fe@Ir) = 0 :40meV . In (b), the spin spiral\npropagates in the Fe@vac-layer of the surface. The magnitudes along the two propagation directions are completely different. The SOC\nreflects the reduced symmetry of the bcc-like stacking. The obtained DMI coefficient is D1(Fe@vac) = 0 :34meV . In (c), we show how the\nIr-Fe DMI (yellow arrows) and Fe-Fe DMI (red arrows) arise from different 3-atom scattering events possible in the two different spin spiral\nconfigurations. Shown are the centered rectangular unit cells (grey solid line) in top view, along with the rhombic unit cell (dotted line) for\norientation. The quantization axis of the SOC is always perpendicular to the spin spiral propagation direction which leads to cycloidals spin\nspirals along q. The scattering partners are connected via dashed lines, in grey for Ir-Fe and in turquoise/dark blue for Fe-Fe interactions. Only\nfor the spin spiral in Fe@vac along the \u0016\u0000\u0000\u0016Mdirection, there is no Ir-Fe DMI possible, which gives rise to the strong asymmetry in (b).\nIn order to confirm that the symmetry of the DMI in the fb\u0003\nstacking depends on the adsorption site of the surface Fe-layer\n(Fe@vac), we calculated the layer dependent SOC for two dif-\nferent spin spiral configurations as in Ref.21. In the first con-\nfiguration, the spin spiral propagates only in the Fe@Ir-layer\nand the Ir-substrate, but not in the Fe@vac-layer, where the\nmagnetic moments are oriented parallel to each other (FM)\nand perpendicular to the rotation plane of the spin spiral. Here\nwe calculate the SOC contribution from the Ir substrate on\nFe@Ir. In the second configuration, the spin spiral propagates\nexclusively in the Fe@vac-layer and in the Ir-substrate, but\nnot in the Fe@Ir-layer to obtain the SOC contribution from the\nIr substrate on Fe@vac. The results are presented in Fig. 14.\nAs each Fe layer taken isolated has 3-fold symmetry, we chose\nthe trigonal reference for the Brillouin zone spanned by \u0016M,\u0016\u0000\nand\u0016K. Fig. 14(a) corresponds to the case where the spin spiral\npropagates in the Fe@Ir layer and Ir substrate, but not in the\nFe@vac layer. It is interesting that indeed a 3-fold symmet-ric SOC is retained, since both \u0016\u0000\u0000\u0016Mand\u0016\u0000\u0000\u0016Kdirections\nexhibit the same magnitude for both the total SOC and the in-\ndividual atomic contributions. The \u0001ESOC(q)is dominated\nby the atomic contribution from the Ir substrate, whereas the\natomic contributions from the two Fe layers are equally strong\nand of opposite sign as compared to the Ir substrate, giving an\noverall DMI coefficient of D1= 0:40meV for this spin spiral\nconfiguration (red solid line).\nHowever, when the spin spiral propagates in the Fe@vac\nlayer and the Ir substrate [Fig. 14(b)], the symmetry is re-\nduced to cm, as was shown in Fig. 2(b). Therefore, we can\nexpect the SOC to have a different amplitude depending on\nthe direction of the propagation vector. Indeed, when the SOC\nis computed along the \u0016\u0000\u0000\u0016Mdirection, the contributions of\nFe@Ir and the Ir-substrate are reduced to zero (turquoise and\ngrey triangles, respectively) while the main contribution arises\nfrom Fe@vac (compare the blue squares and the red dots).\nAlong the perpendicular direction \u0016\u0000\u0000\u0016K, all atoms are con-13\ntributing to the SOC, the dominating contribution originates\nfrom Fe@Ir, whereas the contributions of Fe@vac and the\nIr-substrate counterbalance each other. Overall, the DMI is\ndominated by Fe-Fe interactions in this spin spiral configura-\ntion giving an overall DMI coefficent of D1= 0:34meV .\nWe next analyze how these differences between the two\nspin spiral configurations can be understood. In the model\nof Fert and Levy,18the DMI is associated with a scattering\nprocess among three atoms, arranged in an isosceles triangle,\nwhere the scattering atom at the apex can be non-magnetic.\nThis atomic configuration leads to cycloidal spin spirals which\npropagate in the plane of the ultra-thin film. Therefore, the\navailable scattering partners for the two spin spiral configura-\ntions and propagation directions can be directly identified and\nare presented in Fig. 14(c). The DMI associated with the in-\nteractions between the Ir substrate and Fe@Ir or Fe@vac and\nbetween Fe@Ir and Fe@va are indicated by yellow and red ar-\nrows, respectively, as well as the inter-layer connection lines\nbetween the three scattering atoms (dashed lines). For orienta-\ntion, besides the centered rectangular unit cell (grey solid line)\nalso the rhombic unit cell is indicated (dotted line). In the first\nspin spiral configuration exhibiting the full 3-fold symmetry,\nshown on the left, we have strong Ir-Fe DMI and much weaker\nFe-Fe DMI in both propagation directions of the spin spiral.\nAlong \u0016\u0000\u0000\u0016Kboth DMI vectors are parallel to the quantization\naxis and contribute fully to the DMI. Whereas in the direction\n\u0016\u0000\u0000\u0016M, the two Ir-Fe DMI vectors are rotated by 30\u000ewith re-\nspect to the quantization axis. Therefore, they both contribute\nonly 50% to the DMI, resulting overall in the same magnitude\nfor the Ir-Fe DMI as in the direction \u0016\u0000\u0000\u0016K.\nThe situation is different in the second spin spiral configu-\nration shown in Fig. 14(d). In the direction \u0016\u0000\u0000\u0016M, no DMI\ncontribution from Ir-Fe can be found, as no isoceles triangles\nformed by Fe@vac \u0000Ir@Fe\u0000Fe@vac exist. Therefore, in\nthis direction only the weak Fe-Fe DMI is present. In contrast\nto the \u0016\u0000\u0000\u0016Kdirection, where DMI from both interactions Ir-Fe\nand Fe-Fe can be found.\nTo summarize, the absence of the Ir-Fe DMI in the Fe@vac\nspin-spiral due to the reduced symmetry is the origin of the\nreduced symmetry of the total DMI in the fb\u0003stacking. Thus,\nthe symmetry of the surface Fe-layer indeed determines the\nsymmetry of the DMI, hence, not only the symmetry of the\ninterface matters. This confirms that the DMI is anisotropic\nand could allow for the presence of antiskyrmions in this\nspecific stacking.29\nE. Electronic Structure\nIn order to support the latter statements we present the\norbital-resolved partial densities of states (PDOS) of the\natoms Fe@vac, Fe@Ir and Ir@Fe in the stackings ff and fb\u0003.\nThe PDOS are given in Fig. 15. Presented are the 3dorbitals\nof the Fe atoms and the 5dorbitals of the Ir atom. In each\npanel both the majority (greyscale, negative values) and mi-\nnority spins (color, positive values) are given. We directly\ncompare the PDOS of fb\u0003(filled) and ff (lines) for each atom,orbital and spin channel.\nThe main changes arising from the structural differences\nbetween ff and fb\u0003-stackings occur in the PDOS of the Fe\natoms in both spin channels. The majority spin states of the\ntwo Fe atoms are fully occupied. We observe the largest\ndifferences between the two stackings close to the Fermi\nenergy which is populated by minority spin states. In ff, the\norbitals 3dx2-y2anddxyas well as 3dyzand3dxzof both\nFe@vac and Fe@Ir are degenerate. Moreover, all Fe 3d\norbitals possess a peak at the Fermi energy. We have overall\na three-peak pattern with a small peak at \u00001:5eV (bonding\nstates), and two large peaks around EF(non-bonding states)\nand+1:5eV (anti-bonding states).\nIn fb\u0003, the degeneracy is lifted and the peak structure changes.\nWe find two very broad peaks below and above the Fermi\nenergy. Around EF, the DOS adopts only small values,\nespecially apparent in the 3dz2and3dx2-y2states. Thus, the\nstates with non-bonding character vanish, which explains\nthe enhanced thermodynamic stability of the fb\u0003-stacking\nas compared to the close-packed structures. Moreover, the\ndegeneracy of the 3dyzand3dxzstates is lifted in a particular\nway. While the new 3dxzorbital rather has a three-peak\nstructure with a large peak at the bonding states at \u00001eV ,\nthe3dyzorbital becomes mainly anti-bonding with a large\npeak at +1:3eV . This is a result of the symmetry breaking\nin the centered rectangular unit cell of the fb\u0003structure.\nThe hybridization between the two Fe layers is enhanced\nespecially in the x- and z-direction (i.e. 3dxzand3dz2orbitals\noverlap strongly), where the interlayer atomic distances are\nminimal. Moreover, it is reduced in the 3dyzorbital, where\nthe DOS is largest above the Fermi energy and which corre-\nsponds to an orientation, where interlayer atomic distances\nare maximal. Thus, the changes in the electronic structures\nreflect the symmetry differences between the stackings and\ncan also explain the enhanced thermodynamic stability of the\nfb\u0003-stacking.\nLet us focus next on the electronic structure of the Ir-atoms\nin order to find indications for the reduction of DMI in the\nfb\u0003-stacking. First, we find no effect of the Fe double-layer\nstructure on the 5dx2-y2and5dxystates as they have no out-\nof-plane component and are solely directed towards Ir-atoms\nof the same layer. This is in contrast to the remaining three or-\nbitals, which are either uniquely directed towards the Fe atoms\nlike5dz2or have at least out-of-plane contributions like 5dyz\nand5dxz. Both groups of orbitals are affected differently by\nthe structural changes in the Fe@vac layer. In ff, we find a\nsmall peak at the Fermi energy in 5dz2orbital, whereas fb\u0003\nshows rather a minimum. For the 5dyzand5dxzstates the ef-\nfect is reversed, but to a different degree. We observe a peak\nthree times larger than the PDOS of the ff-stacking at \u00000:3eV\nfor the 5dyzorbital, i.e. a strong increase of the PDOS just be-\nlowEFin fb\u0003. Whereas for the 5dxzorbital, the PDOS is\ndoubled in this energy range as compared to the ff-stacking.\nThus, the electronic states of the atoms in the Ir@Fe-layer\n”sense” the symmetry breaking due to the Fe@vac-layer and\nreact with an unexpected lifting of degeneracy in the 5dyzand\n5dxzorbitals.14\n−0.150.15−0.150.15−0.150.15−0.150.15−0.150.15\n−0.50.5−0.50.5−0.50.5−0.50.5−0.50.5(a) Fe@vac (b) Fe@Ir (c) Ir@Fe\n−5−4−3−2−10123−y23dz2\n3dx2\n3dxy\n3dyz\n3dxzmajority\nspinminority\nspin fffb*\nfb*ffPartial density of states (in eV-1) \nEnergyE−EF(in eV)majority\nspinminority\nspinfffb*\nfb*ff\n−5−4−3−2−10123−y23dz2\n3dx2\n3dxy\n3dyz\n3dxzfffb*\nfb*ffmajority\nspinminority\nspin\n−5−4−3−2−10123−y25dz2\n5dx2\n5dxy\n5dyz\n5dxz\nFIG. 15. (Color online) Orbital and angular momentum-resolved partial densities of states (PDOS) for the 3d- tates of (a) Fe@vac, (b) Fe@Ir\nand for the 5dstates of (c) Ir@Fe in the stackings ff (lines) and fb\u0003(filled). Both majority spins (positive PDOS) and minority spins (negative\nPDOS) are given. The stacking of the Fe@vac layer changes the hybridization in the Ir@Fe layer as indicated by the arrows. Please note the\ndifferent PDOS scales for Fe and Ir atoms.\nF. Magnetic ground state\nFinally, we have determined all quantities in order to study\nthe thermodynamic stability of spin spirals in the different\nstructures at T= 0 K. In Fig. 16 we present the energy dis-\npersions with and without the contributions arising from SOC\nin positive and negative propagation directions along the high-\nsymmetry lines of the BZ.\nAs the DMI coefficients are positive, in all stackings\nclockwise-rotating spin spirals (along positive propagation di-\nrections) are favored. The contribution of the magnetocrys-\ntalline anisotropy amounts to1\n2K1and leads to a constant en-\nergy shift of spin spirals with respect to the FM state ( q=\n0\u0002(2\u0019\na)). Indicated are the energies of the spin spirals rel-\native to the ferromagnetic states and their wave lengths. All\nstackings possess spin spiral ground states.\nThe most isotropic structure is the ff-stacking, given in\n16(a). Here, both energies (about \u00005:8meV) and wavelength\n(1.7 nm) of the spin spirals along the Cartesian \u0000Kand\u0000M\ndirections are quasi identical. Thus, in this structure no pre-\nferred propagation direction exists. As the spin spirals are\nalready stabilized by the exchange interaction, the additional\nDMI gives rise to the asymmetry of the dispersion curve with\nrespect to the chirality (clockwise or anti-clockwise). Here theDMI favors clockwise-rotating cycloidal spin spirals. More-\nover, the DMI shifts the energy minimum to larger q-vectors,\nthus accelerating the spin spiral.\nCompletely different is the result of the fb\u0003-stacking, pre-\nsented in 16(b). Here, the anisotropy of the DMI is also re-\nflected in a strong anisotropy of the magnetic texture. Only\none spin spiral can exist in the fb\u0003structure, which propagates\nalong the cartesian \u0000Kdirection. It has a relatively long wave\nlength of 3.5 nm and a shallow energy minimum at \u00001:2meV .\nThe DMI in the \u0000Mdirection is too small to stabilize a spin\nspiral along this direction. Therefore this stacking is expected\nto exhibit a spin spiral only in three directions in space.\nVery short wavelength spin spirals with low energy result in\nthe fh-stacking shown in 16(c). Spin spirals along the \u0000Kand\n\u0000Mdirection should both have a wavelength of 1.1 nm, but the\nspin spiral along the \u0000M direction is energetically favoured\nby almost 2 meV . Also in the hf-stacking Fig.16(d), we find\na preference for a spin spiral propagating along the \u0000Kdirec-\ntion. It has an energy of \u00001:6meV and a wavelength of 3.0\nnm as compared to the spin spiral along y-direction, which\ngives\u00000:8meV and a much longer wavelength of 4.2 nm.\nThe DMI stabilized spin spirals in fb\u0003and hf have consid-\nerably longer wavelengths than the exchange stabilized spin\nspirals in ff and fh.15\nhf\n−20−15−10−505101520\n−0.4−0.2 00.2 0.4SOC\nno SOC\n−0.4−0.2 00.2 0.4SOC\nno SOCΓ−−Κ−Γ−−Μ−\nλ=3.0 nmλ=4.2 nm\n(q,0,0) (0,q,0)\nSpin spiral vector q(in 2π/a)(a) (b)\n(c)-1.3ff\n−20−15−10−505101520\n−0.4−0.2 00.2 0.4−0.4−0.2 00.2 0.4Γ−−Κ−Γ−−Μ−\nSOC\nno SOCSOC\nno SOC\nλ=1.6 nm λ=1.7 nm\n(q,0,0) (0,q,0)\nSpin spiral vector q (in 2π/a)-5.1 -5.3fb*\n−20−15−10−505101520\n−0.4−0.2 00.2 0.4−0.4−0.2 00.2 0.4Γ−−X−Γ−−Y−\nSOC\nno SOCSOC\nno SOC\nno spin \nspiral λ=3.4 nm\n(q,0,0) (0,q,0)\nSpin spiral vector q (in 2π/a)\n-1.6-0.8(d) fh\n−30−25−20−15−10−50510\n−0.4−0.2 00.2 0.4SOC\nno SOC\n−0.4−0.2 00.2 0.4Γ−−Κ−Γ−−Μ−\nSOC\nno SOC\nλ=1.1 nmλ=1.1 nm\n(q,0,0) (0,q,0)\nSpin spiral vector q(in 2π/a)-15.6\n-17.7Total energy E(q) (in meV) Total energy E(q) (in meV)\nTotal energy E(q) (in meV) Total energy E(q) (in meV)\nFIG. 16. (Color online) Full energy dispersions including SOC of flat cycloidal spin spirals from DFT relative to the FM state of stackings (a)\nff, (b) fb\u0003, (c) fh and (d) hf along high symmetry directions \u0016\u0000\u0000\u0016Kand\u0016\u0000\u0000\u0016M. Given are values with (filled circles, colored solid lines) and\nwithout SOC (open triangles, grey dashed lines). Positive (negative) values of qindicate clockwise (anti-clockwise) rotating spin spirals. In all\nstackings clockwise-rotating spin spirals are favoured. The contribution of the magnetocrystalline anisotropy to spin spirals amounts to1\n2K1.\nThus, all stackings favor spin spiral ground states with dif-\nferent symmetries. They are either stabilized by both mag-\nnetic exchange and DMI, as in ff and fh, or only by DMI, as\nin fb\u0003and hf. There is a more or less strong preference for\nspin spirals along the \u0000Kdirection depending on the stack-\ning. In fb\u0003, along they-axis no spin spiral can be stabilized\nat all, while in ff, spin spirals along \u0000Kand\u0000Mdirection are\ndegenerate.\nLet us compare our results with previous STM experiments.\nIn the supposedly ff-stacked regions a spin spiral of 1.2 nm\nwas identified without a preferred propagation direction.25We\nalso find for this stacking, that the spin spirals along the dif-\nferent propagation directions are degenerate, but the wave-\nlength is 1.6 nm, which corresponds well with the experimen-\ntal value. In the fb\u0003-stacked regions, spin spirals propagat-\ning only along the [100]-direction of the bcc-unit cell with\na wavelength of 1.9 nm were reported. We can confirm this\nfinding. In the fb\u0003-stacking, spin spirals are only found in\nthe\u0000Xdirection, which corresponds to the [100] crystallo-graphic axis of the bcc-unit cell. However, our wavelength\nof 3.4 nm is substantially larger than the experimental value.\nThis is probably due to the differences in strain exposed on the\nFe@vac-layer at the surface. We assumed a pseudomorphi-\ncally strained bcc-like stacking, while SP-STM experiments\nreveal a complex surface reconstruction, which probably oc-\ncurs in order to compensate for the large strain. We recently\naddressed this issue in another publication.26\nIV . CONCLUSION\nWe have investigated the interplay between the Fe double-\nlayer stacking and the stabiliy and the magnetic interactions\nfor 2Fe/Ir (111) ultra-thin films using density functional the-\nory and mapping on an atomistic spin model.\nWe considered in total six different double-layer stackings:\nthe hexagonal close-packed variants ff, fh, hf and hh, and\ntwo structures with a bcc (110)-like surface Fe-layer fb\u0003and16\nhb\u0003. We find that the fb\u0003stacking with a bcc (110)-like sur-\nface layer is the most stable pseudomorphic structure, whereas\nthe hexagonal close-packed stackings ff and fh are not kinet-\nically/mechanically stable. Even if formed during the growth\nprocess, they should transform into the bcc-like stacking at\nconsiderable surface coverages.\nMoreover, we computed the magnetic exchange interaction\nbeyond first nearest neighbors, the DMI beyond first near-\nest neighbors and the magnetocrystalline anisotropy. We find\nconsiderable differences for all magnetic interactions depend-\ning on the Fe double-layer stacking. There are no general\ntrends though. Therefore, every structure has to be analyzed\nindividually.\nIn ff and fh we obtain a clockwise cycloidal spin spirals\nwith periods of 1.6 nm and 1.1 nm, respectively, which are\nstabilized by the frustrated exchange and DMI, while the spin\nspirals in fb\u0003and hf with wavelengths 3.4 nm and 3.0 nm are\nstabilized by the DMI.\nThe reduced symmetry arising from the surface Fe-layer in\nthe bcc-like stackings (plane group c minstead of p 3m1for\nthe hexagonal close-packed structures) has fundamental con-\nsequences for the symmetry of the magnetic interactions. Not\nonly the exchange interaction is affected, but also the DMI.\nWe can explain this finding based on the density of states of\nthe Fe 3dand Ir 5delectrons. We show that an interfacial\nDMI may be influenced by a magnetic overlayer further from\nthe interface.\nThis interaction between the surface Fe-layer and the Ir-substrate provides a new opportunity to tune the DMI by im-\nposing different symmetries on the DMI. In the case of 2Fe/Ir\n(111), the 6-fold symmetric DMI present at the Ir-Fe inter-\nface can be modified by lowering the symmetry of the sur-\nface Fe-layer. The deposition of a bcc 5dtransition metal on\n2Fe/Ir (111) should create a large 2-fold symmetric DMI in\nthe interface between Fe@vac and the 5dtransition metal.\nThe combination of two DMIs of 6-fold and 2-fold symme-\ntry could enable the stabilization of topologically protected\nstates with spatially varying chirality. This variation of chi-\nrality may allow the stabilization of higher order skyrmions\n(such asS=\u00002orS=\u00003) in multilayer geometry.\nACKNOWLEDGEMENTS\nMD and BD are grateful for the computing time granted\non the supercomputers of the North-German Supercomputing\nAlliance (HLRN) and Mogon at Johannes Gutenberg Univer-\nsitt Mainz [55]. MD, BD and JS acknowledge financial sup-\nport from the Alexander von Humboldt Foundation and the\nTransregional Collaborative Research Center (SFB/TRR) 173\nSPIN+X. 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Mandrus2, 3\n1Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA\n2Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA\n3Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA\n(Dated: July 19, 2018)\nThe topologically-protected, chiral soliton lattice is a unique state of matter offering intriguing functionality\nand it may serve as a robust platform for storing and transporting information in future spintronics devices.\nWhile the monoaxial chiral magnet Cr 1=3NbS 2is known to host this exotic state in an applied magnetic field,\nits detailed microscopic origin has remained a matter of debate. Here we work towards addressing this open\nquestion by measuring the spin wave spectrum of Cr 1=3NbS 2over the entire Brillouin zone with inelastic neu-\ntron scattering. The well-defined spin wave modes allow us to determine the values of several microscopic\ninteractions for this system. The experimental data is well-explained by a Heisenberg Hamiltonian with ex-\nchange constants up to third nearest neighbor and an easy plane magnetocrystalline anisotropy term. Our work\nshows that both the second and third nearest neighbor exchange interactions contribute to the formation of the\nhelimagnetic and chiral soliton lattice states in this robust three-dimensional magnet.\nChiral objects, which are non-superimposable on their mir-\nror images, are ubiquitous in nature. These objects appear in\na wide variety of forms, including molecules, biological sys-\ntems, and magnetic materials. They provide a great deal of\nfunctionality [1] and therefore are poised to be the catalyst to\ninnovative technologies.\nThe crystal structures of chiral magnets lack both inver-\nsion centers and mirror planes and therefore the antisymmet-\nric Dzyaloshinskii-Moriya interaction (DMI) is symmetry-\nallowed. Competition between Heisenberg exchange, the\nDMI, magnetocrystalline anisotropy and an applied magnetic\nfield can lead to interesting, non-trivial spin textures such as\nthe skrymion lattice (SkX) originally proposed by Bogdanov\nand Hubert [2]. This topologically-protected phase consists\nof an array of magnetic vortices with particular spin wind-\nings and it has now been established as an accessible phase of\nmatter in the temperature-magnetic field phase diagram of a\nwide variety of cubic, chiral helimagnets [3–6]. The SkX state\nis particularly attractive for many potential spintronics appli-\ncations, including skyrmion-based race track memory, logic\ngates, and transistors [7], due to the mesoscale length scale\n(5-100 nm) of an individual skyrmion [8] and the small cur-\nrent densities ( 106A/m2for MnSi [9]) required to manipulate\nthe lattice through the spin-torque effect.\nMagnetocrystalline anisotropy plays a minor role in the sta-\nbility of the skyrmion lattice state in the phase diagrams of\nMnSi and isostructural helimagnets [10] due to its weak con-\ntribution to the Hamiltonian of cubic systems. This feature\nis advantageous when looking for skyrmion hosts because the\napplied field direction is not critical for the detection of the\nskyrmion state. On the other hand, the stronger magnetocrys-\ntalline anisotropy inherent to lower-symmetry crystal systems\ncan be beneficial for enhancing stability of the SkX state in\ncertain cases [11–13] and it can also lead to entirely unex-\npected magnetic phases. For example, the chiral soliton lattice\n(CSL) was observed recently in the monoaxial chiral helimag-\nnet Cr 1=3NbS 2, which is a topologically-protected phase con-\nsisting of ferromagnetic regions partitioned by helical domainwalls [14]. This state arises when a static magnetic field is\napplied perpendicular to the helical axis and its size and shape\ncan be controlled by varying the field. The high tunability, ro-\nbust nature, and other unique properties of the CSL phase are\nintriguing in the context of spintronics devices with informa-\ntion storage or transportation capabilities [15–17].\nThe detailed microscopic origin of the CSL phase remains\nunknown due to the absence of a robust Hamiltonian for\nCr1=3NbS 2. For this reason, the magnetic properties have\nbeen most commonly studied by the chiral sine-Gordon model\n[18, 19]. In its simplest form, this model consists of Cr\nspins on a 1D lattice that are only coupled by nearest neigh-\nbor FM exchange Jkand a DMI term with ~Dpointing along\nthe chain direction. A helical magnetic ground state arises\nin zero field due to the competition between these two terms,\nwith the turn angle \u000bbetween spins at adjacent sites given\nby\u000b= arctan(D=Jk). This simple model can be used\nto calculate MvsT[18],MvsH[20], and the field-\ndependence of the electron spin resonance spectra [21] that\nare characteristic of the CSL phase and reproduces the corre-\nsponding experimental data well for appropriate choices of\nDandJk[22, 23]. Numerical simulations with the chiral\nsine-Gordon model and an extended version including nearest\nneighbor intraplane coupling J?have also been performed\n[19]. These calculations enabled the authors to estimate Jk,\nJ?, andDand they found that J?>>Jk, which suggests that\nCr1=3NbS 2is a quasi-two-dimensional (2D) magnet. Notably,\nthere is significant variance in the values of JkandDreported\nin these three works. This observation calls into question\nthe simple assumption of the chiral sine-Gordon model for\nCr1=3NbS 2, namely\u000b= arctan(D=Jk). Furthermore, recent\nfirst principles work [24] calculated the exchange constants\nfor Cr 1=3NbS 2and found that Jk>J?, a scenario more con-\nsistent with a robust three-dimensional (3D) magnet. These\ntwo discrepancies have remained unresolved in the literature.\nCr1=3NbS 2is one member of a large family of metallic,\nmonoaxial chiral magnets that have received little attention\noverall [25]. Assuming that a local moment model is appli-arXiv:1807.06665v1  [cond-mat.str-el]  17 Jul 20182\ncable for this family of materials, careful parameterization of\nthe Hamiltonian can be achieved by measuring spin wave dis-\npersions along different crystallographic directions over the\nentire Brillouin zone with inelastic neutron scattering (INS).\nIn this letter, we use INS to measure the magnetic excita-\ntion spectrum of Cr 1=3NbS 2in the helimagnetic state. We\nfind that the spin waves are well-defined throughout the ma-\njority of the Brillouin zone, and therefore we are able to de-\ntermine the Heisenberg exchange constants and magnetocrys-\ntalline anisotropy using linear spin wave theory. Our work\npresents the first step towards developing a complete local\nmoment model that can explain the microscopic origin of the\nhelimagnetic and CSL states observed in this material. This\nmodel may provide insight on how this material can be mod-\nified to optimize the CSL state for future applications or to\nrealize other interesting spin textures such as the SkX state.\nWe note that there is a great opportunity to tune Cr 1=3NbS 2\nthrough uniaxial strain [26] or chemical substitution [25].\nCr1=3NbS 2single crystals were grown at the University of\nTennessee by chemical vapor transport with iodine serving as\nthe transport agent [27]; additional growth details are pro-\nvided in Ref. [28]. Magnetization data was collected with a\nMagnetic Property Measurement System at Oak Ridge Na-\ntional Laboratory (ORNL) to verify that the crystals showed\nevidence for the CSL state in modest magnetic fields.\nINS experiments were performed on the direct-geometry\ntime-of-flight chopper spectrometer SEQUOIA at the Spalla-\ntion Neutron Source at ORNL. An array of 20 single crystals\nwith a total mass of \u00181 g was co-aligned with a 5\u000emosaic\nspread and loaded with the (H0L) plane horizontal in a 8 K\nbase temperature closed cycle refrigerator for this experiment.\nSpectra were constructed by performing 18 minute runs at\nseveral different sample orientations rotating about the [-1 2 0]\ndirection and then combining these runs into a single data file\nfor a given set of instrument conditions. We used an incident\nenergyEi=50 meV , a Fermi chopper frequency of 180 Hz,\nand aT0chopper frequency of 90 Hz, providing an energy\nresolution at the elastic line of approximately 2.7 meV . Ad-\nditional INS measurements with much finer energy resolution\nwere performed on the cold triple axis spectrometer CTAX at\nthe High Flux Isotope Reactor at ORNL. The same co-aligned\narray of crystals was loaded into a cryostat with a base tem-\nperature of 1.5 K for this experiment. The measurements were\nperformed using a fixed final energy Ef=5 meV and colli-\nmation settings of guide-open-80’-open, which generated an\nenergy resolution at the elastic line of 0.3 meV . This fine en-\nergy resolution enabled us to put stringent constraints on an\nenergy gap associated with the spin wave spectrum. No mag-\nnetic field was applied during the INS experiments.\nCr1=3NbS 2crystallizes in the chiral space group P6322;\nthe crystal structure is illustrated in Fig. 1. The Cr atoms oc-\ncupy the 2c Wyckoff site exclusively in the ideal structure.\nThis arrangement results in AB stacking along the c-axis and\nap\n3\u0002p\n3superstructure within the ab-plane with respect to\nthe unit cell of NbS 2. Cr 1=3NbS 2has a helimagnetic ground\nstate with a pitch of 47(1) nm that onsets below Tc=127 K\nFIG. 1: (color online) A schematic of the Cr 1=3NbS 2crystal struc-\nture, depicting the Cr atoms with blue spheres. The three Heisenberg\nexchange pathways considered in this work are J1,J2, andJ3.\nFIG. 2: (color online) (a) MvsHfor a representative Cr 1=3NbS 2\nsingle crystal investigated by INS in this work. (b) M=H vsTfor\nthe same Cr 1=3NbS 2single crystal.\n[14, 28]. The Cr moments are ferromagnetically-coupled in\nthe ab-plane and the magnetic helix propagates along the c-\naxis. This system is very susceptible to Cr disorder since a\nfraction of the 2b and 2d Wyckoff sites may also be occu-\npied instead of their 2c Wyckoff site counterparts. A signifi-\ncant amount of Cr disorder can have profound consequences\non the material properties by modifying both the crystal and\nmagnetic structures, ultimately leading to a centrosymmetric\ncrystal structure hosting a ferromagnetic ground state with a\nTc=88 K [29].\nCr1=3NbS 2crystals with the chiral structure can be identi-\nfied by performing magnetization measurements. Figure 2(a)\ndepictsMvsHdata with~H?^cat 2 K for one of the crys-\ntals in our INS mosaic. The magnetization reaches a satura-\ntion value of 2.7 \u0016BforH=2 kG, which is just under the\nexpected value of 3 \u0016Bfor localized S=3/2 magnetic mo-\nments. The general shape and hysteresis of the MvsHcurve\nare consistent with the development of a chiral soliton lattice\nin finite magnetic fields [18, 20]. Figure 2(b) presents M=H\nvsTdata at 1 kG with ~H?^c. The characteristic cusp ex-\npected atTcwhen cooling into the CSL state is clearly visi-\nble [18, 20] and the value of Tcagrees well with expectations\nfor the chiral crystal structure. These combined magnetiza-\ntion measurements therefore illustrate that our representative\nCr1=3NbS 2single crystal has the desired P6322structure. We3\nFIG. 3: (color online) Energy-momentum color contour plots from SEQUOIA at T=8 K with Ei=50 meV: (a) energy transfer Evs [0\n0 L], (b) Evs [1.5 0 L], (c) Evs [H H 2], and (d) Evs [H 0 -2]. Integration ranges along the two perpendicular momentum axes were 0.15\nreciprocal lattice units (rlu) for directions in the (HK0) plane and 0.3 rlu along [0 0 L]. The black curves superimposed on the data represent the\ndispersion relations calculated using the best fit parameters for the J1-J2-J3-AHeisenberg model. (e)-(h) Energy-momentum color contour\nsimulations calculated with linear spin wave theory using the best fit parameters for the J1-J2-J3-AHeisenberg model .\nrepeated these measurements on other crystals from our mo-\nsaic and obtained identical results.\nOur time-of-flight neutron spectroscopy measurements\nwere particularly useful for mapping out the magnetic dis-\npersions and dynamical structure factors over the entire Bril-\nlouin zone. Representative energy-momentum color con-\ntour plots of the INS spectra at T=8 K along specific\nhigh-symmetry crystallographic directions are presented in\nFig. 3(a)-(d). Well-defined modes appear to originate from the\nBrillouin zone centers (i.e. \u0000points) expected for ferromag-\nnetic spin wave theory and they disappear above Tc=127 K\nas shown in Fig. 4(a) and the Supplementary Material (SM),\nwhich is indicative of a spin-wave origin. The sharp excita-\ntions observed throughout the majority of the Brillouin zone,\ncombined with a saturation moment reduced only slightly\nfrom expectations for localized magnetism, are consistent\nwith a metallic ferromagnet in the strong moment limit and\ntherefore a Heisenberg model should describe the spin dynam-\nics well. Complementary cold triple axis data, in the form\nof a constant Q-scan at the (002) zone center, is shown in\nFig. 4(b). There is no indication of a magnon mode in this\ndata, and therefore the spin wave spectrum is gapless within\nthe 0.3 meV energy resolution of CTAX. This result is consis-\ntent with a magnetically-ordered system characterized by easy\nplane anisotropy, which was initially revealed by magnetiza-\ntion measurements [28].\nA minimal Hamiltonian that can adequately explain the\nlong-range helimagnetic ordering of Cr 1=3NbS 2includes two\nHeisenberg exchange parameters, J1andJ2, an easy plane\nanisotropy term A, and a DMI contribution D[28].J1couples\nnearest neighbor (NN) Cr moments in the ab-plane, while J2\nandDrepresent the NN interplane interactions. We neglect D\nhere because our INS experiments cannot resolve the momen-\nFIG. 4: (color online) (a) Energy-momentum color contour plot\nfrom SEQUOIA for Evs [H 0 -2] with T=150 K, Ei=50 meV ,\nand integration ranges [0 0 L] =[-2.15, -1.85] rlu and [H -2H 0] =[-\n0.075, 0.075] rlu. (b) A constant- Qscan from CTAX at the Brillouin\nzone center (0 0 2) with T=1.5 K and Ef=5 meV . Note that 1\nmonitor count unit (mcu) equals 1 s when Ei=5 meV .\ntum shift away from commensurate reciprocal space points of\nthe magnetic Bragg peaks or of the spin wave dispersion min-\nima that a non-zero Dshould generate [30]. Also, we observe\nno evidence for the helimagnon bands expected to arise for\nnon-zeroD[31]. We used the SpinW software [32] to com-\npute the spin wave dispersions and dynamical structure factors\nwith linear spin wave theory for this simple J1-J2-AHeisen-\nberg model. Our magnetic ground state consists of S=3/2\nCr moments ferromagnetically-aligned along the crystallo-\ngraphic a-axis. The Hamiltonian parameters were determined\nby using SpinW ’s swarm particle optimization routine [33] to\nminimize the sum of the squared residuals obtained from com-\nparing the experimental and model-calculated dispersion rela-\ntions. The experimental dispersion relations were quantified\nby fitting a series of SEQUOIA constant- Qand constant- E\ncuts to Gaussian functions (plus a sloping background) along4\nthe four high-symmetry directions shown in Fig. 3. Represen-\ntative constant- Qcuts with the Gaussian fits superimposed are\nshown in the SM. This fitting routine was run to convergence\n50 times and the interaction parameters reported in this work\ncorrespond to the average values, while the errors correspond\nto the standard deviations of the fit parameters.\nThe best fit parameters for this model are J1=-\n0.51(2) meV , J2=-1.22(1) meV , and A=0.95(9) meV .\nThe calculated scattering intensity using these parameters\nis presented along several high-symmetry directions in the\nSM. Note that negative values for the exchange parame-\nters correspond to ferromagnetic interactions, while the pos-\nitiveAvalue represents an easy plane magnetocrystalline\nanisotropy. Both the Cr3+magnetic form factor and energy\nbroadening were included in these calculations; the latter was\naccomplished by convolving the dynamical structure factor\nwith a Gaussian function. The full-width-half-maximum of\nthis function was fixed to 2.7 meV , which corresponds to\nthe instrumental energy resolution at the elastic line for the\nEi=50 meV data set. There are clear deviations between\nthe calculation and the INS data for this simple model. More\nspecifically, a large gap appears between the low and high en-\nergy modes along the [H 0 -2] direction that is not present in\nthe experimental data and the dispersion along the [1.5 0 L]\ndirection is significantly overestimated by the model.\nIn an effort to improve the overall agreement between the-\nory and experiment, we added a second interplane Heisen-\nberg term to our model, J3. TheJ1-J3exchange paths are\nshown in Fig. 1. We used the same fitting procedure described\nabove to obtain the Hamiltonian parameters in this case, and\nthe calculated scattering intensity using the best fit parame-\nters is presented along several high-symmetry directions in\nFig. 3(e)-(h). We obtain excellent agreement between the INS\ndata and the calculation using this model for the parameter\nset presented in Table I. While there is extra scattering in the\ndata along some of the high-symmetry directions between en-\nergy transfers of 20-25 meV that is not accounted for by the\ncalculation, it likely arises from phonon scattering associated\nwith the Al sample can since the spectral weight appears to\nincrease with Qand persists above Tc.\nOur final parameterization of the Hamiltonian for\nCr1=3NbS 2implies that the RKKY mechanism is mediating\nthe exchange interactions [34, 35], as the long-range cou-\nplings that we find are too strong to have a superexchange\norigin. Notably, the NN and 2NN superexchange pathways\nconsist of two S atoms while the 3NN superexchange path-\nway is even longer and mediated by three S atoms. Our best\nfit Hamiltonian parameters are also inconsistent with some\nof the assumptions and conclusions made in earlier work on\nCr1=3NbS 2. We find that the expression for the helical turn an-\ngle\u000b= arctan(D=Jk)obtained from the chiral sine-Gordon\nmodel is not appropriate here since the Heisenberg exchange\ninteractions between magnetic ions in adjacent ab-planes ex-\ntend beyond NN, which contradicts the simple assumption\nmade in this model. We also find that Cr 1=3NbS 2does not\nconsist of weakly-coupled Cr planes, as this would have ledto a very weak dispersion along the [00L] direction [36] that\nis inconsistent with our INS data. Instead, Cr 1=3NbS 2is\na robust 3D magnet due to the similar values of the intra-\nplane exchange J1and the interplane exchanges J2andJ3.\nThese strong 3D interactions likely work together in a co-\noperative fashion to produce the high ordering temperature\nofTc=127 K. Our findings are in excellent agreement with\nrecent work reporting 3D Heisenberg critical exponents for\nCr1=3NbS 2[37].\nTABLE I: Microscopic parameters determined by fitting the inelastic\nneutron scattering data to the extended Heisenberg model described\nin the main text. Bonds are listed in the form ( x,y,z), where x,y,\nandzare fractions of the direct lattice vectors ~ a,~b, and~ c.\nExchange No. neighbors Bond Distance ( ˚A)Value (meV)\nJ1 6 (1, 0, 0) 5.74 -0.37(1)\nJ2 6 (1/3, -1/3, 1/2) 6.90 -1.02(2)\nJ3 6 (2/3, -2/3, 1/2) 8.97 -0.28(2)\nA - - - 0.8(1)\nIt is interesting to compare our INS results to recent the-\noretical work that determined the Hamiltonian parameters\nfor Cr 1=3NbS 2using first principles calculations [24]. Their\nHeisenberg exchange constants, reported as JijS2, reveal that\nthe first three NN terms in Cr 1=3NbS 2are dominant and the\n2NN term provides the largest contribution to the Hamilto-\nnian. These qualitative findings and the magnitudes reported\nfor these parameters are in excellent agreement with our re-\nsults presented here. The first principles work also finds\nan easy plane magnetocrystalline anisotropy on the order of\n0.1 meV , which agrees well with an earlier experimental esti-\nmate of 0.4 meV using magnetization data [28] and our value\nobtained with INS. Interestingly, the first principles work de-\ntermines values for interplane DMIs that we were unable to\nresolve in our work. They find that the DMIs corresponding\nto several different Cr-Cr interplane distances are comparable.\nThis result implies that the microscopic details leading to the\nestablishment of the long-period helimagnetic and soliton lat-\ntice states in Cr 1=3NbS 2may be rather complicated, but this\nprediction has yet to be verified by experiment.\nIn conclusion, we have presented the full spin wave spec-\ntrum of Cr 1=3NbS 2, as revealed by INS measurements. The\nspin wave dispersions and dynamical structure factors are ex-\nplained well by a Hamiltonian with Heisenberg exchange con-\nstants up to 3rd nearest neighbor and an easy plane magne-\ntocrystalline anisotropy. Our successful parameterization of\nthe Cr 1=3NbS 2INS data using a local moment model is im-\nportant because it implies that this technique can be used for\ndetermining the exchange constants and magnetocrystalline\nanisotropy of other isostructural materials. Elucidating the\nmicroscopic origins of the long-period magnetic structures\nwith effective Hamiltonians should be far more straightfor-\nward than establishing a direct link between this interesting5\nmagnetism and the band structure, which is a necessity for the\nweak itinerant magnet MnSi [38, 39]. Intercalated transition\nmetal dichalcogenides are therefore excellent systems to pur-\nsue optimization of the CSL state for future applications or to\nsearch for CSL/SkX hosts via a systematic approach.\nSUPPLEMENTARY MATERIAL\nSee supplementary material for additional color contour\nplots of the T=150 K SEQUOIA data, representative\nconstant-Qcuts from the T=8 K SEQUOIA data, and spin\nwave simulations for the J1-J2-AHeisenberg Hamiltonian\ndescribed in the main text.\nACKNOWLEDGMENTS\nWe thank M.B. Stone for technical support. MAM ac-\nknowledges support from the US Department of Energy, Of-\nfice of Science, Basic Energy Sciences, Materials Sciences\nand Engineering Division. 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Rev.\nB16, 4956 (1977).\n[39] P. Boni, B. Roessli, and K. Hradil, J. Phys.: Condens. Matter\n23, 254209 (2011)."    },    {        "title": "1807.06884v2.Controlled_anisotropic_dynamics_of_tightly_bound_skyrmions_in_a_synthetic_ferrimagnet_due_to_skyrmion_deformation_mediated_by_induced_uniaxial_in_plane_anisotropy.pdf",        "content": "Controlled anisotropic dynamics of tightly bound skyrmions in a synthetic ferrimagnet due to\nskyrmion-deformation mediated by induced uniaxial in-plane anisotropy\nP. E. Roy,1,\u0003Rub´en M. Otxoa,1, 2and C. Moutafis3\n1Hitachi Cambridge Laboratory, J. J. Thomson Avenue, CB3 OHE, Cambridge, United Kingdom\n2Donostia International Physics Center, Paseo Manuel de Lardizabal 4, Donostia-San Sebastian 20018, Spain\n3School of Computer Science, University of Manchester, Manchester M13 9PL, United Kingdom\n(Dated: July 20, 2018)\nWe study speed and skew deflection-angle dependence on skyrmion deformations of a tightly bound\ntwo-skyrmion state in a synthetic ferrimagnet. We condsider here, an in-plane uniaxial magnetocrystalline\nanisotropy-term in order to induce lateral shape distortions and an overall size modulation of the skyrmions due\nto a reduction of the effective out-of-plane anisotropy, thus affecting the skyrmion speed, skew-deflection and\ninducing anisotropy in these quantities with respect to the driving current-angle. Because of frustrated dipolar\ninteractions in a synthetic ferrimagnet, sizeable skyrmion deformations can be induced with relatively small\ninduced anisotropy constants and thus a wide range of tuneability can be achieved. We also show analytically,\nthat a consequence of the skyrmion deformation can, under certain conditions cause a skyrmion deflection with\nrespect to driving-current angles, unrelated to the topological charge. Results are analyzed by a combination\nof micromagnetic simulations and a compound particle description within the Thiele-formalism from which an\nover-all mobility tensor is constructed. This work offers an additional path towards in-situ tuning of skyrmion\ndynamics.\nI. INTRODUCTION\nMagnetic skyrmions have been extolled as candidates for\nthe constituent information carriers in spintronic devices such\nas racetrack memories and logic circuits1–6. Their advantage\nover conventional domain walls are that they are less sensi-\ntive to edge defects and can be driven at much lower current\ndensities. However, similar to the Hall-effect for an elec-\ntrically charged particle, a magnetic skyrmion viewed as a\nquasi-particle endowed with a topological charge, exhibits a\ndeflection known as the skyrmion Hall-effect7,8, resulting in a\nskew deflection known as skyrmion Hall-angle, \u0002Sk. This can\nlead to detrimental annihilation at device-boundaries at high\nenough driving amplitudes. In order to overcome this, there\nhave been proposals suggesting the usage of both intrinsic9–11\nand synthetic antiferromagnets12,13(SAFs) with identical but\noppositely magnetized sublattices, whereby the direction of\nthe gyroforce on a skyrmion in one sublattice is equal but op-\nposite in direction for the skyrmion on the other sublattice9.\nThis is because, by virtue of satisfying the antiferromagnetic\ncoupling they have opposite topological charge. The two de-\nflection forces then fully cancel out and the compound object\nmoves without deflection. An advantage of antiferromagnetic\nsystems is the robustness against external field perturbations.\nHowever this means that the manipulation and detection of an-\ntiferromagnetic textures is generally a difficult task. Devices\nbased on single layer ferromagnets as the functional compo-\nnent have the advantage of easy detection schemes but are\nsenstitive to external fields. However in ferrimagnetic sys-\ntems, a low net moment is present, offering reasonable de-\ntectibility and at the same time a good degree of robustness\nagainst disturbing stray-fields. A net moment though, means\na finite skyrmion-Hall angle (even if it is much reduced com-\npared to a single layer ferromagnetic system). An additional\nbenefit of synthetic systems is the large tuneability of mate-\nrial properties, but the conclusions extolled herein should bevalid also for instrinsic ferrimagnets. Apart from the desire\nto achieve some degree of control over \u0002Sk, another impor-\ntant dynamical property to tune and/or enhance is the speed\nof skyrmion propagation. Enhancement is desired due to in-\ncreased operating frequency of a device. Tuning in general\nwould find usage wherever the skyrmionic device is a sub-\nset of a bigger multifunctional device, whereby timescales\nneed to be matched at different points of device functional-\nity in space. In order to modulate the dynamical properties\nof a skyrmionic magnetic device, one may consider tailor-\ning of material properties during fabrication and/or affecting\nthe ready-made device by external means during operation.\nWe refer to the former as intrinsic and the latter as extrinsic\ntuning, respectively. In particular, extrinsic tuning offers ma-\nnipulation on the fly. Different approaches have previously\nbeen presented for dynamical control such as: by mismatch-\ning the saturation magnetization of the constituent ferromag-\nnetic (FM) layers in an SFIM14, spatially uniform modulation\nof the perpendicular anisotropy15, perpendicular anisotropy\ngradients16,17and radial magnetic field gradients18. In such\na scenario (for a given topological charge) the skyrmion ra-\ndius is isotropically varied and thus its dynamical behaviour is\nisotropic in the plane of propagation with respect to the driv-\ning current-angle. There have recently been some very in-\nteresting works considering the effect of inducing anisotropic\nDzyaloshinskii-Moriya interaction in single layer FMs as a\ncontrol-knob for skyrmion dynamics (speed and skew-angle),\noffering also due to skyrmion-shape distortion anisotropic dy-\nnamical behaviour with respect to the direction of the driving\nexcitation mechanism,19,20. The anisotropic dynamics adds an\nadditional degree of freedom in tuning the skyrmion dynami-\ncal behaviour.\nIn this work, we focus on synthetic ferrimagnets (SFIMs)\nfor reasons mentioned in the preceeding paragraph and con-\nsider here tuning the dynamics due to deformation of a bound\ntwo-skyrmion texture via only an induced uniaxial in-plane\nanisotropy term, achievable by e.g. an inverse magnetostric-arXiv:1807.06884v2  [cond-mat.mes-hall]  19 Jul 20182\ntive effect whereby a mechanical uniaxial stress induces a uni-\naxial contribution to the magnetocrystalline energy21. The\ndriving mechanism is provided by spin-Hall-effect induced\ntorques. By utilizing a synthetic ferrimagnet, we are able\nto induce large skyrmion deformations for relatively small\nvalues of the induced uniaxial in-plane anisotropy constants\nand predict a wide tuneability of the bound skyrmion skew-\ndeflection, speed and degree of anisotropy of the said quan-\ntities with respect to the driving current-angle. In addition,\nit is shown, that, given a deviation from circularly shaped\nskyrmions, that for driving current angles different from mul-\ntiples of\u0019=2, there is a contribution to the skyrmion deflection\naway from the driving-current direction which is independent\nof the topological charge.\nII. METHODS\nA. Thiele approach and the effective skyrmion mobility tensor\nThe system under consideration is schematically depicted\nin Fig. (1), whereby two FM layers (FM 1and FM 2) are sepa-\nrated by a Ru spacer, magnetically coupled via antiferromag-\nnetic (RKKY-type) and dipolar interactions. Each FM layer\nis also coupled to a heavy metal (HM) which promotes inter-\nfacial Dzyaloshinskii-Moriya (IDMI)-interaction. Each FM\nlayer contains one skyrmion. The two skyrmions are consid-\nered to be strongly antiferromagnetically coupled, i.e. tightly\nbound such that they move as one compound unit without in-\nternal nor relative dynamics. The means of propulsion of this\ncompound skyrmion is supplied by torques exerted by a spin-\naccummulation due to the Spin-Hall-Effect (SHE). The SHE\nis produced by passing a current Ithrough the HMs, generat-\ning current-densities J1andJ2in HM 1and HM 2, respectively\n(see Fig. 1 (a)). We consider an arbitrary in-plane current-\ndirection. Thus we define a global coordinate system and a\nprimed/local system, whereby the current direction defines the\nrotation of primed coordinates with respect to global coordi-\nnates, as shown in Fig. (1 (b)).\nIn order to analytically describe the dynamics of the\nskyrmions, we turn to the Thiele equation22with Spin-Hall\nforces2,7,8. We consider the tightly bound skyrmions in dy-\nnamic equilibrium and sum the forces to zero according to14:\nX\nL\u00160dLML\n\rLf\u0000QL[^z\u0002v]\u0000\u000bLDDDL\u0001v+ 4\u0019b0\nLIIIL\u0001JLg=0:\n(1)\nThe subscript L= 1;2denotes the FM layer num-\nber (Fig. 1 (a)), \rLis the gyromagnetic ratio, \u00160,\nthe permeability in vacuum, MLthe saturation magne-\ntization and dLthe layer thickness. The first term\nin Eq. (1) represents the gyroforce with topological\ncharge23QL=1\n4\u0019RR^mL\u0001[@x^mL\u0002@y^mL]dxdy (the\ntopological charge being a component of the gyrovector\nGLthrough the relation GL= [0 0\u00004\u0019QL]), with\n^mL=ML=ML. The second term constitutes a dissipa-\ntive drag force with an associated dissipation tensor DDDL=\n4\u0019h(Dii)L(Dij)L\n(Dji)L(Djj)Li\n, whose elements are given by (Dij)L=\nFIG. 1. (a) Geometry and constituent layers of the stack consid-\nered in this work. (b): Top view of the global ( xy) and primed/local\n(x0y0) coordinate systems. Here JLis the current density in the L:th\nFM layer whose direction is at an angle \u0012Jwith respect to the global\nsystem. \u0002Skof the bound skyrmion is defined as the angle of devi-\nation from the direction of the current density. Double arrows sig-\nnify the considered directions of induced uniaxial anisotropies (with\nanisotropy constants KxandKydepending on whether the easy di-\nrection is induced along xory, respectively). The velocity compo-\nnents of the effective skyrmion in the global coordinate system are\ndesignatedvxandvy, whereas the speed is denoted as v.\n1\n4\u0019RR\n[@i^mL\u0001@j^mL]dxdy2,22. The last term is the force ex-\nerted by the SHE-induced torque with b0\nL=\rL~\u001eLoL\n\u001602jejMLdL24and\n[Iqr]L=\u00001\n4\u0019RR\n[@q^mL\u0002^mL]s\u000fsrdxdy2,25;q;r2 fx;yg\nand\u000fsris the Levi-Civita symbol. Further, ~is Planck’s re-\nduced constant, \u001eLthe intrinsic spin-Hall angle and eis an\nelectron’s charge. The unit magnitude factor oLtakes into ac-\ncount the effect of HM/FM stacking order, on the direction\nof the resulting spin-accummulation. As HM 1is under FM 1\nand HM 2is above FM 2o1ando2have opposite sign. We set\nhere,o1= 1 ando2=\u00001. For brevity we assign a tensor\nSSSL=b0\nLIIIL. Note that the units of SSSis [m3/As] (velocity per\nunit current density) and as such may be viewed as a SHE-\nrelated mobility tensor in the absence of other forces. We re-\niterate, that the underlying assumptions in stating the problem\nby Eq. (1), are that the internal structures of the skyrmions are\nrigid and that the antiferromagnetic coupling between them is\nstrong enough such that they move together without any rela-\ntive dynamics. Now, intrinsic tuning of \u0002Skand speed of the\ntightly bound system would be to consider a mismatch of the\nsaturation magnetization, gyromagnetic ratio or thickness be-\ntween the two FM layers. This is more easily seen by rewrit-\ning Eq. (1) as\u0000Qe[^z\u0002v]\u0000DDDe\u0001v+wSSS1\u0001J1+SSS2\u0001J2=0,\nwhereQe=wQ1+Q2andDDDe=w\u000b1DDD1+\u000b2DDD2, with\nw=\r2d1M1\n\r1d2M2. Further, we write the current density J2in terms\nofJ1and make the reasonable assumption that J1andJ2\npoint in the same direction such that J2=ksJ1, whereks\nis a real scaling factor. Thus the inclusion of different mag-\nnitudes of current densities in the top and bottom FMs is re-\ntained. Now, we can define; SSSe=wSSS1+ksSSS2. Therefore,\nhere, the SHE-mobility of the compound particle is a function\nof the ratio of current-density amplitudes in the FM layers.\nThe Thiele equation for the single compound particle reads:\n\u0000Qe[^z\u0002v]\u0000DDDe\u0001v+SSSe\u0001J1=0: (2)\nViewing the system in terms of this compound particle de-3\nscription with effective properties, which henceforth is called\neffective skyrmion, it is easily seen that we can modulate Qe,\nDDDeandSSSeby varyingwand thus affect the dynamical prop-\nerties such as speed and \u0002Skof the effective skyrmion. How-\never, a path towards in-situ modulation of the skyrmion dy-\nnamics is rather to utilize the fact that the elements of the ten-\nsors in Eq. (1) and thus in Eq. (2) are determined by the ge-\nometry of the texture15. Provided a suitable material system\nis available, one route towards in-situ manipulation is by an\ninduced in-plane anisotropy in order to deform the magnetic\ntexture (induced by e.g. the inverse magnetostrictive effect,\nwhereby an imposed mechanical stress induces a magnetic\nuniaxial anisotropy). For the type of skyrmion deformation\nconsidered here (expansions along principal axes), we assume\nDe\nxy=De\nyx= 0 andSe\nxy=Se\nyx= 0 (which was verified\nwhen evaluating the tensor components from micromagneti-\ncally obtained magnetization distributions). From Eq. (2) the\nvelocity components are then vx=QeSyyJ1y+De\nyySxxJ1x\n(Qe)2+DexxDeyyand\nvy=De\nxxSyyJ1y\u0000QeSxxJ1x\n(Qe)2+DexxDeyy. Thus we can write the velocity\nof the effective skyrmion in terms of an over-all effective mo-\nbility tensor \u0016\u0016\u0016e. Remembering that the current density in the\nfirst FM is to be used and setting J=J1, we may write:\nvi=\u0016e\nijJj (3)\n\u0016\u0016\u0016e=2\n4De\nyySe\nxx\n(Qe)2+DexxDeyyQeSe\nyy\n(Qe)2+DexxDeyy\n\u0000QeSe\nxx\n(Qe)2+DexxDeyyDe\nxxSe\nyy\n(Qe)2+DexxDeyy3\n5 (4)\nEqs. (3-4) constitute the velocity components of the com-\npound particle with respect to the global coordinate system\n(see Fig. 1(b)) and as such, determining the \u0002Skfrom the\nratiovy=vxwould give an angle with respect to the global x-\naxis. Thus in order to study the \u0002Sk-dependence on current\ndirection we must define it with respect to a rotated coordinate\nsystem defined by the current direction. If we denote the angle\nof the uniform current density with respect to the global x-axis\nby\u0012J, andJ=J[cos (\u0012J) sin (\u0012J)]T, then we can express\nvxandvyin terms of the primed coordinate system as vx0=\ncos(\u0012J)vx+sin(\u0012J)vyandvy0=\u0000sin(\u0012J)vx+cos(\u0012J)vy.\nConsequently, \u0002Sk=atan\u0010vy0\nvx0\u0011\n.\nB. Micromagnetic technique\nIn order to calculate the mobility tensor under various states\nwith different induced in-plane anisotropies, we use equi-\nlibrium configurations determined by micromagnetic simula-\ntions. To stabilize the antiferromagnetic alignment between a\nskyrmion in FM 1and another skyrmion in FM 2, it is neces-\nsary for them to have the same handedness13. The handedness\nis determined by the sign of the IDMI. The sign itself, depends\non; i) intrinsic material properties and ii) whether the HM is\ncoupled below or above a FM, since opposite stacking order\nwill change the sign of the IDMI26,27. For the former, we de-\nnote the intrinsic IDMI strengths by DLfor a given layer L.\nThe latter contribution is taken into account for by a multi-\nplicative factor fLwhere we set f1= +1 andf2=\u00001.Whenf1D1=f2D2then textures in both FMs will have the\nsame handedness13. Thus for the stack considered here, HMs\nneed to be chosen such that D2=\u0000D113. Furthermore, in order\nfor the AFM coupled skyrmions to move in the same direction\nwhen current is passed through both HMs, the intrinsic Spin-\nHall angles, \u001eLof the two HMs need to be opposite in sign\n(\u001e2=\u0000\u001e1)13. One possible HM material pair could be e.g. Pt\nand W13. The total energy density considered of a given FM\nlayerLis:\n\u000fL=AL3X\ni=1jrmi;Lj2+\u001b\ntRu(1\u0000^mL=1\u0001^mL=2)\n+fLDL[mz;L(r\u0001^mL)\u0000(^mL\u0001r)mz;L]\n+K?\nU;L\u0010\n1\u0000(^mL\u0001^z)2\u0011\n+Kk\nU;L\u0010\n1\u0000\u0000^mL\u0001^uk\u00012\u0011\n\u00001\n2\u00160ML^mL\u0001Hm;L:(5)\nIn Eq. (5), ALare the intra-layer exchange stiffness con-\nstants,\u001bthe inter-layer exchange coupling constant (here\nantiferromagnetic) between the layers, tRuis the thickness\nof the Ru spacer, K?\nU;Lare out-of-plane uniaxial magne-\ntocrystalline anisotropy constants, Kk\nU;Lare in-plane uniax-\nial magnetocrystalline anisotropy constants with easy direc-\ntions ^uk. We shall henceforth use the denomination Kxif\n^uk=^xto describe induced in-plane anisotropy along the\nx-direction and Kyif^uk=^yfor they-direction. It is\nalso to be understood that in any state of induced anisotropy\nwe consider, Kk\nU;1=Kk\nU;2. Further, Hm;L is the mag-\nnetostatic field, is evaluated in the the entire computational\ndomain. The interaction terms in Eq. (5) of the main\ntext give rise to an effective field Heff;L=\u00001\n\u00160ML@\u000fL\n@^mLact-\ning on the magnetization. In particular the IDMI field is\nHIDMI;L=\u00002fLDL\n\u00160ML[(r\u0001^mL)^z\u0000rmz;L]28,29and the inter-\nlayer exchange field at a site idue to interaction with a site\njisHRKKY;i=2\u001b\ntRu\u00160Mi^mj. We consider here that this RKKY-\nlike interaction occurs between the computational cells sep-\narated purely along the ^z-axis. Due to the IDMI, Neumann\nboundary conditions on the lateral surfaces of the FMs are\n@^mL\n@^nL=\u0000fLD1;2\n2AL(^z\u0002^nL)\u0002^mL, where ^nLis the unit nor-\nmal vector to the lateral surfaces28,29. For surfaces normal to\nthe plane, homogeneous Neumann conditions are used. The\nmagnetostatic field, Hm;L is computed as the spatial convo-\nlution between a demagnetizing tensor30and the magnetiza-\ntion distributions by Fast Fourier Transform techniques31. In\nthe magnetostatic field computation, for the near-field, we use\nanalytical formulae for the demagnetizing tensor by Newell\nand Dunlop30and for the far field, at relative cell distances\nlarger than 40, the point dipole approximation is used. The\ndynamics of the system is modelled by solving the Landau-\nLifshitz-Gilbert (LLG) equation, with added Spin-Hall-Effect\ntorques28:\n@^mL\n@t=\u0000\rL^mL\u0002Heff;L+\u000bL^mL\u0002@^mL\n@t\n\u0000jb0\nLJLj[^mL\u0002(^mL\u0002^pL)];(6)4\nwhere\rL=2.21\u0002105m/As is the gyromagnetic ratio, \u000bLis\nthe Gilbert damping, b0\nL=~\rL\u001eL\n\u001602eMLdL24with\u001eLbeing the\nintrinsic Spin-Hall angles, ethe electron charge, dLare the\nFM layers’ thicknesses, JLare the current density amplitudes\nand^pLare the directions of the spin-accumulation acting on\nlayerLdue to the SHE at the FM/HM interfaces. The spin-\naccumulation due to a current density along direction ^jL(unit\nvector) present in HM Lis^pL=sgn\u001eL\u0010\n^jL\u0002^nHM-FM;L\u0011\n,\nwhere ^nHM-FM;Lis the unit normal vector directed from a HM\ntowards a FM layer28. For computational simplicity, Eq. (6)\nis cast into explicit form and solved by a fifth order Runge-\nKutta integration scheme32. The lateral space considered is a\nrectangular domain of dimensions 1200 \u0002768 nm2whereas\nthe comprising FM layers and spacer layer all have a thick-\nness of 0.8 nm. The domain is discretized into 1.5 \u00021.5\u00020.8\nnm3cells. Material parameters considered are the following:\nA1=A2=20 pJ/m,M1=0.6 MA/m,M2=0.75 MA/m,\nK?\nU;1=K?\nU;2=0.6 MJ/m3,\r1=\r2=2.21\u0002105m/As,\n\u000b1=\u000b2=0.1,D1=2.8 mJ/m2,D1=\u00002.5 mJ/m2,\n\u001e1=0.1,\u001e2=\u00000.1,\u001b=\u00000.5 mJ/m2andKk\nU;1=Kk\nU;2\nis varied in the range [0,0.09] MJ/m3. For determining static\nequilibrium configurations, initial conditions close to that of\ntwo antiferromagnetically coupled skyrmions were imposed\nand the system let to freely ring down until a convergence was\nreached in the whole computational domain (each layer satis-\nfying1\nMLj^mL\u0002Heff;Lj\u001410\u00006). In the calculation of Qe,\nDeandSefrom the static equilibrium distributions, contribu-\ntions from magnetization-canting at and close to the bound-\naries were excluded by removing 45 computational cells into\nthe computational domain along all lateral normals. For all\ncurrent-driven dynamical calculation we consider J16=J2,\nspecificallyJ1=1011andJ2= 2\u00021011A/m2( i.e.ks= 2).\nFor the evaluation of \u0002Skfrom the micromagnetic simula-\ntions, skyrmion positions versus time in each FM layer were\ntracked by the moments of topological density23. Simulations\nfor each case of induced anisotropy were run until steady\nstate motion of the skyrmion pair was achieved. Care was\ntaken to only extract data for positions sufficiently far away\nfrom the boundaries in order to exclude repulsive boundary-\ninteractions. From the extracted longitudinal and transverse\nspeeds,vxandvythe speedvwas obtained and \u0002Skwas ex-\ntracted from the velocities expressed in the primed coordinate\nsystem according to the preceding section.\nIII. RESULTS AND DISCUSSION\nIn order to quantify the effect of an in-plane anisotropy,\nequilibrium configurations were computed for a range of in-\nduced in-plane anisotropy states. A note however, is in or-\nder for our choice of the particular degree of ferrimagnetism,\ni.e. the choice of M2in relation to M1. Firstly, it is found\nthat for a given in-plane anisotropy constant, the larger the\nratio ofM2toM1, the larger was the resulting skyrmion\ndeformation. Secondly, although the stability of the bound\nskyrmion state ranges over a wide set of M2=M1-ratios in\nour system, 1\u0014M2=M1\u00141:5, the range of applied in-plane anisotropy, whereby the bound skyrmion state is stable\nreduces asM2=M1increases. We found M2=M1=1.25 to\nbe a good compromise between the applicable range of im-\nposed in-plane anisotropies and the degree of skyrmion defor-\nmation (which in turn means range of dynamical tuneability).\nA qualitative argument for the trend of increased skyrmion\ndeformability with increasing M2=M1-ratios can be formed\nby considering that the dipolar interaction between FM 1and\nFM2is frustrated, except in the region where the magnetiza-\ntion lies predominately in plane (within the skyrmion width).\nWhen a uniaxial anisotropy is imposed, the dipolar energy can\nbe somewhat lowered by tilting more moments towards the\nplane. The projection on the plane of this tilting should of\ncourse be in the direction of the induced in-plane anisotropy.\nThus, the higher the dipolar frustration is (meaning as the ra-\ntioM2=M1increases), the higher is the motivation to increase\nthe in-plane portion of the skyrmion state. This would trans-\nlate to a higher degree of skyrmion deformation for a given\nvalue of the in-plane uniaxial anisotropy. Fig. 2 shows the\nequilibrium configurations of the bound skyrmion state for a\nsubset of induced in-plane anisotropies. The effect of a Kxor\nKycan be stated by the following two observations: (i): The\nskyrmions elongate preferentially along the direction perpen-\ndicular to the induced anisotropy, i.e. become elliptical with\nthe major axis perpendicular to the induced anisotropy direc-\ntion. (ii): the over-all skyrmion-size increases with increasing\nin-plane anisotropy. Point (i) is a logical consequence of the\nsystem increasing its number of moments pointing by rotat-\ning the in-plane portion of the skyrmion towards the direction\nof induced anisotropy. Point (ii) is also to be expected, as by\nincluding an in-plane anisotropy, the out-of-plane anisotropy\nis in effect reduced. Test calculations were performed ver-\nifying that the skyrmion size increases as the perpendicular\nanisotropy decreases. Another contribution to skyrmion size-\nenhancement was touched upon above, in terms of lowering\nthe dipolar frustration. If the texture was circularly symmetric\nwith only the radius varying we should expect that the diag-\nonal elements of the effective mobility tensor (Eq. 4) to be\nequal and the absolute values of the off-diagonal elements to\nbe equal, because then the diagonal elements of DDDeare equal\nand the same would apply to SSSe, whileQedoes not depend\non the spin-profile of the skyrmion and thus remains constant\n[compare to circularly symmetric skyrmions in single layer\nFMs7]. In such a case, modulation of values of the tensor el-\nements will indeed alter the speed and magnitude of \u0002Sk, but\nthe dynamics will be isotropic in the plane, i.e. no matter in\nwhich direction the driving current flows, the speed and \u0002Sk\ndo not change. This could be achieved as has been proposed\nin other works; by modulating the perpendicular anisotropy\nconstant14. The situation changes drastically if the skyrmion\nis deformed. Immediately we can expect that the elements of\nthe diagonal tensors DDDeandSSSediffer in magnitude. This is\nshown in Fig. 3, whereby the dependence of DDDe,QeandSSSe,\non induced in-plane anisotropies along xandy-directions are\nshown. The elements were computed by using the micromag-\nnetically obtained configurations with parameters described\nin the caption. It was verified through direct calculations, that\nDe\nxy=De\nyx= 0andSe\nxy=Se\nyx= 0.5\nFIG. 2. (Colour online) Skyrmion mz-profiles along two cuts, XX and YY for various states of induced Kk\nU;L. Below each profile is shown\nthe corresponding vector-plots of the magnetization distributions in the bottom (L=1) and top (L=2) layers with colorcoding corresponding\ntomz; red = +1 and dark blue= -1. The circles with arrows in the magnetization distribution corresponding to Kx=Ky= 0 indicate\nthe in-plane components of the skyrmions. The direction of induced anisotropy is indicated by thick double arrows in the vector-plots : (a):\nKx=Ky=0. (b):Kx=0.05 MJ/m3. (c):Kx=0.09 MJ/m3. (d):Ky=0.05 MJ/m3. (e):Ky=0.09 MJ/m3.\nFIG. 3. (Colour online). The non-zero tensor elements of DDDe,\nSSSeand the effective charge Qeas a function of induced in-plane\nanisotropies along xandy-directions, obtained from micromagnet-\nically computed configurations. Parameters used are: \r1=\r2=\n2.21\u0002105m/As,\u001e1=0.1,\u001e2=\u00000.1,o1=1,o2=\u00001,M1=0.6\nMA/m,M2=0.75 MA/m, \u000b1=\u000b2=0.1 andd1=d2=0.8 nm.\nThe direction of uniaxial in-plane anisotropy is indicated in each plot\nby double arrows. (a): De\nxx,De\nyyandQevs.Kx. (b): Se\nxx,Se\nyyvs.\nKx. (c):De\nxx,De\nyyandQevs.Kyand (d): Se\nxx,Se\nyyvs.Ky.\nWe can see the above discussion of the expectations ver-\nified, that for zero in-plane anisotropy, the elements of the\ndiagonal tensors DDDeandSSSeare equal, signifying circularskyrmion shape. Let us look at Fig. 3 (a) and (b). As Kxin-\ncreases there is a splitting in the dependencies of De\nxxandDe\nyy\nin (a) and the same for Se\nxxandSe\nyyin (b). The charge Qeis as\nexpected unaffected by the presence of the anisotropy-induced\ntexture deformation. As Kxincreases, the skyrmions elongate\nmore alongy(Fig. 2 (b) and (c)). The effect on the drag force\nis then according to Fig. (3 (a)) a rapid increase along the di-\nrection of the elliptically shaped skyrmion’s minor axis and a\nslower increase along the axis of elongation (major axis). If\nwe think of the drag as a resistance to flow then we should\nexpect a larger drag in the direction of propagation with the\nlarger frontal area. Conversely along a direction whereby the\nobject is more stream-lined-shaped, a relatively smaller drag\nis to be expected. This view is consistent with the observa-\ntion made herein concerning sharp increase in De\nxxasKxin-\ncreases. The reason for a noticeable increase also in the De\nyy-\nelement is that the over-all size of the skyrmion also increases\nwith increasing Kx. In terms of Se\nxxandSe\nyy(Fig. 3 (b))\nwhich constitute the spin-Hall effect mobility, the largest in-\ncrease is for Se\nxxas the number of moments along xincreases\nboth as a result of satisfying the direction of anisotropy. The\nreasons for an increase of Se\nyyis similarly to the discussion on\nthe drag force, due to an increase in skyrmion width, mean-\ning an increase also in the number of moments along the y-\ndirection. Since the SHE-induced spin-accummulation is per-\npendicular to the current direction, then we should expect that\ne.g. given a current fixed along x, the speed enhancement\nis greater for an induced anisotropy Kxthan for an induced\nKy. The situation is reversed if the current is injected along y.\nThe same arguments can be applied to the case of an induced\nanisotropyKyalongy. This is shown in Figs. 3 (c) and (d),\nwhereby the situation is reversed with respect to Figs. 3 (a)6\nand (b), because the skyrmion preferential elongation is along\nx. We now evaluate the over-all mobility tensor using DDDe,SSSe\nandQe. In order to be as general as possible we consider the\ncase whereby J16=J2and set here ks=2. Evaluating the\nexpression in Eq. (4) and plotting the tensor elements versus\ninduced in-plane anisotropy along both xandy-directions,\nKxandKy, respectively, we obtain the results shown in Fig.\n4.\nFIG. 4. (Colour online) Effective mobility-tensor elements as a\nfunction of induced in-plane anisotropy. Here, ks=2. (a):\u0016e\nxxand\n\u0016e\nyyvs.Kx. (b):\u0016e\nxyand\u0016e\nyxvs.Kx. (c):\u0016e\nxxand\u0016e\nyyvs.Ky.\n(d):\u0016e\nxyand\u0016e\nyxvs.Ky.\nAs can be seen in Figs. 4 (a) and (c), both mobilities \u0016e\nxx\nand\u0016e\nyyincrease with increasing induced anisotropy. Therate of increase of these elements is greatest in the direction\nof the induced anisotropy. In terms of the off-diagonal ele-\nments, there is also a difference in their behaviour in terms of\ntheir rate of change with respect to the direction of induced\nanisotropy, see Figs. 4 (b), (d). These two behaviours mean\nthat the speed vand skyrmion Hall-angle \u0002Skare tuneable\nby the in-plane anisotropy and that we can have anisotropy in\nthis tuneability by two means; either by, for a given current\ndirection, induce the anisotropy along different directions, or\nfor a fixed induced anisotropy-direction, vary the angle of the\ndriving current. In what manner vand\u0002Skchanges with in-\ncreased magnitude of the induced anisotropy boils down to the\nrelative rate of change and magnitudes of the tensor elements\nin Fig. 3 as the induced anisotropy is varied. The speed for\na given induced anisotropy gets its largest contribution from\nthe diagonal elements of \u0016\u0016\u0016e. In conjunction with the relative\nrate of growth of these elements being large we can expect\nthat for a given fixed current-direction, vwill always increase\nwith increasing magnitude of in-plane anisotropy. In addi-\ntion, it will become clear that the source of effective skyrmion\ndeflection away from the driving current direction can in prin-\nciple stem from two sources (except for current-directions ex-\nactly alongxory-directions): one due to a finite topologi-\ncal charge (the dominant contribution) and the second comes\npurely from lateral shape-distortions of the skyrmions. In fact,\neven for a situation whereby Qe=0 such as in a perfectly bal-\nanced SAF, there can be a deflection away from the driving\ncurrent-angle for \u0012Jdifferent from multiples of \u0019=2. We shall\nnow address these issues in an orderly fashion. Although\nlengthy, it is instructive to write out the full expressions for\n\u0002Skandvgiven an arbitrary current direction. Recalling that\nJ=J[cos (\u0012J) sin (\u0012J)]Twith\u0002Skdefined with respect to\nthe primed coordinate system and vbeing a magnitude which\nis easily stated using the global system as starting point (as the\nmagnitude will be the same in both coordinate systems), one\narrives at:\nv=Jr\nsin (2\u0012J)\u0002\n\u0016exx\u0016exy+\u0016eyx\u0016eyy\u0003\n+ cos2(\u0012J)h\n(\u0016exx)2+\u0000\n\u0016eyx\u00012i\n+ sin2(\u0012J)h\u0000\n\u0016exy\u00012+\u0000\n\u0016eyy\u00012i\n(7)\n\u0002Sk= arctan(1\n2sin (2\u0012J)\u0002\n\u0016e\nyy\u0000\u0016e\nxx\u0003\n+\u0016e\nyxcos2(\u0012J)\u0000\u0016e\nxysin2(\u0012J)\n1\n2sin (2\u0012J)\u0002\n\u0016exy+\u0016eyx\u0003\n+\u0016exxcos2(\u0012J) +\u0016eyysin2(\u0012J))\n(8)\nLet us start with tuneability of vand\u0002Skby inducedKxand\nKyfor a fixed\u0012J: Consider, as an example the case \u0012J=0\n(i.e. current injected purely along the x-direction). Then, from\nEq. (7) and Eq. (4), v(\u0012J= 0) =Jq\n(\u0016exx)2+ (\u0016eyx)2=\nJSe\nxx\n(Qe)2+DexxDeyyq\n(Dyy)2+ (Qe)2. From this expression, it is\nthen clear that the speed will have a different dependency on\nKx(withKy= 0) than onKy(withKx= 0), based on the\nindividual behaviour of the individual tensor elements on the\ninduced anisotropy. Now, for \u0002Sk, then from Eq. (8) and Eq.(4),\u0002Sk(\u0012J= 0) = arctanhvy0(\u0012J=0)\nvx0(\u0012J=0)i\n= arctanh\u0016e\nyx\n\u0016exxi\n=\narctanh\n\u0000Qe\nDeyyi\n, which is the standard result for the skyrmion-\nHall angle [Ref] as a special case of \u0012J= 0, except that\nwe specifically state here the De\nyy-element in the denomina-\ntor. We can intuitively predict the behaviour of \u0002Sk. Let us\nconsider again \u0012J=0 (current flowing along x) and an in-\nduced anisotropy Kx; We know from Fig. 3, that with in-\ncreasingKx, the largest change in dissipation occurs for the\nDe\nxxelement, but De\nyyincreases as well with the result that7\nFIG. 5. (Colour online) speed, vand\u0002Skvs.Kxwith current\ninjected along x(\u0012J= 0). The parameters used are the same as be-\nfore. (a):vvs.Kk(KxandKy) predicted by the effective skyrmion\napproach and computed by full dynamic micromagnetic simulations.\nThe direction of induced in-plane anisotropy is indicated by double\narrows in the legends. (b): \u0002Skvs.Kk(KxandKy).\nthe magnitude of \u0002Skis expected to decrease at some rate.\nNow, keeping the same scenario except now we impose an\nanisotropyKy, we know that the rate of increase of De\nyyis\neven greater with increasing anisotropy Ky. The expectation\nthen of the change in magnitude of \u0002Skshould be expected to\nbe greater for induced anisotropy along ywhen driving with\n\u0012J=0. To illustrate this simple analysis, vand\u0002eas a\nfunction ofKxandKyfor the current-angle \u0012J= 0 is com-\nputed and shown in Fig. 5. We have also compared the ef-\nfective skyrmion approach to full dynamical micromagnetic\nsimulations with an excellent agreement, thus validating the\napproach. The overall range of tuneability is significant and\nis anisotropic with respect to induced-anisotropy direction. In\na real situation we could envision the device mounted on a\npiezoelectric stressor that can transmit strain in two ortogonal\ndirections (one at a time) in order to induce and discriminate\ndifferent speeds. We now address the general case \u0012Jis al-\nlowed to vary whereby anisotropic behaviour of speed and\ndeflection angle are expected to be present. In addition, we\npoint out, that in general, as long as \u0012J6=n\u0019=2(wherenis\nan integer) and the effective skyrmion radius is not isotropic\n(deviation from circular shape) , \u0002Skwill contain a contri-\nbution originating from only the anisotropic deformation of\nthe skyrmion, i.e. independent of the topological charge. In\naddition the speed may also be non-isotropic with respect to\ndriving-current direction. In order to see this, we briefly di-\ngress on this point as it may be of consequence for exper-\niments whereby there is a sizeable inverse magnetostrictive\neffect causing a deviation from circularly shaped skyrmions.\nIf we impose Qe= 0 (in practice this means balancing the\nFM layers such that w=1), such that all off-diagonal el-\nements in Eq. (4) are zero, then From Eqs. (7) and (8),\nv(Qe= 0) =Jq\n(\u0016exx)2cos2(\u0012J) +\u0000\n\u0016eyy\u00012sin2(\u0012J)and\n\u0002Sk(Qe= 0) = arctan\u0014\n1\n2sin(2\u0012J)[\u0016e\nyy\u0000\u0016e\nxx]\n(\u0016exx)2cos2(\u0012J)+(\u0016eyy)2sin2(\u0012J)\u0015\n. It\nis now clear that, provided \u0016e\nxx6=\u0016e\nyy, there will be a finite\n\u0002Skfor all current-angles \u0012J6=n\u0019\n2in the absence of a net\ntopological charge. In terms of the speed, we can also see\nthat the role of a topological charge on vis two-fold; it affects\nboth magnitude and shifting the value of \u0012Jwherebyvhasits maximum or minimum; with Qe= 0,vexhibits maxima\nand minima at \u0012J=n\u0019\n2, but forQe6= 0, thesin (2\u0012J)term\nin Eq. (7) imposes a shift away from n\u0019\n2. We now return to\nour original easily deformable system and make predictions\nfor a case whereby for fixed values of Kx(withKy= 0), we\nsweep the injected current angle \u0012J. The results for the de-\npendence of both vand\u0002Skare shown in Fig. 6. Results are\nalso compared for the highest value of Kxto full dynamic mi-\ncromagnetic simulations as a verification-step of the effective\nskyrmion prediction, with a very good match. The amplitudes\nof oscillations in the dynamical behaviours are quite signifi-\ncant and should be easily detectable in an experiment. Apart\nfrom the additional degree of freedom in terms of modulating\nthe dynamics, this anisotropic behaviour could be used to de-\ntect the possible presence of strain induced anisotropy in the\nsystem and thus of skyrmion deformation.\nFIG. 6. Effective skyrmion prediction of (a): vvs.\u0012Jfor different\nKxand (b): \u0002Skvs.\u0012Jfor different Kx. Solid lines correspond to\nthe effective skyrmion approach and filled circles are results from\nfull dynamical micromagnetic simulations. The value of induced\nanisotropyKx(whose direction in the plane is also indicated by the\ndouble arrows) corresponding to each curve is shown by a number in\nunits of MJ/m3.\nIV . CONCLUSIONS\nIn conclusion, we have shown, by combined micromagnetic\nsimulations and an effective skyrmion analytical model, that\nwe can effectively modulate both speed and skyrmion Hall-\nangle of tightly antiferromagnetically bound skyrmions by in-\nduced in-plane anisotropies. The cause of the said modula-\ntions stem from a deformation of the skyrmion-texture (going\nfrom circular to elliptical shapes). As a further consequence,\nwe showed that this introduces dynamical anisotropy in the\nplane of skyrmion propagation with respect to driving-current\ninjection-angle. In addition, we have shown, given a devia-\ntion from circularly shaped skyrmions, that for driving current\nangles\u0012J6=n\u0019\n2, there is a contribution to the skyrmion de-\nflection away from the driving-current direction independent\nof the topological charge, i.e. even for a perfectly balanced8\nSAF. This may be of consequence for SAF devices whereby\nskyrmions operate on relatively large areas, causing a build-\nup of in time of deviation from the intended target position.Finally, if the uniaxial anisotropies can be induced by me-\nchanical stress, it can possibly lead to less complex device\nstructures as compared to other proposed schemes.\n\u0003per24.ac.uk\n1A, Fert, V . 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Flan-\nnery, Numerical Recipes: The Art of Scientific Computing,\nCambridge University Press, Cambridge, England 1988)."    },    {        "title": "1807.07265v1.Magnetocrystalline_anisotropy_in_the_Kondo_lattice_compound_CeAgAs__2_.pdf",        "content": "arXiv:1807.07265v1  [cond-mat.str-el]  19 Jul 2018Magnetocrystalline anisotropy in the Kondo lattice compou nd CeAgAs 2\nRajib Mondal, Rudheer Bapat, S. K. Dhar, and A. Thamizhavel\nDepartment of Condensed Matter Physics and Materials Scien ce,\nTata Institute of Fundamental Research, Homi Bhabha Road, C olaba, Mumbai 400 005, India.\n(Dated: July 20, 2018)\nWe report on the single crystal growth and anisotropic physi cal properties of CeAgAs 2. The\ncompound crystallizes as on ordered variant of the HfCuSi 2-type crystal structure and adopts the\northorhombic space group Pmca(#57) with two symmetry inequivalent cerium atomic positio ns in\ntheunitcell. The orthorhombic crystal structureof our sin gle crystal was confirmedfrom thepowder\nx-ray diffraction and from electron diffraction patterns obt ained from the transmission electron\nmicroscope. The anisotropic physical properties have been investigated on a good quality single\ncrystal by measuring the magnetic susceptibility, isother mal magnetization, electrical transport and\nheat capacity. The magnetic susceptibility and magnetizat ion measurements revealed that this\ncompound orders antiferromagnetically with two closely sp aced magnetic transitions at TN1= 6 K\nandTN2= 4.9 K. Magnetization studies have revealed a large magnetocry stalline anisotropy due\nto the crystalline electric field (CEF) with an easy axis of ma gnetization along the [010] direction.\nThe magnetic susceptibility measured along the [001] direc tion exhibited a broad hump in the\ntemperature range 50 to 250 K, while typical Curie-Weiss beh aviour was observed along the other\ntwo orthogonal directions. The electrical resistivity and the heat capacity measurements revealed\nthat CeAgAs 2is a Kondo lattice system with a magnetic ground state.\nI. INTRODUCTION\nOne of the most widely investigated topics of research\nin condensed matter physics during the last few decades\nis the competition between the Ruderman, Kittel, Ka-\nsuya and Yosida (RKKY) interaction and the Kondo ef-\nfect in Ce and Yb based intermetallic compounds. The\ninvestigations made on these compounds have discov-\nered a plethora of new phenomena like heavy fermion,\nvalence fluctuation, superconductivity, quantum phase\ntransition etc1–6. In recent years there are many reports\non Ce-based compounds which can be tuned to quan-\ntum critical point (QCP) with pressure as the tuning\nparameter7–9. All these interesting physical properties\nare due to the result of the hybridization of the local-\nized 4felectron with the conduction electrons. A weak\nhybridization usually results in a magnetic ground state\nwhile a stronger hybridization results in a weaker local-\nization of the 4 felectron pushing the system eventually\nto a non-magnetic intermediate valence state. The hy-\nbridization strength and the effects due to the crystal\nelectric field depend on the atomic arrangement of the\nenvironment around the Ce atom in the unit cell, which\nincludes the inter-atomic distances between the Ce atom\nand other atoms and ligandson different crystallographic\nsites. Most of the cerium based compounds possess a sin-\ngle unique atomic position for the Ce-atom in the unit\ncell. However, there are a number of cerium compounds\nwhich have multiple Ce atomic sites in the unit cell10,\nlike for example, Ce 7Ni311, Ce2PdGe312, Ce4Co2Sn513,\nCeRuSn 314, CeRhSn 315, Ce2Rh3Ge16, Ce2RuZn417and\nCeRuSn18etc. The compounds with multiple Ce-sites\nhave been mostly studied in the polycrystalline form.\nHere we report on the magnetic properties of a single\ncrystal sample of CeAgAs 2, which allows us to study the\nanisotropic behavior in detail.The compounds of the form Ce TX2, whereTis a tran-\nsitionmetaland XiseitherSborAs, crystallizemostlyin\nthe HfCuSi 2-type crystal structure with the space group\nP4/nmm(#129) that hosts a unique atomic position for\nCe atom at the 2 csite. The anisotorpic physical proper-\nties of Ce TSb2have been studied quite extensively19–21.\nFor example CeAgSb 2is a ferromagnet with the easy\naxis of magnetization along the [001]-direction and a sat-\nuration moment of only 0.45 µB/Ce, which remains con-\nstant upto fields as high as 45 T. On the other hand,\nthe isostructural CeAuSb 2becomes an antiferromagnet\nat 6 K and exhibits field induced quantum criticality22.\nIn view of the wide variety of magnetic properties ex-\nhibited by the antimonides, the arsenides of this series\nof compounds have also been synthesized and studied\nfor their crystal structure and magnetism. Owing to the\nhugevaporpressureofarsenic,thecompoundsofCe TAs2\nhave been synthesized in polycrystalline form23–26. In\nthe present work, we were successful in growing the sin-\ngle crystals of CeAgAs 2using the flux method. Here\nwe report the interesting anisotropic magnetic properties\nexhibited by this compound.\nII. EXPERIMENT\nWe have employed the high temperature solution\ngrowth or the so called flux growth method to grow the\nsingle crystals of CeAgAs 2. Typically, for flux growth\none of the constituents that has the lowest melting point\nis used as a flux (solvent) to grow the single crystal when\nsupersaturation is achieved on cooling. But in CeAgAs 2\nnone of the elements possess a low melting point and\nhence we have used the eutectic composition of Ag-As,\nas a flux to grow the single crystals27. The binary phase\ndiagram of Ag-As reveals a eutectic composition at 75:252\natomic percent of Ag:As, with a low melting point of\n540◦C. The starting materials of high purity Ce (99.9%\nAmes Lab), Ag (99.99%) and As(99.999%) were taken in\nthe ratio 1 : 37 : 14, which took into account the eutec-\ntic composition of the excess Ag:As flux. The elements\nwere placed in a high quality recrystallized alumina cru-\ncible and subsequently sealed in a quartz ampoule under\npartial pressure of argon gas. The ampoule was loaded\nin a resistive heating box type furnace and the temper-\nature of the furnace was raised at the rate of 15◦C/h\nto 1050◦C and held there for about 24 h for homoge-\nnization of solution. Then the the furnace was cooled\nto 700◦C at the rate of 2◦C/h at which point the am-\npoule was centrifuged to remove the flux. Large, shiny,\nflat platelet like single crystals with typical size of 10 x\n6 x 0.3 mm3were obtained. The flat plane of the crys-\ntal corresponds to the (001)-plane. Powder x-ray, Laue\ndiffraction and transmission electron microscopy (TEM)\nwasperformed on a few growncrystalsto knowthe phase\npurity and the crystallographic orientation and the su-\nperstructure nature of the crystal. The crystal was cut\ninto a bar shaped sample using the spark erosion cut-\nting machine for the anisotropic physical property mea-\nsurements. The electrical resistivity, heat capacity and\nmagnetization measurements were performed in a physi-\ncal property measurement system (PPMS) and magnetic\nproperty measurement system (MPMS) from Quantum\nDesign, USA, respectively.\nIII. EXPERIMENTAL RESULTS\nA. X-ray diffraction\nSince the crystals were grown completely from an off-\nstoichiometric melt, at first we performed a powder x-ray\ndiffraction (XRD) to confirm the phase purity. Figure 1\nshows the powder x-ray diffraction pattern of our sam-\nple. Here it may be pertinent to first recall that a de-\ntailedcrystalstructureanalysisofCeAgAs 2hasbeenpre-\nviously performed by Demchyana et al23which showed\nthat unlike the other members of the Ce TX2(T= Tran-\nsition metal; X= Sb or As) CeAgAs 2does not adopt\nthe tetragonal HfCuSi 2type crystal structure. It crystal-\nlizes in the orthorhombiccrystal structure with the space\ngroupPmca(#57)with latticeparameters a= 5.7441˚A,\nb= 5.7696˚A andc= 21.0074˚A, whicharerelatedtothe\ntetragonalHfCuSi 2structure by√\n2×aTand 2×cT. The\nPmcaspacegrouphastwoatomicpositionsforCeatoms,\nCe1 and Ce2 at 4 dpositions. The crystal structure of\nCeAgAs 2is shown in Fig. 1(b). The nearest distance be-\ntween the Ce1-Ce1 and Ce2-Ce2 atoms is 5 .758˚A while\nthat of Ce1-Ce2 is 4 .078˚A. The Laue diffraction pat-\ntern of the flat plane of the as grown crystal is shown\nin Fig. 1(c), which corresponds to the (001)-plane of the\ncrystal. Well defined spots with clear symmetry pattern\nconfirms the good quality of the grown single crystal.\nFurthermore, the superstructure nature of the unit cell\nFIG. 1: (Color online) Powder x-ray diffraction pattern of\nCeAgAs 2(b) the crystal structure of CeAgAs 2, (c) The Laue\ndiffraction pattern corresponding to (001) plane of the crys tal\nand (d) TEM diffraction pattern depicting the superstructur e\nspots.\nalong the c-axis is observed using the transmission elec-\ntron microscope(TEM) and the observeddiffraction pat-\ntern is shown in Fig. 1(d). The diffused spots confirms\nthe superstructure nature of the unit cell while this is not\nobservedin the TEMdiffractionpatternofCeCuAs 2, not\nshown here for brevity.\nB. Magnetic susceptibility and magnetization\nThe temperature dependence of the dcmagnetic sus-\nceptibility measured along the three principal crystallo-\ngraphicdirectionsfrom1.8to300K,isshowninthemain\npanel of Fig. 2(a). The magnetic susceptibility along the\n[100] and [010] directions increases more rapidly below\nabout 50 K and then exhibits a small cusp at 6 K fol-\nlowed by a sharp drop at 4.9 K indicating two magnetic\ntransitions (see the inset of Fig. 2(a)). From the magne-\ntization measurements vide infra , and from the previous\nneutron diffraction studies26on a polycrystalline sample,\nthe two magnetic transitions are confirmed to be antifer-\nromagnetic in nature. Just below the first antiferromag-\nnetic transition at TN1= 6 K, the magnetic susceptibil-\nity increases and then it drops down more rapidly below\nTN2= 4.9 K. The drop in the magnetic susceptibility3\n/s32/s33/s34/s32\n/s32/s33/s35/s32\n/s32/s36/s37/s38/s39/s40/s41/s42/s40/s43/s44/s45\n/s46/s32/s32 /s35/s32/s32 /s47/s32/s32 /s32\n/s48/s39/s40/s49/s39/s50/s51/s52/s41/s50/s39/s37/s38/s53/s45/s34/s32/s32\n/s35/s32/s32\n/s32/s36/s54/s47/s38/s40/s43/s44/s42/s39/s40/s41/s45\n/s32/s33/s34\n/s32/s33/s35\n/s32/s36/s37/s38/s39/s40/s41/s42/s40/s43/s44/s45\n/s47/s55/s47/s32/s55/s32/s56/s39/s57/s58/s57/s59/s35\n/s36/s60/s37/s61/s37/s47/s55/s32/s37/s62/s47/s32/s32/s63\n/s37/s62/s32/s47/s32/s63\n/s37/s62/s32/s32/s47/s63\n/s37\n/s35/s32\n/s47/s32\n/s32/s36/s37/s38/s64/s37/s47/s32/s54/s46/s37/s39/s40/s41/s42/s40/s43/s44/s45\n/s46/s32/s32 /s35/s32/s32 /s47/s32/s32 /s32\n/s48/s39/s40/s49/s39/s50/s51/s52/s41/s50/s39/s37/s38/s53/s45/s35/s33/s34\n/s35/s33/s35\n/s35/s33/s32/s36/s37/s38/s64/s37/s47/s32/s54/s46/s37/s39/s40/s41/s42/s40/s43/s44/s45\n/s46/s32/s32/s35/s55/s32/s35/s32/s32/s47/s55/s32/s47/s32/s32/s55/s32/s56/s39/s57/s58/s57/s59/s35\n/s65/s37/s32/s33/s33/s32/s37/s62/s32/s32/s47/s63/s38/s51/s45\n/s38/s66/s45\nFIG. 2: (Color online) (a) Temperature dependence of\nanisotropic magnetic susceptibility along the three princ ipal\ncrystallographic directions. A huge drop in the magnetic su s-\nceptibility along [100] and [010] directions is observed. A lso\nshown is the inverse magnetic susceptibility and the modifie d\nCurie-Weiss fit. Inset shows two clear magnetic transitions as\nindicated by the arrow. (b) Zoomed in view of the magnetic\nsusceptibility along [001] direction in the temperature ra nge\n50 - 300 K.\nascertains the antiferromagnetic nature of the transition\nat 4.9 K and confirms the easy axis of magnetization as\n[100] or [010]. For H/bardbl[001], at TN1the susceptibility\nshows a drop followed by a small rise at the second tran-\nsition at TN2, being virtually temperature independent\nat lower temperatures suggesting [001] as the hard axis\nof magnetization. The anisotropy within the abplane\nis weak while the anisotropy between the abplane and\nthec-axis is quite large. The curvilinear behaviour of\nthe inverse magnetic susceptibility along the [100] and\n[010]directions indicates the simple Curie-Weisslaw can-\nnot be used to analyse the magnetic susceptibility data.\nHence, we have employed the modified Curie-Weiss law:\nχ−1= (χ0+C\n(T−θp))−1to fit the inverse magnetic sus-\nceptibility above 100 K for [100] and [010] directions.\nHereχ0is the temperature independent part of the mag-\nnetic susceptibility and C is the Curie constant. We\nhave obtained χ0=−4.434×10−4emu/mol, an effec-\ntive magnetic moment of 2.59 µB/Ce and the paramag-\nnetic Weiss temperature θp=−0.20 K for H/bardbl[100]\nandχ0=−4.797×10−4emu/mol, µeff= 2.66µB/Ce/s32/s33/s34\n/s32/s33/s35\n/s35/s33/s34\n/s35/s36/s37/s38/s39/s40/s41/s42/s43/s37/s41/s42/s44/s39/s45/s46/s47/s48/s49/s50/s40/s51\n/s52/s35 /s53/s35 /s54/s35 /s35\n/s36/s37/s38/s39/s40/s41/s42/s55/s45/s56/s42/s40/s57/s58/s45/s46/s59/s60/s40/s51/s32/s45/s61/s32/s35/s35/s62\n/s45/s61/s35/s32/s35/s62\n/s45/s61/s35/s35/s32/s62/s50/s40/s63/s38/s63/s64/s54 /s32/s33/s34/s33/s54/s45/s65\nFIG. 3: (Color online) Isothermal magnetization measured a t\nT= 2 K along the three principal crystallographic directions .\nA sharp first order like metamagnetic transition is observed\nat 7.5 kOe for H/bardbl[100] and [010] directions.\nandθp=−2.32 K for H/bardbl[010] direction. The obtained\neffective magnetic moment values are very close to the\ntheoretical value of trivalent Ce, suggesting a local mo-\nment behaviour. The modified Curie-Weisslawcouldnot\nresult in a good fitting for H/bardbl[001] direction. However,\nif we fix the effective magnetic moment to 2 .54µB/Ce\na reasonably good fitting is obtained in the temperature\nrange 250 - 300 K with χ0= 7.079×10−4emu/mol\nandθp=−299 K as shown in the main panel of\nFig 2(a). Figure 2(b) shows the magnetic susceptibil-\nity forH/bardbl[001] direction, which is very weak (of the\norder of 10−3emu/mol). It is obvious from the suscepti-\nbility data that the Curie-Weiss behaviour is absent and\na broad hump is observed in the temperature range 50\nto 300 K. This type of behaviour is usually attributed to\nmixed valent or intermediate valent character16,28. How-\never, in the present case, we shall see that it arises due\nto a strong crystal field effect.\nThe isothermal magnetization measured at T= 2 K\nis shown in Fig. 3. In conformity with the mag-\nnetic susceptibility data presented above, the isothermal\nmagnetization exhibits an easy plane magnetocrystalline\nanisotropy. A very sharp spin flip like metamagnetic\ntransitionisobservedatfieldof7.5kOefor H/bardbl[100]and\n[010] directions, and for fields greater than 10 kOe, the\nmagnetization almost saturates. The saturation value of\nthe magnetization is about 1.2 µB/Ce, which is much\nless than that expected for the free-ion value of Ce in\nits trivalent state ( gJJ= 2.14µB/Ce). On the other\nhand the magnetization along [001] increases very grad-\nually and attains a value of only 0.18 µBat 70 kOe, thus\nconfirming it as the hard axis of magnetization. The\nlow saturation value of the magnetic moment for fields\nas high as 70 kOe suggests a strong evidence for crystal\nelectric field (CEF) and Kondo effect. We have also per-\nformed the temperature dependence of isothermal mag-\nnetization and found that the critical field at which the4\nmetamagnetictransitionappearsshiftstolowerfieldsand\nbroadens as the temperature is increased and finally for\ntemperature larger than TN1the magnetization shows an\nalmost linear dependence on field in the paramagnetic\nstate, as expected in antiferromagnetic compounds.\nC. Electrical Resistivity\nThe temperature dependence of electrical resistivity of\nCeAgAs 2for current parallel to [100]-direction or ( ab)-\nplane is shown in the main panel of Fig. 4. We could not\nmeasure the resistivity for J/bardbl[001] because of the small\nthickness of the sample. At room temperature the elec-\ntrical resistivity reads a value of about 700 µΩ·cm which\nis quite large. As the temperature decreases the resistiv-\nity decreases gradually and exhibits a very broad max-\nimum centered around 100 K, which may be attributed\nto the thermal depopulation of the CEF levels. Below\n100 K, the resistivity falls off more rapidly showing a\nbroad minimum around 25 K and then increases grad-\nually. The resistivity data in zero field show a linear\nlogarithmic temperature dependence in the range 20 to\n6 K suggesting that CeAgAs 2is a Kondo lattice system\n[Fig. 4(b)]. At 6 K a sudden change of slope is observed\n(see the inset of Fig. 4(a)) and the electrical resistiv-\nity increases more rapidly, which marks the TN1. At\nTN2= 4.9 K, the electrical resistivity drops due to the\nreduction in the spin disorder scattering, thus confirming\nthe two magnetic transitions observed from the magnetic\nsusceptibility data. The increase in the electrical resis-\ntivity just below TN1is typical of compounds that ex-\nhibit superzone gap29, which occurs due to the difference\nin the lattice periodicity and magnetic periodicity. But\nhere in the present case, the presence of superzone gap is\nruled out because the previous neutron diffraction stud-\nies have revealed the propagation vector k= [0,0,0]26.\nHence the increase in the electrical resistivity in the re-\ngionTN2< T < T N1is mainly attributed to the spin\nfluctuation effect presumably due to the isosceles trian-\ngular arrangements of the Ce1 and Ce2 atoms along the\n[100]-direction. It should be noted that the electrical re-\nsistivity begins to drop at TN2from a value of 634 µΩ·cm\nto about 411 µΩ·cm at 2 K the lowest temperature mea-\nsured. The huge value of the residual resistivity is at-\ntributed to the spin fluctuations as discussed below. The\nupturn in the resistivity below 25 K is reminiscent of a\nKondo lattice system.\nWe have also measured the electrical resistivity un-\nder applied magnetic fields and the plots are shown in\nFig. 4(b). The magnetic field is applied along the easy\naxis of magnetization direction. With the increase in\nthe magnetic field, the two magnetic transition temper-\natures decrease and are not discernible for fields greater\nthan10kOedownto2K.Thistypeofbehaviourisgener-\nallyobservedin antiferromagneticsystems. In anapplied\nfield of 7.5 kOe, the electrical resistivity shows an upturn\nat 3 K and the increasing trend is seen down to 2 K. Thisanomalous behaviour of the electrical resistivity may be\nattributed to the spin flip like metamagnetic transition\nobserved in the isothermal magnetization at 7.5 kOe, it\nis observed only at this field and not at higher or lower\nmagnetic fields. It is interesting to note that the residual\nresistivity decreases substantially with increasing mag-\nnetic fields suggestingasignificantmagnetic contribution\ndue to spin fluctuations which are suppressed by applied\nmagnetic fields.\nThe isothermal field dependent magnetoresistance is\nshown in Fig. 5(a) and the low field part as a zoomed\nin view, with sparse markers, is shown in Fig. 5(b). The\nvariationofmagnetoresistancewith field is in good corre-\nspondence with the magnetizationdatapresented earlier.\nAt the lowest temperature of 2 K, the magnetoresistance\nis slightly positive and a sudden change in magnitude is\nobserved in the negative direction at 7.5 kOe, suggest-\ning the field induced ferromagnetic state. A similar be-\nhaviour is observed for 3 K data as well. At T= 4 K, the\nmagnetoresistance exhibits a sharp peak at 6.3 kOe and\n/s32/s33/s33\n/s34/s33/s33\n/s35/s33/s33\n/s36/s33/s33\n/s33/s37/s38/s39/s40/s41/s38/s42/s43/s44\n/s45/s33/s33 /s36/s33/s33 /s46/s33/s33 /s33\n/s47/s48/s43/s49/s48/s50/s51/s52/s53/s50/s48/s38/s39/s54/s44/s55/s33/s33\n/s34/s33/s33\n/s56/s33/s33\n/s35/s33/s33/s37/s38/s39/s40/s41/s38/s42/s43/s44\n/s46/s33 /s56 /s33/s57/s48/s58/s59/s58/s60/s36 /s61/s38/s62/s62/s38/s63/s46/s33/s33/s64\n/s54\n/s32/s33/s33\n/s34/s33/s33\n/s35/s33/s33\n/s36/s33/s33/s37/s38/s39/s40/s41/s38/s42/s43/s44\n/s46/s36/s35/s34/s32\n/s46/s33/s36/s35/s34/s32\n/s46/s33/s33/s36\n/s47/s48/s43/s49/s48/s50/s51/s52/s53/s50/s48/s38/s39/s54/s44/s57/s48/s58/s59/s58/s60/s36\n/s61/s38/s62/s62/s38/s63/s46/s33/s33/s64/s32/s38/s62/s62/s38/s63/s33/s46/s33/s64\n/s33/s38/s62/s62/s38/s63/s46/s33/s33/s64\n/s32/s33\n/s38/s56/s38/s65/s66/s48\n/s38/s55/s67/s56/s38/s65/s66/s48\n/s38/s46/s33/s38/s65/s66/s48\n/s38/s36/s33/s38/s65/s66/s48\n/s38/s34/s33/s38/s65/s66/s48\n/s38/s35/s33/s38/s65/s66/s48\n/s38/s46/s33/s33/s38/s65/s66/s48\n/s38/s46/s35/s33/s38/s65/s66/s48/s39/s51/s44\n/s39/s68/s44\nFIG. 4: (Color online) (a) Temperature dependence of elec-\ntrical resistivity in the temperature range 1.8 - 300 K. In-\nset shows the low temperature part, where the two magnetic\ntransitions are clearly seen. (b) Magentic field dependence of\nelectrical resistivity for J/bardbl[100] and H/bardbl[100] direction.5ρ(0)\nρ(0)ρ(0)(a) (b)\n(c)\nFIG. 5: (Color online) (a) Isothermal magnetoresistance\nMR=∆ρ\nρ(0)=ρ(B)−ρ(0)\nρ(0)for CeAgAs 2as a function of ap-\nplied magnetic field for J/bardbl[100] and H/bardbl[100]. (b) The\nlow field part of the isothermal magnetoresistance with spar se\nmarkers for clarity and (c) normalized magnetoresistance i n\nthe paramagnetic region plotted as a function B/(T+T∗)\nbecomes negative at 7.5 kOe. This may be attributed\nto the effect of field on spin fluctuations at this tem-\nperature. In the paramagnetic region, the normalized\nmagnetoresistance can be mapped on to a single curve\nif plotted as a function of B/(T+T∗), where T∗is the\ncharacteristic temperature, often identified as the Kondo\ntemperature30. Our experimental data taken at selected\ntemperatures can be mapped on to a single curve with\na characteristic temperature T∗of about -4.5 K. Nega-\ntiveT∗value has been obtained for several antiferromag-\nnetic Kondolattice compounds31,32and it wasattributed\nto the ferromagnetic correlations. From the magnetic\nmeasurements it is clear that CeAgAs 2orders antifer-\nromagnetically. The ferromagnetic correlations in this\ncompound arises due to the parallel alignment of mag-\nnetic moments in the layers of the abplane which are\nantiparallel along the c-axis as inferred from the neutron\ndiffraction studies by Doert et al26.(a)\n(b) (c)\nFIG. 6: (Color online) (a)Temperature dependence of heat\ncapacity of CeAgAs 2and the non-magnetic LaAgAs 2. The\nupper inset shows the low temperature part of C/Tversus\nT2. Thelower inset shows thelow temperaturepartofspecific\nheat in the form of C/TversusT. (b) The magnetic part of\nheat capacity along with the entropy. (c) Field dependence\nof the heat capacity.\nD. Specific heat\nThe temperature dependence of heat capacity of\nCeAgAs 2andthenon-magneticreferenceLaAgAs 2inthe\ntemperature range 2 −200 K is shown in the main panel\nof Fig. 6(a). The heat capacity of CeAgAs 2is larger\nthan that of LaAgAs 2. The bulk nature of the magnetic\ntransitions observed in magnetization and resistivity is\nconfirmed by the two sharp peaks at TN1= 6 K and\nTN2= 4.9 K with appreciable magnitude (lower inset\nof Fig. 6(a)). The low temperature data of LaAgAs 2in\nthe temperature range (2 to ∼6 K) was fitted to the ex-\npression C=γT+βT3, wherethe Sommerfeldcoefficient\nγis the electronic contribution and βis phonon contri-\nbution to the heat capacity. The γandβvalues thus\nobtained are 1.20 mJ/K2·mol and 0 .413 mJ/K4·mol, re-\nspectively. The upper inset of Fig 6(a) shows the low\ntemperature part of CeAgAs 2in the form of C/Tversus\nT2. Similarly, we have obtained the γandβvalues for\nCeAgAs 2as 61.3 mJ/K2·mol and 76.12 mJ/K4·mol re-\nspectively. There is a substantial increase in γdue to the\nKondo effect. The enhanced value of βis attributed to6\nspin waves in CeAgAs 2, which follow a T3dependence in\nantiferromagnets. We have estimated the magnetic part\nof heat capacity by the usual method of subtracting the\nheat capacity of LaAgAs 2from that of CeAgAs 2. The\nobtained C4fcontribution to the heat capacity is shown\nin Fig. 6(b). The magnetic entropy that is released at\n6 K is only 70% of Rln2 while the remaining 30 % of en-\ntropy is released at around 30 K. This indicates that the\nground state is a doublet ground state. The reduction\nin the magnetic entropy is mainly attributed to Kondo\neffect. Similarly the jump in the magnetic part (∆ Cmag)\nof the heat capacity amounts to 9.02 J/K ·mol atTN1\nand 4.53 J/K ·mol atTN2. The reduced jump in the heat\ncapacity, compared to the value expected for S= 1/2\ndoublet state supports the existence of the Kondo effect\nin CeAgAs 2.\nWe have also performed the field dependence of heat\ncapacity of CeAgAs 2. When the field was applied paral-\nlel to the flat plane of the crystal, there was no change\nin heat capacity (not shown here for brevity). Hence\nthe crystals were aligned in such a way that the applied\nmagneticfieldisparalleltotheeasyaxisofmagnetization\nviz., [100]. Figure 6(c) shows magnetic field dependence\nof heat capacity at various applied magnetic fields. It is\nevident from the figure that as the applied magnetic field\nis increased, the N´ eel temperatures shift towards lower\ntemperature. At a field of 7.5 kOe, where the metamag-\nnetic jump is observed in the magnetization data, there\nis a sudden change in the magnitude of the heat capac-\nity and for fields greater than 7.5 kOe, the heat capacity\nshows only a broad hump which shifts towards the right\nside at higher fields. At the lowest temperature mea-\nsured, the in-field heat capacity is lower than its zero\nfield value, which can be attributed to the breakdown of\nKondo coupling, between the localized 4 fand conduc-\ntion electrons with applied field33.\nIV. DISCUSSION\nThe magnetic measurements along the three principal\ncrystallographic directions revealed a strong anisotropy\nbetween the abplane and c-axis. Furthermore, there is a\nstrong suppression of the ordered moment and a reduced\nheat capacity jump together with a Schottky like peak in\ntheC4fheat capacity caused by the CEF effect. Hence,\nwe analysed the magnetocrystalline anisotropy based on\nthe point charge model. For the purpose of the CEF\nanalysis, we have plotted the magnetic susceptibility in\nthe formof1 /(χ−χ0), whereχ0wasdeterminedfromthe\nmodified Curie-Weiss fit as mentioned earlier. There are\ntwo Ce sites occupying the same 4 dWyckoff’s position,\nwith the same xcoordinate while yandzare different23.\nAssuming that the 4 dsite which hosts both the Ce-atoms\nhas identical crystallographic environment, we have per-\nformed the CEF analysis. The point symmetry of the 4 d\nWycoff’s position is mand hence possesses monoclinic\nsite symmetry. For monoclinic site symmetry, the 2 J+1/s32/s33/s33\n/s34/s33/s33\n/s35/s33/s33\n/s33/s36/s37/s38/s39/s38/s37/s33/s40/s39/s41/s38/s36/s42/s43/s44/s45/s46/s42/s47/s40\n/s48/s33/s33 /s35/s33/s33 /s41/s33/s33 /s33\n/s49/s46/s42/s50/s46/s51/s52/s53/s47/s51/s46/s38/s36/s54/s40/s55/s46/s56/s57/s56/s58/s35 /s32/s59/s33/s33/s41/s60\n/s38/s59/s41/s33/s33/s60\n/s38/s55/s61/s62/s38/s62/s63/s53\n/s64/s41/s38/s65/s38/s35/s66/s67/s38/s54/s64/s35/s38/s65/s38/s68/s32/s68/s38/s54\n/s41/s69/s66\n/s41/s69/s33\n/s33/s69/s66\n/s33/s70/s52/s57/s71/s46/s53/s63/s72/s52/s53/s63/s43/s71/s38/s36/s73/s74/s45/s55/s46/s40\n/s32/s33 /s34/s33 /s35/s33 /s33\n/s70/s52/s57/s71/s46/s53/s63/s75/s38/s62/s63/s46/s44/s76/s38/s36/s77/s78/s46/s40/s32/s38/s59/s41/s33/s33/s60\n/s38/s59/s33/s41/s33/s60\n/s38/s59/s33/s33/s41/s60\n/s38/s55/s61/s62/s38/s62/s63/s53/s38/s59/s41/s33/s33/s60\n/s38/s55/s61/s62/s38/s62/s63/s53/s38/s38/s59/s33/s41/s33/s60\n/s38/s55/s61/s62/s38/s62/s63/s53/s38/s59/s33/s33/s41/s60/s55/s46/s56/s57/s56/s58/s35\n/s32/s33/s34/s33/s35/s38/s54/s36/s52/s40\n/s36/s79/s40\nFIG. 7: (Color online) Temperature dependence of the in-\nverse magnetic susceptibility plotted as 1 /(χ−χ0). The solid\nlines in (a) are fits to Eqn. 2. (b) CEF fits to the isothermal\nmagnetization curves of CeAgAs 2at 2 K along the principal\ncrystallographic directions\nground state of Ce-atom splits into three doublets. In\norder to reduce the number of fitting parameters in the\nCEF analysis, we used the CEF Hamiltonian for the or-\nthorhombic site symmetry which is given by,\nHCEF=B0\n2O0\n2+B2\n2O2\n4+B0\n4O0\n4+B2\n4O2\n2+B4\n4O4\n4,(1)\nwhereBm\nnare the CEF parameters and Om\nnare the\nStevens operators34,35. In the above CEF Hamiltonian\nwe have ignored the sixth order terms as they are zero\nfor Ce atom. The magnetic susceptibility including the\nmolecular field contribution λis given by\nχ−1=χ−1\nCEF−λi, (2)\nwhereχCEFis CEF susceptibility. The expression for\nthe CEF susceptibility is given in our previous report33.\nWe have also analysed the isothermal magnetization by7\nusing the following expression:\nMi=gjµB/summationdisplay\nn/angbracketleftn|Ji|n/angbracketrighte−En/kBT\n/summationtext\nne−En/kBT,(3)\nwhere the Enand the eigenfunction |n >are deter-\nmined by diagonalizing the total Hamiltonian\nH=HCEF−gJµBJi(H+λiMi), (4)\nwhereHCEFis given by Eq. 1, the second term is the\nZeeman term and the third is the molecular field term.\nIn Fig. 7(a) we show the calculated CEF curves as\nsolid lines, which reproduce reasonably the observed ex-\nperimental susceptibility along the two principal crystal-\nlographic directions viz.[100] and [001] direction. We\ndid not show the inverse susceptibility plot of [010] direc-\ntion as it overlaps with the [100] direction. CEF analysis\nprovides a goodfit to the experimental data. It nicely re-\nproduces the broad peak in the susceptibility along the\n[001] direction. The CEF parameters and the energy lev-\nels thus obtained from the diagonalization of the CEF\nHamiltonian are given in Table I. The 2 J+1 degenerate\nJ= 5/2 level is split into three doublets at energies 0,\n259 and 767 K. The ground state is predominantly |±1\n2/angbracketright\nwith the mixing from the |±5\n2/angbracketrightand|±3\n2/angbracketrightstates. The\nmolecular field constant is highly anisotropic with posi-\ntive values in the ab-plane along the aandbaxes, while\nit is negative along the c-axis. This is consistent with\nthe alignment of the magnetic moments, where the intra-\nlayer interactions in the ab-plane are ferromagnetic and\nthe inter-layer interaction along c-axis is antiferromag-\nnetic. Figure 7(b) shows CEF analysis of the isothermal\nmagnetizationat 2K.Thesmalldiscrepancybetweenthe\ncalculated CEF curve and the experimental magnetiza-\ntion may be attributed to the fact that the CEF calcula-\ntionsdonottakeintoaccounttheKondoeffect. However,\nthe anisotropy in the magnetization plots is clearly ex-\nplained semi-quantitatively by present set of crystal field\nparameters.\nThevalueofcrystalfieldsplitenergylevelsofthe2 J+1\nground state of the Ce-atom have been used to analyse\nthe heat capacity data. The 4 fcontribution of the heat\ncapacity, in the paramagnetic state can be expressed as\nthe contribution of the electronic, Kondo and the Schot-\ntky terms as given below:\nC4f=Cel+CK+CSch. (5)\nTheCelis the electronic heat capacity defined by\nγT, while the expressions for CKandCSchare given in\nRef [36]. The black solid line in Fig. 7, is the calculated\ncurve based on Eqn. 5. The energy levels obtained from\nthe CEF analysis of the magnetic susceptibility data, re-\nproduce well the Schottky heat capacity. We obtain a\nKondotemperatureof2.5K.An estimation ofthe Kondo/s32/s33\n/s34\n/s33/s32/s35/s33/s36/s37/s38/s39/s40/s36/s41/s42/s43/s44\n/s32/s33/s33 /s34/s33 /s33\n/s45/s46/s41/s47/s46/s48/s49/s50/s51/s48/s46/s36/s37/s40/s44/s32/s32/s41/s49/s52\n/s36/s32/s45/s42/s50/s49/s43\n/s36/s32/s53\n/s36/s32/s40\n/s36/s32/s54/s55/s56\n/s57/s32/s36/s58/s36/s59/s34/s60/s36/s40/s57/s59/s36/s58/s36/s61/s62/s61/s36/s40/s63/s46/s64/s52/s64/s54/s59\nFIG. 8: (Color online) 4 fcontribution the heat capacity of\nCeAgAs 2. The thick solid line is the calculated C4fbased on\nEqn. 5.\ntemperatureinthemean-fieldmodelcanalsobeobtained\nfrom the jump in the C4fof a Kondo system using the\nexpression given by Blanco et al37. The jump in the\nmagnetic part of the heat capacity at TN2amounts to\n9.02 J/K ·mol, which results in a TK/TNvalue of 0.35\nand hence a Kondo temperature of 1.72 K, which is close\nto the value obtained above using Eqn. 5. Although, our\nestimation of the crystal field levels from the magnetic\nsusceptibility data explains the Schottky heat capacity\nwell which may be taken as a consistency check, for a\nprecise determination of these levels, inelastic neutron\ndiffraction has to be performed, which is planned as a\nfuture work.\nV. SUMMARY\nWe were successful in growing a single crystal of\nCeAgAs 2and its non-magnetic analog LaAgAs 2by flux\nmethod using Ag:As eutectic composition as flux. From\nthe x-ray diffraction and the TEM analysis we confirmed\nthe superstructure in CeAgAs 2single crystals which re-\nsults in an orthorhombic variant of the HfCuSi 2type\ntetragonal crystal structure, with two distinct crystal-\nlographic sites for the Ce-atom. The RKKY-type inter-\naction results in two magnetic transitions at TN1= 6 K\nandTN2= 4.9 K. The magnetic susceptibility is highly\nanisotropic and it exhibits a broad hump in the tempera-\nture range 50 to 300 K for field parallel to [001] direction.\nFromthe analysisofmagnetizationbasedonpointcharge\nmodel of crystal electric field combined with the heat ca-\npacitydata, itisconfirmedthatthisbroadhumpisdueto\nthe crystal electric field and not due to the intermediate\nvalence or mixed valence of Ce-atoms. The anisotropy in\nthe magnetic susceptibility can also be clearly explained\nby the crystal field analysis, with the two excited states8\nTABLE I: CEF fit parameters, energy levels and wave functions\nCEF Parameters\nB0\n2\n(K)B2\n2\n(K)B0\n4\n(K)B2\n4\n(K)B4\n4\n(K)λi\n(emu/mol)−1\n5.79−2.49−2.40−1.18 1.74λx= 30\nλy= 30\nλz= -25\nEnergy levels and wave functions\nE (K)|+5\n2/angbracketright |+3\n2/angbracketright |+1\n2/angbracketright |−1\n2/angbracketright |−3\n2/angbracketright |−5\n2/angbracketright\n767 0 0.996 0 0.014 0 0.0901\n767 0.0901 0 0.014 0 0.996 0\n259 0 -0.0865 0 -0.167 0 0.982\n259 0.982 0 0.167 0 0.0865 0\n0 0 -0.0288 0 0.986 0 0.165\n0 0.165 0 0.986 0 -0.0288 0\nat 259K and 767K, respectively. 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Patrikios, and J. R.\nFernandez, Phys. Rev. B 49, 15126 (1994)."    },    {        "title": "1807.09032v2.Impact_of_magnetic_moment_and_anisotropy_of_Co___textrm_1_x__Fe___textrm_x___thin_films_on_the_magnetic_proximity_effect_of_Pt.pdf",        "content": "Impact of magnetic moment and anisotropy of Co 1-xFexthin \flms on the magnetic\nproximity e\u000bect of Pt\nPanagiota Bougiatioti1, Orestis Manos1, Olga Kuschel2, Joachim\nWollschl ager2, Martin Tolkiehn3, Sonia Francoual3, and Timo Kuschel1\n1Center for Spinelectronic Materials and Devices, Department of Physics,\nBielefeld University, Universit atsstra\u0019e 25, 33615 Bielefeld, Germany\n2Department of Physics and Center of Physics and Chemistry of New Materials,\nOsnabr uck University, Barbarastra\u0019e 7, 49076 Osnabr uck, Germany\n3Deutsches Elektronen-Synchrotron DESY, Notkestra\u0019e 85, 22607 Hamburg, Germany\n(Dated: July 26, 2018)\nWe present a systematic study of the magnetic proximity e\u000bect in Pt, depending on the magnetic\nmoment and anisotropy of adjacent metallic ferromagnets. Element-selective x-ray resonant mag-\nnetic re\rectivity measurements at the Pt absorption edge (11565 eV) are carried out to investigate\nthe spin polarization of Pt in Pt/Co 1-xFexbilayers. We observe the largest magnetic moment of\n(0.72\u00060.03)\u0016Bper spin polarized Pt atom in Pt/Co 33Fe67, following the Slater-Pauling curve of\nmagnetic moments in Co-Fe alloys. In general, a clear linear dependence is observed between the Pt\nmoment and the moment of the adjacent ferromagnet. Further, we study the magnetic anisotropy of\nthe magnetized Pt which clearly adopts the magnetic anisotropy of the ferromagnet below. This is\ndepicted for Pt on Fe(001) and on Co 50Fe50(001), which have a 45\u000erelative rotation of the fourfold\nmagnetocrystalline anisotropy.\nThe spin polarization of a nominally paramagnetic\nmaterial generated by the exchange interaction of an ad-\njacent ferro- or ferrimagnetic material, is called magnetic\nproximity e\u000bect (MPE). It is most famous in Pt, which\nis almost ferromagnetic following the Stoner criterion\ndescription [1]. The MPE is a key element in spintronics\n[2] and spin caloritronics [3], since additional e\u000bects\ncan contribute when Pt is employed as a spin current\ndetector. For example, an additional proximity-induced\nanomalous Nernst e\u000bect can occur in spin Seebeck e\u000bect\nexperiments [4{6]. Furthermore, the proximity-induced\nanisotropic magnetoresistance can hamper spin Hall\nmagnetoresistance studies [7].\nIn normal metal (NM)/ferromagnet (FM) bilayer\nsystems, the conductivity of the FM a\u000bects the MPE\nin the NM, as recently systematically investigated by\nexamining the transition from FM metal (FMM) to\nFM insulator (FMI), using various oxygen content in\nPt/NiFe 2Oxbilayers [6]. No MPE in Pt could be found\nexcept for the metallic Pt/Ni 33Fe67case without any\noxygen. Although the MPE varies for di\u000berent material\ncombinations (e.g. FMIs vs. FMMs), general systematic\nstudies of the MPE dependence on material parameters\nwithin one class of material are quite rare. The two\nmost important properties of magnetic materials are\nthe magnetic moment and the magnetic anisotropy.\nTherefore, in this work we systematically investigate\nthe MPE dependence on the FM moment and FM\nanisotropy in NM/FMM systems.\nOne way to investigate the MPE is to use x-ray\nmagnetic circular dichroism (XMCD), allowing to\nextract the absolute magnetic moment per atom of\neach element [8{21]. A much younger technique toinvestigate the magnetic properties of layer systems\nwith element- and depth-sensitivity, is x-ray resonant\nmagnetic re\rectivity (XRMR) that is based on the\nspin-dependent interference of light re\rected from the\ninterfaces in the system [22]. This method even detects\nmagnetic moments at buried interfaces for thicker layers\n[23], when XMCD is not sensitive anymore. In previous\nstudies, we investigated the spin polarization in Pt of\nPt/NiFe 2Ox(4\u0015x\u00150) bilayers [6, 23, 24], in order\nto evaluate the proximity-induced contributions to the\ninverse spin Hall and anomalous Nernst voltages, while\nstudying the transport phenomena on the samples\n[6, 25]. In addition, we examined Pt/FMM bilayers\nproviding information about the spatial distribution of\nthe spin polarization of Pt, across the interface to a\nFMM [23, 24, 26].\nIn this work, we investigate the induced spin polariza-\ntion in Pt on top of a class of material that changes the\nmagnetic moment and anisotropy systematically with\nits content. This material is Co 1-xFexwith a maximum\nmagnetic moment for Co 33Fe67and di\u000berently oriented\nmagnetic anisotropy depending on the Fe content. We\nuse XRMR to extract the Pt magnetic moments and\ncompare the results with the magnetic moment for the\nCo1-xFexlayers. In addition, we detect XRMR magnetic\n\feld loops to study the magnetic anisotropy solely in the\nspin-polarized Pt and to compare to magnetic \feld loops\nof the Co 1-xFexlayer, collected via magnetooptic Kerr\ne\u000bect (MOKE). For both strength of magnetic moment\nand magnetic anisotropy, we \fnd a clear correlation\nbetween the spin polarized Pt and the FMM below.\nWe fabricated Pt/Co 1-xFexbilayers with\nx = 0:00;0:15;0:30;0:50;0:67;1:00, by dc magnetronarXiv:1807.09032v2  [cond-mat.mtrl-sci]  25 Jul 20182\nsputter deposition on top of (001)-oriented MgO sub-\nstrates at room temperature (RT). The FMM layers\nwere prepared with and without Pt in-situ deposited on\ntop, by covering one FMM layer with a mask. The Ar\npressure during the deposition for both Co 1-xFexand Pt\nlayers was equal to 2 \u000110\u00003mbar and the base pressure of\nthe chamber was 3 \u000110\u00009mbar. The appropriate sputter\nparameters were adjusted after evaluating the x-ray\n\ruorescence spectra to achieve the desired composition.\nThe XRMR and XMCD measurements were carried out\nat the resonant scattering and di\u000braction beamline P09\nof the third generation synchrotron PETRA III at DESY\n(Hamburg, Germany) [27]. A fundamental theoretical\nbackground behind XRMR includes the determination\nof the magnetooptic parameters \u0001 \u000eand \u0001\fwhich\ncorrespond to the magnetic change of the dispersion\n\u000eand absorption \fcoe\u000ecients, respectively, of the\ninvestigated material exposed to x-rays of the element's\nabsorption energy. In our case, the x-ray re\rectivity\n(XRR) I \u0006for left and right circularly polarized light,\nrespectively, was detected o\u000b resonance (11465 eV) and\nat resonance at the L 3absorption edge of Pt (11565 eV),\nswitching fast the helicity of incident circular polar-\nization [27]. Afterwards, the XRMR asymmetry ratio\n\u0001I=I+\u0000I\u0000\nI++I\u0000was calculated and the magnetic moment\nper Pt atom extracted using a spin depth pro\fle model\nthat results in a simulated asymmetry ratio \ftting\nthe experimental data. Further details of the XRMR\ntechnique, experiment, data processing, and \ftting can\nbe found in the Supplemental Material [28] (Chap. I).\nThe XMCD spectrum was collected using an energy\ndispersive silicon drift detector synchronized with the\npiezo-actuators underneath the phase plates, allowing\nfor the \ruorescent photons for left and right circular\npolarized incident light to be counted separately at every\npoint of the scan. In order to investigate the magnetic\nanisotropy of the spin polarized Pt layer, we collected\nXRMR magnetic \feld loops for di\u000berent in-plane sample\norientations and a \fxed scattering vector q=4\u0019\n\u0015sin\u0012\nthat corresponds to a maximum asymmetry ratio \u0001 I(\u0015\nis wavelength and \u0012is angle of incidence).\nFigure 1(a) presents the XRMR asymmetry ratio\n\u0001Ifor the Pt/Co 50Fe50bilayer plotted against the\nscattering vector q. The e\u000bect changes sign when the\nmagnetic \feld direction is reversed which con\frms its\nmagnetic origin. Figure 1(b) depicts the experimental\nenergy dependant x-ray absorption spectrum (XAS,\ngreen line) at the Pt L 3edge normalized to the edge\njump, after the subtraction of a linear background.\nThe XMCD intensity ( I+\u0000I\u0000)/2 is also displayed in\nthe \fgure and was extracted to identify the energy\nwith the largest dichroism. The magnetic dichroism of\nthe spin polarized Pt has its maximum slightly below\nthe absorption maximum (dashed line) which is in\nagreement with previous \fndings [23, 24, 26, 29] and,thus, the chosen energy to collect the XRMR data was\nat 11565 eV.\n0.00 .20 .40 .60 .8-0.04-0.020.000.020.04Asymmetry ratio/s32/s916/s921S\ncattering vector q (/s61662-1) positive field \nnegative field( a)(\nb)1\n1535115501156511580115950.00.30.60.91.2E\nnergy (eV)Norm. exp. XAS-\n2.0-1.5-1.0-0.50.00.5 \nXMCD (%)\nFIG. 1. (a) Resonant (11565 eV) asymmetry ratio \u0001 I(q) for\nboth magnetic \feld directions. (b) Experimental energy de-\npendant XAS spectrum (green line) and the XMCD signal\n(red line). All data correspond to the Pt/Co 50Fe50bilayer.\nFigure 2 presents the XRR and XRMR data as well\nas the resulting magnetooptic depth pro\fles for all\nPt/Co 1-xFexbilayers. Figures 2(a)-(e) show the averaged\nresonant magnetic XRR scans, collected at a photon en-\nergy of 11565 eV, plotted against the scattering vector q\nand accompanied by their \fttings. Kiessig fringes appear\nin all scans due to the interference of the re\rected light\nfrom the Pt/Co 1-xFexand Co 1-xFex/MgO interfaces. By\n\ftting the o\u000b-resonant (11465 eV) XRR curves we obtain\nthe thickness (indicated in Figs. 2(k)-(o)) and roughness\n(typical values for the NM/FMM interfaces are between\n0.20 nm and 0.38 nm), using literature \fand\u000evalues\nfor the individual layers. In a second step, we kept the\nstructural parameters \fxed for \ftting the averaged res-\nonant (11565 eV) XRR curves following the description\nof Klewe et al. [26], thus, obtaining the resonant \fand\n\u000evalues. When \ftting the XRMR asymmetry ratios the\nstructural parameters from the o\u000b-resonant XRR \ft and\nthe optical values from the resonant XRR \ft have been\nkept \fxed and just the \u0001 \fdepth pro\fle has been varied.\nThe derived XRMR asymmetry ratios \u0001 I(q) are illus-\ntrated in Figs. 2(f)-(j), plotted together with the cor-\nresponding \fttings. In all cases, pronounced oscillations\nare visible with an amplitude of about 2% comparable to\nour prior studies [6, 23, 24, 26] and additional maxima\nthat can reach up to 4%, unveiling an induced spin po-\nlarization in Pt.\nFigures 2(k)-(o) display the magnetooptic depth pro\fles\nof \u0001\f, which were used to \ft the XRMR asymmetry ra-\ntios in Figs. 2(f)-(j). The magnetooptic pro\fles were\ngenerated by a Gaussian function at the Pt/Co 1-xFex\ninterface, convoluted with the roughness pro\fle of the3\n0.0 0.2 0.4 0.6 0.80.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.810-810-710-610-510-410-310-210-1100\n0.0 0.2 0.4 0.6 0.80.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8-0.02-0.010.000.010.020.030.04(e) (d) (c) (b) Pt/Co33Fe67\n exp. data\n fitPt/Co50Fe50 (a) Pt/Co70Fe30Pt/Co85Fe15Pt/CoXRR intensity (arb. units)\n(g) (f)\nScattering vector q ( -1) exp. data\n fit\n(j) (i) (h)\nScattering vector q ( -1) Scattering vector q ( -1) Scattering vector q ( -1)Asymmetry ratio \nScattering vector q ( -1)\n0 4 812 0 4812 0 4 812 0 4 812 0 4 812Co50Fe50 (8.2)Pt (3.1) Pt (3.2)\nCo33Fe67 (9.6)\n1.3\nCo70Fe30 (9.9)Pt (3.0) Pt (3.0)\nCo85Fe15 (10.0)Pt (2.9)\nCo (9.8)\nΔβ Δβ Δβ Δβ 1.2\n1.2 1.2 1.2\n(o) (n) (m) (l)\nΔβ (k)\nFIG. 2. (a)-(e) Resonant (11565 eV) XRR scans and \fts with (f)-(j) the corresponding determined and \ftted XRMR asymmetry\nratios \u0001I(q) for all Pt/Co 1-xFexbilayers. (k)-(o) Magnetooptic depth pro\fles which were used to \ft the XRMR asymmetry\nratios. The thicknesses of the layers are presented in nm. The dashed lines denote the corresponding interface position between\nPt and Co 1-xFexlayers. The arrows indicate the FWHM of the spin polarized layer, obtained from the structural parameters\nof the XRR \fts.\ncorresponding layer [26]. For all magnetooptic pro\fles,\nwe extracted the full width at half maximum (FWHM)\nwhich represents the e\u000bective thickness of the spin po-\nlarized Pt layer at the Pt/Co 1-xFexinterface. This ef-\nfective spin polarized Pt thickness is between 1.2 nm and\n1.3 nm for all samples, as indicated in Figs. 2(k)-(o). By\ncomparing the experimental \ft values of \u0001 \fwith the\nab initio calculations of Ref. [23], we extracted the mag-\nnetic moment per spin polarized Pt atom at the maxi-\nmum of the magnetooptic pro\fle, as summarized in Ta-\nble I of the Supplemental Materials [28] (Chap. II) for\nall FMM compositions.\nFigure 3(a) presents the Pt magnetic moment for all\nsamples (blue points), plotted against the Fe content x to-\ngether with the magnetic moment values of the Co 1-xFex\nalloys (orange data), taken from Ref. [30]. The error bars\nare estimated by changing the \fvalues until the goodness\nof \ft value \u001f2increases up to 20 %. In our prior studies\n[6, 23, 24, 26] we just roughly estimated the uncertainty,\ntherefore, the previous error values have been slightly\nlarger. The inset depicts a close-up plot of the graph.\nAs visible, the magnetic moments in Pt clearly exhibit a\nsimilar progress as the magnetic moments in the Co 1-xFex\nalloys, which follow the Slater-Pauling curve [31]. BothPt and FMM moments increase with increasing x, peak-\ning at a certain content ratio which is the Co 28Fe72alloy\nfor the literature values and the Pt/Co 33Fe67bilayer for\nour experimental data. For further increase of Fe content,\nboth Pt and FMM moments decrease. Consequently, we\nconclude that the strength of the magnetic coupling be-\ntween the two layers depends on the magnitude of the\nmagnetic moment in the FMM, as indicated by Klewe et\nal. for Pt/Ni 1-xFexbilayers [26] and by Poulopoulos et\nal. [13] for Ni/Pt multilayers. This is valid as long as Pt\nis deposited on FMMs. If Pt is grown on magnetic semi-\nconductors or insulators, the dependence of Pt moment\non FM moment can be di\u000berent or nonexistent due to a\nvanishing MPE [6, 17{20, 23, 24].\nFigure 3(b) exhibits the dependence of Pt mag-\nnetic moment on the FMM magnetic moment for both\nPt/Co 1-xFex(green points) and Pt/Ni 1-xFex(red points,\ntaken from Refs. [23, 26]) bilayers. The dashed lines are\nlinear \fts of the data and indicate the linear dependence\nbetween the Pt and FMM magnetic moments in such bi-\nlayer systems. In addition, the slopes of both curves, as\ndepicted in the graph, are comparable to each other con-\nsidering the errors. The slope of the Pt moment linear\ndependence on the FMM moment might be interpreted4\n0.00.10.20.30.40.50.60.70.80.91.00.00.20.40.60.8\n0.20.40.60.81.00.600.650.700.751\n.952.102.252.402.55 Pt/Co1-xFex \n         Co1-xFexF\ne content xPt moment(\nµB/atom)0\n.00.51.01.52.02.5 \n Co1-xFex moment (µB/atom)0\n.00.30.60.91.21.51.82.12.40.00.20.40.60.8s\nlope: 0.33±0.02 FMM = Co1-xFex \nFMM = Ni1-xFexF\nMM moment of FMM layer (/s956B/atom)Pt moment ofP\nt/FMM bilayer (µB/atom)s\nlope: 0.29±0.03\nFIG. 3. (a) Pt magnetic moment plotted against the Fe con-\ntent x of Pt/Co 1-xFexbilayers (blue points). The orange\ndata is taken from Ref. [30] displaying the magnetic mo-\nment per atom as derived from magnetization measurements\nfor Co 1-xFexalloys. The inset depicts a close-up plot em-\nphasizing the maximum experimental Pt magnetic moment\nin Pt/Co 33Fe67. The data of Pt/Fe is taken from Ref. [23].\n(b) Pt magnetic moment plotted against the FMM magnetic\nmoment for both Pt/Co 1-xFex(green points) and Pt/Ni 1-xFex\n(red points, taken from Refs. [23, 26]) bilayers, respectively.\nThe green (red) dashed line is a linear \ft to the data with a\nslope equal to 0 :33\u00060:02 (0:29\u00060:03), for the Pt/Co 1-xFex\n(Pt/Ni 1-xFex) bilayers. In both graphs, the squares, triangles,\nand circles correspond to FMMs with bcc, fcc, and hcp crystal\nstructure, respectively.\nas the distance to the Stoner criterion. The systematic\nbehaviour for Pt on top of other classes of materials (such\nas semiconductors or slightly oxygen-reduced ferrites [6])\nor for other NM materials (such as Pd) on FMMs, will\nbe part of future work.\nAs a next step, we investigated the magnetic\nanisotropy of Fe and Co 50Fe50samples by performing\nMOKE rotational measurements with di\u000berent in-plane\ncrystal orientation directions (0\u000e\u0015\u000b\u0015360\u000ein steps of\n5\u000e), in the presence of an in-plane magnetic \feld. The\nazimuthal angle \u000bcorresponds to the angle between the\ndirection of the applied magnetic \feld and the [110] di-\nrection of the corresponding alloy, as sketched in the inset\nof Figs. 4(c) and (d). In order to examine the magnetic\nanisotropy of the spin polarized Pt layer, we collected\nXRMR \feld loop measurements, for di\u000berent sample ori-\nentations (0\u000e\u0015\u000b\u001545\u000ein steps of 15\u000e).\nFigures 4(a) and (b) present the squareness which is\nthe ratio between the magnetic remanence and the sat-\nuration magnetization, extracted from the MOKE loops\n(not shown) for the Fe and Co 50Fe50samples and from\nXRMR \feld loops for the Pt/Fe and Pt/Co 50Fe50sam-\nples. For the Fe \flm (cf. Fig. 4(a)), the MOKE measure-\nments reveal magnetic easy axes along the Fe <100>di-\nrections, which correspond to the MgO <110>directions\nαCoFe\n[110]\nFe [110]\nhard axis\nFe [100]\neasy axisFe[110]\nhard axisΗ\nΗCoFe [110]\neasy axis\nCoFe [100]\nhard axisΗ\nCoFe [100]\nhard axisCoFe [110]\neasy axis\nΗFe[100]\neasy axis\n0°\n90°\n180°270°0.60.81.0\n0.6\n0.8\n1.00°\n90°\n180°270°\n-0.06 -0.03 0.00 0.03 0.06-1.0-0.50.00.51.0\n-0.06 -0.03 0.00 0.03 0.06-1.0-0.50.00.51.0CoFe\n[100]Fe[110] Fe[100]Normalized XRMR\nasymmetry ratio Fe (MOKE)\n Pt/Fe (XRMR)CoFe\n[110] Co50Fe50 (MOKE)\n Pt/Co50Fe50 (XRMR)=  = Squareness\nμ0H (T) 0°\n 45°(a) (b)\n(c) (d) Pt/Fe\nμ0H (T)Pt/Co50Fe50FIG. 4. Squareness plotted against the value of the az-\nimuthal angle \u000bfor (a) Fe (MOKE), Pt/Fe (Pt element-\nselective XRMR) and (b) Co 50Fe50(MOKE), Pt/Co 50Fe50\n(Pt element-selective XRMR) samples, collected at RT. The\nangle\u000bcorresponds to the angle between the direction of the\napplied magnetic \feld and the [110] direction of the corre-\nsponding alloy, as sketched in the inset of Figs. 4(c) and (d).\nThe purple (green) square corresponds to the orientation of\nthe MgO substrate with respect to the direction of the ap-\nplied magnetic \feld leading to \u000b= 0\u000e(\u000b= 45\u000e). Detected\nXRMR asymmetry ratio \feld loops with \u000b= 0\u000e;45\u000efor (c)\nPt/Fe and (d) Pt/Co 50Fe50bilayers, collected in resonance\n(11565 eV) and with a qvalue of 0.24 \u0017A\u00001.\n(\u000b= 45\u000e;135\u000e;225\u000e;315\u000e) with high remanence values\nand magnetic hard axes along the the Fe <110>direc-\ntions corresponding to the MgO <100>directions ( \u000b=\n0\u000e;90\u000e;180\u000e;270\u000e) with low remanence values [32, 33].\nOn the other hand, for the Co 50Fe50sample (cf. Fig.\n4(b)) the four-fold magnetic anisotropy is 45\u000erotated\ncompared to Fe. Therefore, the magnetic easy axes for\nthe Co 50Fe50sample are aligned along the CoFe <110>\ndirections which are the MgO <100>directions with high\nremanence values, whereas the magnetic hard axes are\naligned along the CoFe <100>directions corresponding\nto the MgO <110>directions with low remanence values\n[34, 35].\nIn Figs. 4(c) and (d) two normalized XRMR \feld loops\nfor Pt/Fe and Pt/Co 50Fe50samples are illustrated for\n\u000b= 0\u000eand 45\u000e, respectively. For the Pt/Fe bilayer (Fig.\n4(c)), the spin polarized Pt layer showed the magnetic\neasy axis for \u000b= 45\u000e(green curve) and the magnetic\nhard axis for \u000b= 0\u000e(purple curve), which is consis-\ntent with the results extracted from the previously an-\nalyzed MOKE measurements of the pure Fe layer. In\naddition, for the Pt/Co 50Fe50sample (Fig. 4(d)) the\nmagnetic easy axis appeared for \u000b= 0\u000e(purple curve)\nand the magnetic hard axis was found at \u000b= 45\u000e(green\ncurve), which also coincides with the MOKE results of\nthe pure Co 50Fe50\flm. Moreover, the extracted square-5\nness values from the XRMR \feld loops for the Pt/Fe and\nPt/Co 50Fe50samples are included in Figs. 4(a) and (b),\nrespectively. The obtained results clearly show that the\nmagnetic anisotropy of the spin polarized Pt layer on top\nof the FMM (where FMM is Fe or Co 50Fe50) adopts the\nmagnetic anisotropy of the FMM.\nIn conclusion, we investigated the dependence of the\nMPE of Pt on the magnetic moment and anisotropy of\ndi\u000berent FMMs below, performing XRMR and MOKE\nmeasurements. We fabricated Pt/Co 1-xFexbilayers by\ndc magnetron sputter deposition on top of (001)-oriented\nMgO substrates. The XRMR asymmetry ratios were\nquantitatively analyzed and the largest spin polariza-\ntion was found in Pt/Co 33Fe67with a magnetic moment\nequal tomPt= (0:72\u00060:03)\u0016Bper spin polarized Pt\natom. We also found that the magnetic moment in Pt\nfollows the Slater-Pauling curve of magnetic moments\nin Co 1-xFexalloys. In addition, the Pt moment is in-\ncreasing linearly with the FMM moment and the slope\nof this linear dependence is quite similar when we com-\npare the Pt/Ni 1-xFexand Pt/Co 1-xFexbilayers. Further-\nmore, we examined the correlation between the magnetic\nanisotropy in the FMM layers (Fe and Co 50Fe50extracted\nvia MOKE measurements) and the magnetic anisotropy\nof the spin polarized Pt layer (extracted via magnetic\nre\rectivity measurements at the Pt L 3absorption edge)\nof the Pt/Fe and Pt/Co 50Fe50bilayers. The magnetic\nanisotropy of the spin polarized Pt layer re\rects the mag-\nnetic anisotropy of the FMM in the corresponding alloy.\nThus, we can conclude, that the Pt nicely follows the\nFMM magnetization strength and orientation. For Pt\non FMIs this is still an open question beside the general\nquestion of MPE in Pt/FMIs and will be explored in fu-\nture research. Moreover, additional research should be\nconducted in such bilayer systems to create a full FM\nmoment and conductivity mapping of the MPE in var-\nious NMs, to further elucidate the details of the MPE\nmechanism.\nThe authors gratefully acknowledge \fnancial support\nby the Deutsche Forschungsgemeinschaft (DFG) within\nthe Priority Program Spin Caloric Transport (SPP 1538,\nKU 3271/1{1) and the Deutsche Elektronen Synchrotron\n(DESY). Further they are grateful for the opportunity to\nwork at beamline P09, PETRA III at the Deutsche Elek-\ntronen Synchrotron, a member of the Helmholtz Associ-\nation (HGF) and for technical support by David Reuther\nand J org Strempfer. They also thank G unter Reiss from\nBielefeld University, Germany, for making available the\nlaboratory equipment needed for sample preparation and\ncharacterization.\n[1] E. C. Stoner, Proc. R. Soc. 165, 372 (1938).\n[2] A. Ho\u000bmann and S. D. Bader, Phys. Rev. Applied 4,047001 (2015).\n[3] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat.\nMater. 11, 391 (2012).\n[4] S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang,\nJ. Wu, T. Y. Chen, J. Q. Xiao, and C. L. Chien, Phys.\nRev. Lett. 109, 107204 (2012).\n[5] T. Kikkawa, K. Uchida, Y. Shiomi, Z. Qiu, D. Hou,\nD. Tian, H. Nakayama, X.-F. Jin, and E. Saitoh, Phys.\nRev. Lett. 110, 067207 (2013).\n[6] P. Bougiatioti, C. Klewe, D. Meier, O. Manos,\nO. Kuschel, J. Wollschl ager, L. Bouchenoire, S. 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L¨ offler1\n1Laboratory of Metal Physics and Technology, Department of Ma terials, ETH Zurich, 8093 Zurich, Switzerland\n2Condensed Matter Theory Group, Paul Scherrer Institute, CH- 5232 Villigen PSI, Switzerland\n(Dated: July 24, 2018)\nMaterials based on the Sm–Co system exhibit remarkable magnet ic performance due to their high\nCurie temperature, large saturation magnetization, and st rong magnetic anisotropy, which are the\nresult of the electronic structure in Co and Sm and their arra ngement in the hexagonal lattice. In\nthis paper we show, using first-principles calculations, me an-field theory, and atomistic Monte Carlo\nsimulations, that slight modifications of the SmCo 5crystal structure, induced by strain or partial\nsubstitution of Sm by Ce, change the exchange interaction and increase the magnetocrystalline\nanisotropy energy drastically. This finding shows how small changes in local-structure environments\ncan generate substantial changes in macroscopic propertie s and thus enable the optimization of\nhigh-performance materials via tailoring at the atomic leve l.\nI. INTRODUCTION\nThe magnetism in intermetallic phases of rare-earth\n(RE) metals and transition metals (TM), such as the\nhigh-performance magnet SmCo 5, is a result of the syn-\nergy between the 4 fRE electrons, which provide a high\nanisotropy due to spin-orbit coupling, and the 3 dTM\nelectrons, which have large magnetic moments and pro-\nvide strong ferromagnetic exchange interactions, thus en-\nabling long-range order1,2. The synergy between 3 dand\n4felectrons depends crucially on the local atomic envi-\nronments, and thus the net performance of a magnet is\nextremely sensitive to changes of the lattice.\nIn this paper we show that slight modifications of\nthe local atomic arrangements in SmCo 5generate dras-\ntic changes of the magnetic performance, particularly\nof the magnetocrystalline anisotropy (MCA). Specifi-\ncally, we will discuss, based on first-principle electronic -\nstructure calculations and atomistic Monte Carlo simu-\nlations, that tensile strain in SmCo 5leads to enhanced\nmagnetic anisotropy. Hence, we propose a special case\nwhere partial substitution of Sm with Ce generates a\nmaterial with magnetic performance which is superior\nto that of the well-known SmCo 5.\nThe permanent magnet SmCo 5has been intensively\nstudied over the past few decades because it exhibits\nstrong uniaxial MCA, large saturation magnetization\n(Ms), and high Curie temperature ( TC) of about 1000\nK2–10. The high magnetic performance of this sys-\ntem, particularly its spectacularly strong MCA, makes\nit a key component in various precision-sensitive applica-\ntions. In fact, out of all Co- and Fe-based RE-containing\ncompounds SmCo 5has the largest magnetocrystalline-\nanisotropy energy EMCA2,11. It is thus of fundamental\ninterest to investigate the limits of magnetism in this\nhigh-performance system and to find ways of how to en-\nhance the magnetic anisotropies beyond the limit of this\nwell-known compound.\nExperimental and theoretical studies have reported\npossibilities of improving the magnetic performance of\nSmCo 5both in bulk materials12,13and in thin-film\nFIG. 1. Illustration of the crystal structure of SmCo 5and\n(Sm,Ce)Co 5. In the left panel, the ideal hexagonal CaCu 5\nprototype structure with and without strain is shown, and\ntensile and compressive strains are indicated by red and blu e\narrows, respectively. The right panel shows the crystal stru c-\nture of (Sm 1−xCex)Co5forx= 0.25, 0.50 and 0.75. Gray,\nred, light blue and dark blue spheres correspond to Sm, Ce,\nCo(2c) and Co(3g) atoms, respectively. Note that these are\nthe most stable structures obtained by total-energy minimi za-\ntion in the DFT calculations.\nstructures14–17. Promising approaches to enhance the\nmagnetic performance are e.g. the partial substitution of\nCo with other transition metals (TM), such as Cu, Ni,\nFe, or Zr in Sm(Co, TM) 517–19, but in most cases the\nTC, the saturation magnetization ( Ms), and the EMCA\ndecrease with increasing TM content due to the reduced\nexchange coupling parameters when Co is substituted by\narXiv:1807.09257v1  [cond-mat.mtrl-sci]  24 Jul 20182\nother elements17,20.\nConsidering the above, the question remains of\nwhether or not it is possible to increase the EMCA, and\nthus enhance the performance of Sm–Co without substi-\ntuting Co, but by carefully modifying the local atomic\narrangements, i.e., by introducing strain. Strain is a\nparticularly important issue in permanent magnets, be-\ncause their synthesis typically involves sintering, and\nthermal processes inevitably introduce strain on fine-\ngrained materials21,22. Strain can be applied externally\nby mechanical means, but it can also be introduced to\nthe system by partial element substitution, such as e.g.\nof Sm by another RE metal with a different atomic ra-\ndius.\nCe is a particularly promising candidate for substi-\ntuting Sm because it is the most abundant and second-\nlightest among all RE metals, and CeCo 5is isomorhpous\nto SmCo 523,24, suggesting that it is possible to substi-\ntute Sm with Ce through the entire composition, i.e.,\n(Sm 1−xCex)Co5for 0≤x≤1. SmCo 5and CeCo 5crys-\ntallize in the hexagonal CaCu 5structure, where each Sm\n(Ce) occupies a 1a site and Co occupies 2c and 3g sites in\nthe lattice25(see Fig. 1); the unit cell volume decreases\nfrom 84.71 ˚A3in SmCo 5to 83.09 ˚A3in CeCo 5.\nIt is well-known that the intrinsic magnetic material\nparameters of CeCo 5lead to a magnetic performance\nthat is substantially weaker than that of SmCo 511, i.e.,\nreduced TCof 660 K instead of 1000 K, reduced EMCA\nof 5.3 MJ/m3instead of 17.2 MJ/m3, and reduced Msof\n0.95 T instead of 1.15 T. Nevertheless, a partial substitu-\ntion of Sm by Ce can have drastic effects on the magnetic\nperformance, as we discuss below, because it introduces\nstrain in the structure and breaks the lattice symmetry\nin a way that enhances the contribution of the Co atoms\nto the MCA.\nIn the following we briefly discuss the theory and the\ncomputational methodology and then analyze our re-\nsults, which reveal a surprisingly enhanced magnetic per-\nformance of SmCo 5with strain and with partial Sm sub-\nstitution.\nII. COMPUTATIONAL METHOD\nDensity Functional Theory (DFT) calculations were\nmainly performed using plane wave basis sets and pseu-\ndopotentials, implemented in Quantum Espresso (QE)26.\nThis approximation enables time-efficient calculations,\nbut its precision in certain cases might be limited by the\nuse of pseudopotentials. Hence, results from QE were\ncomplemented by more computationally demanding all-\nelectron full-potential augmented plane-wave and local-\norbitals basis sets, implemented in Elk27and WIEN2k28,\nrespectively, which enable more precise calculations.\nThis is particularly important for the precise estimation\nof the MCA energy, which is associated with variations\non the order of meV.\nAll DFT calculations were performed on a 2 ×2×2supercell of the SmCo 5(CaCu 5) structure. For the\nexchange-correlation potential we adapted the local spin-\ndensity approximation plus Hubbard U(LSDA+ U),\nwhich can adequately describe the strongly correlated\nelectronic states of 4 felectrons9,29,30. The LSDA+ U\nmethod requires the Coulomb energy ( U) and the ex-\nchange energy ( J) as input parameters, and we de-\nfined UandJthrough the derivatives of the energy\nlevels ǫfof the f-orbital with respect to their occu-\npancies nf, described as U=∂ǫf↑/∂n f↓andJ=\n∂(ǫf↑−ǫf↓)/∂(nf↑−nf↓) for the majority (minority)\nspin↑(↓), respectively. From these expressions we ob-\ntainU= 6.0 eV and J= 1.0 eV for Sm in our DFT\ncalculations.\nWave functions were expanded by a plane-wave basis\nset with an optimized cutoff energy of 340 Ry, and the\nBrillouin zone was sampled via a 12 ×12×15k-point\nmesh. Different mesh values were tested to ensure the\nnumerical stability of our calculations, with the conver-\ngence criterion being 0 .1µeV. All structures were fully\nrelaxed prior to calculating the magnetic parameters, i.e. ,\nthe atomic arrangements in our calculations correspond\nto the most stable structures, found by minimizing the\ntotal energy.\nThe magnetocrystalline anisotropy energy EMCA was\ncalculated using the force theorem and is defined as\nthe total energy difference between the magnetization\nperpendicular to the c-plane and in the c-plane of the\nSmCo 5structure, i.e., K=E[100]−E[001], where E[100]\nandE[001]are the total energies with the magnetization\naligned along the hard- ([100]) and easy-axis ([001]) of\nthe magnetic anisotropy, respectively, with (001) corre-\nsponding to the cplane. Specifically, EMCA is calcu-\nlated in three steps: (i) collinear self-consistent calcul a-\ntion without spin-orbit coupling (SOC); (ii) global rota-\ntion of the density matrix to consider the magnetization\nalong [100] and [001] to calculate E[100] andE[001], re-\nspectively; and (iii) non-collinear and non-self-consist ent\ncalculation with SOC.\nFor the calculation of the energy-resolved distribution\nof the orbitals within second-order perturbation theory31,\nEMCA can be expressed as\nE↑↓\nMCA∝ξ2×\n/summationdisplay\no↑(↓),u↓(↑)|<o↑(↓)|Lx|u↓(↑)>|2−|<o↑(↓)|Lz|u↓(↑)>|2\nǫo↑(↓)−ǫu↓(↑),\n(1)\nwhere ξis the spin-orbit coupling constant, ↑(↓) repre-\nsent majority (minority) spins, and o(u) and ǫo(u)repre-\nsent the eigenstate and eigenvalue of occupied (unoccu-\npied) states, respectively. For contributions from the d-\norbitals, the matrix elements of the LxandLzoperators\ncan be expressed via the magnetic quantum number m.\nWhen the occupied and unoccupied states have opposite\nspin directions, mo−mu=±1 and mo−mu= 0 con-\ntribute to out-of-plane and in-plane MCA, respectively,\nwhile mo−mu=±2 is treated as zero matrix element.3\nFrom the DFT calculations we obtain the magne-\ntocrystalline anisotropy energy, as described above, and\nfrom the total energy minimization we obtain the atomic\nmagnetic moments and the inter-atomic ferromagnetic\nexchange interactions as the difference between ferro-\nmagnetic and antiferromagnetic spin configuration, i.e.,\nJ= (E↑↑−E↑↓)/2. We then use these intrinsic material\nparameters as inputs for atomistic Monte Carlo simula-\ntions to calculate finite-temperature material properties\nby mapping the material parameters onto a Heisenberg-\ntype model. Here, the Hamiltonian has the form\nH=−1\n2/summationtext\ni/negationslash=jJij/vectorSi·/vectorSj−/summationtext\njKi(Sz)2, (2)\nwhere the spin /vectorSiis the 3-dimensional vector at each lat-\ntice site, Jijis the Heisenberg exchange interaction en-\nergy between neighboring sites, and Kiis the site-specific\nuniaxial anisotropy energy constant. The numerical val-\nues of the interaction and anisotropy energies were taken\ndirectly from the DFT results in absolute values.\nThe simulations were performed by means of the\nMetropolis algorithm, i.e., by single-site updates. For th e\nthermalization the algorithm was run 5000 MCS (Monte\nCarlo steps per site) and another 5000 MCS were run to\nsample the thermal-average magnetic moment /vectorMat the\n1a, 2c, and 3g lattice sites at each temperature.\nWe considered systems with 120000 and 276000 spins,\ncorresponding to real dimensions of 5 nm ×8.5 nm ×39\nnm and 10 nm ×10 nm ×40 nm, respectively, to verify\nthe numerical stability of our results. Additionally, we\nused periodic boundary conditions to eliminate surface\neffects.\nFor the calculation of the Curie temperature we started\nthe simulations with all spins parallel and pointing along\n[001] at T= 0 and gradually increased the tempera-\nture to 1200 K, whereas for the calculation of the finite-\ntemperature domain-wall thickness, we started the sim-\nulations with half the spins pointing along [001] and the\nother half pointing along [00 ¯1] and thermalized the sys-\ntem at each given temperature, in a similar way as de-\nscribed in Ref.32.\nIII. RESULTS AND DISCUSSION\nA. Strain effects in SmCo 5\nWe calculated the total energy of the SmCo 5struc-\nture to optimize the lattice and determined the in-plane\nlattice constant to be a= 4.980˚A and the axis ratio\nto be c/a= 0.792. These values are in excellent agree-\nment with previous first-principles calculations9,29,33,34\nand experiments35, thus confirming our computation of\nthe equilibrium structure.\nFurther, for SmCo 5in equilibrium we obtain an EMCA\nof 9.09 meV/f.u. (f.u. = formula unit), which is/s45/s53 /s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53/s48/s51/s54/s57/s49/s50/s49/s53/s49/s56/s49/s55/s54/s49/s55/s56/s49/s56/s48/s49/s56/s50/s49/s56/s52/s49/s56/s54/s49/s46/s51/s49/s46/s52/s49/s46/s53/s49/s46/s54/s49/s46/s55/s49/s46/s56/s49/s46/s57/s50/s46/s48\n/s32/s112/s108/s97/s110/s101/s32/s119 /s97/s118/s101/s32/s43/s32/s112/s115/s101/s117/s100/s111/s112/s111/s116/s101/s110/s116/s105/s97/s108/s115\n/s32/s112/s108/s97/s110/s101/s32/s119 /s97/s118/s101/s32/s43/s32/s102/s117/s108/s108/s32/s112/s111/s116/s101/s110/s116/s105/s97/s108/s32\n/s32/s111/s114/s98/s105/s116/s97/s108/s32/s98/s97/s115/s105/s115/s32/s115/s101/s116/s115/s32/s43/s32/s102/s117/s108/s108/s32/s112/s111/s116/s101/s110/s116/s105/s97/s108\n/s32/s112/s108/s97/s110/s101/s32/s119 /s97/s118/s101/s32/s43/s32/s111/s114/s98/s105/s116/s97/s108/s32/s98/s97/s115/s105/s115/s32/s115/s101/s116/s115/s32/s43/s32/s102/s117/s108/s108/s32/s112/s111/s116/s101/s110/s116/s105/s97/s108\n/s32/s112/s111/s108/s121/s110/s111/s109 /s105/s97/s108/s32/s102/s105/s116/s69\n/s77 /s67/s65/s32/s40/s109/s101/s86/s32/s47/s32/s102/s46/s117/s46/s41\n/s83/s116/s114/s97/s105/s110/s32/s40/s37/s41/s40/s97/s41\n/s32/s74\n/s67/s111 /s45 /s67/s111 /s74\n/s67/s111/s45/s67/s111/s32/s40/s109/s101/s86/s41\n/s40/s98/s41/s67/s111 /s32/s109/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s32 /s40\n/s66 /s41\n/s32/s67/s111/s32/s40/s50/s99/s41\n/s32/s67/s111/s32/s40/s51/s103/s41\n/s32/s83/s109\n/s40/s99/s41/s51/s50/s51/s52/s51/s54/s51/s56\n/s32/s74\n/s67/s111 /s45 /s83/s109\n/s74\n/s67/s111/s45/s83/s109/s32/s40/s109/s101/s86/s41/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53/s48/s46/s51/s48\n/s83/s109/s32 /s109/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s32 /s40\n/s66 /s41\nFIG. 2. Intrinsic magnetic properties as a function of strain in\nSmCo 5: (a) atomic magnetic moments of Co atoms at sites\n2c (diamonds) and 3g (squares), and Sm atoms (triangles);\n(b) effective total ferromagnetic exchange energy JCo−Cobe-\ntween Co atoms (hexagons) and JCo−Smbetween Co and Sm\natoms (triangles); and (c) magnetocrystalline anisotropy e n-\nergy ( EMCA), calculated by several approximations.\nagain in excellent agreement with experimental data11,36\n(corresponding to 17.18 MJ/m3) and with other DFT\ncalculations19,33. Specifically, we find that in the SmCo 5\nstructure the Co sublattice contributes 2.20 meV and\nthe Sm sublattice contributes 6.88 meV to the to-\ntal anisotropy energy per unit cell (corresponding to\n4.16 MJ/m3and 13.02 MJ/m3, respectively, in good\nagreement with experimental values for the individual\ncontributions36).\nIn order to probe SmCo 5away from equilibrium we ap-\nplied both tensile (+) and compressive ( −) strain along\nthe [001] axis (see Fig. 1), while keeping the unit cell vol-\nume constant, to investigate strain-dependent electronic\nstructure and magnetic properties.\nStrain effects on the magnetism, i.e., individual atomic\nmagnetic moments in SmCo 5, effective ferromagnetic ex-\nchange energy, and MCA energy are presented in Fig. 2.\nWe find that the magnetic moments at each crystal-\nlographic site decrease nearly linearly with increasing\nstrain (see Fig. 2a). The reason for the decrease in the\nmagnetic moments is the decreasing overlap of the d-\norbitals as the caxis grows with strain.\nFurther, we calculated the overall effective exchange\ncoupling parameters in the framework of the multi-\nsublattice mean-field approximation, and we obtained4\nJCo−CoandJCo−Sm, which are the sum of the exchange\ncoupling constants within a sphere of radius R= 5a.\nFrom these calculations we observe that the total fer-\nromagnetic exchange interaction increases monotonically\nwith increasing strain (see Fig. 2b), in contrast to the\nmagnetic moments. Note that the Curie temperature\ndoes not change significantly because the increase in the\neffective ferromagnetic exchange is offset by the decrease\nin the magnetic moments.\nWhile the overall exchange interactions increase with\nincreasing strain, the intra-plane interaction strengths\nJCo(2c) −Co(2c) andJCo(3g) −Co(3g) increase while the inter-\nplane interaction strength JCo(2c) −Co(3g) decreases with\nincreasing tensile strain (data not shown). This illus-\ntrates the dependence of the ferromagnetic exchange\non the inter-atomic distance: as the c–c and g–g dis-\ntances decrease the corresponding exchange interaction\nincreases, whereas as the c–g distance increases with in-\ncreasing tensile strain the exchange interaction decrease s.\nSimilar strain effects were calculated for YCo 5in\nRef.29,33, but with significantly weaker impact. It was,\nhowever, pointed out that the inter- vs intra-plane hy-\nbridization plays a key role in the development of the fer-\nromagnetic exchange, the magnetic moments, and EMCA.\nConsidering the effects of strain on EMCA in SmCo 5,\nwe find that the magnetocrystalline anisotropy energy\nchanges monotonically with strain. While compressive\nstrain decreases EMCA, increasing tensile strain has the\nopposite effect and EMCA along the c-axis exhibits a re-\nmarkable strengthening (see Fig. 2c). In fact, even with\nonly +1% strain EMCA increases by ∼30%, whereas with\n+5% strain EMCA increases, strikingly, by 80% compared\nto that in equilibrium at zero strain. Importantly, we\nfind that the results based on the use of pseudopotentials\nare in excellent agreement with full-electron calculation s,\nand they only start to deviate by 3–5% with increasing\nstrain, as the values of EMCA become larger. Specifically,\ncalculations based on orbital basis-sets and plane-waves\nconsistently tend to slightly underestimate EMCA.\nWe examined the cause of this remarkable enhance-\nment of EMCA and found that the major contribution\ncomes from the Co(2c) sites. Note that the 2c sites also\nplay a vital role in the MCA of the SmCo 5in equilibrium,\nas indicated by nuclear magnetic resonance studies37, and\nthat calculations of strained YCo 5also indicated the im-\nportance of 2c contributions to the anisotropy33.\nTo obtain a deeper understanding of why the lattice\ndistortion affects the MCA, we plot tensile and com-\npressive strain-induced changes in the electronic density\nof states (DOS), i.e., ∆DOS = DOS(+5%)–DOS(0%)\n(Fig. 3a) and ∆DOS = DOS( −5%)–DOS (0%) (Fig. 3b).\nWe observe that, especially in the range near the Fermi\nlevel, the changes in the DOS with m(with up and down\nspins) are mirrored depending on strain directions, i.e.,\nthem= 0,↓andm=±2,↓states lose electrons while\nthem=±1,↑state gains electrons with tensile strain,\nand vice versa with compressive strain.\nThe concurrence of the increased occupied m=±1,↑/s45/s48/s46/s51 /s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s45/s48/s46/s50/s53/s45/s48/s46/s50/s48/s45/s48/s46/s49/s53/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53\n/s32/s109 /s61/s48/s44 /s32/s32/s32 /s32/s109 /s61/s48/s44 /s32\n/s32/s109 /s61 /s49/s44 /s32 /s32/s109 /s61 /s49/s44\n/s32/s109 /s61 /s50/s44 /s32 /s32/s109 /s61 /s50/s44 /s32/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s32/s101/s86/s45/s49\n/s97/s116/s111/s109/s45/s49\n/s115/s112/s105/s110/s45/s49\n/s41\n/s69 /s32/s40/s101/s86/s41/s40/s97/s41\n/s45/s48/s46/s51 /s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51\n/s69 /s32/s40/s101/s86/s41/s40/s98/s41\nFIG. 3. Changes in DOS of the Co(2c) atom induced by (a)\n+5% (tensile) and (b) −5% (compressive) strain. Positive\nchange indicates gain of electrons, whereas negative chang e\nindicates loss of electrons for each state. Black, blue, and r ed\nlines denote m= 0,m=±1, and m=±2, respectively, while\nsolid and dashed lines represent majority and minority stat es,\nrespectively. The Fermi level is set to zero.\nstate and decreased unoccupied m= 0,↓andm=±2,↓\nstates around the Fermi level lead to larger EMCA be-\ncause of the stronger matrix elements of < m =±1,↑|\nLx|m= 0,↓>and< m =±1,↑|Lx|m=±2,↓>.\nMeanwhile, for the compressive strain shown in Fig. 3(b),\nthis process is reversed and EMCA decreases.\nThese results show that tensile stain along the c-\ndirection enhances the effective ferromagnetic exchange\nenergy notably but most importantly it causes a striking\nincrease of the MCA energy. This also suggests that off-\nequilibrium properties that exceed those of the equilib-\nrium state so drastically may be exploited in applications\nwhere materials are mechanically strained, either contin-\nuously or in pulses. Importantly these off-equilibrium\nmaterial parameters provide a deeper understanding of\nthe fundamental mechanisms of magnetic anisotropy and\nferromagnetic exchange in RE–TM systems and thus may\nenable the design of materials with enhanced properties.\nInspired by these findings, we now turn to a case study\nwhere we partially substitute Sm by Ce to induce local\nstrain in the cell and to predict changes in magnetic per-\nformance.5\n/s49/s51/s46/s56/s49/s51/s46/s57/s49/s52/s46/s48/s49/s52/s46/s49\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s86/s111/s108/s117/s109/s101/s32/s40 /s197/s51\n/s41/s40/s97/s41/s83/s116/s114/s97/s105/s110/s32/s40/s37/s41\n/s120 /s32/s40/s97/s116/s111/s109/s115/s47/s102/s46/s117/s46/s41/s32/s40/s98/s41\nFIG. 4. Change of (a) unit-cell volume and (b) effective strain\nin (Sm 1−xCex)Co5as a function of Ce concentration x.\nB. Partial substitution of Sm by Ce\nWe performed calculations of the (Sm 1−xCex)Co5(x\n= Ce/Sm) system, where we considered seven different\ncompositions: x= 0, 0.125, 0.25, 0.375, 0.5, 0.75, and\n1. For each xthe total energy was minimized to obtain\nthe most stable atomic arrangement among all possible\nconfigurations and the corresponding unit-cell volume.\nExamples of the systems with x= 0.25, 0.5, and 0.75 are\nshown in Fig. 1, where gray, red, light blue and dark blue\nspheres correspond to Sm, Ce, Co(2c) and Co(3g) atoms,\nrespectively.\nIn order to obtain a direct comparison between the\neffects of Ce-substitution on the magnetic properties with\nthose of strain, we show in Fig. 4(a) and (b) the unit-cell\nvolume and the effective strain, respectively, as function\nof the Ce concentration x. The volume decreases nearly\nlinearly with increasing x, which is due to the smaller\natomic radius of Ce compared to that of Sm, whereas\nthe strain increases correspondingly monotonically with\nincreasing x. This substitution-induced tensile strain can\nhave drastic effects on the magnetic state, which do not\nnecessarily correspond to the electronic properties of Ce\nthat would tend to decrease the anisotropy energy.\nThe calculated magnetic moments of the Co, Ce,\nand Sm atoms decrease monotonically with increasing\nCe-substitution x, which correlates with the decreasing\natomic volume, following Vergard’s law38(see Fig. 5a).\nAlso, the calculated JCo−CoandJCo−RE, which is the\naveraged interaction between Co and Sm or Ce atoms,\ndecrease strongly with increasing x, as seen in Fig. 5(b).\nFor the magnetocrystalline anisotropy, however, we\nfind a surprising non-monotonic behavior. In fact, for\nx= 0.125 the anisotropy energy increases to 10.49\nmeV/f.u. (19.8 MJ/m3) and for x= 0.25 it becomes/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s50/s48/s49/s52/s48/s49/s54/s48/s49/s56/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s32/s112/s108/s97/s110/s101/s32/s119 /s97/s118/s101/s32/s43/s32/s112/s115/s101/s117/s100/s111/s112/s111/s116/s101/s110/s116/s105/s97/s108/s115\n/s32/s112/s108/s97/s110/s101/s32/s119 /s97/s118/s101/s32/s43/s32/s102/s117/s108/s108/s32/s112/s111/s116/s101/s110/s116/s105/s97/s108/s32\n/s32/s111/s114/s98/s105/s116/s97/s108/s32/s98/s97/s115/s105/s115/s32/s115/s101/s116/s115/s32/s43/s32/s102/s117/s108/s108/s32/s112/s111/s116/s101/s110/s116/s105/s97/s108\n/s32/s112/s108/s97/s110/s101/s32/s119 /s97/s118/s101/s32/s43/s32/s111/s114/s98/s105/s116/s97/s108/s32/s98/s97/s115/s105/s115/s32/s115/s101/s116/s115/s32/s43/s32/s102/s117/s108/s108/s32/s112/s111/s116/s101/s110/s116/s105/s97/s108/s69\n/s77 /s67/s65/s40/s109/s101/s86/s32/s47/s32/s102/s46/s117/s46/s41\n/s120 /s32/s40/s97/s116/s111/s109/s115/s47/s102/s46/s117/s46/s41/s40/s97/s41\n/s32/s74\n/s67 /s111/s45/s67 /s111/s74\n/s67/s111/s45/s67/s111/s32/s40/s109/s101/s86/s41/s40/s98/s41/s77/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s32 /s40\n/s66 /s41\n/s32/s67/s111/s40/s50/s99/s41/s32/s32 /s32/s67/s111/s40/s51/s103/s41\n/s32/s83/s109/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s67/s101\n/s40/s99/s41/s50/s48/s50/s52/s50/s56/s51/s50/s51/s54\n/s32/s74\n/s67 /s111/s45/s82 /s69\n/s74\n/s67/s111/s45/s82/s69/s32/s40/s109/s101/s86/s41\nFIG. 5. Magnetic properties as a function of Sm-substitution\nby Ce: (a) atomic magnetic moments for Co on sites 2c (di-\namonds) and 3g (squares), Sm (triangles), and Ce (inverse\ntriangles); (b) effective total ferromagnetic exchange inte rac-\ntion energy JCo−Cobetween Co atoms (hexagons) and JCo−RE\nbetween Co and RE atoms (triangles), averaged over Co–Sm\nand Co–Ce pairs; and (c) magnetocrystalline anisotropy en-\nergy ( EMCA). The size of the symbols is larger than the error\nbars. In panel (c) the crossed circle corresponds to the EMCA\nof metastable systems with x= 0.5.\n11.63 meV/f.u. (22.0 MJ/m3), which is 30% larger than\nthat of the SmCo 5system in equilibrium (see Fig. 5c).\nAsxincreases further, however, EMCA decreases and for\nx= 1, i.e., CeCo 5, it becomes 2.96 meV/f.u. (corre-\nsponding to 5.6 MJ/m3). This EMCA for CeCo 5is also\nin excellent agreement with experimental data11.\nNote that for x= 0.5 the equilibrium structure has\nEMCA = 6.18 meV/f.u., which corresponds to only 68%\nof that of the SmCo 5compound, but we found metastable\nconfigurations that yield a magnetic anisotropy energy of\n7.56 meV/f.u., which is only slightly smaller than that of\nSmCo 5despite the fact that half of the Sm is substituted\nby Ce.\nThe 30% enhancement in EMCA forx= 0.25 is un-\nexpected, given that the electronic structure of Ce and\nits contribution would tend to substantially lower the\nanisotropy. We find, however, that this increase corre-\nlates with the strain in the unit-cell of SmCo 5induced\nby the substitution of every 4th Sm atom by a Ce atom,\nwhich corresponds to 1.13% strain (see Fig. 4b). For\nthe non-substituted SmCo 5system, 1% of strain corre-\nsponds to 22% increase in EMCA, as shown in Fig. 2c.6\nHence, this means that with 1/4 Ce substitution the ef-\nfects of the strained Sm–Co structure dominate over the\nelectronic contributions of Ce.\nAgain we find excellent agreement between the dif-\nferent approximations, with an exception at x= 0.25,\nwhich exhibits the highest EMCA and is slightly underes-\ntimated by the approximation based on orbital basis-sets\nand plane waves.\nThe main contribution to the enhancement of EMCA\ncomes from the Co(2c) sites, which is in direct analogy\nto the effects of strain (see Fig. 3). In order to in-\nvestigate in detail the origin of the enhanced EMCA of\n(Sm 0.75Ce0.25)Co5and the decreased value in high Ce\nconcentration cases, we present in Fig. 6 the DOS de-\ncomposed into dxy,x2−y2,dyz,xz anddz2orbitals, which\ncorrespond to magnetic quantum numbers of m=±2,\nm=±1, and m= 0, respectively. To provide a clear\nexplanation, only the DOS of Co(2c) atoms that drive\nthe change of EMCA is presented.\n/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54/s83/s109/s67/s111\n/s53\n/s83/s109\n/s48/s46/s55/s53/s67/s101\n/s48/s46/s50/s53/s67/s111\n/s53/s32/s109 /s32/s61/s32\n/s32/s109 /s32/s61/s32\n/s32/s109 /s32/s61/s32\n/s67/s101/s67/s111\n/s53/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s32/s101/s86/s45/s49/s32\n/s97/s116/s111/s109/s45/s49/s32\n/s115/s112/s105/s110/s45/s49\n/s41\n/s69 /s40/s101/s86/s41\nFIG. 6. Comparison of the d-projected DOS of Co(2c) atoms\nin SmCo 5, (Sm 0.75Ce0.25)Co5and CeCo 5. Shaded, solid, and\ndotted lines denote m= 0, m=±1, and m=±2, respec-\ntively. The Fermi level is set to zero. Arrows emphasize\norbital shifting.\nAs shown in Fig. 6(a), in the DOS of SmCo 5we find\n< m =±1,↑|Lx|m= 0,↓>and< m =±1,↑|\nLx|m=±2,↓>couplings, as the peaks of m=±1\nandm= 0 lie directly below and above EF, respec-\ntively, and the energy difference between occupied and\nunoccupied states is reduced (see Eq. 1), leading to\nthe strong MCA along the caxis. In the substitutedsystem with (Sm 0.75Ce0.25)Co5the unoccupied minor-\nitym= 0 and m=±2 orbitals, indicated by arrows,\nshift toward the Fermi level. As a result, EMCA is en-\nhanced because the < m =±1,↑|Lx|m= 0,↓>and\nthe< m =±1,↑|Lx|m=±2,↓>coupling become\nstronger. For the CeCo 5case, however, the peaks of mi-\nnority m= 0 and m=±2 orbitals move to the occupied\nstates, and therefore couplings which contribute to the\nEMCA are strongly weakened.\nGiven the above, first-principles calculations provide\ncompelling evidence that strain in SmCo 5, either in\nthe form of mechanical strain or due to partial Sm-\nsubstitution by Ce in (Sm 1−xCex)Co5, can push the lim-\nits of magnetic anisotropy in this system. The agreement\nbetween our results and existing literature for the equilib -\nrium properties of SmCo 5, both structural and magnetic,\nconfirms our calculations and makes our predictions for\nthe strained structures very promising.\n/s45/s50/s48 /s45/s49/s53 /s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48 /s49/s53 /s50/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s32/s77\n/s122/s32/s32/s67/s111/s32/s40/s51/s103/s41\n/s32/s77\n/s122/s32/s32/s67/s111/s40/s50/s99/s41/s32/s38/s32/s82/s69/s40/s49/s97/s41/s77\n/s122/s32/s40\n/s66/s41\n/s68/s105/s115/s116/s97/s110/s99/s101/s32/s102/s114/s111/s109/s32/s100/s111/s109/s97/s105/s110/s32/s119/s97/s108/s108/s32/s40/s110/s109/s41\nFIG. 7. Monte Carlo simulation at finite temperature showing\nthe domain-wall profile of the sublattice containing Co(3g)\natoms and the sublattice containing RE(1a) and Co(2c)\natoms. A snapshot of the system containing a domain wall is\npresented at the corner.\nAll calculated material parameters so far correspond\nto the intrinsic magnetic properties at zero temperature.\nHence, we extend now our computational analysis to fi-\nnite temperature to predict both the intrinsic material\nparameters and the resulting relevant length scales, i.e.,\nthe domain-wall thickness δdw, of Ce-substituted Sm–Co\nmaterials. The computation of finite-temperature length\nscales takes into account the effects of the microstructure\nand is thus particularly relevant to high-temperature ap-\nplications.\nTaking the DFT results shown in Fig. 5 as inputs, we\nperformed Monte Carlo simulations and computed the\nlength scale of a domain wall by simulating domain-wall\nbehavior in the Sm–Ce–Co system. An example is illus-\ntrated in Fig. 7, which shows the spatially-resolved sub-\nlattice magnetization Mzas a function of the distance\nfrom the domain-wall center. The spins rotate around the7\n/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s49/s50/s51/s52/s48/s46/s55/s53/s48/s46/s57/s48/s49/s46/s48/s53/s49/s46/s50/s48/s54/s48/s48/s55/s53/s48/s57/s48/s48/s49/s48/s53/s48/s49/s50/s48/s48\n/s32\n/s32/s100/s119/s32/s40/s110/s109/s41\n/s120/s32 /s40/s97/s116/s111/s109/s115/s47/s102/s46/s117/s46/s41/s32/s51/s48/s48/s32/s75\n/s32/s53/s48/s48/s32/s75/s32\n/s32/s77 /s40/s84/s41/s32 /s40\n/s66/s41\n/s40/s99/s41/s40/s98/s41/s32\n/s32/s84\n/s67/s32/s40/s75/s41/s40/s97/s41\nFIG. 8. Effect of the Ce-substitution on the finite-\ntemperature magnetic properties: (a) Curie temperature; ( b)\naverage magnetic moments at T= 300 K and 500 K; and (c)\ndomain-wall thickness at T= 300 K and 500 K, as a function\nof the Ce concentration x. The size of the symbols is larger\nthan the error bar.\ncenter of the domain wall and the profile is proportional\nto tanh ( r/λ), with rthe distance from the domain-wall\ncenter and λ=δdw/2. Note that we distinguish between\nthe magnetization of the 3g sites and the 1a and 2c sites.\nIn order to obtain a comparison between all the Ce-\nsubstituted systems, we computed the Curie Tempera-\nture TC, the average atomic magnetic moment M, and\nδdwatT= 300 K and 500 K (= 0 .5TSmCo 5\nC ). Figure 8\nshows these results as a function of the Ce concentration.\nBoth TCandMgradually decrease with increasing\nCe concentration. For SmCo 5(x= 0) we obtain\nTC≈1100 K, whereas for CeCo 5(x= 1) we obtain\nTC≈700 K, both of which are in good agreement with\nexperimental data ( ∼1050 K for SmCo 5and∼660 K\nfor CeCo 5).\nFor the domain-wall thicknesses, we find that for\nSmCo 5at room temperature δdw= 1.61(1) nm, which\nis in very good agreement with previous theoretical\ncalculations39(δdw= 1.6 nm) and experiments40(δdw=\n1.3±0.3 nm), supporting the validity of our first-\nprinciples calculations and MC simulations, consideringthat SmCo 5without Ce is our reference system.\nTheδdwreflects the tendency of EMCA shown in Fig. 5,\nwith the smallest domain wall being at x= 0.25 and\nthe largest at x= 1. Note that the decreasing domain-\nwall thickness in the Ce-substituted system suggests that\na domain wall could fit in smaller structures and be\nmore easily trapped, thus potentially increasing the co-\nercivity of a nanostructured magnet, and hence ensuring\nthat the same microstructure that makes SmCo 5a high-\nperformance magnet would also make (Sm 0.75Ce0.25)Co5\na permanent magnet with even higher performance, es-\npecially with regards to its MCA, which is the crucial\ncomponent for precision applications.\nIV. CONCLUSIONS\nWe found, based on electronic structure calculations,\nthat changes in the lattice spacing of SmCo 5have a\nstrong effect on the density of states close to the Fermi\nlevel, causing dramatic changes in the magnetic prop-\nerties. Specifically, we observed that tensile strain in\nSmCo 5monotonically enhances the magnetocrystalline\nanisotropy energy ( EMCA) and the effective ferromag-\nnetic exchange interaction ( Jeff), while reducing the\natomic magnetic moments of Sm and Co. In contrast,\ncompressive strain reduces the EMCA and the Jeff, and\nincreases the atomic magnetic moments. Further, we in-\nvestigated the effects of strain induced by partial sub-\nstitution of Sm by Ce and discovered that it has also\ndrastic effects on the EMCA. In fact, when Ce substi-\ntutes every 4th Sm atom in the lattice the EMCA in-\ncreases by more than 30%, compared to that of SmCo 5,\nbecause the density of states increases close to the Fermi\nlevel. In order to predict the potential of this system\nfor high-performance applications, we investigated the\nfinite-temperature properties of Ce-substituted SmCo 5\nand found that the magnetic length scales, the atomic\nmagnetic moments, and the Curie temperature are com-\nparable to those of the parent compound SmCo 5, suggest-\ning that the Ce-substituted material is a very promising\ncandidate for high-performance high-temperature appli-\ncations with enhanced magnetocrystalline anisotropy and\nwith 25% reduced heavy-rare earth content. Further, we\nhave shown that multi-scale modeling is a powerful strat-\negy for the detailed analysis and prediction of functional\nmaterials properties.\nV. ACKNOWLEDGMENTS\nThe authors acknowledge funding from the Swiss Na-\ntional Science Foundation via Grant No. 200021–165527\nand from the ETH Grant ETH-47 17-1, and thank P. 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Phys. 64, 1338 (1988)."    },    {        "title": "1807.10649v3.Magnetocrystalline_anisotropy_of_Fe5PB2_and_its_alloys_with_Co_and_5d_elements__a_combined_first_principles_and_experimental_study.pdf",        "content": "arXiv:1807.10649v3  [cond-mat.mtrl-sci]  30 Nov 2018Magnetocrystalline anisotropy of Fe 5PB 2and its alloys with Co and 5 delements:\na combined first-principles and experimental study\nMiros/suppress law Werwi´ nski∗\nInstitute of Molecular Physics, Polish Academy of Sciences ,\nM. Smoluchowskiego 17, 60-179 Pozna´ n, Poland\nAlexander Edstr¨ om\nMaterials Theory, ETH Z¨ urich, Wolfgang-Pauli-Str. 27, 80 93 Z¨ urich, Switzerland\nJ´ an Rusz\nDepartment of Physics and Astronomy, Uppsala University, B ox 516, SE-751 20 Uppsala, Sweden\nDaniel Hedlund, Klas Gunnarsson, and Peter Svedlindh\nDepartment of Engineering Sciences, Uppsala University, B ox 534, SE-751 21 Uppsala, Sweden\nJohan Cedervall and Martin Sahlberg\nDepartment of Chemistry – The ˚Angstr¨ om Laboratory,\nUppsala University, Box 538, SE-751 21 Uppsala, Sweden\n(Dated: December 3, 2018)\nThe Fe 5PB2compound offers tunable magnetic properties via the possibi lity of various combi-\nnations of substitutions on the Fe and P-sites. Here, we pres ent a combined computational and\nexperimental study of the magnetic properties of (Fe 1−xCox)5PB2. Computationally, we are able\nto explore the full concentration range, while the real samp les were only obtained for 0 ≤x≤0.7.\nThe calculated magnetic moments, Curie temperatures, and m agnetocrystalline anisotropy energies\n(MAEs) are found to decrease with increasing Co concentrati on. Co substitution allows for tuning\nthe Curie temperature in a wide range of values, from about si x hundred to zero kelvins. As the\nMAE depends on the electronic structure in the vicinity of Fe rmi energy, the geometry of the Fermi\nsurface of Fe 5PB2and the k-resolved contributions to the MAE are discussed. Low tempe rature\nmeasurements of an effective anisotropy constant for a serie s of (Fe 1−xCox)5PB2samples determined\nthe highest value of 0.94 MJ m−3for the terminal Fe 5PB2composition, which then decreases with\nincreasing Co concentration, thus confirming the computati onal result that Co alloying of Fe 5PB2\nis not a good strategy to increase the MAE of the system. Howev er, the relativistic version of the\nfixed spin moment method reveals that a reduction in the magne tic moment of Fe 5PB2, by about\n25%, produces a fourfold increase of the MAE. Furthermore, c alculations for (Fe 0.95X0.05)5PB2(X\n= 5delement) indicate that 5% doping of Fe 5PB2with W or Re should double the MAE. These\nare results of high interest for, e.g., permanent magnet app lications, where a large MAE is crucial.\nPACS numbers: 71.20.Be, 75.30.Gw, 75.50.Bb, 75.50.Cc, 75. 50.Ww\nI. INTRODUCTION\nMany sectors of modern technology depend on mag-\nnetic materials, which are used in such ubiquitous ap-\nplications as electric motors, power generators, trans-\nformers, and recording media. Hence, magnetic materials\nare crucial, not only for the digital technology revolution\nobserved in past decades, but also for the green energy\nrevolution expected within the years to come. The fun-\ndamentally and technologically most important intrin-\nsic parameters of magnetic materials include the Curie\ntemperature ( TC), saturation magnetization ( Ms), and\nmagnetocrystalline anisotropy energy (MAE). These pa-\nrameters are important in a wide variety of applications,\nincluding hard and soft magnetic materials for energy\nconversion, spintronics, and information storage. Thus,\nthe ability to predict these basic magnetic parameters\nfrom first principles is of utmost importance, and accu-rate modern electronic structure calculations provide an\nindispensable tool for exploring new materials with de-\nsired properties. In parallel, experimental synthesis and\ncharacterization retains its fundamental importance and\na close interplay between computational and experimen-\ntal work is of ever increasing value in modern materials\ndiscovery.\nOne example of an area in which the search for new\nmagnetic materials, with specific combinations of prop-\nerties, has been intense in recent years is that of per-\nmanent magnets. In this field it is typically desirable to\nhave large Ms,TCand MAE. This combination is ob-\ntained in the commonly used rare-earth transition metal\ncompounds, such as NdFe 14B2. However, the so called\nRare-Earth Crisis1triggered immense international re-\nsearch initiatives in search for new substitute permanent\nmagnet materials with reduced amounts of, or no, rare-\nearth elements2–6. The main challenge in this context\nis obtaining a sufficiently large MAE in transition metal2\ncompounds, where a uniaxial (e.g. tetragonal or hexag-\nonal) crystal structure is a crucial prerequisite. Other\nareas of applications depend upon other combinations of\nproperties. For example, for magnetocaloric solid state\ncooling, it is desirable to be able to tune the ordering\ntemperature such that it coincides with the operating\ntemperature (often room temperature)7,8.\nVarious works have shown how strain engineering or\nalloying can be used to carefully tune the properties of\nmagnetic materials to obtain desired functionality. For\nexample, it was shown that a careful control of strain and\nalloy concentration allows for a large MAE in bct FeCo\nalloys9–12. The potential route to FeCo-based perma-\nnent magnets offered by that work inspired subsequent\nstudies aiming to stabilize tetragonality in FeCo by B\nor C-impurities13–15. Also the tetragonal (Fe 1−xCox)2B\ncompound has been carefully studied due to its tunable\nMAE as function of x16–20which, furthermore, has an\nintriguing temperature dependence19,21,22. It was also\nshown, in both calculations and experiments, that small\namounts of 5 dsubstitutions on the Fe/Co site allowed a\nlarge increase in the MAE of this material19.\nThe tetragonal family of compounds with compositions\n(Fe1−xCox)5P1−ySiyB2has also been the subject of nu-\nmerous recent studies23–28. Additionally, other chemical\nsubstitutions, including Mn on the Fe/Co site23, have\nbeen considered. Due to the broad range of chemical\ncompositions available, this material offers wide tunabil-\nity of its magnetic properties. Furthermore, the tetrag-\nonal crystal structure could potentially allow for a large\nMAE and, thus, make the compounds interesting within\nthe context of permanent magnet applications. The ma-\nterials also exhibit other interesting aspects, such as the\ntemperature dependent spin-reorientation transition in\nFe5SiB225.\nThe aim of the work is to investigate the effect of the\nCo and 5 ddopants on the tunable magnetic properties\nof the technologically promising semi-hard Fe 5PB2com-\npound. Fe 5PB2crystallizes in the Cr 5B3-type structure\nwith a body-centered tetragonal (bct) unit cell, space\ngroup I4/mcm29(see Fig. 1). The unit cell of Fe 5PB2\nconsists of 4 formula units (32 atoms). Fe atoms occupy\ntwo inequivalent sites Fe 1(16l) and Fe 2(4c). Fe 1atoms\nare distributed on the 16-fold position, Fe 2and P on the\n4-fold, and B on the 8-fold position.\nOne of the motivations to investigate the\n(Fe1−xCox)5PB2system are our previous results\nfor isostructural (Fe 1−xCox)5SiB2system (with Si in\nplace of P), for which we have predicted the highest\nMAE = 1.16 MJ m−3for Co concentration x= 0.3.24\nNext, the (Fe 0.8Co0.2)5SiB2sample (with Co concentra-\ntionx= 0.2) was synthesized by McGuire and Parker23\nand their magnetic measurements showed an increase of\nthe anisotropy field after Co substitution, which supports\nour prediction. All the previous experimental studies\nconducted on the (Fe 1−xCox)5SiB2system are limited\nto the Fe-rich compositions, while Co 5SiB2is not known\nto form.23For melt-spun samples Fe 5(Si0.75Ge0.25)B2\nFIG. 1. The crystal structure of Fe 5PB2, space group\nI4/mcm (no. 140).\nLejeune et al. were determined a relatively high\nanisotropy constant K1of about 0.5 MJ m−3at room\ntemperature, which is about double the value for\nFe5SiB2.24,30Recently, we also presented a combined\nexperimental and theoretical study of the Fe 5Si1−xPxB2\nsystem, which showed the highest anisotropy constant\nfor the terminal Fe 5PB2composition.27\nFe5PB2has high TCof about 655±2 K, magnetic mo-\nment of 1.72 µB/Fe atom (8.60 µB/f.u.), and anisotropy\nconstant K1of 0.50 MJ m−3measured at 2 K for single\ncrystal.26The value of an effective anisotropy constant\nKeffof Fe 5PB2obtained in our previous work is however\nsignificantly higher and equal to ∼0.9 MJ m−3at 10 K.27\nAn important parameter, in context of permanent mag-\nnets, is magnetic hardness, defined as:\nκ=/radicalBigg\n|K|\nµ0M2\nS, (1)\nwhere Kis the magnetic anisotropy constant and MSis\nthe saturation magnetization. An empirical rule κ >1\nspecifies whether the material have a chance to resist\nself-demagnetization.4From the experimental values of\nKeff∼0.65 MJ m−3andMS= 0.87 MA/m27, we deter-\nmined for Fe 5PB2κ= 0.69 (at 300 K). It implies, that\nwithout a further engineering of the anisotropy constant,\nFe5PB2will stay in a category of semi-hard magnets.4\nIn this work we consider alloying of Fe 5PB2with Co\nand 5 delements. In our recent study of (Fe 1−xCox)5PB2\nalloys we observed a reduction in magnetization and\nCurie temperature with an increase of Co concentra-\ntion.28McGuire and Parker also found that 20% Co al-\nloying in (Fe 1−xCox)5PB2leads to decrease in magneti-\nzation, Curie temperature and anisotropy field.23Previ-\nously we showed also that increase of the MAE of 3 dal-\nloys can be achieved through doping with 5 delements.19\nIn this work we follow this idea and calculate the resul-\ntant MAEs of Fe 5PB2-based alloys with 5% substitutions3\nof each 5 delement in place of Fe.\nII. COMPUTATIONAL AND EXPERIMENTAL\nDETAILS\nA. Computational Details\nThe electronic band structure calculations for\n(Fe1−xCox)5PB2and (Fe 0.95X0.05)5PB2(X = 5 del-\nement) systems were carried out with use of the\nfull-potential local-orbital electronic structure code\nFPLO14.0-4931using a fixed atomic-like basis set. The\nFPLO was an optimal choice for the accurate calcula-\ntions of MAE due to the full potential and fully rela-\ntivistic character of the code. To model the Co and\n5dalloying we used the supercell method. The gener-\nalized gradient approximation (GGA) was used in the\nPerdew-Burke-Ernzerhof form (PBE).32A 16 ×16×16\nk-mesh was found to lead to well converged results of the\nMAE. For k-point integration, the tetrahedron method\nwas used.33The energy and charge density convergence\ncriteria of ∼10−7eV and 10−6, respectively, were applied\nsimultaneously. The lattice parameters and Wyckoff po-\nsitions were optimized for Fe 5PB2and Co 5PB2within\na spin-polarized scalar-relativistic approach. The crys-\ntallographic parameters for compositions with interme-\ndiate Co concentrations were taken from calculations of\nfull lattice relaxation carried out previously in virtual\ncrystal approximation.28For the (Fe 0.95X0.05)5PB2su-\npercells we used the same crystallographic parameters as\nfor the Fe 5PB2. The MAE was evaluated as a difference\nbetween the fully relativistic total energies calculated f or\nquantization axes [100] and [001]. In the adopted sign\nconvention the positive sign of MAE corresponds to an\neasy magnetization axis along the [001] direction. The\nFermi surface (FS) of Fe 5PB2was calculated on a 283\nk-mesh in a boundary of the first Brillouin zone. Using\nthe fully relativistic fixed spin moment (FSM) scheme34\nwe study the MAE as a function of total magnetic mo-\nment ( m) for Fe 5PB2. A supercell method was used to\nmodel the chemical disorder.18,35To build a supercell,\nmultiplication of the basal unit cell and replacement of\nan appropriate amount of atoms of one type by atoms of\nthe other type were made. The Fe atoms were replaced\nby Co or 5 datoms forming the (Fe 1−xCox)5PB2and\n(Fe0.95X0.05)5PB2compositions (X = 5 delement). For\n(Fe1−xCox)5PB2the considered intermediate composi-\ntions were: x= 0.2, 0.4, 0.6, and 0.8. The MAE calcula-\ntions based on the supercell method18are uncommon, as\nthey are time-consuming even for relatively simple alloys.\nThe reason for that is significant increase in the number\nof inequivalent atomic positions generated for the super-\ncell model. Additionally, accurate results require aver-\naging over several different large supercells.18,36It limits\nthe size of supercells which we can use for MAE calcu-\nlations. Hence, we study only the supercells including\nsymmetry operations and consisting of up to 16 inequiv-\nFIG. 2. The crystal structures of the (Fe 1−xCox)5PB2super-\ncells. (a-c) Three configurations of Fe 4Co1PB2and (d) single\nconfiguration for Fe 3Co2PB2.\nalent atoms. The considered crystal structures are pre-\nsented in Fig. 2. Three configurations were considered\nfor Fe 4Co1PB2and Fe 1Co4PB2and one for Fe 3Co2PB2\nand Fe 2Co3PB2compositions. For the considered super-\ncell models, the energy convergence with a number of\nk-points was carefully tested. The supercell method was\nalso employed to calculate the MAE of (Fe 0.95X0.05)5PB2\ncompositions with various 5 delements X. To construct\nthe models, one of twenty Fe atoms in the basal Fe 5PB2\nsupercell was replaced by the dopant. It led to the crystal\nstructures containing 10 inequivalent atomic positions.\nFor calculations of the systems with 5 ddopants a rela-\ntively dense 20 ×20×10k-mesh was used, in order to get\nthe well converged results of MAE.\nTo compute the Curie temperatures within the\nmean-field theory ( TMFT\nC) for the whole series of\n(Fe1−xCox)5PB2compositions the FPLO5.00 version of\nthe code was used.37TheTMFT\nCis proportional to the to-\ntal energy difference between the ferromagnetic and para-\nmagnetic configurations38–40according to:\nkBTMFT\nC =2\n3EDLM −EFM\nc, (2)\nwhere EDLM andEFMare total energies for the paramag-\nnetic and ferromagnetic configurations, kBis Boltzmann\nconstant, and cis total concentration of magnetic atoms.\nIn case of Fe 5PB2containing five Fe atoms (considered\nas magnetic ones) within a formula unit consisting of\neight atoms, the concentration parameter cis equal 5/8.\nTo model the paramagnetic state the disordered local\nmoment (DLM) method was used,41in which the ther-\nmal disorder among the magnetic moments is modeled4\nby using the coherent potential approximation (CPA).42\nThe FPLO5 is the latest public version of the code al-\nlowing for the CPA calculations and does not have im-\nplemented the GGA. Thus, the local density approxi-\nmation (PW92)43form of the exchange-correlation po-\ntential had to be chosen. For the calculations within\nFPLO5, a scalar-relativistic mode and a 12 ×12×12k-\nmesh were used. In the FPLO5 the magnetically ordered\nstate (resulting in EFM) was artificially modeled within\nthe CPA, to avoid numerical discrepancies between the\nordered (in principle non-CPA) and DLM (CPA) mod-\nels. In calculations using the FPLO5 code, the minimum\nbasis have been optimized for the terminal compositions\nFe5PB2and Co 5PB2, subsequently the resultant com-\npression parameters were used for intermediate composi-\ntions modeled with CPA. The VESTA code was used for\nvisualization of crystal structure.44\nB. Experimental Details\nThe samples in the series (Fe 1−xCox)5PB2(xfrom\n0.0 to 0.7) were synthesized by mixing stoichiometric\namounts of the master alloys Fe 5PB2and Co 5PB2. The\nmaster alloys were prepared, in accordance with previ-\nous studies27, from pure elements of iron (Leico Indus-\ntries, purity 99.995%, surface oxides reduced in H 2-gas),\ncobalt (Johnson Matthey, purity 99.999%), phosphorus\n(Cerac, purity 99.999%), and boron (Wacher-Chemie,\npurity 99.995%). This was done by forming first the\nTM 2B (TM = Fe, Co), using a conventional arc fur-\nnace, and subsequently dropping the phosphorus in a\nmelt of the metal boride in an induction furnace using the\ndrop synthesis method.45All samples were subsequently\ncrushed, pressed into pellets, and heat treated in evac-\nuated silica ampules at 1273 K for 14 days after which\nthey were quenched in cold water. At xhigher than 0.7\nthe correct crystalline phase could not be produced, all\nattempts resulted in a decomposition to other crystalline\nphases.\nTo study the phase content and to perform crystal\nstructure analysis of all samples, a powder X-ray diffrac-\ntion (XRD) was used. The measurements were done us-\ning a Bruker D8 diffractometer equipped with a LynxEye\nposition sensitive detector (4° opening) using CuK α1ra-\ndiation ( λ= 1.540598 ˚A) at 298 K in a 2 θrange of 20°–\n90°. The crystal structures were evaluated with the soft-\nware FullProf46using refinements according to the Ri-\netveld method.47The unit cell parameters were precisely\nstudied using the least square refinements of the peak\npositions, employing the software UnitCell.48\nThe synthesized samples were magnetically studied us-\ning a Quantum Design PPMS 6000. Samples were im-\nmobilized in gelatin capsules with varnish. The magne-\ntization at 3 K was measured between applied magnetic\nfields of 0 and 7.2 MA m−1. The magnetization in SI\nunits was calculated from magnetic moment using the\nsample weight and the crystallographic volume obtainedfrom the XRD measurements at 298 K. When approach-\ning magnetic saturation the magnetization process is de-\nscribed by the law of approach to saturation (LAS).49\nLAS has been formulated in several ways49–52, but it\ntakes a general form\nM\nMS=/summationdisplay\njajHj, (3)\nwhere jis usually an integer, ajare coefficients, Mand\nMSare magnetization and saturation magnetization, and\nHis the applied magnetic field. The LAS was used to\ndetermine an effective anisotropy constant |Keff|in the\nsame implementation as we used before.25,27The inter-\nval 93%–98% of the magnetic saturation was used. The\napplied formula was\nM\nMS= 1 + aH+b\nH+c\nH2. (4)\nThe experimental data was fit with four models in which\naandbcoefficients can be zero or non-zero and since1\nH2\nterm is used to extract |Keff|this part is always consid-\nered as non-zero. |Keff|is given here by\n|Keff|=/radicalbigg\n15c\n4µ0MS. (5)\nThe difference in results between all four models are rel-\natively small (max. 0.20 MJ m−3), thus in the experi-\nmental section we present only the |Keff|for the simplest\nmodel with the coefficients a=b= 0.\nIII. RESULTS AND DISCUSSION\nThe results of first-principles calculations of technolog-\nically important magnetic parameters for the considered\nsystems are shown. For (Fe 1−xCox)5PB2theMS,TC,\nand MAE are presented. For (Fe 0.95X0.05)5PB2(X =\n5delement) the results are limited to MAE and partial\nmagnetic moments. For the main phase – Fe 5PB2– a\ndetailed analysis of electronic structure, magnetic mo-\nments, Fermi surface, and MAE is given. The theoretical\nefforts are complemented by experimental synthesis and\nmeasurements of the considered (Fe 1−xCox)5PB2com-\npositions.\nA. Crystal Structure and Electronic Structure of\nFe5PB 2and Co 5PB 2\nThe optimized crystallographic parameters of Fe 5PB2\nand Co 5PB2are compared in Table Iwith the results\nof measurements. For Fe 5PB2the agreement between\nthe GGA and experiment is good and for the Co 5PB2\nthe GGA underestimates aand overestimates c. The\ndisagreement may originate from both theory and exper-\niment. The lattice parameters of Co 5PB2were last re-\nfined by Rundqvist back in 1962.29Unfortunately, we did5\nTABLE I. The optimized crystallographic parameters for\nFe5PB2and Co 5PB2as calculated with the FPLO14 code, us-\ning the GGA(PBE) functional, with (SP) and without (NM)\nspin polarization. Space group I4/mcm , no. 140. The Wyck-\noff positions are: Fe 1/Co 1(x,x+1/2, z), Fe 2/Co 2(0, 0, 0), P\n(0, 0, 1/4), and B ( x,x+1/2, 0). For comparison the values\nmeasured in this work at room temperature for Fe 5PB2and\nthe literature values for Co 5PB2are also reported.\nsystem a[˚A]c[˚A]xFe1/Co 1zFe1/Co 1xB c/a\nFe5PB2(GGA-SP) 5.456 10.296 0.170 0.139 0.381 1.887\nFe5PB2(expt.) 5.492 10.365 0.170 0.141 0.381 1.887\nCo5PB2(GGA-SP) 5.284 10.541 0.169 0.142 0.376 1.995\nCo5PB2(GGA-NM) 5.309 10.406 0.169 0.141 0.376 1.960\nCo5PB2(expt.)295.42 10.20 - - - 1.882\nnot manage to synthesize the Co 5PB2sample. According\nto comprehensive study of Haas et al. the PBE remains\nthe best GGA functional for most of the solids containing\n3dtransition elements.53However, it has a tendency to\noverestimate the lattice constants.53The presented PBE\nresults for Fe 5PB2go against this trend. PBE underesti-\nmates also a volume of Co 5PB2. The observed underes-\ntimation of lattice parameters/volumes is similar to the\nresults obtained from GGA for bcc Fe54and fcc Co55, for\nwhich the use of GGA leads to about 0.5 - 1.0% under-\nestimation of the lattice parameters (which is equivalent\nto about 1.5 - 3.0% underestimation in volume). In case\nof Fe 5PB2and Co 5PB2the calculated (with spin polar-\nization) PBE volumes are 2.2 and 1.8% underestimated,\nrespectively. It is surprising, however, that when the c/a\nratio for Fe 5PB2is in agreement with experiment (both\nvalues are equal to 1.887), the corresponding result for\nCo5PB2from (spin polarized) GGA is significantly dif-\nferent (1.995 against the experimental value 1.882). The\nnon-magnetic GGA calculations leads for Co 5PB2toc/a\nequal to 1.960 – also significantly different from the mea-\nsured value. We can only give a very general explanation\nfor this discrepancy as coming from the insufficient treat-\nment of corrections in PBE functional.\nThe spin projected partial and total densities of states\n(DOS) for Fe 5PB2and Co 5PB2are presented in Fig. 3.\nThe valence bands of these two metallic systems start\naround -9 eV. In a range from -9 to -3 eV the main con-\ntributions to a valence band come from the P 3 pand\nB 2porbitals, while from -5 eV up to above EFthe\ndominant role play the 3 dorbitals. The observed spin\nsplitting (proportional to the magnetic moment) is big-\nger for Fe 5PB2than for Co 5PB2, which is related to a\nhigher filling of the valence band for Co 5PB2than for\nFe5PB2. The majority spin channels of the two com-\npounds are similar and nearly completely occupied. The\nadditional electrons in the Co 5PB2fill mainly the minor-\nity spin channel, reducing the magnetic moment. The\nweak spin polarization of the P 3 pand B 2 porbitals is\ninduced by the 3 dorbitals. The spin polarization on the\nFermi level is defined as P=|Du−Dd\nDu+Dd|, where Duis the\ndensity of states at the Fermi level of the majority spin0510\n0510\n-10-50\n-10-50\n012\n012\n-2-10\n-2-10\n012\n012\n-2-10\n-2-10\n0.00.5\n0.00.5\n-0.50.0\n-0.50\n0.00.5\n0.00.5\n-8 -6 -4 -2 0 2-0.50.0\n-8 -6 -4 -2 0 2-0.50\nE - EF (eV)Fe5PB2\nFe1 3d \nFe2 3d DOS ( states  eV-1 spin-1  (f.u.-1 or atom-1) )\nB 2pP 3p\nE - EF (eV)P 3pCo5PB2\nB 2pCo1 3d \nCo2 3d \nDOS ( states  eV-1 spin-1  (f.u.-1 or atom-1) )\nFIG. 3. The spin projected partial and total densities of\nstates (DOS) for Fe 5PB2and Co 5PB2. Calculations were\ndone within the FPLO14 code using the PBE functional and\ntreating the relativistic effects in a full 4-component form al-\nism (including spin-orbit coupling).\nchannel, and Ddfor the minority spin channel. The cal-\nculated spin polarization on the Fermi level (a total value\nincluding Fe, Co, P, and B contributions) is about 0.46\nfor Fe 5PB2and 0.60 for Co 5PB2.\nB. Magnetic Moments of (Fe 1−xCox)5PB 2\n0.0 0.2 0.4 0.6 0.8 1.00246810\nTheory (GGA + supercells)\nExpt. (3 K)\nCo concentration xTotal magnetic moment ( µB f.u.-1 )\nFIG. 4. The Co concentration dependence of total magnetic\nmoment for the (Fe 1−xCox)5PB2system. The results cal-\nculated with supercell method are denoted by red circles,\nthe results measured at 3 K by blue squares.28Linear fits\nare drawn for a better perception. Calculations were done\nwith the FPLO14 code, using the GGA functional (PBE),\nand treating the relativistic effects in a full 4-component f or-\nmalism (including spin-orbit coupling).6\nTABLE II. The spin, orbital, and total magnetic moments\n(µB(atom or f.u.)−1) for Fe 5PB2and Co 5PB2as calculated\nalong the quantization axis [001] (easy axis) with the FPLO1 4\ncode using the PBE functional and treating the relativistic\neffects in a full 4-component formalism (including spin-orb it\ncoupling). The saturation magnetization MS(MA m−1) is\nevaluated based on the total magnetic moments mand theo-\nretical lattice parameters.\nFe5PB2 Co5PB2\nsite msmlmsml\n3d11.78 0.033 0.41 0.011\n3d22.11 0.052 0.64 0.013\nP-0.13 0.002 -0.02 0.001\nB-0.21 0.001 -0.05 0.000\nm 8.85 2.20\nMS 1.07 0.28\nThe calculated Co concentration dependence of the to-\ntal magnetic moment (a sum of spin and orbital con-\ntributions) for the (Fe 1−xCox)5PB2system is presented\nin Fig. 4together with the experimental results at low\ntemperature (3 K).28Whereas the results presented here\nare based on the supercell approach28, in our previous\nwork one can find the corresponding m(x) plots based\non the virtual crystal approximation (VCA) and coher-\nent potential approximation (CPA). The calculated and\nexperimental m(x) curves presented in Fig. 4stay in good\nqualitative agreement, showing a linear decrease of mag-\nnetic moment with Co concentration. Nevertheless, they\ndiffer by about 0.5 – 1.0 µB/f.u., where the lower values\ncome from measurements. The reasons for this discrep-\nancy should be sought on both experimental and theoret-\nical sides. Looking at the experiment, it is worth noting\nthat the samples produced in this work are slightly non-\nstoichiometric and with a small amount of impurities.28\nOur measurements at 3 K for a powder sample of Fe 5PB2\nshowed a total magnetic moment equal to 8.29 µB/f.u.\nin comparison to 8.6 µB/f.u. obtained by Lamichane et\nal.for a Fe 5PB2single crystal at 2 K.26It leads us to\nthe conclusion that the magnetic moments we have mea-\nsured may be slightly underestimated. The calculated\ntotal magnetic moment of Fe 5PB2(8.85 µB/f.u.) using\nGGA is closer to the result obtained for the single crystal\nthan for the powder sample. The discrepancy between\nthe result of GGA calculations and single crystal mea-\nsurements can then be attributed to the insufficiency of\nthe GGA in description of correlations, however the cal-\nculations still provide an acceptable level of agreement\nwith experiment.\nThe calculated spin, orbital, and total magnetic mo-\nments ( ms,ml,m) for Fe 5PB2and Co 5PB2are collected\nin Table II. For Fe 5PB2the calculated magnetic moments\non Fe 1and Fe 2sites are equal to 1.81 (1.62) and 2.16\n(2.16) µB, respectively, where in parentheses are given\nestimations from the magnetic hyperfine fields.56The\ninduced spin magnetic moments on P and B are rela-\ntively small and oriented antiparallel to the dominant3dmoments on Fe/Co. The total magnetic moments of\nFe5PB2and Co 5PB2are almost entirely of spin charac-\nter, where the 3 dorbital magnetic moments ( ml’s) are\nnearly quenched. The ml’s of Fe 1and Fe 2of the Fe 5PB2\n(calculated for the [001] quantization axis) are equal to\n0.033 µBand 0.052 µB, respectively. These values sur-\nround the mlvalue calculated for bcc Fe (0.043 µB) and\nare reduced in comparison to the experimental value for\nthe bcc Fe (0.086 µB).57The underestimation of the or-\nbital magnetic moment in transition metals is recognized\nas a general weakness of the LDA and GGA. Finally, al-\nmost no orbital contributions are observed for P and B\natoms ( ml∼10−3µB). The calculated mof Co 5PB2\nis equal to 2.20 µB/f.u. (0.44 µB/Co atom). For com-\nparison, the experimental magnetic moment of hcp Co is\nequal to 1.67 µB/atom.58The calculated ml’s of Co 1and\nCo2of the Co 5PB2are equal to 0.011 µBand 0.013 µB,\nrespectively, and are one order of magnitude smaller than\nthemlmeasured for hcp Co (0.13 µB).59Although the\ntheoretical values of magnetic moments for Co 5PB2have\nbeen presented above, the magnetic ground state of this\nsystem has not been unambiguously resolved, which will\nbe discussed in the next section.\nC. Curie Temperature of (Fe 1−xCox)5PB 2\n0.0 0.2 0.4 0.6 0.8 1.00200400600800\nTCExpt.\nTCMFT(LDA)\nb2TCMFT(LDA)\nCo concentration xTC (K)\nFIG. 5. The Curie temperatures as functions of Co concentra-\ntionxin (Fe 1−xCox)5PB2. The theoretical TMFT\nC (LDA) are\ncalculated in the mean-field approximation with the FPLO5\ncode, using LDA functional, treating the chemical disorder\nwith CPA, and modeling the paramagnetic state with DLM.\nThe experimental TCwas defined from the inflection point of\nfield cooled magnetization versus temperature measurements\nin a field of µ0H = 0.01 T.28\nThe Curie temperatures ( TMFT\nC(LDA)) calculated for\nthe whole concentration range of the (Fe 1−xCox)5PB2\nsystem within the mean-field theory and with the LDA\nfunctional are presented in Fig. 5. The observed over-\nall decrease of the calculated TMFT\nCwith increase of Co\nconcentration is consistent with experimental observa-\ntions.23,28However, in the whole range in which it is pos-7\nsible to compare the MFT-LDA results with the experi-\nment (0 .0≤x≤0.7), theoretical values are smaller. For\nexample, the calculated TMFT\nC(LDA) of Fe 5PB2is equal\n547 K, whereas the corresponding experimental value is\n622 K for the powder sample28, or 655±2 K for the single\ncrystal.26This difference is due to the limitations of the\nMFT approach and insufficiency of the LDA in descrip-\ntion of correlations. By calculating Heisenberg exchange\ninteractions, one could extract accurate critical temper-\natures using the random phase approximation (RPA) or\nMonte Carlo simulations.60,61The insufficiency of the\nLDA manifests in underestimated values of the calcu-\nlated magnetic moments of Fe 5PB2; 7.30 µB/f.u. versus\n8.6µB/f.u. from experiment for a single crystal.26As it\nhas been shown in the previous subsection, a much better\ndescription of magnetic moments of Fe 5PB2in relation\nto the experimental result can be obtained by using the\nGGA functional instead LDA. Thus, we suggest that the\nnegative effect on TMFT\nCcoming from the limitations of\nthe LDA can be partially corrected by using the correc-\ntion parameter based on the magnetic moments obtained\nfrom GGA. In Heisenberg model, TMFT\nCis proportional\nto squared effective moment ( m2\neff). Defining b=mGGA\nmLDA\nthecorrected Curie temperature is b2TMFT\nC(LDA), where\nin case of (Fe 1−xCox)5PB2bis about 1.2. Figure 5shows\nthat for the region of intermediate Co concentrations the\nb2TMFT\nC(LDA) curve is in a better agreement with ex-\nperiment than the uncorrected MFT-LDA results.\nUnfortunately, we were unable to get experimental re-\nsults of TCfor Co concentrations x >0.7. Because of\nthat, we can not unambiguously resolve the issue of the\nmagnetic ground state of terminal composition Co 5PB2.\nLinear extrapolation of experimental magnetic moments\nfor (Fe 1−xCox)5PB2system suggests non-zero moment\nfor Co 5PB2, see Fig. 4. On the contrary, linear ex-\ntrapolation of the measured Curie temperature suggests\na transition from ordered to disordered magnetic state\nat about x= 0.9, and therefore a non-magnetic ground\nstate of Co 5PB2, see Fig. 5. Furthermore, experimen-\ntal results reported by McGuire and Parker suggested\nabsence of magnetic ordering for Co 5PB2.23From the-\noretical point of view, both uncorrected and corrected\napproaches show the non-zero values of TCfor Co-rich\nregion ( TMFT\nC(LDA) = 37 K for Co 5PB2). Taking into\naccount (1) the problems with synthesis of the Co 5PB2\nphase, (2) preliminary character of the measurements re-\nported by McGuire and Parker, (3) issues mentioned in\nprevious subsection regarding optimization of the struc-\ntural model of Co 5PB2, and (4) limitations of LDA/GGA\nin description of correlations of Co-rich phases, we con-\nclude that based on existing data the magnetic ground\nstate of Co 5PB2can not be definitively determined.\nFIG. 6. (a)–(i) The nine sheets of the Fermi surface of\nFe5PB2. (i) The k-path used to calculate the band structure\nplot. Inside the visible tubes the sheets (e) and (f) contain the\ninvisible pockets centered at Γ. Calculations were done wit h\nthe FPLO14 code using the PBE functional and treating the\nrelativistic effects in a full 4-component formalism (inclu ding\nspin-orbit coupling).\nD. Fermi Surface of Fe 5PB 2\nFigure 6presents the calculated Fermi surface (FS) of\nFe5PB2in a boundary of the first Brillouin zone. The\nFS of Fe 5PB2reflects the tetragonal symmetry of the\ncrystal. The FS consists of nine sheets and is relatively\ncomplex. The states at the Fermi level ( EF) have a Fe 3 d\ncharacter, as can be read from the DOS plots in Fig. 3.\nThe observed FS sheets can be divided into two groups.\nThe first group consists of four nested sheets of hole-\ntype, see panels (a)–(d) of Fig. 6, and the second group\nincludes the remaining five sheets of electron-type nested\nin a multiwalled way around the high symmetry point Z,\nsee panels (e)-(i) of Fig. 6. While the sheets (c)–(f) form\nrather tubular shapes, allowing for open orbits along the\nsymmetry axis, the remaining sheets, (a)–(b) and (g)–\n(i), take the form of pockets enabling only for closed FS\norbits.62Because the band structure was calculated with\nspin-orbit coupling, the FS sheets cannot be unambigu-\nously attributed to a particular spin channel.\nE. Magnetocrystalline Anisotropy of Fe 5PB 2\nThe results of investigating the MAE of Fe 5PB2carried\nout in this work are: the band structure in vicinity of the\nFermi level, one- and two-dimensional k-resolved MAE\nplots, and the cross-section of FS. Our inquiry is com-\nplemented by considerations of MAE engineering, as for\nexample reduction of total magnetic moment. The calcu-8\nlated MAE of the Fe 5PB2is 0.52 MJ m−3. It indicates a\nuniaxial magnetocrystalline anisotropy with an easy axis\nalong the tetragonal axis. This result stays in a good\nagreement with the experimental value of anisotropy con-\nstant measured at 2 K (0.50 MJ m−3) and with the pre-\nvious theoretical findings (0.46 MJ m−3).26Previously\nreported results for Fe 5PB2show that K1first increases\nwith temperature starting from 2 K up to about 100 K\nand then decreases to zero at TC.26The well known ori-\ngin of the magnetocrystalline anisotropy is the spin-orbit\ncoupling, which is taken into account in the fully relativis -\ntic full potential calculations. In comparison with scalar\nrelativistic approach, the fully relativistic one results in\nadditional splitting of the electronic bands. Since the\nspin-orbit coupling constant of 3 d-metals is of the order\nof 0.05 eV, the spin-orbit splitting also does not exceed\nthis value. The spin-orbit splitting leads to slightly diffe r-\nent band structures for different quantization axes (e.g.\nfor the orthogonal [001] and [100] axes). Figure 7presents\nthe band structures calculated for Fe 5PB2in the prox-\nimity of EF, together with the MAE contributions per\nk-point obtained with the magnetic force theorem63–65\nfrom the formula:\nMAE = E(θ= 90◦)−E(θ= 0◦) =\n=/summationdisplay\nocc’ǫi(θ= 90◦)−/summationdisplay\nocc”ǫi(θ= 0◦),(6)\nwhere θis an angle between the magnetization direction\nand the caxis, E(θ) is a total energy for a specific direc-\ntion; and ǫiis the band energy of the ith state. The spin-\norbit splitting is most easily observed for the energy win-\ndow of a tenth eV around EF. The k-point resolved MAE\ntakes positive and negative values, depending on the spin\nand orbital character of the bands near the Fermi energy.\nGenerally, negative MAE-contributions coincide with oc-\ncupied bands for a [100] spin quantization axis (solid red\nline) being pushed below corresponding bands for a [001]\nspin quantization axis (dashed blue line), and vice versa\nfor positive contributions. For example, at the Z-point,\nthere is a negative MAE contribution and at approxi-\nmately -0.3 eV one can observe a solid red line below the\ndashed blue line. A more detailed analysis of the MAE\ncontributions is in principle straight forward but some-\nwhat complicated due to the complex band structure.\nNevertheless, one can clearly observe the characteristic\njumps where the bands cross EF, confirming the usual\nbehavior that the MAE is determined by the electronic\nstructure around the Fermi energy. Thus, controlling the\nMAE around EFalso allows for control of the MAE, as\nis practically possible, for example, via alloying.\nThe same form of presentation of the k-resolved MAE,\nas we have shown in Fig. 7, dominates in literature. How-\never, it is possible to plot the MAE( k) data within a three\ndimensional Brillouin zone, similar like the FS. Recently,\nthe 3D MAE( k) maps were presented for (Fe 1−xCox)2B\nand FeNi.17,66In Fig. 8(a) we show a cross-section of the\nMAE( k) (single plane going trough the Γ-point). Theselected profile is perpendicular to the easy axis [001],\ncrosses the high symmetry point Γ, and is limited by the\nBrillouin zone boundaries. The MAE( k) cross-section is\na relatively complicated map of symmetric regions con-\nsisting of positive and negative contributions. The MAE\ncontributions observed in Fig. 8along the orthogonal\naxes [100] and [010] are not equal, because the [100] di-\nrection is distinguished as quantization axis resulting in\nbreaking of the four-fold symmetry. As the EFis an up-\nper integration boundary of total MAE, the FS sheets co-\nincide with sharp changes in the k-resolved MAE contri-\nbutions. It can be seen in Fig. 8(b), where the MAE( k)\n2D plot is overlapped by the corresponding section of the\nFS. As many of k-resolved MAE contributions is in or-\nder of 10−3eV per k-point, the total MAE value of about\n10−4eV/f.u. (83 µeV/f.u. or 0.52 MJ m−3) indicates a\nfine compensation of many bigger components. Unfortu-\nnately, this extra fine compensation and the complexity\nof the MAE( k) makes the ways to increase the MAE of\nthe material difficult to predict.\nF. Fully Relativistic Fixed Spin Moment\nCalculations for Fe 5PB 2\nThe MAE value for Fe 5PB2(0.52 MJ m−3) is calcu-\nlated with the equilibrium value of the magnetic mo-\nment (8.85 µB/f.u.). In the fixed spin moment (FSM)\nmethod34the value of spin magnetic moment is consid-\nered as a parameter. The fully relativistic implementa-\ntion of FSM method allows to calculate the MAE as a\nfunction of spin magnetic moment. Previously, we pre-\nsented the MAE results as a function of FSM and Co\nconcentration for the (Fe 1−xCox)2B alloys.19Figure 9\npresents the evolution of the MAE with the total mag-\nnetic moment mfor the Fe 5PB2, together with the pre-\nvious results for Fe 5SiB2.24The two MAE( m) plots are\nsimilar in shape. Going down from an equilibrium m\nthe corresponding MAE first increases, then it reaches\nmaximum, to decrease finally to zero at mequals zero.\nFor Fe 5PB2the maximum MAE( m) is 1.94 MJ m−3for\na fixed total magnetic moment of 6.7 µB/f.u., which\nmeans that the optimal magnetic moment has to be re-\nduced by about 25% with respect to the equilibrium value\n(8.85 µB/f.u.). Thus, the question arises, how to stabi-\nlize this reduction. A simple solution would be alloying\nthe magnetic Fe by a non-magnetic element, which of-\nten results in a linear decrease of magnetization. How-\never, alloying with a new element can severely affect the\nband structure, which would change also the expected\nvalue of the MAE. The smallest impact on the electronic\nstructure should have substitutions chemically most sim-\nilar to Fe and for this purpose we suggest Ru and Os of\nthe Fe group. Another strategy could be alloying of Fe\n(ZFe= 26) with two elements at the same time, e.g. Cr\n(ZCr= 24) and Ni ( ZNi= 28), keeping a constant num-\nber of the valence electrons, which should affect the band\nstructure the least. The above considerations, however,9\n-0.4-0.20.00.20.4\nΓ X P Γ3 Z Γ Γ1Γ2E - EF (eV)\n-0.4-0.20.00.20.4\n0.0\n-0.02-0.010.000.010.02K-point resolved MAE (eV)\nFIG. 7. The band structure of Fe 5PB2calculated for quantization axes [100] (solid red lines) an d [001] (dashed blue lines),\ntogether with the MAE contribution of each k-point (thick green line) as obtained by the magnetic force t heorem. The high\nsymmetry points are presented within Brillouin zone in Fig. 6(i). Calculations were done with the FPLO14 code using the\nPBE functional and treating the relativistic effects in a ful l 4-component formalism (including spin-orbit coupling).\nFIG. 8. (a) The cross-section of the k-resolved MAE with (b)\nthe overlapped cross-section of the Fermi surface (black li nes)\nfor the Fe 5PB2. The results of MAE( k) are obtained by the\nmagnetic force theorem within the FPLO14 code using the\nPBE functional and treating the relativistic effects in a ful l\n4-component formalism (including spin-orbit coupling).\ntake into account only the band structure and neglect\nfurther issues like the crystal structure and size of the\natoms, for example.\nG. Magnetocrystalline Anisotropy of\n(Fe 1−xCox)5PB 2\nThe effect of Fe/Co alloying on the MAE is not\nobvious in advance, whereby the first-principles cal-\nculations are of great value in predicting the results,\nas has been shown previously for the (Fe 1−xCox)2B17\nand (Fe 1−xCox)5SiB224alloys. Figure 10presents the\nMAE( x) dependence for the (Fe 1−xCox)5PB2system as0 2 4 6 8 10-0.50.00.51.01.52.0\nFe5PB2\nFe5SiB2\n9.208.85MAE ( MJ m-3 )\nTotal magnetic moment ( µB f.u.-1 )\nFIG. 9. MAE as function of total magnetic moment\n(mS+mL) for Fe 5PB2and Fe 5SiB224as calculated with fixed\nspin moment (FSM) method with the FPLO14 code using the\nPBE functional and treating the relativistic effects in a ful l\n4-component formalism (including spin-orbit coupling). T he\nequilibrium values of magnetic moments are denoted with dot -\nted lines.\ncalculated with use of the supercell method. The MAE\ncalculations based on the supercell method proved to\nbe one of the most accurate method for evaluation the\nMAE.18However, our calculations were limited by com-\nputational challenges of the supercell method. Thus, in\npractice we were able to consider only a relatively small\nnumber of configurations, see Sec. II A. The scattering\nof individual data points for x= 0.2 and x= 0.8 is in a\nsimilar range as observed by D¨ ane et al.18or Steiner et\nal.36and shows that an averaging for several configu-\nrations is needed for accurate results. In Fig. 10the\nregions of positive and negative MAE (of perpendicular\nand in-plane anisotropy) are separated at Co concentra-10\n0.0 0.2 0.4 0.6 0.8 1.0-1.0-0.50.00.51.01.5\nTheory (GGA+supercells) \nTheory (GGA+supercells) - average\nExpt. (3 K) - this work\nExpt. (2 K)  - Lamichhane et al. 2016|Keff| / MAE (MJ m-3)\nCo concentration x\nFIG. 10. The experimental effective anisotropy constant\n|Keff|of the (Fe 1−xCox)5PB2system at 3 K (the sign of\nKeffis not considered), together with the magnetocrystalline\nanisotropy energy MAE values as calculated with the FPLO14\ncode. In calculations the supercell method for modeling of\nchemical disorder and the PBE functional were used. The\nrelativistic effects were treated in a full 4-component form al-\nism (including spin-orbit coupling). (for xequal to 0.2 and\n0.8 several inequivalent supercells are considered). For c om-\nparison the value of K1measured by Lamichhane et al. for\nFe5PB2.26\ntionx≃0.5. The calculated MAE is equal 0.52 MJ m−3\nfor Fe 5PB2and -0.51 MJ m−3for Co 5PB2. Whereas, the\nanisotropy value close to zero, observed for x≃0.5, indi-\ncates a good soft magnetic material. Figure 10presents\nalso the low temperature measurements of the effective\nanisotropy constant |Keff|carried out at 3 K for sev-\neral (Fe 1−xCox)5PB2compositions within the bound-\naries of 0 .0≤x≤0.7. The value of |Keff|is the highest\n(0.94 MJ m−3) for Fe 5PB2and the the lowest for a Co\nconcentration x∼0.6.|Keff|measured for Fe 5PB2is sig-\nnificantly larger than the K1= 0.5 MJ m−3measured at\n2 K for the single crystal.26The decrease of |Keff|with\nxis in agreement with the previous measurements for\n(Fe0.8Co0.2)5PB2suggesting that 20% Co substitution\nreduces the anisotropy field.23Previously we also showed\nthe corresponding |Keff|results for the Fe 5Si1−xPxB2\nsystem.27The presented values of |Keff|for Fe 5PB2were\n∼0.9 MJ m−3at 10 K and ∼0.65 MJ m−3at 300 K.27\nNotice that LAS is unable to determine the sign of |Keff|\nand thus the negative values of MAE predicted for x/greaterorsimilar0.6\ncannot be confirmed by this method. Other methods,\nsuch as magnetometry measurements in different direc-\ntions for single crystals or torque magnetometry would\nbe preferable. Here, single crystals were not available,\nand up to 10 wt% of impurities were present in the sam-\nples. Therefore, given the limitation in the model and the\nstarting material the results presented from these should\nbe seen as semi-quantitative. Taking into account the\nlimitations of the LAS and the supercell method, the dif-\nferences between theoretical and measured MAE( x) re-\nsults are acceptable. We conclude, that Co alloying of\nFe5PB2is not a good strategy to increase the MAE of0 1 2 3 4 5 6 70.00.20.40.60.81.0Magnetization (MA m-1)\nApplied field (MA m-1)0.1 0.2 0.3 0.4 0.50.920.940.960.981.00M/MS\u0000 ✁ ✂✄ ☎ ✆ ✝\n✞ ✟ ✞ ✠ ✟ ✟ ✞ ✟\nFIG. 11. Magnetization ( M) as a function of applied field\n(H) measured for Fe 5PB2at 3 K. The inset shows a normal-\nized magnetization ( M/M S) as a function of 1/ H2.\nthis system.\nA typical magnetization ( M)versus applied field ( H)\ncurve measured at 3 K is shown in Fig. 11. The inset\nof Fig. 11presents a plot of M/M Sversus 1/H2as used\nto determine the |Keff|within the LAS method. More\ndetails on the implementation of the LAS method can be\nfound in Sec. II B.\nH. Doping Fe 5PB 2with 5 dElements\nOne of the methods of tailoring the MAE is doping\nwith 5 delements.19,67Previously, we have confirmed that\nthe 5 delements can significantly affect the MAE due to\na large spin-orbit coupling.19From the Fe 5Si1−xPxB2\nand (Fe 1−xCox)5PB2systems, the highest MAE is found\nin the Fe 5PB2phase.27Thus, it is considered as the\nparental compound for a further MAE engineering. The\nMAE of (Fe 0.95X0.05)5PB2compounds (X = 5 dele-\nments) is calculated using the supercell method. The\nresults are shown in Fig. 12, with the 5 delement marked\non the xaxis and dashed line indicating the MAE of un-\ndoped Fe 5PB2. The 5 ddoping has sometimes beneficial\nand sometimes adverse effect on MAE.35,68,69Significant\nincrease of MAE is observed for W or Re doping, simi-\nlar like in the case of (Fe 1−xCox)2B alloys investigated\nexperimentally in our previous work19. The MAE grows\nfrom 0.52 MJ m−3for Fe 5PB2to about 1.1 MJ m−3for\nthe compositions with W or Re, with 5% Fe substitution.\nPreviously we have shown, that the increase in MAE ob-\nserved for W and Re dopants is mainly due to the strong\nspin-orbit coupling of the 5 datoms, however other vari-\nations in electronic structure also affect the MAE.19Al-\nthough in our calculations the 5 delements are initially\nconsidered as non-magnetic, the dopants undergo spin11\n72737475 76 7778 79 80\n-101atomic numberMAE (MJ m-3)\nHf    Ta    W    Re   Os     Ir    Pt     Au    Hg\nFIG. 12. MAE for various 5 delements X in\n(Fe0.95X0.05)5PB2as calculated with supercell method. Cal-\nculations were done with the FPLO14 code using the PBE\nfunctional and treating the relativistic effects in a full 4-\ncomponent formalism (including spin-orbit coupling). The\ndashed line indicates the MAE of Fe 5PB2(0.52 MJ m−3) for\ncomparison.\n72737475 76 7778 79 80\n-0.2-0.10.00.10.20.30.4\nms\n10mlatomic number Magnetic moment ( µB )\nHf    Ta    W    Re   Os     Ir    Pt     Au    Hg\nFIG. 13. Spin ( ms) and orbital ( ml) magnetic moments\nof 5dtransition metal impurities X in (Fe 0.95X0.05)5PB2as\ncalculated for spin quantization axis along the c-axis. Super-\ncell calculations were done with the FPLO14 code using the\nPBE functional and treating the relativistic effects in a ful l\n4-component formalism (including spin-orbit coupling).\npolarization in a ferromagnetic medium and contribute\nto the total magnetic moment of the system. The calcu-\nlated spin and orbital magnetic moments on 5 dimpurity\nshow clear trend along the increasing atomic number of\n5delement, see Fig. 13. The spin magnetic moment of\n5dimpurities are antiparallel to the Fe moments in the\nearly 5 dseries, while they are parallel in the late 5 dse-\nries. Corresponding trends for 5 datoms in magnetic 3 d\nhosts have been found previously computationally70,71\nand experimentally.72\nIV. SUMMARY AND CONCLUSIONS\nOur considerations began with a detailed theoretical\nanalysis of the Fe 5PB2compound. The Fe 3 dorbitalsare dominant in the valence band and responsible for the\nformation of large magnetic moments. For the Fe 5PB2\nthe fully relativistic band structure in the vicinity of\nFermi level was considered to better understand the ori-\ngin of the high value of magnetocrystalline anisotropy\nenergy (MAE). The calculated Fermi surface requires ex-\nperimental confirmation. The results of fully relativistic\nfixed spin moment calculations suggested that reduction\nof the magnetic moment of Fe 5PB2should induce about\nfourfold increase of the MAE. For practical realization\nof magnetic moment reduction it is suggested to alloy Fe\nwith a non-magnetic element Ru or Os from the Fe group,\nor to partially replace Fe with two elements at once, Cr\nand Ni, for example, keeping constant number of valence\nelectrons.\nThree critical parameters for technological applica-\ntions: saturation magnetization ( MS), Curie tempera-\nture ( TC), and MAE were calculated for the whole con-\ncentration range between Fe 5PB2and Co 5PB2. The\ncalculated MSandTCdecreased with Co concentration\nand for the terminal composition Co 5PB2a weakly or-\ndered magnetic ground state was predicted. The calcu-\nlated M(x) and TC(x) were in decent agreement with\nthe measurements, although the ferromagnetic ground\nstate of Co 5PB2is questionable. The Co doping in\n(Fe1−xCox)5PB2system gives the possibility of tuning\ntheTCin a range from about six hundred kelvins to al-\nmost down to zero. The calculated MAE was positive\nfor Fe 5PB2, negative for Co 5PB2, and went through zero\naround 50% Co concentration. This picture of MAE( x)\nbehavior was in overall agreement with the experimen-\ntal study of the effective anisotropy constant |Keff|for\nthe (Fe 1−xCox)5PB2alloys. The measurements showed\nthe highest |Keff|value for stoichiometric Fe 5PB2which\ndecreased with Co doping. We concluded then that Co\nalloying is not a good strategy to increase the MAE of\nFe5PB2alloy. The measured |Keff|of about 0.94 MJ m−3\nat 3 K was, however, the highest value obtained so far\nfor Fe 5PB2, giving a hope for potential application of its\nother alloys. It was also calculated how the 5% doping of\nFe with 5 delements affects the MAE of the Fe 5PB2. 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Costa4,1,5\n1Brazilian Nanotechnology National Laboratory (LNNano), C NPEM, 13083-970 Campinas, Brazil.\n2Department of Physics and Department of Chemistry, Univers ity of North Texas, Denton TX, USA.\n3Center for Materials Genomics, Duke University, Durham, NC 27708, USA.\n4Instituto de Telecomunica¸ c˜ oes, Physics of Information a nd Quantum Technologies Group, Lisbon, Portugal.\n5Instituto de F´ ısica, Universidade Federal Fluminense, 24 210-346 Niter´ oi, RJ, Brazil.∗\nWe study the spin excitation spectra and the dynamical excha nge coupling between iron adatoms\non a Bi bilayer nanoribbon. We show that the topological char acter of the edge states is preserved\nin the presence of the magnetic adatoms. Nevertheless, they couple significantly to the edge spin\ncurrents, as witnessed by the large and long-ranged dynamic al coupling we obtain in our calcula-\ntions. The large effective magnetocrystalline anisotropy o f the magnetic adatoms combined with the\ntransport properties of the topologically protected edge s tates make this system a strong candidate\nfor implementation of spintronics devices and quantum info rmation and/or computation protocols.\nINTRODUCTION\nTopological quantum matter has been recognized as a\nfundamental concept in physics as well as a huge promise\nforfuturequantumtechnologies[1–3]. Thepeculiarprop-\nerties of topologically protected edge states (TPES) have\nbeen hailed as a “Holy Grail” for quantum information\nprocessing [4]. Heterostructures combining topological\ninsulators and other “non-trivial” matter, such as su-\nperconductors, hold promises for even more intriguing\nstates, such as Majorana fermions [5, 6].\nMagnetic systems of nanoscopic dimensions have also\nbeen considered as candidates for implementation of\nquantum technologies. Proposals range from controlling\nelectronic spins in quantum dots [7] to manipulate indi-\nvidual atomic magnetizations on surfaces [8].\nThe idea of depositing magnetic atoms on two-\ndimensional topological insulators has particularly en-\nticing aspects. From the fundamental point of view, it\nis intriguing to enquire about the strength and nature\nof the coupling between magnetic units and the topo-\nlogically protected edge states. From the technological\nside, functionalization with magnetic adatoms may be\nan efficient route to tap into the attractive properties\nof TPES [9, 10]. As an added bonus, the high spin-\norbit coupling strength necessary to produce topologi-\ncallynon-trivialstatesmayalsoendowmagneticadatoms\nwith magnetocrystalline anisotropy large enough to sta-\nbilize the magnetization direction against quantum and\nthermalfluctuations. Twogeneralquestionsimmediately\ncome to mind: How does the presence of magnetic impu-\nrities affect topological protection, considering it breaks\ntime-reversal symmetry? and How does the topological\nnature of the substrate affect the properties of the mag-\nnetic unities and its interactions? To try and answer\nthese questions is the main motivation of this work.\nThe study of spin excitations provide a prolific source\nof information on the properties of magnetic systems.\na)\nb)\nFIG. 1. Schematic pictureof thetwomagnetic impurities con -\nfiguration, at the (a) center and (b) edge of the nanoribbon.\nThe nanoribbon is wide enough to minimize the presence of\nthetopological edgestates initsmiddleregion(center). T here\nis a remarkably difference in their interaction ( W) at the two\nconsidered positions.\nIt can reveal the nature and magnitudes of individ-\nual contributions to the system’s energy (such as ex-\nchange, anisotropy, etc.) as well as relaxation and ab-\nsorption properties. Spin excitations in individual mag-\nnetic adatoms have been probed experimentally with in-\nelastic tunneling spectroscopy [11, 12], revealing the pe-\nculiarities of the magnetization dynamics at the atomic\nscale. Theoretical knowledge of the spin excitation spec-\ntra of such systems has been instrumental in the design\nand interpretation of those experiments.\nIn this letter we investigate theoretically, employing\na combination of combination of ab initio DFT calcula-\ntions, realistictight-bindingmodelsandmany-bodytech-\nniques, the ground state and magnetic excitations of Fe2\nadatoms on a Bi bilayer nanoribbon. The Bi bilayer is\na promising TI with a 500 meV gap [13–18], which is\noriginated from the large Bi SOC. We consider isolated\nadatoms and pairs of adatoms deposited on an otherwise\npristine ribbon. The ribbon’s width is large enough to\nguarantee the existence of topologically protected edge\nstates [16]. We investigate the indirect exchange cou-\npling (sometimes dubbed RKKY coupling) through the\nsubstrate and show that, for the smallest distances con-\nsidered, it is essentially zero. We focus our attention\non the dynamical coupling that appears as a result of\nthe exchange of pure spin currents between the out-of-\nequilibrium magnetizations of the Fe adatoms. We show\nthat it is very long ranged as a result of the peculiar\nproperties of the TPES.\nMETHODOLOGY\nWe use a multi-orbital tight-binding Hamiltonian\nto describe the electronic structure of the decorated\nnanoribbon. The full hamiltonian can be written as the\nsum of band energy H0, effective intra-atomic Coulomb\nrepulsion HIand the atomic spin-orbit coupling HSOC.\nThe band energy H0is given by,\nH0=/summationdisplay\nll′/summationdisplay\nµν/summationdisplay\nσTµν\nll′c†\nlµσcl′νσ. (1)\nwhereTµν\nll′is the hopping matrix, σis a spin index, l,l′\nare atomic site indices and µ,νare atomic orbital in-\ndices. The hopping matrix is obtained directly from a\nDFT calculation using the pseudo atomic orbital projec-\ntion method [19–23]. The method consists in project-\ning the Hilbert space spanned by the plane waves onto\na compact subspace composed of the pseudo atomic or-\nbitals (PAO). These PAOs functions are naturally built\ninto the pseudo potential used in the DFT calculation.\nThe atomic spin-orbit coupling (SOC) is introduced in\na effective approximation,\nHSOC=/summationdisplay\nl/summationdisplay\nµν/summationdisplay\nσσ′ξµν\nl/an}bracketle{tlµσ|/vectorL·/vectorS|lνσ′/an}bracketri}htc†\nlµσclνσ′(2)\nwhere/vectorLand/vectorSare the orbital and spin angular momen-\ntum operators, respectively. The SOC strength is given\nbyξµν\nland is also obtained from the DFT calculations.\nThe magnetism of the Fe adatoms is driven by the\neffective intra-atomic Coulomb repulsion\nHI=/summationdisplay\nl/summationdisplay\nµνµ′ν′/summationdisplay\nσσ′Uµνµ′ν′\nlc†\nlµσc†\nlνσ′clν′σ′clµ′σ.(3)\nThe spin-polarized ground state is obtained from a self-\nconsistent mean-field treatment of the effective Coulomb\nrepulsion term, Eq. (3). Both magnitude and direction\nof the magnetization at each adatom are found as solu-\ntions of the self-consistency equations. After a solutionis found, a calculation of the spin excitation spectrum\nreveals whether it is a stable or unstable solution. In the\nlatter case, we search for a new solution, usually start-\ning from a high symmetry configuration. The compari-\nson between the band structures from the DFT and the\nmulti-orbital tight-binding Hamiltonian are shown in the\nsupplemental material [24] along with the procedure to\nobtain the SOC strength ξµν\nl.\nThe spin excitation spectrum is obtained from the spin\nsusceptibility matrix χab\nµν(l,l′), where a,b∈ {+,−,↑,↓\n}[25, 26]. For instance, if the equilibrium magnetization\nlies along the zdirection, the transverse spin excitation\nspectrum is given by χ+−\nµν(l,l′). For a general magneti-\nzation vector, however, a more complicated combination\nof the matrix elements χabwill result from applying to it\nthe appropriate rotation transformations.\nThe spin susceptibility matrix elements are the propa-\ngators\nχab\nµν(l,l′|t−t′) =−iθ(t−t′)/an}bracketle{t/an}bracketle{tSa\nlµ(t),Sb\nl′ν(t′)/an}bracketri}ht/an}bracketri}ht.(4)\nDue to the interaction term Eq. 3, the equations of mo-\ntion forχabare coupled to higher order propagators, in\nan infinite chain. We employ a decoupling scheme that\nis frequently dubbed “time dependent Hartree-Fock”. It\nis equivalent to the summation of the infinite series for\nthe electron-hole pair ladder diagrams, and leads to the\ncollective spin excitation modes identified with the poles\nofχab. In translation-invariant systems those modes are\nthe spin waves; when the magnetic entities are confined\nin space the spin excitation modes are akin to normal\nmodes of coupled oscillators. In the presence of spin-\norbit coupling the transverse spin excitations are cou-\npled to longitudinal excitations and charge excitations.\nThis is reflected in the fact that the equations of motion\nfor all the elements of χabhave to be solved simultane-\nously. The solution has been presented in detail in refer-\nence [25]. The physical consequences of this dynamical\nspin-charge coupling have been little explored hitherto,\nbut we intend to divulge the results of our investigations\non this matter in future publications.\nWhen considered as a matrix in site indices, the trans-\nverse susceptibility χ+−\nll′yields two very relevant sets\nof informations: the imaginary part of its diagonal el-\nements represent the local (atomic site) projection of the\nmagnon spectral density, or the local magnon density of\nstates. From it we can extract the magnon’s energiesand\nlifetimes. The off-diagonal terms represent non-local re-\nsponse functions, and inform about the excitation of a\nmagnon at site ldue to the application of a transverse\nfield atsite l′. It isimportant tonotice that this “dynam-\nical coupling” may exist even when |/vectorRl−/vectorRl′|is so large\nthat the (RKKY) indirect exchange coupling is utterly\nnegligible [27, 28]. Its physical origin is the spin current\nshed by the spin excitation at l′, which flows through the\nsupporting medium and reaches site l, thus imparting a3\n12318 17 16 ……… 123\n4526 WLPeriodic direction 25 24 \n23 \n22 \n-0.2 0 0.2 0.4 0510 \npristine \nadatom @5 \nadatom @26 \nFermi Level (eV) -0.2 0.0 0.2 0.4 Conductance (2e 2/h) \n0510 pristine \nadatom @5 \nadatom @26 \nFIG. 2. Top panel shows the Bi zigzag nanoribbon structure\nwith 18 zigzag units and a width (W) of 65 ˚A. The length\n(L) is equal to 13.05 ˚A along the periodic direction. The\nadsorption sites are labeled, from right to left, as showed b y\nthe blue numbers. In bottom panel the pristine, adatom @26\nand adatom @5 conductance is showed.\ntorque on the magnetization there. We chose to repre-\nsenttheintensityofthe dynamicalcouplingbetweensites\nlandl′as the amplitude of the magnetization precession\natsiteldue toatransverse,circularlypolarizedmagnetic\nfield applied at site l′.\nRESULTS AND DISCUSSION\nWe begin by showing that the topologically protected\nedge states (TPES) of the Bi bilayer nanoribbon are neg-\nligibly perturbed by the presence of the adatoms. In fig-\nure 2, bottom panel, we compare the conductance of the\npristine nanoribbon with that of the ribbons decorated\nwith a single Fe adatom. It is clear that the conductance\nof the TPES is preserved over most of the bulk gap for\nall adatoms positions considered. The behavior of the\nedge spin currents as a function of energy are modified\nby the adatoms, but their overall order of magnitude is\npreserved.\nThe magnetic moments of the adatoms have been de-\ntermined selfconsistently, both in size and direction. For\na single adatom at the center of the ribbon the magnetic\nmoment is mostly perpendicular to the plane of the rib-\nbon, with a small in-plane component perpendicular to\nthe ribbon’s length. For the adatom close to the edge the\nmagnetic moment becomes almost parallel to the plane\nof the ribbon.\nThe spectral densities for transverse spin excitations\nfor single adatoms at different positions across the rib-0 20 40 60 80\nEnergy (meV)-1500\n-1000\n-500\n0Im{χ+-}\nFIG. 3. Spectral densities for transverse spin excitations of\nsingle adatoms adsorbed to the Bi bilayer nanoribbon at two\ndistinct sites: close to one edge (@5, black line) and at the\ncenter of the ribbon (@26, red line).\nbon are shown in figure 3. From those spectral densities\nwecanextractan effectivemagnetocrystallineanisotropy\nenergy (eMAE) [29] and the damping rate (or equiva-\nlently, the lifetime) of the transverse spin excitations.\nFor the adatom at the center of the ribbon the eMAE\nis 38 meV, which corresponds to a precession frequency\nof approximately 9.2 THz. The eMAE for the adatom\nclose to the edge is 19 meV ( ν0≈4.6 THz). Although\nthe nominal excitation lifetime for the adatom close to\nthe edge is larger, the quality factors for both positions\nare very similar ( Q= 12 for the center and Q= 13 close\nto the edge). Notice that the MAE we obtained is com-\npletely due to the huge spin-orbit coupling strength of\nBi (ξBi= 1.475 eV), since the tiny SOC strength of Fe\n(ξFe= 0.08eV)wasnotevenincludedinourcalculations.\nIt is interesting to notice that transverse spin excita-\ntionsofadatomsoninsulatingsurfacesusuallyhavemuch\nlonger lifetimes than those we obtained. [11] The large\ndamping is certainly due to the huge SOC strength of\nBi. However, we speculate that the spin angular mo-\nmentum associated with the spin excitation has first to\npenetrate into the Bi substrate in order to be then trans-\nferred to the orbitaldegreesof freedom and ultimately be\ndissipated by the lattice vibrations. The fact that spin\ncurrents can easily flow into a non-magnetic insulating\nsubstrate may seem somewhat counterintuitive, but in\nwhat follows we will show that this is indeed the case.\nWe now turn to the spin excitations of pairs of Fe\nadatoms. We are mainly interested in the behavior of\nthe transverse spin excitation spectra as the distance be-\ntween the adatoms is varied. As we will show, for al-\nmost all distances considered there is no Heisenberg-like\nindirect exchange coupling between the adatoms. How-\never, a large dynamical coupling appears as soon as the\nmagnetic moments are driven away from equilibrium in\na time-dependent manner. We choose as a measure of4\nthe dynamical coupling the norm of the transverse mag-\nnetization induced at adatom 2 due to a transverse field\napplied to adatom 1. This is proportional to the non-\nlocal susceptibility χ+−\n12, as detailed in reference [30]. In\nfigure 4 we show the dynamical coupling as a function\nof energy for various distances between the adatoms. In\neach case the adatoms are placed in the nanoribbon on\nthe same position relative to the edge. In the top panel\nthe adatoms are close to the edge and in the bottom\npanel they are adsorbed at the center of the nanorib-\nbon. It is clear that the dynamical coupling decreases\nabruptly as a function of distance for the adatoms at the\ncentral sites, but only slightly for the adatoms close to\nthe edge. This isan indicationthat the topologicallypro-\ntected edge states have an essential role in mediating the\ndynamical coupling between the adatoms. To highlight\nthe importance of the topologically protected edge states\nas mediators of the dynamical coupling, we reduced by\nhand the spin-orbit coupling in Bi to 0.8 eVand repeated\nthe calculations. For this value of the SOC strength the\nBi bilayer nanoribbon is a non-topological insulator and\nthere are no TPES. In figure 5 we compare the behavior\nof the dynamical coupling as a function of the distance\nbetween the adatoms in the two cases, “real” Bi, and Bi\nwith an artificially reduced SOC strength. For both the\nadatom close to the edge and at the center of the ribbon,\nthe dynamical coupling decays exponentially with the\ndistance for the reduced SOC case. This is markedly dif-\nferent from the behavior of the real system. There, only\na slight, polynomial decay is observed when the adatoms\nare close to the edge. For the adatoms at the center\nthe decay is faster, although definitely not exponential,\nand the dynamic coupling reaches a plateau for distances\nlarger than about 8 nm. We interpret these results as ev-\nidence that the topologically protected edge states play a\ncrucial role in carrying the spin currents responsible for\nthe dynamical coupling between the adatoms’ magnetic\nmoments. There is a curious interplay here, because the\nstrong SOC of Bi would be expected to scramble spin\ninformation rapidly for spin currents flowing through a\nnon-topological material made of Bi. In the language\ncommonly used in the spin transport community, Bi ma-\nterials would be considered almost ideal spin sinks. The\nexistence of topologically protected edge states, however,\nrenderBibilayeranalmostidealspincurrentconduit, en-\nabling spin information to be transported through large\ndistances with very small losses.\nCONCLUDING REMARKS\nWe used a combination ofDFT calculations and multi-\norbital tight-binding hamiltonians to study the collec-\ntive spin excitations of magnetic adatoms on a two-\ndimensional topological insulator. We have shown that\nthelargespin-orbitcouplinginBiinducesunusuallylarge10 20 300200400600800dynamical coupling α|S⊥|                      d = 2.6 nm\n10 20 303.9 nm\n10 20 306.5 nm\n10 20 3011.4 nm\n20 40\nEnergy (meV)010203040\n20 40\nEnergy (meV)20 40\nEnergy (meV)20 40\nEnergy (meV)\nFIG. 4. Dynamical coupling as a function of excitation fre-\nquency for selected inter-adatoms distances. The top panel\ncorresponds to the adatoms adsorbed close to one of the\nribbon’s edges (@5). The bottom panel corresponds to the\nadatoms adsorbed to the center of the ribbon (@26).\n10 20 30\ndistance (nm)0.011100dynamical coupling (| S⊥|)\nFIG. 5. Dynamical coupling as a function of the inter-adatom\ndistance at resonance. Circles and squares correspond to\nthe adatoms adsorbed to the center and close to the edge of\nthe ribbon, respectively. The values depicted using triang les\nand diamonds were obtained by artificially suppressing the\nstrength of spin-orbit coupling in Bi to 0.8 eV, such that no\ntopologically protected edge states exist in the nanoribbo n.\neffective magnetocrystalline anisotropy energies in the\nFe adatoms, which result in spin excitation frequencies\nwithin the terahertz range. Most importantly, we have\nshown that the topologically protected edge states of the\nBi bilayer nanoribbon mediate a long range dynamical\ncoupling between the adatoms’ magnetic moments. This\ncoupling is a consequence of the almost unperturbed flow\nofspincurrentsbetweentwodistantadatoms. Webelieve\nthese findings indicate an effective way to couple exter-5\nnal probes to topologically protected edge states. They\nalsohighlighttheunusualandpotentiallyveryusefulspin\ntransport properties of two-dimensional topological insu-\nlators. Moreover,the findings we reportedshow unequiv-\nocallythat thereismuchtobe learnedbybuilding hybrid\nstructures from topological insulators and magnetic ma-\nterials.\nACKNOWLEDGEMENT\nThis work was supported by the Brazilian agencies\nFAPESP, through the Grant TEM ´ATICO (2017/02327-\n2), and CNPq. We would like to acknowledge com-\nputing time provided by Laborat´ orio de Computa¸ c˜ ao\nCien´ ıfica Avancada (Universidade de S˜ ao Paulo). MBN\nacknowledgessupport by DOD-ONR (N00014-13-1-0635,\nN00014-11-1-0136,N00014-15-1-2863), the High Perfor-\nmance Computing Center at the University of North\nTexas, and the Texas Advanced Computing Center\nat the University of Texas, Austin. ATC wishes to\nacknowledge partial financial support from Funda¸ c˜ ao\npara a Ciˆ encia e a Tecnologia (Portugal), namely\nthroughprogrammesPTDC/POPH/POCHandprojects\nUID/EEA/50008/2013, IT/QuNet, partially funded by\nEU FEDER, from the QuantERA project TheBlinQC,\nand from the JTF project NQuN (ID 60478), and\nFAPESP (Brazil). ATC is also indebted to R. Bechara\nMuniz and F. S. M. Guimar˜ aes for enlightening discus-\nsions.\n∗antoniocosta@id.uff.br\n[1] F. D. M. Haldane, Rev. Mod. Phys. 89, 040502 (2017).\n[2] M. Z. Hasan and C. L. Kane,\nRev. Mod. Phys. 82, 3045 (2010).\n[3] X.-L. Qi and S.-C. Zhang,\nRev. Mod. Phys. 83, 1057 (2011).\n[4] G. J. Ferreira and D. Loss,\nPhys. Rev. Lett. 111, 106802 (2013).\n[5] L. Fu and C. L. Kane,\nPhys. 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This means\nthat the excitation energy is determined by the ground\nstate electronic structure, and may differ from the MAE\ncalculated via total energy differences.\n[30] A. T. Costa, R. B. Muniz, M. S. Ferreira, and D. L.\nMills, Phys. Rev. B 78, 214403 (2008)."    },    {        "title": "1808.00988v1.Exchange_interactions_and_Curie_temperature_of_Ce_substituted_SmCo5.pdf",        "content": "Exchange interactions and Curie temperature of Ce-substituted SmCo 5\nS.-Y. Jekal1, 2,∗\n1Laboratory of Metal Physics and Technology, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland\n2Condensed Matter Theory Group, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland\n(Dated: August 2, 2018)\nA partial substitution of Sm by Ce can have drastic effects on the magnetic performance, because\nit will introduce strain in the structure and breaks the lattice symmetry in a way that enhances the\ncontribution of the Co atoms to magnetocrystalline anisotropy. However, Ce substitutions, which\nare benefit to improve the magnetocrystalline anisotropy, are detrimental to enhance the Curie\ntemperature ( TC). With the requirements of wide operating temperature range of magnetic devices,\nit is important to quantitatively explore the relationship between the TCand ferromagnetic exchange\nenergy. In this paper we show, based on mean-field approximation, that Ce substitution-induced\ntensile strain in SmCo 5leads to enhanced effective ferromagnetic exchange energy and TCwhile Ce\natom itself reduces TC.\nI. INTRODUCTION\nOwing to the large magnetocrystalline anisotropy, high\nCurie temperature ( TC) and saturation magnetization\n(Ms), Sm–Co compounds have drawn attention for the\nhigh-performance magnet1–12. The net performance of\nthe magnet crucially depends on the inter-atomic inter-\nactions, which in turn depend on the local atomic ar-\nrangements. In order to improve the intrinsic magnetic\nproperties of Sm–Co system, further approaches5,7,9,12–19\nhave been attempted in the past few decades.\nA promising scheme is the partial substitution of Co\nwith other transition metals (TM), such as Cu, Ni, Fe,\nor Zr in Sm(Co, TM) 515,19–22. The partial TM substi-\ntution enhances magnetic anisotropy energy (MAE) due\nto slight modifications of the atomic structure and the\nbreaking of the lattice symmetry. In most cases, how-\never, due to the reduced exchange coupling parameters,\nbothTCand theMsof the Sm(Co, TM) 5compounds\ndecrease with increasing TM content.15,19,20.\nImportantly, any substitution of Co with another TM\nreduce the effective ferromagnetic exchange coupling and\nconsequently reduce TC. This is highly undesirable be-\ncause the most important application field of Sm-Co-\nbased magnets is that of high temperature. Hence, it is\nvital that we find a way to enhance the magnetic perfor-\nmance in Sm-Co magnets without substituting away the\nCo, but by carefully modifying the local atomic arrange-\nments, by introducing strain. Strain is a particularly im-\nportant issue for permanent magnets, because their syn-\nthesis typically involves sintering, and thermal processes\ninevitably introduce strain on fine-grained materials23,24.\nStrain effects are also associated directly to partial ele-\nment substitution, such as e.g. of Sm with another RE\nmetal.\nCe is a particularly promising candidate because it is\nthe most abundant and second lightest among all RE\nmetals, and CeCo 5is isomorhpous with SmCo 525,26. It\nis well-known that the magnetic material parameters of\nCeCo 5are substantially weaker than those of SmCo 527,\ni.e., reduced TCof 660 K instead of 1000 K, reduced\nFIG. 1. Illustration of the crystal structure of Sm 1−xCexCo5\nforx=0, 0.125, 0.25, 0.375, 0.5, 0.75, and 1. Gray, red, and\nblue spheres correspond to Sm, Ce, and Co atoms, respec-\ntively.\nMAE of 5.3 MJ/m3instead of 17.2 MJ/m3, and reduced\nMsof 0.95 T instead of 1.15 T. Nevertheless, in recent,\nwe found a partial substitution of Sm by Ce can have\ndrastic effects on the magnetic performance, because it\nwill introduce strain in the structure and breaks the lat-\ntice symmetry in a way that enhances the contribution of\nthe Co atoms to magnetocrystalline anisotropy (MCA)28.\nHowever, Ce substitutions, which are benefit to improve\nthe MCA with slight modifications of the local atomic\narrangements, are detrimental to enhance the Curie tem-\nperature. With the requirements of wide operating tem-\nperature range of magnetic devices, it is very important\nto quantitatively explore the relationship between the\nCurie temperature TCand ferromagnetic exchange en-\nergy.\nIn this paper we will report, based on mean-field ap-\nproximation (MFA), that Ce substitution-induced tensile\nstrain in SmCo 5leads to enhanced effective ferromagnetic\nexchange energy and TC, however Ce atom itself reduces\nTC.\narXiv:1808.00988v1  [cond-mat.mtrl-sci]  2 Aug 20182\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s120/s32 /s40/s67/s101/s47/s83/s109/s41/s83/s116/s114/s97/s105/s110/s32/s40/s37/s41\n/s49/s51/s46/s56/s53/s49/s51/s46/s57/s48/s49/s51/s46/s57/s53/s49/s52/s46/s48/s48/s49/s52/s46/s48/s53/s49/s52/s46/s49/s48/s49/s52/s46/s49/s53\n/s32/s86/s111/s108/s117/s109/s101/s32/s40 /s197 /s51\n/s41\nFIG. 2. Modification of the effective strain (square) com-\npared to the SmCo 5cell, and the atomic volume (circle) in\nSm1−xCexCo5, as a function of Ce substitution x.II. RESULTS AND DISCUSSION\nA. Crystal Structure\nThe intermetallic magnetic compounds SmCo 5and\nCeCo 5crystallize into the hexagonal CaCu 5structure,\nwhere each Sm (Ce) occupies a 1 asite and Co occu-\npies 2cand 3gsites in the lattice that Co 3and SmCo 2\nsub-layers are alternatively along the c-axis29. To un-\nderstand the Ce substitution effect on crystal structure\nand magnetic properties, we performed calculations of\nthe Sm 1−xCexCo5(x= Ce/Sm) system, where we con-\nsidered seven different compositions: x=0, 0.125, 0.25,\n0.375, 0.5, 0.75, and 1. Configurations shown in Fig. 1 are\nthe most stable structures obtained by total energy min-\nimization in the density functional theory calculations28.\nThe atomic volume decreases linearly from 14.12 ˚A3\nin SmCo 5to 13.84 ˚A3in CeCo 5with increasing x, which\nis due to the smaller atomic radius of Ce compared to\nthat of Sm, whereas the strain increases correspondingly\nmonotonically with increasing x(see Fig. 2). It has\nbeen known that substitution-induced tensile strain on\nSm1−xCexCo5has drastic effects on the magnetic state28.\nB. Exchange Interactions\nMean-field Hamiltonian for the 3-sublattice system\ncould be written as\nHMFA =−Jc−cmcNc/summationtext\niSc,i−2Jc−gmcNg+Nc/summationtext\niSg,i−Jg−gmgNg/summationtext\niSg,i−2Jg−amgNg+Na/summationtext\niσa,i−2Jc−amcNc+Na/summationtext\niσa,i,\n(1)\nwhereJij,mi, andSiare exchange coupling between\nsitesiandj, magnetization, and classical spin vector at\ncrystal site i, respectively. Labels of candgcorrespondto 2cand 3gsites of Co atom, while a represents 1 asite\nof Sm.\nSublattice magnetization is\nmi=2Si+1\n2Sicoth(2Si+1\n2Si1\nT/summationtext\nijzijJi−jmj)−1\n2Sicoth(1\n2Si1\nTΣijzijJi−jmj)(2)\nwherezijis the number of neighboring sites in jth\nsublattice to ith sublattice.\nWhile the inter-atomic ferromagnetic exchange inter-\nactions can be obtained as the difference between ferro-magnetic and antiferromagnetic spin configuration, i.e.,\nJ= (E↑↑−E↑↓)/2 by using density functional theory\ncalculations28, the effective exchange energies of Co and\nSm are calculated as\nJCo\neff= (zgg+zgc+zcc+zcg)JCo−Co\nJR\neff= (zac+zca+zag+zga)JCo−R(3)\n, respectively. Due to the smallness of the de Gennes factor of Sm,3\nFIG. 3. Intrinsic magnetic properties as a function of strain\nin SmCo 5: (a) atomic magnetic moments of Co atoms at sites\n2c(triangles) and 3 g(inverse-triangles) and Sm atoms (cir-\ncles); (b) exchange coupling between Co sites iandj; and (c)\neffective total ferromagnetic exchange energy.\nthe Curie temperature is dominantly determined by the\ninteratomic exchange interaction of the Co atoms9,30.\nTherefore the calculations on the exchange interaction\nare focused on the Co-Co couplings. To understand the\nrole of Ce substitution-induced strain and Ce substitu-\ntion itself, magnetic moment, inter-site exchange cou-\npling (Jij) and effective exchange energy ( Jeff) are calcu-\nlated as a function of the strain in SmCo 5(Fig. 3) and\nCe concentration in Sm 1−xCexCo5(Fig. 4).\nIn Fig. 3(a), the magnetic moments at each crystal-\nlographic site decrease nearly linearly with increasing\ntensile strain. The reason for the decrease of the mag-\nnetic moments is the decreasing overlap of the d-orbitals\nas thec-axis grows with strain. Further, we calculate\nthe inter-site exchange coupling and overall effective ex-\nchange coupling parameters. The intra-plane interac-\ntion strengths JCo(2c)−Co(2c)andJCo(3g)−Co(3g)increase\nwhile the inter-plane interaction strength JCo(2c)−Co(3g)\ndecreases with increasing tensile strain (Fig. 3b). This\nillustrates the dependence of the ferromagnetic exchange\non the inter-atomic distance: as the c–candg–gdis-\ntances decrease the corresponding exchange interaction\nincreases, whereas as the c–g distance increases with\nFIG. 4. Magnetic properties as a function of Ce-substitution\nof Sm: (a) atomic magnetic moments of Co atoms at sites\n2c(triangles) and 3 g(inverse-triangles) and Sm atoms (cir-\ncles); (b) exchange coupling between Co sites iandj; and (c)\neffective total ferromagnetic exchange energy.\nincreasing tensile strain the exchange interaction de-\ncreases. Then we obtain effective exchange interactions\nJeff, which are the sum of the exchange coupling con-\nstants within a sphere of radius R= 5a. From these\ncalculations we observe that the total ferromagnetic ex-\nchange interaction increases monotonically with increas-\ning strain (see Fig. 3c), in contrast to the magnetic mo-\nments.\nFor Ce substituted Sm 1−xCexCo5, the calculated mag-\nnetic moments of the Co, Ce, and Sm atoms decrease\nmonotonically with increasing Ce-substitution x, which\ncorrelates to the decreasing atomic volume, following\nVergard’s law31(see Fig. 4a). Also, the calculated\nJCo−CoandJeffdecrease strongly with increasing x, as\nseen in Fig. 4(b) and (c).\nC. Calculations of Curie temperature\nThe corresponding Curie temperatures are also derived\nfrom the calculated exchange parameters based on the\nMFA.\nThe MFA is based on the notion of single-spin excita-4\ntions, and the Hamiltonian is\nˆhi=−/vector µi·/vectorHi=−gµB/vectorSi·/vectorHi (4)\nwheregandµBare the Lande factor and Bohr mag-\nneton, respectively. The molecular field could be thus\ndefined as\nˆHi=−1\ngµB/summationtext\njJij</vectorSj>(5)\nwith\n<Siz>=SiBSi(xi),xi=gµBHi\nkBT(6)\nwhereBSi(xi) is the Brillouin function, kBis the Boltz-\nmann constant, and Tis the temperature in K. When Tis very high, such as kBT >> µ BHi, theBSi(xi) and\n<Siz>are rewritten as\nBs(x) =(S+1)xi\n3,<Siz>=Si(Si)µBHi\n3kBT. (7)\nBy using the exchange integral Jijpreviously obtained,\nwe get\n<Siz>=−Si(Si+1)\n3kBT/summationtext\njJij<Sjz>,\nT <Sjz>+Si(Si+1)\n3kB/summationtext\njJij<Sjz>= 0(8)\nwhich has nonzero solution only if the determinant\n\na11−T ... ... a 1n\n... ... ... ...\nan1an2... ann=T\n= 0,aij=Si(Si+1)\n3kBΣjJij. (9)\nFIG. 5. Variation of the Curie temperature with respect to\n(a) strain and (b) Ce concentration x.\nAmong solutions in Eq. (9), the highest positive Tis\nthe desired Curie temperature TC.\nAs shown in Fig. 5(a), TCincreases from 1085 K to\n1110 K with increasing tensile strain along c-axis from 0%\nto 5%. For non-strained SmCo 5(0%), the calculated TC\n(1085 K) is comparable with the experimental values ( TC= 1020 K)8, considering the MFA Curie temperature is\ngenerally overestimated by about 10-20%32. On the other\nhand,TCrapidly reduces with increasing concentration\nof Ce substitutions, and eventually CeCo 5exhibit only\nTC=660 K. Because for Sm 1−xCexCo5the decrease of\nthe effective ferromagnetic exchange is reinforced by the\ndecrease of the magnetic moments. The appearance of\nthe value of TCresults from the strain dependence of the\neffective exchange parameters Jeff.\nIII. CONCLUSIONS\nUsing mean-field theory approximation, we investi-\ngated the Ce substitution and strain effect on ferromag-\nnetic exchange energy and corresponding Curie tempera-\nture of SmCo 5compounds. It is found that tensile strain\nalongc-direction improves TCwhich is key property for\nhard magnets. In comparison with an equilibrium state\nof SmCo 5, about 40 K higher TCis observed when 5%\ntensile strain is applied while Ce substitutions rapidly re-\nduce theTC. The enhancement and reduction of TCcan\nbe explained by responses of exchange energy parameters\nto the lattice strain and Ce substitution, respectively.\nIV. 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Tomczak\u0003\nQuantum Physics Division\nFaculty of Physics, Adam Mickiewicz University ul. Umultowska 85,\n61-614 Pozna\u0013 n, Poland\nH. Puszkarski\nSurface Physics Division\nFaculty of Physics, Adam Mickiewicz University ul. Umultowska 85,\n61-614 Pozna\u0013 n, Poland\n(Dated: August 7, 2018)\nThe new method of numerical analysis of experimental ferromagnetic resonance (FMR) spectra in\nthin \flms is developed and applied to (Ga,Mn)As thin \flms. Speci\fcally, it starts with the \fnding\nof numerical solutions of Smit-Beljers (SB) equation and continues with their subsequent statistical\nanalysis within the cross-validation (CV) approach taken from machine learning techniques. As a\nresult of this treatment we are able to reinterpret the available FMR experimental results in diluted\nferromagnetic semiconductor (Ga,Mn)As thin \flms with resulting determination of magnetocrys-\ntalline anisotropy constants. The outcome of CV analysis points that it is necessary to take into\naccount terms describing the bulk cubic anisotropy up to the fourth order to reproduce FMR exper-\nimental results for (Ga,Mn)As correctly. This \fnding contradicts the wide-spread conviction in the\nliterature that only \frst order cubic anisotropy term is important in this material. We also provide\nnumerical values of these higher order cubic anisotropy constants for (Ga,Mn)As thin \flms resulting\nfrom SB-CV approach.\nI. INTRODUCTION: THE EXPERIMENTAL\nDATA\nGallium manganese arsenide, (Ga,Mn)As, is probably\none of the most thoroughly studied diluted ferromagnetic\nsemiconductors. Simultaneous presence of magnetism\nand conductivity in this material makes it possible to\ncontrol of both the charge and the spin degrees of free-\ndom of the charge carriers. This creates potential spin-\ntronic applications. Another reason for the intense re-\nsearch on (Ga,Mn)As are its remarkable magnetic prop-\nerties between which magnetic anisotropy plays an im-\nportant role. It determines, among others, the orienta-\ntion of magnetization in the absence of an applied mag-\nnetic \feld1. Although its understanding is important for\nprospective applications such as e.g., memory devices, its\norigins are far from being fully explained. Magnetocrys-\nralline anisotropy in (Ga,Mn)As, usually described by\nthe single-domain model2,3, is being investigated by var-\nious experimental techniques, such as ferromagnetic reso-\nnance (FMR) and spin-wave resonance (SWR)4. Most of\nthese methods have been used to obtain information on\nanisotropy bulk properties of this material. Recently we\nhave proposed5that one can use the SWR to get infor-\nmation on such magnetic properties as surface anisotropy\nand surface pinning energy of (Ga,Mn)As thin \flms and\ntheir dependence on the orientation of magnetization in\nthe material.\nThe ferromagnetic resonance spectroscopy has long\nbeen a good tool for examining magnetocrysralline\nanisotropy, see e.g., recent review on FMR in (Ga,Mn)As\nthin \flms4. In FMR experiment, since the equilibriumposition of the total magnetic moment Mof the sample\ndoes not coincide with the direction of magnetic \feld H\ndue to the presence of magnetocrysralline anisotropy, M\nprecesses around its equilibrium position with a speci\fc\n(microwave) frequency !. By changing the magnetic \feld\nHone hits a resonance \feld Hr: the precession frequency\nofMis equal to the frequency of the spectrometer. The\nvalue of the resonance \feld Hrstrongly depends on its\norientation with respect to the examined sample, which\nis determined by angles \u0012Hand\u001eH, see Fig. 1, due to\nmagnetocrysralline anisotropy.\nFIG. 1: The coordinate system in which an orientation\nof the applied magnetic \feld His described with\nrespect to the sample in the FMR experiment. The \feld\ndirection is characterized by angles #Hand'H\nmeasured relative to the sample [001] and [100] axes.\nThe equilibrium direction of the sample\nmagnetization Mis represented by angles #and'.\nThis article presents the results of the analysis of bulk\nmagnetocrystalline anisotropy in (Ga,Mn)As based on\nexamination of the uniform mode in SWR resonance inarXiv:1808.01347v1  [cond-mat.mtrl-sci]  3 Aug 20182\nFIG. 2: Resonance \feld6Hrof the uniform SWR mode\nas a function of the magnetic \feld orientation for the\nout-of-plane con\fguration (plane Hr-#H) and for the\nin-plane con\fguration (plane Hr-'H).\n(Ga,Mn)As thin \flm6. The motivation to carry out this\nanalysis was two-fold: First { on the basis of examina-\ntion of surface mode in the same SWR experiment6we\nhave schown5that magnetocrystalline surface anisotropy\nin (Ga,Mn)As thin \flms contains cubic terms up to third\norder, which is not commonly found among ferromagnets.\nWe wonder if this is also true for bulk magnetocrystalline\nanisotropy. Second { it was originally shown6that in or-\nder to reproduce the experimental dependence of the res-\nonance \feld on the orientation of the magnetic \feld with\nrespect to the sample, only the \frst order term of cubic\nanisotropy should be taken into account, which, to some\nextent, is contradictory to the analysis carried out for the\nsurface5. In the meantime, numerical tools have been de-\nveloped that allow a thorough analysis of this problem.\nThat is why we have considered the old problem again.\nAt the begining let us recall the angular dependencies\nof resonance \feld for the uniform mode in ferromagnetic\n(Ga,Mn)As thin \flm6. They are shown in Ref. [6] in\nFig. 5 for the out-of-plane geometry and in Fig. 6 for the\nin-plane geometry, respectively. We show them again in\nFig. 2, to clearly emphasize the di\u000berence between reso-\nnance \feld resulting from the uniaxial anisotropy (plane\nHr-#H) and that resulting from the cubic anisotropy\n(planeHr-'H). We focus on the interpretation of this\nexperiment because the authors very carefully identi\fed\nresonance from uniform SWR modes and distinguished\nit from that for surface modes.\nII. PHENOMENOLOGICAL FREE ENERGY\nThe starting point for the interpretation of experimen-\ntal data of ferromagnetic resonance in (Ga,Mn)As is the\nphenomenological formula for the free energy of the in-\nvestigated sample. We assume that there exists a single\nhomogeneous magnetic domain within the sample and\nthat the free energy of unit volume of the sample con-\nsists of Zeeman term FZ, demagnetization term FD, and\nmagnetocrystalline anisotropy terms (cubic FCand uni-axiallFU):\nF=FZ+FD+FC+FU: (1)\nExpressing free energy in terms of \fctitious \felds one\nobtains\nf(#H;'H;#;' ) =F\nM=HZ+HD+HC+HU;(2)\nMstands here for the value of homogeneous magnetiza-\ntion. Dependence of \fctitious anisotropy \felds in Eq. (2)\non the direction in space is expressed by unit vectors\nalong the applied magnetic \feld Hand along the mag-\nnetization of the sample M. Their spatial orientation is\ndetermined by angles #H,'Hand#,'with respect to\nthe [001] and [100] axes, see Fig. 1. The unit vectors are\ngiven by\nH\nH= [nH\nx;nH\ny;nH\nz] = [cos'Hsin#H;sin'Hsin#H;cos#H];\n(3a)\nM\nM= [nx;ny;nz] = [cos'sin#;sin'sin#;cos#]:\n(3b)\nZeeman \feld is given by\nHZ(#H;'H;#;' ) =\u0000H(nxnH\nx+nynH\ny+nznH\nz):(4)\nDemagnetization \feld of a thin \flm may be approxi-\nmated by the expression describing demagnetization \feld\nof an in\fnite plane\nHD(#) = 2\u0019Mn2\nz: (5)\nThe \feldHC(#;') should be invariant under the cubic\nsymmetry transformations. It follows that it is possible\nto expand it into basis functions with the same symme-\ntry. Typically this expansion is limited to some low-order\nterms of systematically decreasing basis functions. The\nway of constructing such basis functions is presented, e.g.,\nin Ref. [7]: they are chosen from all terms of the expan-\nsion of the identity ( n2\nx+n2\ny+n2\nz)n= 1 (n= 2;3:::) which\nare invariant under permutation of the indices x,y, and\nz. Expansion up to n= 7 is used in what follows:\nHC(#;') =Hc1(n2\nxn2\ny+n2\nyn2\nz+n2\nzn2\nx)+\nHc2(n2\nxn2\nyn2\nz)+\nHc3(n4\nxn4\ny+n4\nyn4\nz+n4\nzn4\nx)+\nHc4(n4\nxn4\nyn2\nz+n4\nxn2\nyn4\nz+n2\nxn4\nyn4\nz)+\nHc5(n4\nxn4\nyn4\nz)+\nHc6(n6\nxn6\nyn2\nz+n6\nxn2\nyn6\nz+n2\nxn6\nyn6\nz):(6)\nAll terms included in the expansion of cubic \feld\nHC(#;') are shown in Fig. 3 (a)-(f). Let us emphasize\nthat every next term is smaller than the previous one -\nthe expansion (6) is convergent. Let us note, however,\nthat the set of six basis functions used in expansion (6)3\n(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFIG. 3: The basis functions up to 6thorder (a-f) used\nin the expansion (6) of the cubic magnetocrystalline\n\feld. Although each next function of a higher order is,\nfor given#;', smaller than the preceding one, they are\nshown here as being comparable in size.\n(a)\n (b)\n (c)\n (d)\nFIG. 4: Terms representing contributions to uniaxial\nmagnetocrystalline anisotropy of (Ga,Mn)As thin \flm\nin spherical coordinate system. Uniaxial anisotropy\nalongzaxis 1stto 3rdorder:n2\nz- (a),n4\nz- (b),n6\nz- (c).\nUniaxial anisotropy along [110] axis ( ny\u0000nx)2- (d).\nis neither orthogonal nor complete. Consequently, the\nexpansion may not be unique8. Nevertheless, it is widely\nused, at least in the cases of analyzing systems with lower\norder magnetocrystalline anisotropies. One can also ex-\npand the cubic magnetocrystalline \feld HC(#;') into\nother basis functions, such as, e.g., spherical harmonics,\nremembering however, to use only those with the appro-\npriate symmetry9.\nIt is recognized4,7that uniaxiall anisotropy \feld\nHU(#;') in (Ga,Mn)As consist of two terms H1[001]n2\nz\nandH[110](ny\u0000nx)2representing uniaxial anisotropy\nalongzaxis and uniaxial anisotropy along [110] axis, re-\nspectively. We add, however, two additional terms of 4th\nand 6thorder:\nHU(#;') =\u00001\n2H1[001]n2\nz\u00001\n2H2[001]n4\nz\u00001\n2H3[001]n6\nz\n\u00001\n2H[110](ny\u0000nx)2:\n(7)\nNote that two terms with cos2#Hare present in expan-\nsion 2: 2\u0019M and\u00001\n2H1[001] . Usually they are grouped\ntogether and referred to as e\u000bective anisotropy \feld:\nHeff\n[001]= 2\u0019M\u00001\n2H1[001] ). Four terms entering the ex-\npansion of uniaxial \feld HU(#;') are shown in Fig. 4.The right side of Eq. (2) depends on 15 variables:\nHr;#H;'H;#;';Hc1;:::;Hc6;Heff\n[001];H2[001];H3[001];\nH[110]. The \frst three will be considered as independent\nvariables that are measured in the experiment and are\nshown in Fig. 2. The pairs of angles ( #H;'H) and\n(#;') are not independent. The angles determining\nthe equilibrium orientation of M,#and', should\nminimize free energy. For the set of \fxed parameters\nHr;#H;'H;Hc1;:::;Hc6;Heff\n[001];H2[001];H3[001];H[110],\none \fnds them from the equilibrium condition\n@f\n@#= 0;@f\n@'= 0: (8)\nThus the right-hand side of Eq. (2) re-\nally depends on 10 \fctitious anisotropy \felds\nHc1;:::;Hc6;Heff\n[001];:::;H 3[001];H[110] which we col-\nlectively denote by the vector\nh\u0011(Hc1;:::;Hc6;Heff\n[001];H2[001];H3[001];H[110]):(9)\nThe questions should then be asked: how to \fnd\nanisotropy \felds and how many of them are necessary\nto reproduce the experimental dependence Hr(#H;'H)\nwell? These questions are answerd in the next section.\nIII. WHICH ANISOTROPY FIELDS ARE\nIMPORTANT FOR (GA,MN)AS?\nThe resonance condition (in the case of uniform mag-\nnetization) is given by10,11\n\u0012!\n\r\u00132\n=1\nsin2#(f##f\u001e\u001e\u0000f2\n#\u001e); (10)\nwheref#\u001e=@f\n@#@f\n@',\r=g\u0016B\n~withgbeing the spectro-\nscopic splitting factor, \u0016Bthe Bohr magneton and ~- the\nPlanck constant. The 9.46 GHz spectrometer was used\nin the considered experiment6, thus Eq. (10) reads\n45:6834 =g2\nsin2#(f##f\u001e\u001e\u0000f2\n#\u001e);[kOe2]: (11)\nAt resonance Eq. (11) should be met for any given\ndirection of Hr(#H;'H), i.e., in the case under consider-\nation, for all points shown in Fig. 2. Let us treat gand\ncomponents of the vector has not known parameters and\ndenote the right-hand side of the Eq. (11) calculated at\ni-th experimental point by Ri,\nRi(g;h) =g2\nsin2#(fi\n##fi\n\u001e\u001e\u0000(fi\n#\u001e)2): (12)\nNote that free energy derivatives (e.g., fi\n##) are calcu-\nlated ati-th the experimental point | for given values\nHr;#H;'H.\nOne should \fnd such values of unknown coe\u000ecients\ng;hthat the Eq. (11) is met as accurately as possible for4\nTABLE I: Models of cubic magnetocrystalline\nanisotropy in (Ga,Mn)As for which cross-validation was\ncarried out. In the second column the terms are given\nincluded in the expansion of the free energy (6) for\nmodels C1 - C6. Uniaxial anisotropy for C1 - C6 models\ndoes not change | only \felds Heff\n[001]andH[110]are\npresent.\nModel Cubic anisotropy Uniaxial anisotropy\nC1 Hc1 Heff\n[001]; H[110]\nC2 Hc1; Hc2 Heff\n[001]; H[110]\nC3 Hc1\u0000Hc3 Heff\n[001]; H[110]\nC4 Hc1\u0000Hc4 Heff\n[001]; H[110]\nC5 Hc1\u0000Hc5 Heff\n[001]; H[110]\nC6 Hc1\u0000Hc6 Heff\n[001]; H[110]\neach experimental point. So the following error function,\nbeing the positive square root of the sum of squares of\nresiduals,\nEN\nRMS(g;h) =s\n1\nNX\ni\u0000\nRi(g;h)\u000045:6834\u00012;(13)\nshould be minimized in 11-dimensional parameter space\n(g;h). The sum in Eq. (13) runs over all Nexperimen-\ntal points shown in Fig. 2. This least squares approach\nto \fnding the unknown parameters represents a speci\fc\ncase of maximum likelihood approach12,13. Actual values\nof the spectroscopic splitting factor gand magnetocrys-\nralline anisotropy \felds are those for which Eq. (13) has\na minimum close to zero.\nWe have included 10 magnetocrysralline anisotropy\n\felds into the formula for free energy. Now we will check\nwhich ones are really essential to describe the experimen-\ntal results well using a simple cross-validation scheme12.\nFor this purpose anisotropy models are de\fned in Tables\nI and II. For example, in the model C3 (third row of Ta-\nble I) the cubic anisotropy is expanded up to 3rdorder,\nthe uniaxial anisotropy along zaxis up to 1storder, the\nuniaxial anisotropy along [110] axis up to 1storder and\nsimilarly for other models.\nThe cross-validation, within leave-one-out technique ,\nruns as follows: We divide the N(= 55) element set\nof experimental data into two subsets: the training one\nand the test one. The \frst one contains N\u00001 elements,\nthe second one | 1 element. One can do it in Npos-\nsible ways. Subsequently the Nsubsets obtained in this\nway are used to train, i.e., to determine the values of\nthe unknown parameters ( g;h) by minimizing the error\nfunctionEN\u00001\nRMS(g;h), de\fned in Eq. (13) for each model\nunder consideration. Simultaneously the error function\nE1\nRMS(g;h) is calculated for one left test point for eachTABLE II: Models of uniaxial magnetocrystalline\nanisotropy in (Ga,Mn)As for which cross-validation was\ncarried out. In the second column the terms are given\nincluded in the expansion of the free energy (4) for\nmodels U1 - U3. Cubic anisotropy for U1 - U3 models\ndoes not change | only \felds Hc1\u0000Hc4are present.\nModel Uniaxial anisotropy Cubic anisotropy\nU1 Heff\n[001]; H[110] Hc1\u0000Hc4\nU2 Heff\n[001]; H2[001]; H[110] Hc1\u0000Hc4\nU3 Heff\n[001]; H2[001]; H3[001]H[110] Hc1\u0000Hc4\nFIG. 5: The values of error functions\nEN\u00001\nRMS\u000b\nand\nE1\nRMS\u000b\nde\fned in Eq. (12) for all models de\fned in\nTables I and II. The values error functions were\ncalculated for the corresponding \feld values taken from\nthe Tables III and IV.\nmodel. Note that its value informs us how well we are\ndoing in predicting the values of anisotropy \felds for\na particular model. After Nminimizations one exam-\nines how averages\nEN\u00001\nRMS\u000b\nand\nE1\nRMS\u000b\ndepend on the\nmodel, i.e., on the order od expansion (6) or (7). We use\nthe following criterion to assess the quality of the model:\nthe model describes magnetocrystalline anisotropy well if\ntaking into account higher order terms in the expansions\n(6) and (7) does not improve its predictive ability given\nby the average\nE1\nRMS\u000b\n.\nThe procedure described above allowed to \fnd the av-\nerage values of anisotropy \felds, g-factor and error func-\ntions\nEN\u00001\nRMS\u000b\nand\nE1\nRMS\u000b\nfor each model after N mini-\nmizations for the sample in the experiment under consid-\neration. They are collected in Tables III and IV. The pre-\ndictive ability, measured by the error function\nE1\nRMS\u000b\n,\nfor all models de\fned in Tables I and II is shown in Fig.\n5.\nE1\nRMS\u000b\ndecreases for C1 - C4 models and remains5\nTABLE III: Anisotropy \felds [Oe] in bulk (Ga,Mn)As related to cubic and uniaxial symmetry and values of g-factor\ncalculated for models C1-C6 according to the procedure described in the text. In the last two columns error\nfunctions\nEN\u00001\nRMS\u000b\nand\nE1\nRMS\u000b\nare shown.\nModel Hc1 Hc2 Hc3 Hc4 Hc5 Hc6 Heff\n[001]H[110] g\nEN\u00001\nRMS\u000b \nE1\nRMS\u000b\nC1 91.81 4765 65.53 1.978 0.95 0.69\nC2 92.21 -87.94 4774 70.72 1.979 0.88 0.68\nC3 77.06 -4.241 57.00 4776 61.70 1.982 0.75 0.61\nC4 78.07 -534.1 43.93 1405 4811 66.29 1.985 0.59 0.50\nC5 79.57 -583.4 41.68 1790 2080 4814 64.50 1.984 0.58 0.48\nC6 78.78 -419.0 41.82 436.1 2841 2700 4809 63.81 1.984 0.57 0.51\nLiu et. al6197a4588 77 1.98\naNote that de\fnition of cubic anisotropy in Ref. [6] is slightly di\u000berent. It takes into account tetragonal distortion in (GaMn)As thin\n\flms. The value of cubic anisotropy \feld calculated from this de\fnition should be approximately equal to 2 Hc1.\nTABLE IV: Anisotropy \felds [Oe] in bulk (Ga,Mn)As related to cubic and uniaxial symmetry and values of g-factor\ncalculated for models U1-U3 according to the procedure described in the text. In the last two columns error\nfunctions\nEN\u00001\nRMS\u000b\nand\nE1\nRMS\u000b\nare shown.\nModel Heff\n[001]H2[001] H3[001] H[110] Hc1 Hc2 Hc3 Hc4 g\nEN\u00001\nRMS\u000b \nE1\nRMS\u000b\nU1 4811 66.28 78.07 -534.1 43.93 1405 1.985 0.59 0.50\nU2 4779 21.83 66.08 80.12 -449.3 40.76 1222 1.987 0.59 0.51\nU3 4778 21.99 6310 66.13 80.15 -449.3 40.59 1221 1.987 0.59 0.50\nroughly constant for the C4 - C6 models. It means that\nhigher order terms used in free energy expansion (2) for\nC5 and C6 models do not improve their ability to pre-\ndict cubic magnetocrystalline anisotropy on the basis of\nexperimental data from Fig. 2. Similarly we see that tak-\ning into account higher order terms in uniaxial anisotropy\nexpansion for models U2 and U3 does not improve their\npredictive ability. It follows that the correct description\nof magnetocrystalline anisotropy in (GaMn)As requires\nthat the free energy expansion should include terms de-\nscribing cubic anisotropy up to fourth order and that is\nenough to take into account uniaxial anisotropies up to\n\frst order.\nOne can also see the result of minimization in Fig. 6\nin the form of a collapse: for the real minimum of error\nfunction at g\u0003;h\u0003its valuesRi(g\u0003;h\u0003) for all experimen-\ntal points fall onto a line 45.6834. By comparing the\nscattering of points for C1 model (a) and C4 model (b)\nwe note the important thing: the addition of the higher\norders of cubic anisotropy \felds improves \ftting not only\nforin-plane experimental points but also for out-of-plane\nexperimental points. This is due to the occurrence of par-\ntial mixed derivatives of the free energy in the determi-\nnant from Hessian (representing a local curvature of free\nenergy in#H;'Hspace) in Smit-Beljers equation (10)\nFIG. 6: The values of function Ri(g;h) de\fned in\nEq. (12) for model C1 - (a) and for model C4 - (b) for\nall experimental points from Fig. 2. Squares { in-plane\ngeometry:#H= 90\u000e,'Hchanges; circles { out-of-plane\ngeometry:#Hchanges,'H=\u000045\u000e. The values of\nRi(g;h) were calculated for the corresponding \feld\nvalues taken from the Table III.\nwhich we solve numerically14for all experimntal points\nsimultaneously treating them on equal footing. There-\nfore, to get as accurate as possible anisotropy \feld values\nit is important to measure resonance \felds in di\u000berent\ngeometries.6\nFul\flling condition (8) while solving Eq. (13) leads to\n\fnding dependences #(#H;'H) and'(#H;'H). for all\nmodels from Tables III and IV. They are shown for C4\nmodel in Figs. 7 and 8. Function #(#H;'H) for a given\n'Hit always is a concave function. For angle \u001eH=\n45\u000eand 135\u000ewe see a ripple, which is the result of the\npresence of cubic symmetry. Note also, that function\n'(#H;'H) for a given #His for all#Ha linear function\n'/'H(does not depend on #H).\nFIG. 7: Equilibrium magnetization angle #versus the\nexternal \feld angles #H;'Hdetermined from the\ncondition (8) for the (Ga,Mn)As thin \flm studied in\nRef. 6.\nFIG. 8: Equilibrium magnetization angle 'versus the\nexternal \feld angles #H;'Hdetermined from the\ncondition (8) for the (Ga,Mn)As thin \flm studied in\nRef. 6.\nLet us summarize this section by stating that for the\ncorrect description of magnetocrystalline anisotropy in\n(GaMn)As, the free energy expansion should include\nterms describing cubic anisotropy up to fourth order and\nthat is enough to take into account uniaxial anisotropies\nup to \frst order.IV. BACK TO THE EXPERIMENT: WHAT IS\nTHE EFFECT OF INCORPORATING OF\nANISOTROPY FIELDS OF HIGHER ORDERS?\nLet us now examine the dependence of the resonance\n\feldHron#Hand'H. The problem can be stated in\nthe following way: Given the values of Hron the bound-\nary of the box presented in Fig. 2, determine the reso-\nnance \feld inside the box. One \fnds a solution in two\nstages. First, one determines the anisotropy \felds, for\nwhich Eq. (11) is satis\fed on the boundary of the box.\nThis stage has been described in the Section III. Second,\nto get the resonance \feld for each #Hand each'Hone\nshould solve Eq. 11 numerically for the anisotropy \felds\ndetermined in the \frst stage (collected in Table III) with\ncondition (8) met for each tentative point obtained dur-\ning the numerical solving procedure. In Figs. 9 and 10\nFIG. 9: Comparison of the Hr-values calculated\nnumerically from Eq. 10 with experimental data for\nout-of-plane geometry for C1-C4 models. The resonance\n\feld for in-plane geometry measurement is visible as\na small rectangle on the vertical axis for #H= 90\u000e.\none can see dependencies Hr(#H;\u000045\u000e), i.e, for out-of-\nplane geometry and Hr(90\u000e;'H) { for in-plane geome-\ntry, respectively. Uniaxial anisotropy is the most visible\nfor out-of-plane geometry (Fig. 9) and although the use\nof higher order cubic terms does improve the agreement\nof the calculated Hr-values with the experimental data,\nthis improvement is not particularly visible in the scale\nof Fig. 9 because the cubic anisotropy is much smaller\nthan the uniaxial one. The improvement, however, can\nbe seen for in-plane geometry: the use of higher order\nterms in of cubic anisotropy expansion given by Eq. (6)\nbecomes necessary to describe dependence Hr(90\u000e;'H)\nmore precisely.\nSpatial dependence of the resonance \feld on angles\n#Hand'His shown in Fig. 11. For small angle #H\nwe see resonance \feld whose source is mainly uniaxial\n[001] anisotropy (with two-fold symmetry), whereas for\nangle#H\u001990\u000ethe resonance \feld with four fold sym-7\nFIG. 10: Comparison of the Hr-values calculated\nnumerically from Eq. 10 with experimental data for\nin-plane geometry for C1-C4 models. Taking into\naccount higher order terms of cubic anisotropy one\nimproves the agreement of calculated Hr-values with\nexperimental data.\nFIG. 11: Spatial angular dependence of resonance \feld\nHr(#H;'H) resulting from our theory. Experimental\ndata (squares) in the plane ( Hr;#H) correspond to\nout-of-plane geometry while those in the ( Hr;'H) plane\n{ to in-plane geometry. The surface in the \fgure is the\nnumerical solution of Eq. (11) with condition (8).\nmetry becomes more noticeable. It is the result of cubic\nanisotropy, although Hr(90\u000e;45\u000e)< Hr(90\u000e;135\u000e) due\nto small uniaxial [110] anisotropy, see also Fig. 10.\nThe spatial dependence of the cubic anisotropy \feld is\nshown in the Fig. 12. We see that really the magnetic\n\feld determines hard/easy directions, not hard/easy\naxes: The largest cubic anisotropy \feld is for #H= 23:7\u000e\nand'H= 45\u000ewhereas the position of hard axis direction\nis given by #H= 54:7\u000eand'H= 45\u000e.\nThe values of \fctitious anisotropy \felds are impor-\ntant in that they allow to reproduce the spatial de-\npendence of the energy of the sample in a magnetic\nFIG. 12: Cubic anisotropy \feld for C4 model versus #H\nand'H.\n\feld due to magnetocrystalline anisotropy and to \fnd\neasy and hard axes. To \fnd this spatial dependence\none needs to know the saturation magnetization. Then\nthe magnetic anisotropy constants can be easily ex-\npressed by corresponding anisotropy \felds, see e.g., very\nclearly written Ref. [15]. We have found the sat-\nuration magnetization16of the considered sample: it\namountsMs= 30:5 emu/cm3, which is a typical value\nfor (Ga,Mn)As containing a few percent of Mn atoms.\nReturning to the Eq. (6) and multiplying it by Msone\nobtains the spatial distribution of energy FC(#;') stored\nin bulk (Ga,Mn)As and related to its cubic magnetocrys-\ntalline anisotropy for the C4 model\nFC(#;') =MsHc1(n2\nxn2\ny+n2\nyn2\nz+n2\nzn2\nx)+\nMsHc2(n2\nxn2\nyn2\nz)+\nMsHc3(n4\nxn4\ny+n4\nyn4\nz+n4\nzn4\nx)+\nMsHc4(n4\nxn4\nyn2\nz+n4\nxn2\nyn4\nz+n2\nxn4\nyn4\nz)\u0011\nKc1(n2\nxn2\ny+n2\nyn2\nz+n2\nzn2\nx)+\nKc2(n2\nxn2\nyn2\nz)+\nKc3(n4\nxn4\ny+n4\nyn4\nz+n4\nzn4\nx)+\nKc4(n4\nxn4\nyn2\nz+n4\nxn2\nyn4\nz+n2\nxn4\nyn4\nz);(14)\nand similarly for the C1 model. Kc1-Kc4stand in\nEq. (14) for cubic anisotropy constants. Taking the nu-\nmerical values of anisotropy \felds Hc1-Hc4from Ta-\nble III one obtains the numerical values of anisitropy\nconstants for C1 and C4 models { they are collected\nin Table V. Let us note that the values of \frst order\ncubic anisitropy for (Ga,Mn)As are several dozen to sev-\neral hundred times smaller than the corresponding values\nfor such ferromagnets as Ni or Fe. Perhaps this is why\nanisotropies of higher orders become visible in resonance\nexperiments only for weak ferromagnets.\nTo assess the accuracy of the present method we used\nthe bootstrap method to evaluate errors for C1 and C4\nmodels. To do this we assumed that the error probabil-\nity distribution of experimental results was normal, and8\n(a)\n (b)\nFIG. 13: Spatial dependence of cubic\nmagnetocrystalline energy in spherical coordinate\nsystems for C1 (a) and C4 (b) models.\nFIG. 14: Spatial dependence of the di\u000berence of cubic\nmagnetocrystalline energy between C1 and C4 models\nin spherical coordinate systems.\nconsequently the errors of solution of Smit-Beljers equa-\ntion had also a normal distribution. Then we could de-\ntermine the approximated errors of obtained anisotropy\nconstants. They are listed in the Table V.\nFinally, let us show, how taking into account the higher\norders of anisotropy \felds changes the cubic anisotropy\nenergy surface. It might seem that correction will be\nof little importance. However, this is not the case: the\nSmit-Beljers equation (10) describing the curvature of\nenergy surface is nonlinear with respect to the second\nderivatives. This leads to signi\fcant corrections in en-\nergy values. Figure 13 shows the spatial dependence of\nenergy from cubic anisotropy on the same scale for mod-\nels C1 and C4. Axes [100], [010] and [001] are easy axes\nwith respect to cubic anisotropy and axis [111] is a hard\none. For example, for hard axis we have FC1([111]) =933\u00066,FC4([111]) = 769\u00064 [erg/cm3]. The C1 model\nthus overestimates the anisotropy energy along hard axis\nby about 20%. Figure 14 shows the spatial dependence\nof energy di\u000berence between C1 and C4 models.\nV. SUMMARY AND OUTLOOKS\nThe article presents how to determine bulk magne-\ntocrysralline anisotopy in (Ga,Mn)As thin \flm by nu-\nmerical solution of the Smit-Beljers equation for all data\ncollected in one FMR experiment, i.e., for di\u000berent spa-\ntial orientations of the magnetic \feld with respect to the\nsample, on equal footing. To avoid essential drawbacks\nof \ftting procedures (lack of information which \ftted\nconstants are relevant and possibility of over\fting) by\n\fnding anisotropy constants we cross-validated the nu-\nmerical solutions of Smit-Beljers equation for six models\n(C1-C6). The results of this cross-validation, i.e., the val-\nues of the function\nE1\nRMS\u000b\ndisplaying predictive ability\nfor models C1-C6 point that it is necessary to expand\nbulk cubic anisotropy up to the fourth order to repro-\nduce spatial dependence of the resonance \feld correctly\n| that is increasing the order of expansion of anisptropy\ndoes not change predictive ability of the model under\nconsideration. Such cubic anisotropy (up to fourth or-\nder) is visible in the resonant experiment. It means that\nthe models of \frst order cubic anisitropy applied so far\nto (Ga,Mn)As overstimated the value of this anisotropy.\nLet us stress that this description of the bulk anisotropy\nis consistent with the presented earlier description5of the\nsurface anisotropy (both descriptions require higher or-\nder expansion of cubic anisotropy). We also have shown\nthat FMR data allow one to \fnd the spectroscopic split-\nting factor with high accuracy. We intend to con\frm the\nusfullness of this new approach by applying it to other\navailable resonance experiments in the near future.\nAcknowledgment The authors would like to thank\nMarcin Tomczak for stimulating discussions. This study\nis a part of a project \fnanced by Narodowe Centrum\nNauki (National Science Centre of Poland), grant no.\nDEC-2013/08/M/ST3/00967. Numerical calculations\nwere performed at Pozna\u0013 n Supercomputing and Net-\nworking Center under Grant no. 284.\n\u0003Corresponding author: ptomczak@amu.edu.pl\n1U. Welp, V. K. Vlasko-Vlasov, X. Liu, J. K. Furdyna, and\nT. Wojtowicz, Phys. Rev. Lett. 90, 167206 (2003).\n2X. Liu, Y. Sasaki, and J. K. Furdyna, Phys. Rev. B 67,\n205204 (2003).\n3G. Nieuwenhuys, T. Prokscha, A. Suter, E. Morenzoni,\nD. Chiba, Y. Nishitani, T. Tanikawa, F. Matsukura,\nH. Ohno, J. Ohe, S. Maekawa, and Y. J. Uemura, Na-\nture Materials 9, 299 (2010).\n4X. Liu and J. K. Furdyna, Journal of Physics: Condensed\nMatter 18, R245 (2006).5H. Puszkarski and P. Tomczak, Surface Science Reports\n72, 351 (2017).\n6X. Liu, Y. Y. Zhou, and J. K. Furdyna, Phys. Rev. B 75,\n195220 (2007).\n7J. Zemen, J. Ku\u0014 cera, K. Olejn\u0013 \u0010k, and T. Jungwirth, Phys.\nRev. B 80, 155203 (2009).\n8E. Callen and H. Callen, Journal of Physics and Chemistry\nof Solids 16, 310 (1960).\n9H. Puszkarski, Progress in Surface Science 9, 191 (1979).\n10J. Smit and H. G. Beljers, Philips Res. Rep. 10, 113 (1955).\n11L. Baselgia, M. Warden, F. Waldner, S. L. Hutton, J. E.9\nTABLE V: Cubic anisotropy \felds ( Hc) in Gaussian units [Oe] and SI units [kA/m] and cubic anisotropy constants\n(Kc) in [erg/cm3] and [J/m3] for bulk (Ga,Mn)As calculated for models C1 and C4.\nHc1 Hc2 Hc3 Hc4 Kc1 Kc2 Kc3 Kc4\nModel C1\nGaussian 91.81 \u0006.55 2800 \u000617\nSI 7.306 \u00060.044 280.0 \u00061.7\nModel C4\nGaussian 78.07 \u0006.40 -534 \u000628 43.9 \u00061.2 1410 \u000670 2381 \u000613 -16280 \u0006860 1330 \u000637 42900 \u0006220\nSI 6.213 \u00060.032 -42.5 \u00062.3 3.493 \u00060.096 112.2 \u00065.6 238.1 \u00061.3 -1628 \u000686 133.0 \u00063.7 4290 \u000622\nDrumheller, Y. Q. He, P. E. Wigen, and M. Mary\u0014 sko,\nPhys. Rev. B 38, 2237 (1988).\n12C. M. Bishop, Pattern Recognition and Machine Learning\n(Springer, 2006).\n13C.-H. W. A. G. D. C. R. C. K. F. D. J. S. Pankaj Mehta,Marin Bukov, arXiv:1803.08823v1.\n14We use Python packages scipy.optimize and Numdi\u000btools .\n15M. W. Gutowski, arXiv:1312.7130v1.\n16Details of this calculation will be published in a separate\npaper."    },    {        "title": "1808.05785v2.Temperature_Dependence_of_Magnetic_Properties_of_an_18_nm_thick_YIG_Film_Grown_by_Liquid_Phase_Epitaxy__Effect_of_a_Pt_Overlayer.pdf",        "content": "Temperature Dependence of Magnetic Properties of\nan 18-nm-thick YIG Film Grown by Liquid Phase\nEpitaxy: Effect of a Pt Overlayer\nNathan Beaulieu\u0003;y, Nelly Kervarecz, Nicolas Thieryx, Olivier Kleinx, Vladimir Naletovy;x;{,\nHerv ´e Hurdequinty, Gr´egoire de Loubensy, Jamal Ben Youssef\u0003and Nicolas Vukadinovick\n\u0003LabSTICC, CNRS, Universit ´e de Bretagne Occidentale, 29238 Brest, France\nySPEC, CEA, CNRS, Universit ´e Paris-Saclay, 91191 Gif-sur-Yvette, France\nzPlateforme technologique RMN-RPE, Universit ´e de Bretagne Occidentale, 29238 Brest, France\nxUniv. Grenoble Alpes, CEA, CNRS, Grenoble INP, INAC-Spintec, 38000 Grenoble, France\n{Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation\nkDassault Aviation, 92552 Saint-Cloud, France\nAbstract —Liquid phase epitaxy of an 18 nm thick Yttrium\nIron garnet (YIG) film is achieved. Its magnetic properties are\ninvestigated in the 100 – 400 K temperature range, as well as the\ninfluence of a 3 nm thick Pt overlayer on them. The saturation\nmagnetization and the magnetocrystalline cubic anisotropy of\nthe bare YIG film behave similarly to bulk YIG. A damping\nparameter of only a few 10\u00004is measured, together with a\nlow inhomogeneous contribution to the ferromagnetic resonance\nlinewidth. The magnetic relaxation increases upon decreasing\ntemperature, which can be partly ascribed to impurity relaxation\nmechanisms. While it does not change its cubic anisotropy, the Pt\ncapping strongly affects the uniaxial perpendicular anisotropy of\nthe YIG film, in particular at low temperatures. The interfacial\ncoupling in the YIG/Pt heterostructure is also revealed by an\nincrease of the linewidth, which substantially grows by lowering\nthe temperature.\nI. I NTRODUCTION\nYttrium Iron garnet (Y 3Fe5O12, YIG) is the marvel material\nfor ferromagnetic resonance (FMR) [1], with the lowest known\nGilbert damping parameter, \u000b= 3\u000110\u00005for bulk [2],\n[3]. Since the 1970s, liquid phase epitaxy (LPE) has been\nthe reference method to grow micrometer-thick YIG films\nwith bulk-like dynamical properties [4], [5], and numerous\nmicrowave devices based on the propagation of spin-waves in\nsuch films have been developed [6]. In the past years, thin\nfilms of YIG have become highly desirable in the context of\nmagnonics [7], [8] and its coupling to spintronics [9] for three\nmain reasons. Firstly, in magnonics, one wishes magnetic films\nwith low damping to ensure large propagation length and with\nthickness limited to a few tens of nanometers so that fun-\ndamental spin-wave modes do not interact with higher order\nthickness modes, two conditions met in nanometer-thick YIG\nfilms [10]. Secondly, coupling YIG magnonics to spintronics\nwas made possible by the discovery that pure spin currents\ncan be transmitted at the interface between YIG (a magnetic\ninsulator) and an adjacent normal metal [11], [12]. In these\nhybrid bilayers, ultrathin YIG layers are required to enhance\nthe interfacial effects. For instance, the damping of nanometer-thick YIG can be controlled by the spin-orbit torque produced\nby an electrical current flowing in an adjacent Pt layer [13],\nleading to the generation of coherent spin-waves above a\nthreshold instability [14]. Thirdly, due to the high resilience\nof YIG, only nanometer-thick films can be patterned through\nstandard nanofabrication techniques, which is an important\nasset to design magnonic crystals [15] and engineer the spin-\nwave spectrum of individual YIG nanostructures [16].\nDue to this renew of interest for YIG thin films, lots of\nefforts have been put in the last few years to produce ultrathin\nfilms with high epitaxial and dynamical qualities. Pulsed laser\ndeposition (PLD) is a technique of choice to deposit ultrathin\nYIG layers ablated from stoichiometric targets on Gd 3Ga5O12\n(GGG) substrates [17], [18], [19], [20], [21], [22]. Magnetic\nproperties close to bulk ones [19] and Gilbert damping pa-\nrameters below 10\u00004have been reported for films thinner\nthan 50 nm [22], even though such low intrinsic damping\noften comes at the detriment of the full linewidth due to in-\nhomogeneous broadening [20], [21]. Another method to grow\nepitaxial nanometer-thick YIG films is off-axis sputtering [23],\nwhich can be used to control the strain-induced anisotropy on\nlattice-mismatched substrates [24]. Finally, even though LPE\nis not well suited to grow sub-micron thick films, it has been\nsuccessfully used to produce YIG films with thicknesses in\nthe 100 – 200 nm range and damping parameters approaching\n10\u00004[25], [26].\nHere, we demonstrate that the growth of a YIG film as\nthin as 18 nm and of high quality can be achieved by LPE.\nWe present a detailed investigation of its magnetic properties,\nincluding relaxation, as a function of temperature, as well as\nthe effect of a 3 nm thick Pt overlayer on them.\nII. S AMPLE PREPARATION AND PRELIMINARY\nCHARACTERIZATIONS AT ROOM TEMPERATURE\nThe ultrathin YIG film under consideration was grown by\nLPE from PbO and B 2O3flux on a 1 inch (111)-oriented\nGGG substrate. High-purity oxides (6N) were used. The keyarXiv:1808.05785v2  [cond-mat.mtrl-sci]  24 Aug 2018Fig. 1. (a) X-ray reflectometry of the 18 nm thick LPE grown YIG film.\nThe inset shows the AFM surface topography. (b) Resonance linewidth vs.\nfrequency of the bare YIG layer measured by broadband FMR at room\ntemperature. The inset displays the FMR line at 10 GHz.\nparameter is the growth rate that depends on the growth\ntemperature. To get an ultrathin YIG film, a very low growth\nrate is required which means that the growth temperature has\nto be close to the saturation temperature (small supercooling).\nFor the selected sample, the growth rate was around 1 nm/s.\nAnother important parameter is the speed of the substrate\nrotation during the growth process that must be controlled.\nThe quality of the utrathin YIG film was checked by atomic\nforce microscopy (AFM) measurements from which a surface\nroughness of about 0.25 nm was determined (see inset in\nFig.1(a)). The film thickness of 18 nm was confirmed by X-ray\nreflectivity measurements (Fig.1(a)). From the experimental\nin-plane magnetization curves recorded along the [110]and\n[112]axes (not shown here), several conclusions can be drawn:\n(i) the saturation magnetization 4\u0019MSat room temperature is\nequal to 1700 G, close to the bulk value, (ii) the magnetization\ncurves are isotropic in the film plane, and (iii) the film has a\nvery weak coercive field ( Hc'0:3Oe).\nBroadband FMR on a millimeter-size slab of the grown\nfilm was used to investigate its magnetic relaxation at room\ntemperature. The full width at half maximum (FWHM) of the\nresonance line measured in in-plane magnetized configuration\nas a function of the excitation frequency is shown in Fig.1(b).\nFig. 2. (a) Dependence of the magnetization of the 18 nm thick YIG film on\ntemperature. (b) Resonance field vs.polar angle measured at X-band at three\ndifferent temperatures. (c) Cubic and (d) uniaxial perpendicular anisotropies\nextracted as a function of temperature. The red dashed lines in (a) and (c)\nare the dependences for bulk YIG from literature, the dotted line in (d) is a\nguide to the eye.\nThe linear dependence of the linewidth on frequency allows to\nfit a Gilbert damping parameter \u000b= (3:4\u00060:2)\u000110\u00004and an\ninhomogeneous contribution \u0001H0= 2:44\u00060:24Oe. These\nvalues, which are close to those reported on PLD grown YIG\nfilms of similar thickness [18], highlight the good dynamical\nquality of our film.\nIII. T EMPERATURE DEPENDENCE OF MAGNETIC\nPROPERTIES FOR THE BARE YIG FILM\nThe temperature dependence of the saturation magnetization\nfor the bare YIG film measured by means of a supercon-\nducting quantum interference device (SQUID) magnetometer\nand a vibrating sample magnetometer (VSM) is displayed inFig.2(a). The 4\u0019MS(T)profiles are consistent between the\ntwo methods and are in very good agreement with the one\nreported for a bulk YIG sample [27] for T\u0015200 K. For\nlower temperatures, the saturation magnetization for the bulk\nsample exceeds the one found for our ultrathin YIG film.\nNext, the anisotropy constants were extracted from FMR\nmeasurements performed in a X-band cavity (magnetic field\nswept at fixed frequency f= 9:3GHz). We note that the slab\nused in this experiment has a slightly larger inhomogeneous\ncontribution to the linewidth than the one measured by broad-\nband FMR. Using the Makino’s procedure [28], the first-order\ncubic anisotropy constant K1, the first-order uniaxial perpen-\ndicular anisotropy constant KUand the gyromagnetic ratio \r\nwere determined as a function of temperature by exploiting\nthe polar angle variation of the polarizing magnetic field, the\n4\u0019MS(T)profile being known for that very same slab of\nfilm (red stars in Fig.2(a)). Examples of such variations are\nreported in Fig.2(b) for three temperatures: T= 140 K, 300 K,\nand 405 K, where \u0012His the polar angle of the polarizing\nmagnetic field. \u0012H= 0\u000ecorresponds to the film normal ( [111]\ncrystallographic axis) and \u0012H= 90\u000eis the in-plane [112]axis.\nTwo remarks can be made. First, the amplitude of the angular\nvariation increases for decreasing temperature. Second, this\nvariation becomes more and more asymmetrical with respect\nto the film plane as the temperature is lowered. This last point\nmeans that the absolute value of K1is enhanced for decreasing\ntemperature. The experimental temperature dependence of K1\nis displayed in Fig.2(c). The increase of the absolute value of\nK1for decreasing Tsatisfactorily matches with the variation\nfound for a bulk YIG sample [29]. On the other hand, the\ntemperature dependence of KUis reported in Fig.2(d). It\nappears that the sign of KUis positive (easy axis along the\nfilm normal) and KUincreases with T.\nThe origin of KUin micrometer-thick LPE grown YIG\nfilms has been discussed in detail in the eighties. Two main\ncontributions were identified, namely, the growth and the stress\nanisotropies. For pure YIG films, the growth anisotropy is\nmainly induced by the Pb impurities from solvent. It has\nbeen established that it raises with the Pb content [30].\nIn addition, the Pb content increases with the supercooling\n[31]. In our case, the growth of the ultrathin YIG film with\na small supercooling prevents a large contribution of the\ngrowth anisotropy. On the other hand, the uniaxial stress\nanisotropy arises from the lattice parameter mismatch between\nthe GGG substrate ( as= 12:383 ˚A) and the YIG film\n(af= 12:376 ˚A). Introducing \u0001a?= (as\u0000a?\nf)=aswherea?\nf\nis the lattice parameter of the YIG film in the growth direction,\nthe uniaxial stress anisotropy constant Ks\nUis expressed by:\nKs\nU= (\u00003=2)\u0015111\u001bwhere\u001b=E=(1\u0000\u0017)\u0001a?, whereE\nis the Young’s modulus and \u0017the Poisson’s ratio. For our\nfilm grown under tension, \u001bis positive,\u0015111is negative and\na positive value of Ks\nUis expected in agreement with the\nexperimental data. An estimate of Ks\nUbased on the values of\nE,\u0017and\u0015111for a bulk YIG sample leads to about 40% of the\nKUvalue deduced from FMR measurements. The remaining\ndifference can be ascribed to a surface induced contribution to\nFig. 3. (a) FMR lines measured at X-band on the bare YIG layer for three\ndifferent temperatures. (b) FMR linewidth vs.temperature. The inset shows\nthe comparison of the data to the slowly relaxing impurity model (Eq.1).\nthe uniaxial perpendicular anisotropy, which is also present in\nultrathin films. In fact, we have observed at room temperature\nthatKUvaries with the YIG thickness, and becomes negative\n(easy plane) above about 50 nm.\nMoreover, a value of the gyromagnetic ratio nearly inde-\npendent of temperature was extracted from the angular fit of\nthe resonance field, j\rj= 1:7685\u00060:002s\u00001Oe\u00001. This value\nis consistent with the one deduced from the broadband FMR\nmeasurements,j\rj= 1:771\u00060:0007 s\u00001Oe\u00001.\nAnother interesting feature is the temperature dependence of\nthe magnetic relaxation. Fig.3(a) shows the absorption spectra\n(derivative form) versus magnetic field recorded at 9.3 GHz\nin the parallel configuration for three temperatures. These\nspectra reveal that the FMR linewidth increases for decreasing\ntemperature. The value of \u0001HFWHM at 140 K is nearly four\ntimes greater than the one at 405 K. Such a behavior has been\nrecently reported on ultrathin YIG films grown either by a\nspin coating method [32] or by off-axis sputtering [33], and\non YIG spheres [34]. The broader temperature range probed\nin these studies allows to evidence a peak-like maximum\nin the temperature dependence of the FMR linewidth. The\nposition of this peak is frequency dependent and is located\naroundT= 25 K at X-band. Based on these observations,\nthe origin of the resonant temperature dependent linewidth\nwas ascribed to the slowly relaxing impurity mechanism\nextensively investigated on bulk YIG samples in the sixties\n[2], [3]. The existence of rare earth or Fe2+impurities induced\nduring the growth process was put forward. In both cases, the\ncontribution of the slowly relaxing impurity mechanism to theFig. 4. Dependences on temperature of (a) cubic anisotropy, (b) uniaxial\nperpendicular anisotropy and (c) resonance linewidth for both the bare YIG\nlayer (black dots) and the YIG layer capped with 3 nm Pt (red squares). The\ninset shows the FMR lines measured at 300 K on both samples. All dashed\nlines are guides to the eye.\nlinewidth can be described by the following expression [33]:\n\u0001HSR=A(T)!\u001c\n1 + (!\u001c)2; (1)\nwhereA(T)is a frequency-independent prefactor and \u001ca\ntemperature-dependent time constant. The total FMR linewidth\n\u0001HFWHM = \u0001H0+4\u0019\u000bf\nj\rj+ \u0001HSRis displayed in the\ninset in Fig.3(b) (red dashed curve). It can be seen that the\nexperimental temperature dependence of the FMR linewidth\ncannot be fully reproduced by the model, where only \u0001HSRis\na function of temperature. Other relaxation mechanisms seem\nto contribute to the FMR linewidth. As highlighted in Ref.[26],\nthe existence of a transition layer between LPE grown YIG\nfilms and the GGG substrate, whose thickness is estimated to\nbe around 5 nm (nearly a third of our film thickness), could\nproduce additional relaxation channels. For our ultrathin film,\na sizeable surface induced contribution to the linewidth can\nalso be expected.\nIV. T EMPERATURE DEPENDENCE OF MAGNETIC\nPROPERTIES FOR THE YIG/P T HETEROSTRUCTURE\nNext, we are interested in the influence of a thin overlayer\nof Pt on the magnetic properties of our ultrathin YIG film.\nUsing electron beam evaporation, a 3 nm thick Pt layer was\nevaporated on top of the YIG film after a very soft in situdry etching of its surface. Similarly to the bare YIG film,\nthe obtained YIG/Pt heterostructure was then studied at X-\nband as a function of temperature. The main results of this\ncharacterization are compared to those obtained on the bare\nYIG film in Fig.4. In order to extract the anisotropy constants\nof the YIG/Pt bilayer, we used the same 4\u0019MS(T)profile\nas in Fig.2(a), since it is known that no magnetic moment\nis induced in Pt by proximity effects at ferrites/Pt interfaces\n[35]. As can be seen in Fig.4(a), the Pt overlayer does not\nchange the cubic anisotropy of the bare YIG film, which is\nexpected from its bulk magnetocrystalline origin. In contrast,\nit does strongly affect the uniaxial perpendicular anisotropy,\nwhich becomes negative (easy plane) at low temperature, as\nshown in Fig.4(b). It means that an interfacial mechanism [36]\nof spin-orbit origin changes the surface anisotropy component\nofKUdiscussed above for the bare YIG film. The gyromag-\nnetic ratio remains unaffected by the Pt overlayer within our\ndetermination uncertainty.\nAs expected, the interfacial coupling between YIG and Pt\nalso leads to the increase of the FMR linewidth [37], [38],\nwhich is obvious from the comparison presented in Fig.4(c).\nAt 300 K, the measured linewidth is nearly doubled by the\ndeposition of Pt on top of YIG. To prove its interfacial\norigin, we have checked that this enhancement is inversely\nproportional on the YIG thickness by varying the latter in\nthe 15 – 200 nm range (not shown here). It is interesting\nto note that the increase of the linewidth due to the Pt\noverlayer is not constant as a function of temperature. The\nincrement in the damping depends both on the strength of\nthe interfacial coupling (whose microscopic origin is rarely\ndiscussed, most often, it is parametrized by the so-called spin\nmixing conductance of the hybrid interface) and on the spin\ntransport parameters in the Pt layer (the thickness of the\nlatter being here comparable to the spin diffusion length in\nPt at room temperature [39]). These two contributions can be\ntemperature dependent, leading to a non trivial behavior of the\nincrement in the damping versus temperature. As a matter of\nfact, the latter seems to be minimum around 250 K, and its\nrapid increase below this temperature should be confirmed by\nmeasurements at even lower temperature [32].\nV. C ONCLUSION\nIn sum, this study demonstrates that the LPE growth method\ncan be used to produce YIG films with thickness below 20 nm\nand good structural parameters. 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B , vol. 97, p. 064422, 2018."    },    {        "title": "1809.00801v3.Enhanced_perpendicular_magnetocrystalline_anisotropy_energy_in_an_artificial_magnetic_material_with_bulk_spin_momentum_coupling.pdf",        "content": "Enhanced perpendicular magnetocrystalline anisotropy energy in\nan artificial magnetic material with bulk spin-momentum coupling\nAbdul-Muizz Pradipto,1, 2,∗Kay Yakushiji,3Woo Seung Ham,1Sanghoon\nKim,1Yoichi Shiota,1Takahiro Moriyama,1Kyoung-Whan Kim,4, 5\nHyun-Woo Lee,6Kohji Nakamura,2Kyung-Jin Lee,7, 8and Teruo Ono1\n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan\n2Department of Physics Engineering, Mie University, Tsu, Mie 514-8507, Japan\n3National Institute of Advanced Industrial Science and\nTechnology (AIST), Tsukuba, Ibaraki 305-8568, Japan\n4Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany\n5Center for Spintronics, Korea Institute of Science and Technology, Seoul 02792, Korea\n6Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea\n7Department of Materials Science and Engineering,\nKorea University, Seoul 02841, Korea\n8KU-KIST Graduate School of Converging Science and Technology,\nKorea University, Seoul 02841, Korea\nWe systematically investigate the perpendicular magnetocrystalline anisotropy\n(MCA) in Co−Pt/Pd-based multilayers. Our magnetic measurement data shows\nthat the asymmetric Co/Pd/Pt multilayer has a significantly larger perpendicular\nmagnetic anisotropy (PMA) energy compared to the symmetric Co/Pt and Co/Pd\nmultilayer samples. We further support this experiment by first principles calcula-\ntions on the CoPt 2, CoPd 2, and CoPtPd, which are composite bulk materials that\nconsist of three atomic layers in a unit cell, Pt/Co/Pt, Pd/Co/Pd, Pt/Co/Pd, re-\nspectively. By estimating the contribution of bulk spin-momentum coupling to the\nMCA energy, we show that the CoPtPd multilayer with the symmetry breaking\nhas a significantly larger perpendicular magnetic anisotropy (PMA) energy than the\nother multilayers that are otherwise similar but lack the symmetry breaking. This\nobservation thus provides an evidence of the PMA enhancement due to the struc-\ntural inversion symmetry breaking and highlights the asymmetric CoPtPd as the\nfirst artificial magnetic material with bulk spin-momentum coupling, which opens aarXiv:1809.00801v3  [cond-mat.mtrl-sci]  22 Apr 20192\nnew pathway toward the design of materials with strong PMA.\nThe interplay between magnetism, electronic structure and spin-orbit coupling (SOC)\nin materials has led to the possibility to utilize both the charge and spin degrees of free-\ndom of electrons for spintronic devices [1–4]. Among the most important effects emerging\nfrom the SOC is the magnetocrystalline anisotropy (MCA), resulting in a preferred mag-\nnetization direction with respect to the crystallographic structure of materials [5, 6]. For\nthe practical design of devices, perpendicular magnetic anisotropy (PMA) with respect to\nsurface/interface planes is strongly desired [7–9], as manipulation of the magnetic moment\ndirections and/or the magnetic domain walls can be done more efficiently [10, 11]. The\norigin of MCA has initially been attributed to the orbital localization as a result of reduced\ndimensionality [12, 13]. It was also shown that the MCA is strongly related to the SOC of the\nelectronic states near the Fermi level [14]. Consequently, manipulation of the band structure\naround the Fermi level provides a natural way to tune the MCA. This can be achieved by\nthe modification of the orbital occupation, for example via the application of an external\ngate voltage [15, 16], or by direct chemical manipulations of the band structure. The latter\nis commonly done by doping or impurities [17, 18] or by engineering the materials interfaces\n[19, 20]. Another milestone in understanding the MCA comes from the proportionality of\nMCA and the anisotropy of SOC-induced orbital magnetic moment, which was proposed by\nBruno [21], and has been confirmed in different materials [22–24].\nAnother mechanism of MCA, which depends on broken inversion symmetry, has been\nproposed [25, 26]. In systems with broken inversion symmetry, the SOC becomes odd in\nthe momentum /vectorkspace [27]. The oddness of SOC in the /vectorkspace is visible for instance from\nthe Rashba model of SOC [28] which has the form HR=αR/parenleftBig\n/vectork׈z/parenrightBig\n·/vector σthat depends on\nlinear term of /vectork. Here theαRis the Rashba parameter, and zis the direction of inversion-\nsymmetry-breaking-induced potential gradient. We note that the broken inversion symmetry\nresults not only in the linear-in- kcontribution but also in higher-odd-order contributions.\nFrom now on, we use the term “Rashba” to describe all odd-order-in- kcontributions for\nsimplicity. Since the Rashba interaction only develops in non-centrosymmetric systems, the\nstrength of Rashba parameter can provide an indication to the degree of structural inversion\nsymmetric breaking. Although it was originally proposed in nonmagnetic materials [28–\n30], Rashba-type spin splitting was later observed also in magnetic systems, such as on the3\nTABLE I: The MCA energy EMCA per unit volume (in meV/ ˚A3) obtained from a self-consistent\ntreatment of the SOC; the ky−dependentE±ky\nMCAevaluated within half of the Brillouin zone; and\nthe even and odd contributions to the k−dependentEMCA (see text).\nSystem CoPt 2CoPd 2CoPtPd\nEMCA (SCF) 0 .028 0.027 0.043\nky−dependentEMCA:\nE+ky\nMCA 0.015 0.013−0.072\nE−ky\nMCA 0.015 0.013 0.113\nEeven\nMCA (from Eq. (2)) 0 .030 0.026 0.041\nEodd\nMCA (from Eq. (3)) 0 .000 0.000 0.185\nGd(0001) grown on an oxide substrate [31]. Recently, the MCA has been analyzed using\nthe SOC model of Rashba [25, 26] and it was shown that MCA changes with the increasing\nRashba parameter strength, i.e. the more asymmetric the system is, the stronger it develops\nthe MCA. Such description is very insightful for the understanding of MCA, however, its\nverification is still lacking.\nWe report in this work our analysis on the effects of asymmetric stacking of Co −Pt/Pd-\nbased multilayer systems by combining experimental magnetic measurements and first-\nprinciples calculation. Co/Pt and Co/Pd-based layered structures have been well known\nto exhibit strong PMA of about 3 −10 Merg/cm3[10, 32–36]. We show that the broken in-\nversion symmetry plays a significant role in the appearance of PMA in this material. These\nresults present an evidence to the PMA driven by structural asymmetry as suggested in pre-\nvious theoretical analyses [25, 26] and may provide a useful guide in the design of materials\nwith strong PMA.\nIn order to study the effect of the symmetry of the stacking structure on the magnetic\nbehavior, three variations of the multilayer structures were fabricated as follows: [Co(0.2\nnm)/Pt (0.2 nm)] 10, [Co (0.2 nm)/Pd(0.2 nm)] 10and [Co(0.2 nm)/Pd(0.2 nm)/Pt(0.2 nm)] 10.\nOur films have been fabricated with a sputtering apparatus (Canon-Anelva C-7100) on\nthermally oxidized Si substrates at room temperature. A Ta(4 nm)/Ru(1 nm)/Pt(0.5 nm)\nseed/buffer layer was first deposited on the substrate. Then a Co-based multilayer was\ngrown by alternate deposition of Co, Pd and Pt at room temperature [35]. Fig. 1a shows\ntheirM−H(magnetization vs. magnetic field) loops measured at room temperature.4\nAll of them display typical perpendicularly magnetized properties as the sharp reversals\nin the out-of-plane (op) loop and the gradual saturation in the in-plane (ip) loop with a\nsubstantial perpendicular magnetic anisotropy field ( Hk). The values of the effective PMA\n(Keff=HkMs/2) were estimated to be 5.7, 4.8 and 10.8 Merg/cm3derived from Hkand\nthe saturation magnetization ( Ms), for [Co (0.2 nm)/Pt (0.2 nm)] 10, [Co (0.2 nm)/Pd (0.2\nnm)] 10and [Co (0.2 nm)/ Pd (0.2 nm)/Pt (0.2 nm)] 10, respectively. It is obvious that\nthe asymmetric stacking, the Co/Pd/Pt-multilayer, exhibited twice larger Keffthan the\nsymmetric stackings, the Co/Pt- and Co/Pd-multilayers. Fig. 1b shows the cross-sectional\nhigh-resolution transmission electron microscopy image for the full stack of the [Co(0.2\nnm)/Pd(0.2 nm)/Pt(0.2 nm)] 10multilayer. It suggests that an fcc(111) oriented Co/Pd/Pt-\nmultilayer part was formed on the hcp-c-plane oriented Ru buffer layer with a flat interface.\nLargerKeffvalues of the asymmetric samples with respect to those of the symmetric ones\n[Co/Pt]10 \n[Co/Pd]10 \n[Co/Pd/Pt ]10 \n5 nm  \nRu (1 nm)  \nTa (4 nm)  Ru (1 nm)  Ta \nCo/Pd/Pt (6 nm)  \nPt (0.5 nm)  (a) (b) \nFIG. 1: (a) M−Hloops of [Co(0.2 nm)/Pt(0.2 nm)] 10, [Co(0.2 nm)/Pd(0.2 nm)] 10and [Co(0.2\nnm)/Pd(0.2 nm)/Pt(0.2 nm)] 10films. The op and ip denote M−Hloops with out-of-plane and\nin-plane magnetic fields, respectively. (b) High-resolution TEM image for the full stack of the\n[Co(0.2 nm)/Pd(0.2 nm)/Pt(0.2 nm)] 10film.5\nare consistently observed in various samples with different thicknesses, see Supplemental\nMaterial [37].\nWe now turn to our first-principles calculations in the basis of Density Functional Theory\n(DFT) approach to CoPt 2, CoPd 2, and CoPtPd model systems in order to understand\nthe origin of this behavior (see Supplemental Material [37] for the detail of calculations).\nThe structures resemble that of nonmagnetic noncentrosymmetric semiconductor BiTeI [38],\nwhich exhibits bulk Rashba spin-momentum coupling. The calculated EMCAper unit volume\nare summarized in the first row of Table I. From the positive EMCA it is clearly visible that\nall systems possess PMA. Secondly, while CoPt 2and CoPd 2bulk systems have comparable\nMCA energies of around 0.027 −0.028 meV/ ˚A3, we find stronger MCA energies of more than\n0.040 meV/ ˚A3for the CoPtPd system, in qualitative agreement with our experiment. If\nMCA arose entirely from each individual interface, the expected MCA for CoPtPd would be\n0.028/2+0.027/2 = 0.0275 meV/ ˚A3considering that there are two types of interfaces, Co/Pt\nand Co/Pd. This value is smaller than the calculated value of 0.043 meV/ ˚A3(see Table I) by\n0.0155 meV/ ˚A3, which is sizable. Other contribution apart from the interfacial effect should\ntherefore play a role in the larger MCA in CoPtPd system compared to the other cases,\nand it may be related to one of the following scenarios: (i) the difference in the number of\nvalence electrons in all unit cells [6]; (ii) the difference in the SOC strength [24]; (iii) the\nanisotropy of orbital magnetic moment, as proposed by Bruno [21]; (iv) the different orbital\nhybridization that occurs in the considered systems [22]; and/or finally (v) the structural\ninversion symmetry breaking along z[25, 26] that leads to the bulk Rashba-type splitting\n[38] in the CoPtPd case, compared to the other two systems.\nThe trivial scenarios (i) and (ii) can immediately be ruled out, as discussed in the Sup-\nplemental Material [37], due to the same total number of valence electrons in the unit cell,\nwhich is 29, and the presence of Pd in CoPtPd which has larger PMA than CoPt 2. Addi-\ntionally, the anisotropy of the orbital magnetic moment, induced as a direct consequence of\nSOC, can be considered by defining µanis\norb=µip\norb−µop\norb. From our fully self-consistent SOC\ncalculations, we obtain µanis\norbto be−0.025µBfor CoPt 2, which is larger than −0.023µBfor\nCoPtPd. Therefore, a larger MCA does not trivially correspond to a larger orbital moment\nanisotropy in the CoPtPd system here, thus making scenario (iii) inapplicable. Indeed, it\nhas been pointed out that an extra care should be taken when considering the Bruno model\nfor systems with strong spin-orbit coupling such as those containing 5 dtransition metal6\nelements [39].\nTo assess the relevance of scenario (iv), we calculated the Densities of States (DOSs)\nof these systems, as shown in Fig. S1 of the Supplemental Material [37]. The calculated\nmagnetic moments in all systems are found to be more than 2.0 µBfor Co and around 0.3\nµBfor both Pt and Pd, hence the MCA should be driven by Co. We additionally calculate\nthe relative contribution of each atomic layer to the PMA, by artificially switching on and\noff the SOC of the atoms. When the SOC of Co is switched off, in all considered cases\nthe MCA vanish, confirming the crucial role of Co moment to drive the MCA. In Table I,\nhowever, both CoPt 2and CoPd 2systems give comparable perpendicular MCA despite the\nnonnegligible difference in the Co −dbandwidth in both cases. This observation suggests\nthat the role of Co −Pt and Co−Pd orbital hybridizations in driving the PMA, as implied by\nthe scenario (iv), is not significant. The remaining scenario which might explain the origin\nof the enhanced MCA in CoPtPd is therefore the structural inversion symmetry breaking\nalongzdue to the presence of both Pt andPd layers sandwiching the Co layer, as suggested\nby scenario (v).\nAs in the case of nonmagnetic systems [28–30], the broken inversion symmetry in magnetic\nmaterials also results in the band splitting in the presence of SOC, as described by the\nRashba model. This effect has previously been demonstrated in Gd(001) magnetic surfaces\n[31]. We note however that the Rashba effect manifests differently in the nonmagnetic and\nmagnetic systems, as can be summarized in the following [31]: In nonmagnetic systems, the\npresence of Rashba interaction splits the otherwise degenerate up- and down-spin states.\nIn ferromagnetic systems, the degeneracy of the up- and down-spins are already lifted, and\nRashba interaction enhances/reduces the splitting depending on the sign of /vectork·(ˆz×/vector m). The\nresulting splitting is thus asymmetric with respect to /vectorkand the sign of asymmetry gets\nreversed when /vector mis reversed. The asymmetric splitting is most pronounced along the /vectork\ndirection parallel to ˆ z×/vector m.\nTherefore, in order to consider the role of structural asymmetry in the present model\nsystems, we first visualize the spin splitting by following the prescription introduced in\nprevious works [27, 31, 40]. We choose two different directions of in-plane magnetization,\ni.e. along + xand−xdirections. The results are presented in Fig. 2 for the CoPt 2system\nrepresenting the symmetric case and CoPtPd in which the inversion symmetry is broken.\nThe band structures are presented in Figs. 2a-2d along the ( kx,ky,kz)=(0,−1\n2,0)→(0,1\n2,0)7\n(a) (b) \n(c) (d) \n(e) (f) \nFIG. 2: Band structure along the kx=kz= 0, i.e. along the (0, −1\n2,0)→(0,1\n2,0) path, without spin-\norbit coupling, (a) and (b), and with SOC, (c) and (d); and the two dimensional ( kx,ky) in-plane\nFermi surface, i.e. at kz= 0, together with the contour map of k−dependent MCA energy within\nthis surface, (e) and (f). Left and right panels show the plots for CoPt 2and CoPtPd, respectively.\nThe m +x, m−x, and m zlabels indicate the magnetization along respectively the + x,−x, andz\ndirections. Likewise, E(m α) indicates the energy of αdirection-imposed magnetization.8\npath. Near the Fermi level, the band structures are mainly dominated by the Co minority-\nspin states, as shown Figs. 2a and 2b without SOC. The majority-spin bands, on the other\nhands, are largely dispersive, and dominated by the Pt and Pd bands. When the SOC is\nswitched on, the characteristic Rashba-type splitting for ferromagnetic system [31] emerges\nin the asymmetric CoPtPd case (Fig. 2d), but is absent in the band structure of CoPt 2case\n(Fig. 2c).\nThe Rashba-type splitting also manifests further in the distortion of Fermi contour (Figs.\n2e and 2f). The Fermi surface of CoPt 2does not show any shift in the ( kx,ky) plane for\nboth in-plane magnetization directions. On the other hands, the Fermi surface of CoPtPd is\nshifted towards positive kydirection when the magnetization is oriented along + xdirection,\nas indicated by the red arrow in Fig. 2f. Additionally, switching the magnetization direction\nalong the−xdirection, which is an opposite direction, gives a mirror symmetric Fermi\ncontour along the ky[27], i.e.\nE(m+x,+ky) =E(m−x,−ky), (1)\nas indicated by the blue arrow in Fig. 2f. Such mirror symmetry is disappearing along kx\nfor this particular magnetization direction, as pointed out by Grytsyuk, et al. [27]. The\ntwo-dimensional contour map of EMCA within the ( kx,ky) plane is fully symmetric along ky\nfor CoPt 2, in contrast to the nonsymmetric CoPtPd case.\nWe can proceed further by defining the EMCAas theE(m+x)−E(mz) orE(m−x)−E(mz).\nBoth definitions give the same total MCA energy as reported in Table I when the evaluation\nis done within the whole Brillouin zone. However, the k−dependentEMCA can provide an\nadditional insight, since it can be virtually decomposed into E+ky\nMCA andE−ky\nMCA, in which\nE±ky\nMCA =E(mx,±ky)−E(mz,±ky). By integrating the E±ky\nMCA within half of the Brillouin\nzone, i.e. within all ( kx,kz) space for each + kyand−ky, respectively, we obtain the results\nas reported in Table I. In this regard, one can define an even contribution\nEeven\nMCA=E+ky\nMCA+E−ky\nMCA, (2)\nwhich is exactly the EMCAwithin the whole Brillouin zone. The estimation of this contribu-\ntion to each model system is also shown in Table I. The small difference that occurs between9\nEMCA(SCF) and Eeven\nMCAclearly comes from the different SOC treatment in both cases, since\nthe evaluation of E+ky\nMCA andE−ky\nMCA cannot be done self-consistently. Interestingly, another\nquantity,\nEodd\nMCA=E+ky\nMCA−E−ky\nMCA, (3)\ncan also be defined. This quantity provides an estimation on the shift of the Fermi surface,\nthus the estimation of the degree of inversion symmetry breaking. As summarized in the last\nrow of Table I, Eodd\nMCA is zero for both symmetric CoPt 2and CoPd 2. TheEodd\nMCA in CoPtPd\nis on the other hand very large, implying that the system is highly nonsymmetric.\nTo gain an additional insight on the role of the broken inversion symmetry, one may use\na simple argument in which the MCA energy is expressed as EMCA=λ\nεuoin the second-order\nperturbation theory, where εuodenotes the energy gap between the occupied and unoccupied\nstates, while the λcontains the spin-orbit interaction between these states and depends on\nthe SOC coupling constant. The E+ky\nMCA andE−ky\nMCA can be given by E+ky\nMCA =1\n2λ\nεuo−∆εand\nE−ky\nMCA =1\n2λ\nεuo+∆ε, in which±∆εdenotes the widening or narrowing of the energy gap due\nto the departure from the symmetric band structure, and hence is absent (∆ ε= 0) in the\nsymmetric cases (Fig. 2c). The ±∆εis therefore only present in the asymmetric systems,\nand in the case of CoPtPd (Fig. 2d), ±∆εdescribes for instance the gap widening and\nnarrowing at ky=−0.2 andky= 0.2. The total contribution is nothing but the Eeven\nMCA in\nEq. (2) as summarized in Table I. However, in the specific cases, where the broken inversion\nsymmetry is present (∆ ε/negationslash= 0), the total MCA energy will be given by\nEeven\nMCA≈/parenleftbiggλ\nεuo/parenrightbigg1\n1−(∆ε\nεuo)2· (4)\nSince (∆ε\nε0)2is always positive, Eq. (4) indicates that the presence of ∆ εdue to the asym-\nmetry will lead to an increase of MCA. In other words, the enhancement of Eeven\nMCA due to\nthe inversion symmetry breaking is given by ∆ Eeven\nMCA =/parenleftBig\n∆ε\nεuo/parenrightBig2\nEeven\nMCA, showing that such\nmodification indeed occurs only in asymmetric systems, thus capturing the enhancement of\nMCA energy due to the spin-momentum coupling, as proposed recently [25, 26].\nAt this stage, it is appealing to consider the origin of Rashba-type splitting in the CoPtPd.\nReturning to the band structure of CoPtPd (with SOC, Fig. 2d), the splitting can be seen\nfor instance near the Fermi level around ky=±0.2 and at the energy of −1 eV around the Γ10\npoint. Furthermore, this splitting is likely to be induced by the Pt and Pd states. Around the\nFermi level, for example, the Rashba-type splitting at around ky=±0.2 is clearly dominated\nby the states indicated by the red broken rectangle in Fig. 2b. We further switched off the\nSOC of Co and we obtained an Eodd\nMCA of 0.193 meV/ ˚A3, resembling closely the Eodd\nMCA in\nTable I. Additionally, Eeven\nMCA vanishes, showing the significance of the SOC of Co for the\nMCA. On the other hands, when only the SOC of Co is maintained while switching off the\nSOC of both Pt and Pd, the Eodd\nMCA diminishes down to −0.010 meV/ ˚A3, confirming the\ncrucial role of the SOC of Pt and Pd to drive the Rashba-type splitting. Interestingly, the\nEeven\nMCA also becomes practically zero in the latter case, implying that despite Co being the\nmagnetic moment carrier, the SOC of Co alone is not sufficient to induce large MCA.\nIn real materials, interlayer mixing is likely to occur during the growth process. Such\nmixing can influence the MCA, especially for multilayer systems with small monolayer thick-\nness as in our cases. We have therefore performed the calculation to estimate the effect of\nthe intermixing (see Supplemental Material [37]). We found that although the calculated\nMCA is quantitatively altered by the intermixing, the perpendicular MCA of CoPtPd is\nconsistently larger than that of CoPt 2case, indicating that the effect of inversion symmetry\nbreaking on MCA is still effective in multilayers with interlayer mixing, although detailed\neffects of the structural disordering to the MCA require a further study.\nFinally we note the similarity between the structure of CoPtPd system in the present\nwork and that of the nonmagnetic noncentrosymmetric semiconductor BiTeI [38]. The\ncrystal structure of BiTeI consists of triangular network of single layers of Bi, Te, and I. In\nthis system, a giant bulk Rashba-type splitting has also been recently observed [38]. Our\nwork thus highlights the presence of bulk Rashba-type effect due to the broken inversion\nsymmetry in magnetic materials, as well as a possible direct consequence in terms of the\nenhanced MCA.\nIn summary, from our combined first-principles calculation and experimental results, we\nprovide for the first time an evidence for the asymmetric-structure-driven enhancement of\nPMA in the transition metal multilayers. In agreement with previous prediction on the MCA\nmodulation due to the change in the Rashba parameter [26], we show that the breaking of\ninversion symmetry does not only lead to a modification, but also to an enhancement, to\nthe PMA. While the PMA is realized through Co which carries the magnetic moments, the\nbreaking of inversion symmetry can significantly enhance the perpendicular MCA strength11\ndue to the presence of Pt and Pd with strong SOC. This work may provide a guideline on\nthe design of materials with strong PMA, as well as a suggestion to the investigation of\npossible enhancement of other SOC-related properties, such as the anomalous and spin Hall\neffect, due to the broken structural inversion symmetry.\nA.-M. P. was supported by the Japan Society for the Promotion of Science (JSPS) KAK-\nENHI Grant No. 15H05702. H.-W. L. acknowledges financial support from the National\nResearch Foundation of Korea (NRF, Grant No. 2018R1A5A6075964). K.-J. L. and K.-W.\nK. acknowledge the KIST Institutional Program (Projects No. 2V05750 and No. 2E29410).\nK.-W. K. also acknowledges financial support from the German Research Foundation (DFG)\n(No. SI 1720/2-1). Work was also in part supported by JSPS KAKENHI Grant No.\n16K05415, the Cooperative Research Program of Network Joint Research Center for Mate-\nrials and Devices, and Center for Spintronics Research Network (CSRN), Osaka University.\nComputations were performed at Research Institute for Information Technology, Kyushu\nUniversity.\n∗Electronic address: a.m.t.pradipto@gmail.com\n[1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln´ ar, M. L. Roukes,\nA. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001).\n[2] C. Chappert, A. Fert, and F. Nguyen Van Dau, Nat. Mater. 6, 813 (2007).\n[3] S. D. Bader and S. S. P. Parkin, Annu. Rev. Condens. Matter. Phys. 1, 71 (2010).\n[4] H. Ohno, Nat. Mater. 9, 952 (2010).\n[5] L. M. Falicov, D. T. Pierce, S. D. Bader, R. Gronsky, K. B. Hathaway, H. J. Hopster, D. N.\nLambeth, S. S. P. Parkin, G. Prinz, M. Salamon, et al., J. Mater. Res. 5, 1299 (1990).\n[6] G. H. O. Daalderop, P. J. Kelly, and F. J. 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B 87, 041301(R) (2013).Supplemental Material for “Enhanced perpendicular\nmagnetocrystalline anisotropy energy in an artificial magnetic\nmaterial with bulk spin-momentum coupling”\nAbdul-Muizz Pradipto,1, 2,∗Kay Yakushiji,3Woo Seung Ham,1Sanghoon\nKim,1Yoichi Shiota,1Takahiro Moriyama,1Kyoung-Whan Kim,4, 5\nHyun-Woo Lee,6Kohji Nakamura,2Kyung-Jin Lee,7, 8and Teruo Ono1\n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan\n2Department of Physics Engineering, Mie University, Tsu, Mie 514-8507, Japan\n3National Institute of Advanced Industrial Science and\nTechnology (AIST), Tsukuba, Ibaraki 305-8568, Japan\n4Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany\n5Center for Spintronics, Korea Institute of Science and Technology, Seoul 02792, Korea\n6Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea\n7Department of Materials Science and Engineering,\nKorea University, Seoul 02841, Korea\n8KU-KIST Graduate School of Converging Science and Technology,\nKorea University, Seoul 02841, Korea\nI. LAYER THICKNESS DEPENDENCE OF EFFECTIVE\nMAGNETOCRYSTALLINE ANISOTROPY CONSTANTS\nA series of samples representing the repetitive [Co/Pt] n, [Co/Pd] n, and [Co/Pd/Pt] nmul-\ntilayer structures have been fabricated with various thicknesses according to the procedure\ndescribed previouslyS1. In Fig. S1a, we show the dependence of effective magnetocrystalline\nanisotropy (MCA) values, i.e. Keff=HkMs/2, as functions of Co layer thickness. While\ntheKeffof [Co/Pt] nmultilayers shows an optimum value at Co thickness of around 0.4 nm,\ntheKeffvalues of [Co/Pd] nand [Co/Pd/Pt] nmultilayers tend to keep increasing with the\ndecreasing Co thickness, hence no optimum Co thickness have been implied for [Co/Pd] n\nand [Co/Pd/Pt] nmultilayes. We therefore consider 0.4 nm to be the optimum thickness of\nCo layer and fix the Co thickness to 0.4 nm to see further the behavior of Keffon the various\nheavy metal layer thicknesses, as summarized in Fig. S1b.\nThe different behaviors of Keffin [Co/Pt] nand [Co/Pd] nmultilayer with the decreasing\nCo thicknesses as shown in Fig. S1a are in accordance with previous works on the sputtered\nmultilayersS2,S3. In [Co/Pd] nmultilayer, the absence of peak in the Keffat various Co\nthicknesses was also shownS3. On the other hands, an optimum Co thickness of about\n4−6˚A in Co/Pt multilayer has also been reportedS2, which is very similar to our results,\ndespite the absence of multilayer repetition. It turns out that the behavior of [Co/Pd/Pt] narXiv:1809.00801v3  [cond-mat.mtrl-sci]  22 Apr 20192\n-5051015202530\n0 0.2 0.4 0.6 0.8 1 1.2 1.4Keff(Merg/cm3)\ntCo(nm)Co(t nm)/Pt(0.4 nm)\nCo(t nm)/Pd(0.4 nm)\nCo(t nm)/Pd(0.4 nm)/Pt(0.4 nm)\nCo/Pt(0.4)Co/Pt(0.6)Co/Pt(0.8)\nCo/Pd(0.4)Co/Pd(0.6)Co/Pd(0.8)\nCo/Pd(0.2)/Pt(0.2)Co/Pd(0.3)/Pt(0.3)Co/Pd(0.4)/Pt(0.4)\n0246810121416\n0 0.2 0.4 0.6 0.8 1Keff(Merg/cm3)\ntPd/Pt (nm)Co(0.4 nm)/Pt(t nm)\nCo(0.4 nm)/Pd(t nm)\nCo(0.4 nm)/Pd(t nm)/Pt(t nm)(a) (b)\nFIG. S1: (a) Co thickness dependence, with fixed thickness of heavy metal layers of 0.4\nnm, and (b) heavy metal thickness dependence, with fixed Co thickness of 0.4 nm, of the\nmagnetocrystalline anisotropy (MCA) of [Co/Pt] n, [Co/Pd] n, and [Co/Pd/Pt] n\nsuperlattices.\nis resembling [Co/Pd] nmultilayer in this regard.\nIn Fig. S1b, all Keffvalues are visibly enhanced with the increasing heavy metal thick-\nnesses. However, if one compares the Keffof samples with the same thickness of heavy\nmetals, e.g. [Co(0.4)/Pt(0.4)] n, [Co(0.4)/Pd(0.4)] n, and [Co(0.4)/Pd(0.2)/Pt(0.2)] nmulti-\nlayers, an important behavior can be discerned. As discussed in the main text, if only the\ninterfacial effect is crucial to determine the MCA, the expected value of Keffof [Co/Pd/Pt] n\nwould be the average of those of [Co/Pd] nand [Co/Pt] n. However, in samples with same\nheavy metal thicknesses, the MCA constant of the asymmetric system [Co/Pd/Pt] nis always\nlarger than those of the other systems, even when an optimum Co thickness in [Co/Pt] nhas\nbeen assumed, with visibly larger Keffthan the average values of [Co/Pd] nand [Co/Pt] n.\nA consistently larger perpendicular MCA in the [Co/Pd/Pt] ncase should therefore sup-\nport the crucial role of bulk spin-orbit coupling effect in driving the enhanced MCA in the\nasymmetric multilayer.\nII. FIRST-PRINCIPLES CALCULATIONS\nOur calculations employ the Generalized Gradient Approximation (GGA)S4functional as\nimplemented in the Full-Potential Linearized Augmented Plane Wave (FLAPW) codeS5–S7.\nThe in-plane lattice constant has been fixed to the optimized value of bulk Pt, but the out-of-\nplane atomic positions and lattice constants, i.e. normal to the layer plane, have been relaxed\nfor all systems. The MCA energy EMCA has been defined in the calculations as EMCA =\nEip−Eop, whereEip(Eop) refers to the total energy of the in- (out-of-)plane magnetization,\nimplying that positive EMCA indicates the preferred out-of-plane magnetization. The SOC3\nyz\nxyCoPt\nPd\nSide view Top view\nyz\nxyCoPt\nPd\nSide view Top view(a) (b) \nFIG. S2: (a) The model crystal structure of CoPtPd multilayer system. For CoPt 2\n(CoPd 2) model system, the Pd (Pt) layer is replaced by another Pt (Pd) atomic layer. (b)\nThe Density of States (DOS) of CoPt 2, CoPtPd, and CoPd 2systems.\nhas been included self-consistently, so that full relaxation of the charge and spin densitites is\nallowed upon the treatment of SOC. The MCA energies obtained in this self-consistent field\n(SCF) manner are indicated by EMCA (SCF) in Table 1 of the main text. Our convergence\ntest calculations show that the MCA energies do not vary larger than 10−4meV/ ˚A3after\n75×75×45k−point mesh which gives a total of 253125 k−points in the Brillouin zone.\nIn order to model the multilayers, three different bulk systems have been chosen in the\ncalculations so that the unit cells contain three layers of Pt/Co/Pt (referred to hereafter as\nCoPt 2model system), Pd/Co/Pd (referred as CoPd 2), and Pt/Co/Pd (referred as CoPtPd)\nstacked in an fcc(111) pattern, such that the Co atoms are sandwiched between 2 Pt layers,\n2 Pd layers, or between a Pt layer and a Pd layer, respectively (see Fig. S2a).\nScenario (i), as discussed in the main text, for explaining the different MCA on the basis\nof the difference in electron numberS8can immediately be ruled out since all systems have\nthe same total number of valence electrons in the unit cell, which is 29. Furthermore, in\norder to discuss scenario (ii) which is based on the SOC strengthS9, we notice that in CoPt 2\nunit cell there are two Pt, i.e. 5 d, atoms with a SOC strength of about 320 meV. On the\nother hands, CoPtPd contains Pd, a 4 delement, whose SOC strength is around 104 meV,\ni.e. three times weaker than the 5 datom. The other system, CoPd 2, does not even contain\nPt in the unit cell. However, both CoPt 2and CoPd 2have equally strong MCA and CoPtPd\nhas decidedly the largest perpendicular MCA among these systems, hence a na¨ ıve view just4\nPt \nCo1-xPtx \nPd Pt \nCo1-xPtx \nPt Pt \nPt \n Pt \nPt \nPt1-xCox \nCo \nPd Pt1-xCox \nCo \nPt Pt \nPt \n Pt \nPt \n(a) (b) (c) (d) \nFIG. S3: The structural disordering models: (a) and (b) Pt 1−xCoxstructures to model the\ncases where some Co amounts mix within the Pt monolayer of CoPtPd and CoPt 2\nsystems, respectively; (c) and (d) Co 1−xPtxstructures to describe the mixing of Pt in the\nCo monolayer of CoPtPd and CoPt 2systems, respectively.\nby considering the SOC strength as in scenario (ii) does not hold in this case. The Density\nof States of these systems, relevant for the discussion of scenario (iv) based on the different\norbital hybridization in the main text, are shown in Fig. S2b. The majority- and minority-\nspin features of the DOSs suggest that the magnetic moments are mainly carried out by\nCo.\nIII. STRUCTURAL DISORDERING\nTo consider the effects of structure disordering during the synthesis of the materials to the\nMCA, we generate supercells by doubling the unit cells of CoPtPd and CoPt 2to represent\nrespectively the asymmetric and the symmetric structures along both in-plane directions. We\nfurther introduce the in-plane alloying between Co and Pt, as depicted in Fig. S3. Due to the\nincrease of the system sizes, we only attempt the second-variationalS10,S11, instead of the self-\nconsistent, treatment of the SOC for the calculation of MCA. The results are summarized in\nTable S1. As can be seen for the calculated MCA values of the ideal structures, i.e. without\ndisorder, our second-variational SOC treatment has reproduced reasonably well the MCA\nenergy obtained from the self-consistent treatment for the unit cell (Table 1 of the main\ntext).\nWhen the layer intermixing is introduced, in all systems, the MCA is visibly affected by\nthe composition of Co and Pt in the layer. One may notice that the introduction of certain\namounts of Co composition to the Pt layer (i.e. the Pt 1−xCoxstructures in Fig. S3b) clearly\ndestroys the inversion symmetry on the otherwise symmetric CoPt 2. Therefore, while the\nperpendicular MCA of CoPtPd is still larger than that of CoPt 2, discussing the difference5\nTABLE S1: The MCA energy EMCA per unit volume (in meV/ ˚A3) obtained from the\nsupercell calculations for the disordered structures, as depicted in Fig. S3.\nSystem CoPtPd CoPt 2\nIdeal structures 0 .042 0.030\nPt0.75Co0.25(Figs. S3a and S3b) 0 .027 0.024\nCo0.75Pt0.25(Figs. S3c and S3d) 0 .043 −0.007\nof MCA in terms of inversion symmetry breaking would become irrelevant. On the other\nhand, the inversion symmetry may be maintained in the Co 1−xPtxstructures (Fig. S3d),\ni.e. when some Pt amounts mix within the Co monolayer. When a Pt atom intermixes\nwithin the Co layer, a larger spin moment of about 0.39 µBis induced at the heavy metal\natom, compared to 0.35 µBand 0.37µBfor the Pt atoms at the ideally ordered structures of\nCoPtPd and CoPt 2, respectively. In such cases, the calculated MCA is quantitatively altered\nby the intermixing, with a much more pronounced effect on the CoPt 2case. One may also\nnote that in the present calculations, the mixing of Co and Pt with x= 0.25 is rather\nlarge. But even though it can significantly reduce the sample quality, a general trend is\nqualitatively maintained, i.e. the larger perpendicular MCA in CoPtPd than that in CoPt 2.\nQuantitatively, a modeling that requires huge unit cells with many configurations inducing\ndifferent atoms and vacancies or a different theoretical frameworkS12may be desirable, which\nis beyond the scope of the present study.\n∗Electronic address: a.m.t.pradipto@gmail.com\n[S1] K. Yakushiji, T. Saruya, H. Kubota, A. Fukushima, T. Nagahama, S. Yuasa, and K. Ando,\nAppl. Phys. Lett. 97, 232508 (2010).\n[S2] C. L. Canedy, X. W. Li, and G. Xiao, Phys. Rev. B 62, 508 (2000).\n[S3] T. Kato, Y. Matsumoto, S. Kashima, S. Okamoto, N. Kikuchi, S. Iwata, O. Kitakami, and\nS. 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B 43, 10335 (1991)."    },    {        "title": "1809.05055v1.Single_layer_antiferromagnetic_semiconductor_CoS2_with_the_pentagonal_structure.pdf",        "content": "Single-Layer Antiferromagnetic Semiconductor CoS 2with the Pentagonal Structure\nLei Liu, Immanuella Kankam, and Houlong L. Zhuang\u0003\nSchool for Engineering of Matter Transport and Energy,\nArizona State University, Tempe, AZ 85287, USA\n(Dated: September 14, 2018)\nStructure-property relationships have always been guiding principles in discovering new materials.\nHere we explore the relationships to discover novel two-dimensional (2D) materials with the goal\nof identifying 2D magnetic semiconductors for spintronics applications. In particular, we report a\ndensity functional theory + Ustudy of single-layer antiferromagnetic (AFM) semiconductor CoS 2\nwith the pentagonal structure forming the so-called Cairo Tessellation. We \fnd that this single-\nlayer magnet exhibits an indirect bandgap of 1.06 eV with light electron and hole e\u000bective masses\nof 0.03 and 0.10 m0, respectively, which may lead to high carrier mobilities. The hybrid density\nfunctional theory calculations correct the bandgap to 2.24 eV. We also compute the magnetocrys-\ntalline anisotropy energy (MAE), showing that the easy axis of the AFM ordering is out of plane\nwith a sizable MAE of 153 \u0016eV per Co ion. We further calculate the magnon frequencies at di\u000berent\nspin-spiral vectors, based on which we estimate the N\u0013 eel temperatures to be 20.4 and 13.3 K using\nthe mean \feld and random phase approximations, respectively. We then apply biaxial strains to\ntune the bandgap of single-layer pentagonal CoS 2. We \fnd that the energy di\u000berence between the\nferromagnetic and AFM structures strongly depends on the biaxial strain, but the ground state\nremains the AFM ordering. Although the low critical temperature prohibits the magnetic applica-\ntions of single-layer pentagonal CoS 2at room temperature, the excellent electrical properties may\n\fnd this novel single-layer semiconductor applications in optoelectronic nanodevices.\nI. INTRODUCTION\nIn the book Thank You for Being Late by the renowned\nPulitzer Prize winner Thomas Friedman,1Moore's law\nrepresenting the technology is deemed as one of the three\nlargest forces|the other two being the market refer-\nring to globalization and mother nature alluding to cli-\nmate change and biodiversity loss|which are changing\nour planet at a breathtaking pace. For more than four\ndecades, Moore's law has accurately predicted that the\nnumber of transistors in an integrated circuit doubles ev-\nery two years. But as the size of the circuit keeps shrink-\ning, it is expected that Moore's law will come to an end in\nthe foreseeable future. A road map for addressing these\nchallenges is to make use of magnetic materials at the\nnanoscale by utilizing spin instead of charge properties\nof electrons.2As such, atomically thin two-dimensional\n(2D) magnets are promising candidate materials to keep\nMoore's law alive.\n2D magnets are critical building components for future\ngenerations of computers that rely on the usage of spin\n\feld-e\u000bect transistors. Experimental e\u000borts have been\nexpended to obtain 2D magnets. 2D magnets such as\nCrI3,3Cr2Ge2Te6,4and Fe 2GeTe 35have recently been\nobtained in experiments. It is expected that such a list\nof 2D magnets will grow longer.\nMost of the above-mentioned, cutting-edge 2D mag-\nnets adopt hexagonal structures. Recently, 2D materials\nwith pentagonal, corrugated structures have attracted in-\ntense attention due to a wealth of exotic physical prop-\nerties and potential applications that are associated with\nthe pentagonal structure. For example, Sn X2(X= S, Se,\nand Te) consisting entirely of pentagonal rings is quan-\ntum spin Hall insulator that produces sizable nontrivialgaps and maintain robust band topology.62D pentagonal\nSnX2also present promising applications in low-power-\nconsuming electronic devices. More recently, Akinola et\nal.show that 2D pentagonal PdSe 2possess similar in-\ndirect and direct bandgaps (1.30 and 1.43 eV, respec-\ntively), useful for optoelectronic applications. Moreover,\n2D pentagonal PdSe 2exhibits high electron mobility and\nair stability, making it a candidate for \feld-e\u000bect tran-\nsistors applications.7,8\nOther examples of recently predicted 2D pentagonal\nmaterials with the AB2formula include B 2C,9B2N4,10\nB4N2,10CN2,11SiC2,12{14and SiN 2.14The structure-\nproperty relationships are also manifested in these 2D\nmaterials with pentagonal, buckling structures. For in-\nstance, 2D pentagonal B 2C possesses in-plane structural\n\rexibility. It transforms from a buckled structure to\na planar structure under biaxial tensile strains, with\nthe bandgaps reduced from 2.28 eV to 0.06 eV, allow-\ning for potential applications in \rexible and stretchable\nelectronics.9In addition to the AB2formula, 2D pen-\ntagonal materials also assume other chemical formulae\nsuch asABandAB3. Transition-metal borides/carbides\n(TMB/Cs) with the chemical formula of ABhave been\nintensively studied. Shao et al. predicted pentago-\nnal TMB/Cs (TaB, WB, ZrC, HfC, and TaC) to be\nthermodynamically stable, among which WB and ZrC\nshow substantial performance for the hydrogen evolution\nreaction.15\nInspired by the geometries of the existing 15 types of\nconvex pentagons that can tessellate a plane without cre-\nating a gap or overlap, we recently combined these pen-\ntagonal geometries and density functional theory (DFT)\ncalculations to predict novel 2D materials.16,17For exam-\nple, we discovered a hidden pattern of pentagons called\nthe Cairo tessellation in a group of bulk materials witharXiv:1809.05055v1  [cond-mat.mtrl-sci]  13 Sep 20182\n(b)(a)\nabc ab\ncCoS\nFIG. 1. (a) A sketch of the Cairo tessellation formed from\ntype 2 pentagons. (b) Top and side views of a 3 \u00023 supercell\nof single-layer CoS 2adopting the pentagonal structure.\nthe pyrite structure, having a chemical formula of XY2\nand the space group pa\u00163. Figure 1 illustrates the Cairo\ntessellation resulted from tessellating type 2 pentagons\nin a plane. We used single-layer PtP 2as an example\nand predicted it to exhibit a completely planar pentago-\nnal structure with a direct band gap.18In this work, we\naim to computationally identify a single-layer pentago-\nnal material with a magnetic ordering suitable for spin-\ntronics applications. Because CoS 2has the same crystal\nstructure as PtP 2, and a combination of Co with another\nelement could lead to a 2D magnet, single-layer pentago-\nnal CoS 2becomes a natural candidate for this theoretical\nstudy.\nII. METHODS\nWe perform the DFT calculations using the Vi-\nenna Ab-initio Simulation Package (VASP, version\n5.4.4).19We also use the Perdew-Burke-Ernzerhof (PBE)\nfunctional for approximating the exchange-correlation\ninteractions.20The standard PBE version of the potential\ndatasets for Co and S generated based on the projector-\naugmented wave (PAW) method are used for describing\nthe electron-ion interactions.21,22We choose the plane\nwaves with the kinetic cuto\u000b energy below 550 eV to ap-\nproximate the total electron wave function. Figure 1(b)\nillustrates a 3 \u00023\u00021 supercell of single-layer CoS 2and\neach unit cell consists of two formula units. We use a \u0000-\ncentered 12 \u000212\u00021k-point grid for the integration inthe reciprocal space.23We additionally use an e\u000bective U\nparameterUe\u000bof 3.32 eV with the Dudarev method24to\ntreat thedorbitals of Co atoms. This Ue\u000bparameter is\ntaken from Ref. 25 and used in the Materials Project for\nmany compounds containing Co.26We create a surface\nsupercell model of single-layer CoS 2, which is essentially\na single-layer surface slab of the CoS 2(001) surface. The\nvacuum spacing of the surface slab is set to 18.0 \u0017A to\navoid image interactions between the monolayers. VASP\nfully optimizes the in-plane lattice constants along with\natomic coordinates in all the three directions until the\nresidual Hellman-Feynman forces are smaller than 0.01\neV/\u0017A.\nIII. RESULTS AND DISCUSSION\nWe \frst determine the ground-state magnetic order-\ning of single-layer pentagonal CoS 2. Because Co ions in\nthis single-layer material adopt the d5con\fguration in a\nnearly square planar crystal \feld, we consider three pos-\nsible spin states: high-spin (HS), intermediate-spin (IS),\nand low-spin (LS). The three spin states are distinguished\nby setting the initial magnetic moments to 5.0, 3.0, and\n1.0\u0016B(\u0016B: Bohr magneton), respectively, in the VASP\ncalculations. For each spin state, we consider two pos-\nsible magnetic orderings: antiferromagnetic (AFM) and\nferromagnetic (FM). For comparison, we also perform a\nnon-magnetic (NM) calculation, where the magnetic mo-\nments of Co ions are not taken into account. Table I\ncompares the optimized total energies, the in-plane lat-\ntice constants, and the total magnetic moments of the six\nspin states and of the NM state. We observe that the LS\nAFM state is the ground state. The optimized in-plane\ngeometry is nearly a square with the lattice constant b\n(5.43 \u0017A) slightly larger than a(5.34 \u0017A). The net magneti-\nzation is zero and the two Co ions in the unit cell possess\nTABLE I. Optimized in-plane lattice constants aandb(in\n\u0017A), energies with reference to LS-AFM (low spin and antifer-\nromagnetic) state \u0001 E( in meV per formula unit), band gaps\nEg(in eV), and magnetic moments M(in\u0016Bper Co ion) of\nsingle-layer CoS 2with di\u000berent spin states. HS: high spin;\nIS: intermediate spin; LS: low spin; AFM: antiferromagnetic;\nFM: ferromagnetic; NM: non-magnetic. All these results are\nobtained from the PBE + U(Ue\u000b= 3.32 eV) calculations.\nSpin state \u0001 E a b m E g\nHS-AFM 101.17 5.78 5.78 0.00 1.41\nHS-FM 252.91 5.75 5.76 2.34 Metal\nIS-AFM 0.00 5.34 5.43 0.00 1.06\nIS-FM 11.99 5.33 5.43 1.00 1.97\nLS-AFM 0.00 5.34 5.43 0.00 1.06\nLS-FM 11.99 5.33 5.43 1.00 1.97\nNM 663.45 5.40 5.40 0.00 Metal3\n0100200300400500Frequency (cm-1)\nГXM YM ГГXYM\nFIG. 2. Predicted phonon spectrum of single-layer pentagonal\nCoS 2. The inset rectangle illustrates the \frst Brillouin Zone\nwith the high-symmetry phonon qpoints denoted.\nanti-parallel magnetic moments with the same magnitude\nof about 1.0 \u0016B. Unlike single-layer PtP 2with a planar\nCairo tessellation,16single-layer pentagonal CoS 2with\nthe AFM ordering displays a buckled pentagonal struc-\nture as shown in Fig. 1(b). Each Co ion is surrounded by\nfour S ions, and the Co-S bond length is 2.20 or 2.22 \u0017A.\nThe sublayer of Co ions is sandwiched between the top\nand the bottom sublayers of S ions; the interlayer dis-\ntance of the Co and S sublayers is 0.54 \u0017A. The energy of\nthe FM state of the LS con\fguration is higher than the\nground state by around 12.0 meV per formula unit. We\nalso \fnd that the IS AFM and FM states are relaxed to\nthe LS AFM and FM states, respectively. By contrast,\nthe HS AFM and FM states are optimized into the nearly\nIS state with the magnetic moment of 2.34 \u0016B. The NM\nstate exhibits the highest energy, showing the importance\nof considering spin polarization in the calculations.\nTo con\frm the dynamic stability of single-layer pentag-\nonal CoS 2structure with the AFM ordering, we calculate\nits phonon spectrum with the force information obtained\nfrom VASP calculations for 3 \u00023\u00021 supercells followed\nby post-processing calculations via Phonopy.27Figure 2\nshows the calculated phonon spectrum. Although there\nare some negligibly small imaginary frequencies near the\n\u0000 point due to the translational invariance ( i:e:, the\nacoustic sum rule), the absence of imaginary modes in\nthe otherqpoints throughout the Brillouin zone shows\nthe dynamic stability of single-layer pentagonal CoS 2.\nThe pentagonal structure of single-layer CoS 2leads to\nthe orbital-resolved band structure illustrated in Fig. 3.\nAs can be seen, single-layer pentagonal CoS 2is a semi-\nconductor with an indirect bandgap of 1.06 eV|we also\ncompute the bandgaps of the other spin con\fgurations\nand the band gaps are listed in Table I. The orbitals at\nthe valence band maximum (VBM) are from the contri-\nbutions of mixed dorbitals of Co atoms and porbitals of S\natoms. The conduction band minimum (CBM) is located\nat theXpoint. The top valence band near the Ypoint\nhas almost the same energy as the CBM, but slightly\nsmaller. The band dispersions near the CBM and VBM\n-4-2024Energy (eV)\nГXMГ MYCo-d S-p\nГXYMFIG. 3. Orbital-resolved band structure of single-layer pen-\ntagonal CoS 2calculated with the PBE + U(Ue\u000b= 3.32 eV)\nmethod. The valance band maximum is set to zero.\n020406080100120140160 MAE (μeV/Co ion)ab\nbcac\nRotation angle (o)03060 90120 150180 210240 270300 330360\nFIG. 4. Magnetocrystalline anisotropy energy (MAE) of\nsingle-layer pentagonal CoS 2with the spins of the two Co\nions antiparallelly oriented in di\u000berent directions in the ab,\nbc, andacplanes.\nboth show parabolic relationships, indicating that the\ncarriers resulted from doping exhibit a two-dimensional\nelectron/hole gas behavior and that the electron/hole ef-\nfective masses are nearly isotropic. We calculate the elec-\ntron e\u000bective mass m\u0003\neat the CBM as 0.03 m0(m0: the\nelectron rest mass) and the hole e\u000bective mass m\u0003\nhat the\nVBM as 0.10 m0. These masses are much lighter and\nalso less anisotropic than the masses in other popular\nsingle-layer semiconductors such as molybdenum disul-\nphide (m\u0003\ne= 0.34m0;m\u0003\nh= 0.46m0)28and black phos-\nphorene (m\u0003\ne\u0019m\u0003\nh\u00190:30m0).29Such small e\u000bective\nmasses may lead to high carrier mobilities. We also use\nthe HSE06 hybrid density functional30to correct the pos-\nsibly underestimated bandgap of single-layer pentagonal\nCoS 2with the PBE + U(Ue\u000b= 3.32 eV) method. In-\ndeed, the calculated HSE06 bandgap (2.24 eV) is nearly\ntwice as wide as the PBE + Ubandgap (1.06 eV). The4\nHSE06 functional also con\frms the ground state is the\nAFM ordering with the energy lower than that of the FM\nordering by 8.45 meV per formula unit.\nThe occurrence of the AFM ordering in single-layer\nCoS 2follows the Goodenough-Kanamori rules,31,32which\nstate that the type of magnetic ordering due to the su-\nperexchange interactions between two metal ions bridged\nby a nonmetal ion depends on the metal-nonmetal-metal\nbond angle. If this angle is near 90\u000e, the resulting mag-\nnetic ordering is FM; if the angle is greater than 90\u000e\nthe corresponding ordering is AFM. The calculated Co-\nS-Co angle is 118.5\u000e. We therefore suggest the mecha-\nnism for the AFM ordering is due to superexchange in-\nteractions, where the Co-Co ions are far away from each\nother, and the interactions between these two ions must\nbe bridged by the non-metallic S ions. An intuitive phys-\nical picture33of this superexchange interactions is that\none electron in one of the dorbitals (e:g:, spin-updz2) of a\nCo ion overlaps with one of the porbitals (e:g:, spin-down\npz) of the bridging S ion to form a \u001bbond. This leads\nto a remaining spin-up electron in the pzorbital overlap-\nping with the spin-down dz2orbital in another Co ion.\nThe net e\u000bect of these superexchange interactions is the\nAFM ordering.\nTo identify the easy axis of single-layer pentagonal\nCoS 2, we compute the magnetocrystalline anisotropy en-\nergy (MAE) with the torque method as implemented in\nVASP.34Figure 4 displays the variations of the MAEs\nwith the spin orientations in the ab,bc, andacplanes.\nIn the ac plane, the MAE is almost negligible while the\nMAEs in the other two planes are similar and much\nstronger. The zero rotation angle in Fig. 4 refers to the\nspin con\fguration where the spin axes of the two Co ions\nare parallel and anti-parallel to the caxis, respectively.\nThe positive MAEs at the other rotation angles therefore\nshow that the easy axis for the AFM ordering of single-\nlayer pentagonal CoS 2is thecaxis. The highest MAE\n(153\u0016eV/Co ion) occurs when the spin axes are along\nthebdirection.\nWe also calculate the magnon spectrum of single-layer\npentagonal CoS 2using the frozen magnon method.35We\nset a small polar angle 3.0\u000eas suggested in Ref. 36. Fig-\nure 5 shows a magnon spectrum along the high-symmetry\nqpoint path. We also use a 5 \u00025\u00021q-point grid of spin-\nspiral vectors in the \frst Brillouin Zone (BZ) to compute\nthe magnon energies at these qvectors. We then calcu-\nlate the N\u0013eel temperatures via the mean \feld and random\nphase approximations ( TMFA\nN andTRPA\nN) written as36\nkBTMFA\nN =M\n3\u00141\nNBZX\nq=0!q\u0015\n; (1)\nand\nkBTRPA\nN =M(N\u00001)\n3\u0014BZX\nq6=01\n!q\u0015\u00001\n; (2)\nwherekBis the Boltzmann constant, and Nis the to-\ntal number of qvectors, i.e.,N= 25. The calculated\n024681012 Energy (meV)\nΓXM YM ΓFIG. 5. Predicted magnon spectrum of single-layer pentago-\nnal CoS 2using the PBE + U(Ue\u000b= 3.32 eV) method. The\nopen circles represent our calculated data; the solid red line\nis used only to aid the view of the magnon dispersions.\nkBTMFA\nN andkBTRPA\nN are 20.4 and 13.3 K, respectively,\nexcluding the practical magnetic applications of single-\nlayer pentagonal CoS 2at ambient conditions. This issue\nof low critical temperature seems to be a common chal-\nlenge faced by 2D magnets ( e.g., the Curie temperature\nof CrI 3is merely around 45 K),3requiring joint experi-\nmental and theoretical e\u000borts in the future work.\nThe low, predicted critical temperatures of single-layer\npentagonal CoS 2are due to the weak superexchange in-\nteractions between Co ions. To estimate the strength\nof these superexchange interactions, we use the energy\ndi\u000berence approach to derive the exchange integral J1\nbetween the nearest-neighboring Co ions in a unit cell.\nBy mapping the magnetic interactions between Co ions\nonto the Heisenberg Hamiltonian,37we calculate J1as\nJ1=EFM\u0000EAFM\n8M2; (3)\nwhereMdenotes the dimensionless magnitude ( M= 1.0)\nof the magnetic moment of a Co ion. EFMandEAFM are\nthe energies of CoS 2unit cell with FM and AFM con\fg-\nurations, respectively. The calculated J1from Eq. 3 and\nusing the energy di\u000berence shown in Table I is 3.01 meV,\nwhich is much smaller than that of other predicted 2D\nmagnets such as Co 2S2with theJ1of 58.7 meV.38\nStrain engineering has been widely used to tune the\nstructural and electrical properties of single-layer ma-\nterials, o\u000bering an important degree of \rexibility.39{41\nWe apply in-plane biaxial strains ( \u000faa=\u000fbb) to single-\nlayer pentagonal CoS 2. We \frst examine whether the\nstrains induce a transition in the AFM ordering to the\nFM ordering. Figure 6 shows the energy di\u000berence of\nsingle-layer pentagonal CoS 2with the FM and AFM or-\nderings under the biaxial stains ranging from -4% to 4%\nat an incremental step of 0.5%. We observe that the en-\nergy di\u000berence remains positive in the range of applied5\n1012141618EFM-EAFM (meV/Co ion)\n-4 -2\nBiaxial strain (%)0 2 4\nFIG. 6. Energy di\u000berences between single-layer pentagonal\nCoS 2with the ferromagnetic and antiferromagnetic orderings\nunder biaxial strains ranging from -4% to 4%.\n0.40.60.81.01.21.41.6Eg  (eV)\nBiaxial strain (%)0 2 4 -2 -4\nFIG. 7. Bandgaps of single-layer pentagonal CoS 2under bi-\naxial strains ranging from -4% to 4%.\nbiaxial strains, showing that the AFM ground state is\nuna\u000bected by the strains. Furthermore, the energy dif-\nference increases with the increasing tensile strains and\ndecreases with the increasing compressive strains. Ac-\ncording to this trend, one probably needs to apply an\nextremely large compressive strain to turn the AFM to\nthe FM ordering.\nThe bandgap size of single-layer pentagonal CoS 2is\nsigni\fcantly a\u000bected by the biaxial strains. Figure 7\nshows that the bandgaps decrease almost linearly from\n1.51 eV at the maximum compressive strain ( \u000faa=\u000fbb=\n-4%) to 0.52 eV at the maximum tensile strain ( \u000faa=\n\u000fbb= 4%), displaying the tunability of -0.12 eV per 1%\nof biaxial strain. The trend of the bandgaps with the\nbiaxial strains|Namely, the bandgap decreases with in-\ncreasing bond lengths|has been typically observed in\nsingle-layer ionic materials such as BN.42,43The bandgap\ntype of single-layer pentagonal CoS 2under any strain re-\n-4-2024Energy (eV)-4-2024Energy (eV)\nГ X M MYГεaa = ε bb= -4%(a)\n(b)\nεaa = ε bb= 4%FIG. 8. Band structures of single-layer pentagonal CoS 2un-\nder biaxial strains of (a) -4% and (b) 4% calculated with the\nPBE +U(Ue\u000b= 3.32 eV) method. The valance band max-\nima are set to zero.\nmains indirect , as can be seen from the band structures\ndisplayed in Fig.8 for the biaxial strains of -4% and 4%.\nIV. CONCLUSIONS\nIn summary, we have predicted a new single-layer AFM\nsemiconductor CoS 2via surveying structure-properties\nrelationships for 2D materials. The predicted single-\nlayer material exhibits a pentagonal structure forming\nthe Cairo Tessellation. Our DFT+ Ucalculations also\nshow that single-layer pentagonal CoS 2is an indirect\nbandgap semiconductor with the bandgap of 1.06 eV.\nThe more accurate hybrid density functional corrects this\nbandgap to 2.24 eV, within the visible light range, indi-\ncating potential energy-related applications of this novel\nsemiconductor. We also found that single-layer pentag-\nonal CoS 2exhibits a sizable MAE with the easy axis\nalong thec-axis. We further applied the frozen magnon\nmethod to compute magnon frequencies and the N\u0013 eel\ntemperatures using the MFA and RPA. The resulting\nN\u0013eel temperatures are much smaller than room temper-6\nature, make the material not suitable for practical ap-\nplications. 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Hennig, Applied Physics Letters\n101, 153109 (2012)."    },    {        "title": "1810.00253v1.Micromagnetic_Simulations_Study_of_Skyrmions_in_Magnetic_FePt_Nanoelements.pdf",        "content": "Micromagnetic Simulations Study of Skyrmions in Magnetic FePt\nNanoelements\nLeonidas N. Gergidis∗1, Vasileios D. Stavrou1, Drosos Kourounis2, and Ioannis\nPanagiotopoulos1\n1Department of Materials Science and Engineering, University of Ioannina, 45110 Ioannina,\nGreece\n2NEPLAN AG, CH-8700 Küsnacht (ZH), Switzerland\nOctober 2, 2018\nAbstract\nThe magnetization reversal in 330nmtriangular pris-\nmatic magnetic nanoelements with variable magne-\ntocrystalline anisotropy similar to that of partially\nchemically ordered FePt is studied using micromag-\nnetic simulations employing Finite Element discretiza-\ntions. Several magnetic properties including the eval-\nuation of the magnetic skyrmion number Sare com-\nputed in order to characterize magnetic configurations\nexhibiting vortex-like formations. Magnetic vortices\nand skyrmions are revealed in different systems gen-\nerated by the variation of the magnitude and relative\norientation of the magnetocrystalline anisotropy direc-\ntion, withrespecttothenormaltothetriangularprism\nbase. Micromagnetic configurations with skyrmion\nnumber greater than one have been detected for the\ncase where magnetocrystalline anisotropy was normal\nto nanoelement’s base. For particular magnetocrys-\ntalline anisotropy values three distinct skyrmions are\nformed and persist for a range of external fields. The\nsimulation-based calculations of the skyrmion number\nSrevealed that skyrmions can be created for magnetic\nnanoparticle systems lacking of chiral interactions such\nasDzyaloshinsky-Moriya, butbyonlyvaryingthemag-\nnetocrystalline anisotropy.\n1 Introduction\nMagnetic nanoparticles and nanostructures find nu-\nmerous applications in a wide variety of scientific\nfields including information signal processes, spin de-\nvices, high density storage media, magnetic sensors,\nmedicine and biology [41, 20, 35, 2, 14]. Nowadays\nprogress in nanoscale material growth have allowed\nthe synthesis and fabrication of nanoparticles in a\nwide range of shapes and sizes [37, 44, 16]. Their\nmagnetic response is of paramount importance and\ncouldbeassociatedwith geometrical andmaterialsfac-\ntors; thus, a lot of effort has been devoted to exper-\nimental, simulation and theoretical studies. In par-\n∗lgergidi@uoi.gr; lgergidis@gmail.comticular, the process of magnetization reversal in mag-\nnetic nanoparticles could be exploited, engineered for\ntechnological applications but necessitates the knowl-\nedge and control over the formation of rather complex\nmicromagnetic structures, such as multiple domain\nwalls, vortices, skyrmions-antiskyrmions and merons\n[38, 48, 42, 46, 45, 43, 28, 15].\nMagnetic skyrmions are vortex-like magnetization\nconfigurations. They have been predicted theoretically\nby Bogdanov et al. [3],[4] long before their experimen-\ntal detection and discovery [26, 24]. In recent years\nmagnetic skyrmions have attracted a lot of theoretical\n[13, 22, 23, 39, 5, 21], simulation [9, 18, 40, 32], and\nexperimental [13, 34, 17, 11, 27] attention due to their\nthermodynamic and topological stability, their small\nsize, and their inherent property of easy movement\nand repositioning under the application of low or even\ntiny in-plane electric currents. They seem promising\nfor use in next generation spintronic devices [33, 50]\nas information carriers, giving the credentials of ul-\ntra dense low-cost power storage and the capability\nto perform logical operations [12]. The next genera-\ntion magnetic memory devices would rely on the effi-\ncient creation and control of magnetic textures, such\nas magnetic skyrmions using rather tiny electric cur-\nrents [35, 29, 7, 19]. The aforementioned topological\nstability is related to the confined skyrmion magnetic\nconfiguration, which is predicted to be stable because\nthe individual atomic spins, oriented opposite to those\nof the surrounding thin-film cannot perform flipping\nmotion. Spins hindered to align themselves with the\nrest of atoms in confined geometry without overcom-\ning an energy barrier. The origin of the energy bar-\nrier is attributed to the “topological protection”. The\nthermodynamic stability of skyrmions is considerably\nstrong and can be attributed to the particular mag-\nnetic configuration which can be characterized by a to-\ntal topological charge described by skyrmion number S\n[22, 23]. This skyrmion number Sis an integer and to\nthis point is being considered having quantized values\nthat cannot be changed continuously. The skyrmionarXiv:1810.00253v1  [physics.comp-ph]  29 Sep 20182 MICROMAGNETIC MODELING\nnumberSis defined as\nS=1\n4π/integraldisplay\nAqSkdA (1)\nwhereqSkis given by the following relation\nqSk=m·(∂m\n∂x×∂m\n∂y). (2)\nThe quantity mis the unit vector of the local mag-\nnetization defined as m=M/MswithMbeing the\nmagnetization and Msthe saturation magnetization.\nThe skyrmion number Sis a physical and topologi-\ncal quantity that measures how many times mwraps\nthe unit sphere [47]. The integrated quantity describes\nthe topological density qSkand has units of nm−2,\nwhichareimpliedthroughoutthemanuscript. Inmany\ninstances the integrated quantity is also referred as\n“topologicalcharge”. Surface Aisthesurfacedomainof\nintegration and corresponds to the upper or the lower\ntriangular bounding surface of the FePt nanoelement\nunder investigation.\nThe magnetization reversal in 330 nm triangular\nprism magnetic nanoelements with variable magne-\ntocrystalline anisotropy (as that of partially chemically\nordered FePt) has been studied using Finite Elements\nmicromagnetic simulations in [38]. The simulation re-\nsults showed that a wealth of reversal mechanisms is\npossible sensitively depending on the uniaxial mag-\nnetocrystalline anisotropy values and directions; the\nlatter may explain the different Magnetic Force Mi-\ncroscopy patterns obtained in such magnetic systems.\nIn addition, the micromagnetic simulations revealed\nthatinterestingvortex-likeformationscanbeproduced\nand stabilized in large field ranges and in sizes that can\nbe tuned by the magnetocrystalline anisotropy (MCA)\nof the material. The aforementioned spontaneous\nground states of skyrmion-like configurations were ob-\ntained and in other magnetic systems without the im-\nplication of chiral interactions such as Dzyaloshinsky-\nMoriya (DMI) [45, 39, 6, 51, 36, 10, 49].\nIn the present work a quantitative description is\ngiven for the vortex-skyrmionic configurations by cal-\nculatingtheskyrmionnumberat 0◦KobtainedforFePt\ntriangular magnetic nanoislands having variable mag-\nnetocrystalline anisotropy. The aforementioned calcu-\nlation can reveal information not readily recognizable\nby simple visual-inspection of the micromagnetic con-\nfigurations. Furthermore, the skyrmion number as a\nfunction of the applied field along a hysteresis curve\ncan give quantitative information of the reversal mech-\nanisms and energy barriers involved. In what follows\nwe present examples on the numerical calculation of\nskyrmion number as means of characterizing magnetic\nconfigurations and reversal in nanoelements including\nthin film asymmetric triangular nanoislands.2 Micromagnetic modeling\n2.1 FEM solution of Landau-Lifshitz-\nGilbert (LLG) equation\nTherateofchangeofthedynamicalmagnetizationfield\nMis governed by a nonlinear equation of motion, the\nLandau-Lifshitz-Gilbert (LLG) equation\ndM\ndt=γ\n1 +α2(M×Heff)−αγ\n(1 +α2)|M|M×(M×Heff).\n(3)\nIn the aforementioned LLG equation α > 0is a phe-\nnomenological dimensionless damping constant that\ndepends on the material and γis the electron gyro-\nmagnetic ratio. The effective field that governs the dy-\nnamical behavior of the system has contributions from\nvarious effects that are of very different nature and can\nbeexpressedas Heff=Hext+Hexch+Hanis+Hdemag.\nRespectively, these field contributions are the exter-\nnal magnetic field Hext, the exchange field Hexch,\nthe anisotropy field Hanisand the demagnetizing field\nHdemag.\nFor the solution of the LLG equation We have per-\nformed micromagnetic finite element calculations us-\ning the Nmag software [8]. The dimensionless damp-\ning constant αwas set to 1 in order to achieve fast\ndamping and reach convergence quickly as we are in-\nterested in static magnetization configurations. The\ndefault convergence criterion for each applied exter-\nnal field step Hextwas that the magnetization should\nmove slower than 1 degree per nanosecond, globally\nor on average for all spins. The sample was described\nas a triangular prism with equilateral triangle base of\n330 nm and 36 nm in height and is shown in Fig. 1.\nThe 36 nm thickness of the film matches that reported\nin [25] while Okamoto in [31] suggests 24 nm, which is\ncomparable and not expected to lead to qualitatively\ndifferent behavior. The following frame of reference\naxes assignment convention was used: x along the tri-\nangle height, y along the side perpendicular to x, and\nz perpendicular to the film plane. The considered fi-\nnite element mesh for the triangular film under study\nwas generated using the automatic three dimensional\n(3D) tetrahedral mesh generator Netgen [30]. We\nhave used a 3.4 nm maximum distance between nodes\nwhich is lower than the value of the exchange length\nlex=/radicalBig\n2A\nµ0M2s≈3.5nm. This resulted in 488874 vol-\nFigure 1: Model geometry and the generated mesh.\n22.2 Skyrmion number computation 3 RESULTS\nume elements (94800 points) per triangular magnetic\nisland. The material parameters were chosen similar to\nthose typical for bulk FePt with a saturation polariza-\ntion ofµ0Ms= 1.43T(MS=1.138MA/m), and an ex-\nchange constant Aexch=11pJ/m which has been found\nto be independent of the degree of ordering Okamoto\n[31]. The magnetocrystalline anisotropy (MCA) con-\nstantKuwas varied between Ku= 100kJ/m3and\n500kJ/m3, as forKuexceeding this value the reversal\nmode was simply homogeneous rotation. The demag-\nnetization factor Nhas been estimated to N= 0.71\nby the saturation field perpendicular to the triangle\nfor the case of Ku= 0kJ/m3. We must note that\nfor the range of these MCA values the easy axis re-\nmains in-plane as the shape anisotropy contribution\n−1\n2Nµ0M2\ns=−580kJ/m3exceedsKufor all the\ncases presented here. Four different directions of the\nmagneto-crystallineanisotropyweretestedalong x,y,z\nas well as along the [111]direction. The latter is of in-\nterest as in many instances FePt and CoPt films tend\nto grow with their <111>crystallographic directions\nalong the film normal resulting an angle of 54.7◦to the\nfilm normal [1].\nThe total duration of the micromagnetic simulation\nwas ranging for 1 to 10 days on Intel i7 4770K de-\npending on the relative orientation of the applied mag-\nnetic field with respect to magnetic anisotropy. The\nmagnetization curves for every production run were\ninvestigated by applying external magnetic fields Hext\nwith fixed orientation running parallel to z-direction\n(the normal to the triangular base). The range values\nofHextwere +1000kA/m (maximum) and -1000kA/m\n(minimum) introducing an external magnetic field step\nδHext=4kA/m [38].\n2.2 Skyrmion number computation\nThe calculation of skyrmion number Snecessitates the\nknowledge of the computed normalized magnetization\nvector mobtained from the solution of LLG equa-\ntions. Once the finite element approximation of the\nnormalized magnetization mhhas been computed, the\nskyrmion number Sh, following (1), is approximated\nby\nSh=1\n4πNe/summationdisplay\ne=1me\nh·/parenleftbigg∂me\nh\n∂x×∂me\nh\n∂y/parenrightbigg\n|Ae|.(4)\nwheremh,Share the discrete representations of m,S\nrespectively with edenoting the element and Nethe\ntotal number of elements used for the surface domain\ndiscretization. It should be noted that throughout the\nmanuscript the symbol Sinstead ofShwill be used\nfor the computed value of the skyrmion number. The\nintegration takes place over the top or bottom surface\nboundary of the prism A. Since we are using tetra-\nhedral P1 elements for the discretization of the prism\nvolume, the top (or bottom) surface boundary is com-\nprised of triangles with outwards pointing normal par-\nallel to the ˆzunit vector. Magnetizations are extracted\nfor the top or bottom surface of the magnetic triangu-\nlar prismatic nanoelement. These particular magneti-zationslocatedonthesurfaceelementsofthetwobases\nare used for the actual computations of Sand of the\nrelative topological quantities. It is possible for a mag-\nnetic configuration to include more than one skyrmion.\nInevitably, the total skyrmion number would be the al-\ngebraicsumofitsindividualskyrmionicconfigurations.\nIt follows that a structure that locally includes several\nskyrmions of different polarity or chirality may yield a\nzero total skyrmion number. The physical significance\nof this situation may be attributed to the fact that\nstructures with opposing Smay be easier to mutually\nannihilate. Therefore, it is of interest to monitor fol-\nlowing [9] the integral of the absolute value of the topo-\nlogical density symbolized with Sabsas it describes the\nexistence of topological entities that are masked and\nwashed out when Sis calculated through the integral\nover all surface domain A. The quantity Sabsis defined\nby the relation\nSabs=1\n4π/integraldisplay\nA/vextendsingle/vextendsingle/vextendsingle/vextendsinglem·/parenleftbigg∂m\n∂x×∂m\n∂y/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingledA (5)\nThe scalar quantity Sabsis injective and provides the\nnecessary distinctness for different skyrmionic states\n[9]. The stabilization of such magnetic skyrmions\nis usually linked to the existence of some kind of\nanisotropic Dzyaloshinskii-Moriya interaction (DMI).\nIts discrete estimation follows similarly to (4). It is\ninteresting the calculation of skyrmion numbers for\nmicromagnetic configurations in nanoelements since it\ncan reveal information not readily recognizable by sim-\nple visual inspection of the micromagnetic configura-\ntions. Furthermore, the skyrmion number Sas a func-\ntion of the applied field Hextalong an hysteresis curve\ncan provide quantitative information of the reversal\nmechanisms and energy barriers involved in the pro-\ncess.\n3 Results\n3.1 In-plane MCA\nIn the first system studied the magnetocrystalline\nanisotropy (MCA) lies within the plane of the trian-\ngular nanoelement parallel to x-direction ( K//[100])\nwith the external field Hextapplied along the width\nof the nanoelement which is parallel to z-direction\n(Hext//[001]). The Hextdirection is being fixed par-\nallel toz-direction throughout this work for all sys-\ntems studied. We have calculated the half-hysteresis\nloop (descending branch of the loop) for the triangu-\nlar prismatic nanoelements for different MCA values.\nTheKu= 100kJ/m3is presented in Fig. 2along with\nqSk. Some micromagnetic configurations being formed\nalong the path of the reversal are shown in Fig. 3. We\npresent results only for a declining external field Hext\nsince the full hysteresis diagram does not contribute\nany additional information regarding the magnetiza-\ntion reversal process [38]. Presenting the full hysteresis\ndiagram would inevitably doubled the required compu-\ntational effort.\n33.1 In-plane MCA 3 RESULTS\nMagnetization reversal depicted in Fig. 2 starts\nwhen the normalized magnetization decreases from the\nthe saturation value M/Ms= 1, and passing through\nthe nucleation field it reaches to M/Ms= 0. Then\nafter passing the annihilation field, it attains finally\nthe valueM/Ms=−1indicative that all spins have\nreversed their magnetization vector orientation. Dur-\ning this reversal process the values of SandSabsare\ncomputed in order to provide a quantitative descrip-\ntion of the skyrmion-like localized configurations in\nconjunction with the qualitative actual visualization\nof the normalized magnetization vector mof the in-\ndividual spins. Values of Shave been calculated for\nthe top and bottom surface of the triangular prismatic\nnanoelement for various Kuvalues shown in Fig. 4\nfor all magnetic systems in the present investigation.\nThe values of Shave the same quantitative and qual-\nitative behavior on top and bottom surface of the na-\nnoelement ensuring the consistency of the numerical\ncalculations. Throughout the manuscript the reported\nvalues of skyrmion numbers S,Sabsrefer to the com-\nputed values on the top surface of the nanoelement.\nFor the MCA value Ku= 100kJ/m3represented in\nFig. 2the process of the reversal of spins gives birth\nto vortex like formations. The value of Semerges\nfrom zero and gradually increases; attaining value 0.5,\ncharacteristic of magnetic vortex for external mag-\nnetic field Hext= 4.0×103A/m, it reaches the max-\nimum value of S≈0.75that can be considered\nas an incomplete skyrmion for field value close to\nHext=−2.1×105A/m. Note that similar non-integer\nvalues of skyrmion number have been reported in con-\nfined helimagnetic nanostructures [9] and in thin con-\nfined polygonal nanostructures [32]. It is anticipated\nthatinthepresentfinitemagneticsystemtheskyrmion\nnumber can be non-integer due to the restricted area\nof integration AinEq. 1andEq. 5and the essen-\ntial contribution of magnetostatic energy. The value of\nS= 0.5describes a vortex like micromagnetic configu-\nration.\nAround the external field value of\nHext= 9.2×105A/ma small step can be detected\nin the variation of M/Msreflected also in Sand\nFigure 2: Normalized magnetization ( M/Ms) and\nskyrmion numbers S,Sabsduring the magnetization re-\nversal process for in plane MCA (along x-direction)\nwithKu= 100kJ/m3andHext//[001].\nFigure 3: Top view of qSkatKu= 100kJ/m3for MCA\nbeing parallel to x-direction and Hext//[001]. From\nleft to right and from top to bottom different values\nofHext= 384,256,−200,−296 (×103A/m)are repre-\nsented referring to points of A, B, C, D of Fig. 2.\nLocal magnetization vectors are shown with arrows\nsuperimposed with qSk. The actual value ranges of\nqSk/nm−2are represented in color bars with maxi-\nmum and minimum values respectively: left top (A\nmax : 0.00394−min :−0.00174) , right top (B\nmax : 0.00693−min :−0.00203), left down (C\nmax : 0.0272−min :−0.00037), right down (D\nmax : 0.00185−min :−0.00057).\nFigure 4:Sas a function of Kufor MCA parallel to\n[100], [001] and [111] directions with Hext//[001].Sis\nbeing calculated on the top and bottom surface of the\ntriangular prismatic nanoelement for Hext= 0A/m.\nmagnified in the injective property Sabsindicating\nthe origin of magnetization reversal through vortex\nor skyrmion type mechanisms. In addition, around\nHext= 2.6×105A/ma jump discontinuity is evident\nboth forS,Sabsnot captured by the magnetization\ncurveM/Ms. A second jump discontinuity on S\nis evident around Hext=−2.7×105A/mcausing\nthe change of Svalue approximately from 0.7 to -\n0.4 (relative variation 157%). The aforementioned\nvariation is followed by a change of the sign of S\ncharacteristic for a change in the polarity of the vortex\ntype micromagnetic configuration. Further decrease\n43.1 In-plane MCA 3 RESULTS\nFigure 5: Sas a function of HextforKu=\n150−350kJ/m3with MCA parallel to x-direction and\nHext//[001].\nof the external magnetic field causes the continuous\ndecay ofSand its annihilation following the final\nstage of the reversal process. The magnification of\njump discontinuities in Sabsis anticipated since it is\ninjective and provides the necessary distinctness for\ndifferent skyrmionic and consequently energy states.\nTherefore, it is being used in order to detect and\nexplore possible energy barriers during the reversal\nprocess. Additionally, in Fig. 2 differentiation of\nshapes describing S,Sabsreveals that there are local\ntopological density qSkregions-domains which sum up\nto zero and therefore mutually annihilate.\nVisualizations of micromagnetic configurations are\nshownin Fig. 3fortherepresentativeselectedexternal\nfield values depicted in Fig. 2and designated as A, B,\nC, D. The actual topological density qSkdefined in Eq.\n2represents the \"local\" skyrmion number of the surface\nelement and is also visualized in Fig. 3. This enriched\nrepresentation gives both qualitative and quantitative\ndescription of the actual magnetization configuration.\nThe existence of domains with augmented local topo-\nlogical density is observed close to upper and lower\ncorners of the nanoelement along y-direction for A, B\nand D points. In addition, an elliptical domain can be\nobserved along the x-axis which is also the direction\nof the in plane MCA Ku= 100kJ/m3, at the center of\nthe triangular base present in all characteristic points\nA, B, C, D. These three magnetic entities can be con-\nsidered as incomplete skyrmions and are present for a\nwide range of external field values. In Fig. 3(point\nA) the aforementioned magnetic entities expose differ-\nent magnetization circulations. Those in the corners\nhave a counter clockwise circulation while the domain\nwith augmented qSkat the center of the triangle has a\nclockwise circulation.\nInFig. 5, skyrmion number Sas a function of\nthe external field Hextfor different values of Kuis\nshown. As the magnitude of MCA increases grad-\nually starting from Ku= 100kJ/m3a similar behav-\nior with Fig. 2 is observed regarding the qualita-\ntive and quantitative characteristics of S. In the cases\nofKu= 150,250,350kJ/m3the skyrmion number at-\ntains lower values as Kuincreases exhibiting jump dis-\ncontinuities. The calculated maximum (or negative\nminimum) values denoted as Smaxfor the aforemen-\nFigure 6:Smaxas a function Kufor MCA being in\nplane (//x), perpendicular (//z) and parallel to [111]\ndirection with respect to the surface base of the na-\nnoelement with the external field Hext//[001].\ntioned cases do not exceed the value Smax= 0.7for\nKu= 100kJ/m3represented in Fig. 5. ForKu=200\nand300kJ/m3Sattains maximum values 0.55–0.50 re-\nspectively but at different external magnetic field val-\nuesHext. It is evident that the magnetization rever-\nsal mechanism necessitates the formation of incom-\nplete skyrmions ( 0.5< S < 1), vortex-like states\n(S= 0.5) not only for low Ku=100 and 150kJ/m3val-\nues but also for the considered as intermediate values\nofKu= 250,350kJ/m3(Fig. 6). Further increase of\nMCA’s values to Ku= 400,450,500kJ/m3expose sim-\nilar reversal characteristics with Smaxestablishing a\nplateau region at 0.5 shown in Fig. 6characteristic\nfor vortex like reversal process.\nThe discontinuities detected in case of\nKu= 100kJ/m3are present not only for different\nvalues ofKubut for different orientation of the MCA\nwith respect to the surface of the nanoelement e.g.\nparallel to z-direction. They need further clarification\nand can be associated to the rich energetic envi-\nronment having contributions from demagnetization\nEdemag, exchange Eexchange and anisotropic Eanis\nenergies which can be computed for the systems\nstudied in the present work. Complicated and rich\nenergetic landscapes on the surface of the nanoelement\nare anticipated. The skyrmion formation is related to\nthe interplay of these energetic contributions.\nIn order to measure the effect of each individual en-\nergy type on the micromagnetic configuration during\nthe magnetization reversal process the absolute rela-\ntive energy difference ∆Erel\ntype=|Ei+1\ntype−Ei\ntype\nEi\ntype|×100(%)\n(wheretypestands foranis,exch,demag ) is computed\nbetween the consecutive external magnetic field values\nHi\next,Hi+1\next. As mentioned in Section 2 the external\nmagnetic field step between consecutive field values is\nδHext= 4kA/m. The values of the relative differences\nof anisotropy, demagnitization and exchange energies\nare shown in Fig. 7 as functions of Hextfor MCA\nvalues ofKu= 100,150kJ/m3superimposed with Sabs.\nIt is clear that even in the first steps of the magne-\ntization reversal around values 8.5−9.5×105A/mof\nthe external field for Ku= 100kJ/m3small steps of the\nskyrmion number are induced by jump discontinuities\non the relative energies and vice versa. These jump\n53.2 Perpendicular MCA 3 RESULTS\nFigure 7: Relative values in %percentages for energeti-\ncal types of demagnitization, exchange and anisotropic\nforKu= 100for MCA parallel to [100].\ndiscontinuities represent the actual energy barriers are\npronounced for all types of energies at the beginning of\nthe reversal process with Eanisbeing the more promi-\nnent compared to Edemag,Eexch. The magnetic system\nand its associated energy should overcome a significant\nenergy barrier in order to start the reversal process in\nthe confined triangular prismatic nanomagnet. In par-\nticular, ∆Erel\nanisattains values close to 42%. Further\nreduction of the external magnetic field shows contin-\nuous behavior up to the value of Hext= 2.5×105A/m\nforSabs. The decrease of the external field Hextdrives\nthe continuous and gradual formation of an incom-\nplete skyrmion ( S < 1). This continuous behaviour\nis dictated from the continuous behaviour of the to-\ntal magnetization energy and its energetic components\nEanis,Edemag,Eexch. A new energy barrier is observed\nforSabsaroundHext<2.5×105A/mfor the exchange\nenergyEexchnot followed by the other calculated en-\nergies. The value ∆Erel\nexchof this discontinuity is close\nto 10%. It is clear that this sharp change at Eexchin-\nduces the incomplete skyrmion discontinuity not only\nforHext<2.5×105A/mbut also for the symmetric\nwith respect Hext= 0A/msecond maximum around\nHext=−2.5×105A/m.\nIn particular, for external fields in the vicinity of\nHext= 0 A/ma steep descent is evident for ∆Edemag.\nEnergy component ∆Eanisshows a smoother behavior\nbetween the first and last discontinuities having a de-\nclining behavior in the first half of the magnetization\nreversal process where the external field approaches\nzerovalue. Acontinuousincreaseoftherelativechange\n∆Eanisis profound in the second half and final stage of\nthe reversal process. Relative energydifference ∆Eexch\nexhibits an interesting non-continuous behavior hav-\ning a significant number of sharp discontinuities in\nthe range of Hext= (−2.5 to + 2.5)×105A/mexter-\nnal field values. Calculation of energies and relative\nenergies differences for varied Kushowed similar qual-\nitative and quantitative characteristics with the case of\nKu= 100kJ/m3. Vortex like or incomplete skyrmion\nstates are triggered by abrupt energy changes and en-\nergy barrier crossings.\nFurther calculations have performed by changing the\ndirection of MCA from xtoydirection. This direc-\ntional change of MCA gives similar physical results re-garding the formation of vortices despite the fact that\nMCA’s direction remains in plane. This situation with\nMCA along y-direction differs from MCA lying along\nx-direction only with respect to the subtle effects of the\nedge curling on the nucleation process.\n3.2 Perpendicular MCA\nIt is challenging to investigate the effects associated\nwith MCA when is set to z-direction which is the direc-\ntion normal to the basis surface of the triangular FePt\nnanoelement. As the vortex-like formations in this sys-\ntem arise from the competition of exchange and mag-\nnetostatic energies it is expected that these phenomena\nwill be more pronounced when there is a perpendicu-\nlar MCA which leads to strong demagnetizing fields.\nThe numerical calculations showed that the depen-\ndence ofSfor the lowest MCA value Ku= 100kJ/m3\nis quite similar to the case of in-plane MCA direc-\ntions exhibiting the same magnetization reversal pro-\ncess. The magnetic system case with Ku= 150kJ/m3\nis presented in Fig. 8. Since the Kuvalue is not\nhigh enough to give perpendicular anisotropy as the\nHextis reduced the system departs from saturation\nthrough a series of topologically non-trivial forma-\ntions giving a gradual increase of the skyrmion num-\nber which attains the maximum value S/similarequal0.9in\na plateau region between Hext=−2.5×105A/mand\nHext=−5×105A/mis evident. The aforementioned\ngradual increase of the skyrmion number Sfor MCA\nparallel to z-direction for all Kuvalues studied starts\nat lower external field values compared to systems hav-\ning in plane MCA direction. This external field val-\nues retardation can be attributed to the higher energy\nbarrier needed to overcome in order to start the re-\nversal process through skyrmion formation. The criti-\ncal field signaling skyrmion formation for the different\nKu= 150,250,350kJ/m3values for MCA parallel to z-\ndirection is located around Hext= 3.5×105A/m(Fig.\n8) and is considerably lower compared to the external\nfield value of Hext= 7.5×105A/mfor in-plane MCA\n(Fig. 2).\nThe behavior is strikingly different when MCA at-\ntainsthevaluesof Ku= 250,350kJ/m3depictedalsoin\nFig. 8. Forthecasewhere Ku= 350kJ/m3skyrmionic\nconfigurations having negative values emerge for exter-\nnal field values lower than 5.0×105A/m. The micro-\nmagnetic configuration in Fig. 8hosts one skyrmion\nlocated at the center of the nanoparticle. A counter-\nclockwise circulation (positive circulation) can be iden-\ntified on the top view of the nanoparticle. The central\nspins of the skyrmion point inwards (negative polarity)\nas clearly presented at the bottom view of the spin con-\nfiguration. Consequently, chirality which is the prod-\nuct of circulation and polarity is negative dictating in\nthis manner the negativity of the skyrmion number.\nFor the particular case of Ku= 250kJ/m3regions\nhosting skyrmions are evident having skyrmion num-\nber values close to S= 3in a range of applied external\nfields from Hext=−0.3×105A/mto−5.3×105A/m.\nInFig. 9micromagnetic configurations with topolog-\n63.2 Perpendicular MCA 3 RESULTS\nFigure 8: Sas a function of Hextfor\nKu= 150,250,350kJ/m3with MCA running par-\nallel to z-direction alongside with the micromagnetic\nconfiguration in top and bottom view (rotated with\nrespect to top view by 60 degrees) for Ku= 350kJ/m3\nwhen skyrmion number attains negative values\n(S=−1) at representative point P.\nFigure 9: Top view of micromagnetic configuration\nshowing topological density qSksuperimposed with the\nlocal magnetization for Ku= 250kJ/m3. Different di-\nagrams depict the four points A, B, C, D of Fig. 8\n(from left to right and from top to bottom) describ-\ning the reversal process. The actual value ranges of\nqSk/nm−2are represented in color bars with maxi-\nmum and minimum values respectively: left top (A\nmax : 3.419×10−5−min :−1.117×10−6) , right\ntop (B max : 0.00063−min :−0.00012), left down\n(Cmax : 0.00346−min :−0.00020), right down (D\nmax : 0.00063−min :−0.00012).\nical density qSksuperimposed with the local magneti-\nzation vector are presented. Different diagrams depict\nfour representative points A, B, C, D in four character-\nistic regions during the reversal process of the Svari-\nation with the external field Hext. InFig. 9(point\nA inFig. 8 forKu= 250kJ/m3) skyrmion is under\ndevelopment. Regions of augmented qSkare evident\nexactly at the triangle’s corners. Lower values of qSk\ncan be seen at the center of the triangular base of the\nnanoelement. Point B is located at a skyrmion num-berSdiscontinuity with actual value close to S= 1.\nThe respective isosurfaces of qSkfor point B reveal the\nexistence of one skyrmion located at the center of the\ntriangular prismatic nanoelement.\nPoints C, D represent micromagnetic configura-\ntions hosting three skyrmions. The centers of the\nskyrmions define an equilateral triangle in both mi-\ncromagnetic configurations. The essential difference of\nthe skyrmions present in C and D points is their ac-\ntual size. Skyrmions of point C are larger compared to\nthe respective three skyrmions in D although they have\nthe same circular shapes and location. This can be at-\ntributed to the fact that while the sizes in skyrmions of\nC (Fig. 9(C) ) are larger than those of D ( Fig. 9(D)\n) the topological densities qSkare higher in D in order\nto compensate the lower skyrmion surface contributing\ntothetotaltriangularsurfacebaseintegrationgivingin\nbothcasesCandDatotalskyrmionnumberequalto3.\nRepresentative values of the maximum qSkon the sur-\nface element are qSk,C = 0.003462<qSk,D = 0.02181.\nInFig. 6 the maximum values of Sdenoted as\nSmaxare reported for MCA parallel to z-direction.\nMoreover, representative configurations depicted for\nexternal field values corresponding to Smaxshowing\nqSksuperimposed with magnetization are presented\ninFig. 10 . For the case where Ku= 200kJ/m3at\nthe center of the triangular base a clear and com-\nplete skyrmion emerges. It has an almost perfect cir-\ncular shape with the higher values of qSklocated at\nthe center mitigating when moving away but along\nthe circle’s radius. The magnetization vectors have\na counter-clockwise circulation around the skyrmion.\nIn case where Ku= 250kJ/m3andSmax≈3.25ex-\ntended regions located at the triangle’s vertices run-\nning along the edges of the triangular base are ev-\nident. This magnetic configuration significantly dif-\nfers from the configuration having three well developed\nskyrmions defining segregated and distinct regions pre-\nsented in Fig. 9(point C). These multiple skyrmions\nor clusters of skyrmions on the surface of the trian-\ngular disk are evident and the actual calculation of S\ngives values during the reversal process in the range of\n1≤S < 4for all MCA values studied having perpen-\ndicular anisotropy. The maximum value Smax= 3for\nthe case ofKu= 300kJ/m3is also depicted in Fig. 10\nexposing three skyrmions located in extended circular\nregions.\nThe increase of the MCA value from Ku= 150kJ/m3\nto500kJ/m3plays a dramatic effect in the magneti-\nzation reversal process and the creation of skyrmion\nregions. Particularly, in the cases where Kuattains\nvalues beyond 300kJ/m3reaching 500kJ/m3the max-\nimum of skyrmion number Sestablishes a plateau at\nS= 2indicative of the formation of two skyrmions\non the surface of the nanoelement. The case of\nKu= 350kJ/m3whereSmax= 2is representative and\nis shown in Fig. 10 . The two skyrmions are exactly\nlocated along the height of the triangle (parallel to x-\ndirection) having different shapes resembling different\nperturbations of the circular shape. The formation and\nactual detection of skyrmions can be revealed by visu-\n73.3 MCA parallel to [111] 3 RESULTS\nFigure 10: Top view of micromagnetic configurations\nreferring to SmaxforKu= 200,250,300,350kJ/m3\n(from left to right and from top to bottom) for MCA\nbeing parallel to z-direction. Magnetization shown\nwith arrows superimposed with qSk. The actual value\nranges ofqSk/nm−2are represented in color bars with\nmaximum and minimum values respectively: left top\n(Ku= 200kJ/m3,max : 0.00342−min :−0.00643)\n, right top ( Ku= 250kJ/m3,max : 0.00175−\nmin :−0.00043), left down ( Ku= 300kJ/m3,\nmax : 0.00135−min :−0.00020), right down\n(Ku= 350kJ/m3,max : 0.00322−min :−9.61×\n10−5).\nalizing the actual reversal process. A quantitative and\ncoherent picture can be provided by the calculation of\nS.\nComputations of relative energy differences ac-\ncompanied by Sabsare depicted in Fig. 11 for\nKu= 250kJ/m3which is the case showed the forma-\ntion of three skyrmions. As it is already commented\nthe first steps of the magnetization reversal for MCA\nnormal to nanoelement’s surface start at external field\nvaluesHextsignificantly lower ( <5×105A/m) com-\npared to the case of in-plane MCA. Following Sabsand\nrelativeenergiesvariationwithrespecttoexternalfield,\nenergy barriers are present around Hext= 5×105A/m\nfor all the components of energy. Particularly in Fig.\n11forKu= 250kJ/m3,∆Erel\nexchis close 90%and is\nthe most prominent jump discontinuity compared to\n∆Erel\nanis,∆Erel\ndemagwhose jumps are around 10%. It\nshould also noted that Erel\nanisexhibits a 90%jump,\nwhich is twice as much as the one observed in the case\nof in plane anisotropy at Ku= 100kJ/m3which was\ncalculated to 42%. The reduction of the external mag-\nnetic field affects in a continuous manner the different\nmagnetic energies up to the value of Hext= 0A/m.\nIn the vicinity of the zero for negative external fields\nnew jump discontinuities are evident for all energies.\nThe decrease of external field Hextdrives the contin-\nuous and gradual formation followed by energetic and\ntherefore skyrmionic discontinuity events. Complete\nand incomplete skyrmions are present having values1<S < 4. Rich energy patterns and textures are clear\nhaving similar behaviour not only for Ku= 250kJ/m3\nbut for allKuvalues studied when MCA is normal to\nthe nanoelements’ base.\nFig. 12 represents the relative energies for\ncharacteristic points selected from Fig. 11 at\nKu= 250kJ/m3. Energies Eanis,Eexch,Edemagare\ngrouped and shown in five different points. The\ngrouped energies {Ai,Di,Ei}associated with point i\n(i= 1,..,5) are depicted at the same or approximately\nthe sameHext,ivalue. At point A1before crossing\nthe energy barrier, Eanisdevelops an inner triangu-\nlar region with values lower compared to the maxi-\nmum values located at the sides of the nanoelement’s\nbase. AtA2point where the barrier is being crossed\nand skyrmion has been formed the energy distribution\nfollows the skyrmion’s location with increasing values\nofEanisradially moving from the center towards the\nouter circular domain of skyrmion. Energy isosurfaces\nfor pointA3expose three new developed regions in the\nvicinity of the nanoelement’s base vertices. Point A4\nrepresents a local energy mimimum of Eanisin which\naround the skyrmion domain at the center well devel-\noped stripe-like energy domains running parallel to na-\nnoelement’s edges are evident. Point A5is the location\nwhere the highest energy barrier is detected. From the\ncontour plot is evident that the energy follows exactly\nthe location of the multiple skyrmions formed. Calcu-\nlation ofEdemagprovides the same qualitative behav-\nior with the subtle difference on point D1at almost\nidenticalHextvalue with A1whereEdemaghas a uni-\nform distribution on the nanoelement’s base. The cal-\nculated exchange energy Eexchdistribution follows the\nformation process of skyrmions. At point E1high en-\nergy regions are located at the vertices of the nanoele-\nment in contrast to Eanis. The competition between\nthese energies and energy barrier crossings gives birth\nto skyrmion magnetic configurations.\n3.3 MCA parallel to [111]\nIn addition to MCA lying on the surface or being nor-\nmal to the surface of the nanoelement the magneto-\ncrystalline anisotropy is set parallel to [111] direc-\ntion.The latter case is of interest as in many instances\nFigure 11: Values in %percentages for relative en-\nergy differences of demagnitization, exchange and\nanisotropic for Ku= 250kJ/m3for MCA parallel to\n[001].\n83.3 MCA parallel to [111] 3 RESULTS\nFigure 12: Demagnitization, exchange, and anisotropic energies for Ku= 250kJ/m3for selected points during\nthe reversal process chosen from Fig. 11.\nFePt and CoPt films tend to grow with their [111]crys-\ntallographicdirectionsresultinganangleof 54.7otothe\nfilm normal [1]. The numerical simulation results are\npresented for SinFig. 4as a function of Kufixing ex-\nternal field value at Hext=0A/m. At Ku= 100kJ/m3\nskyrmion number is close to value S= 0.7. By in-\ncreasing MCA up to Ku= 250kJ/m3skyrmion num-\nber reduces attaining negative but non-zero values for\nexternal field Hext=0A/m. This is an interesting fact\nand significantly differentiates compared to the cases\nwhere MCA is parallel to x and z (in the majority of\nKuvalues studied) directions where positive skyrmion\nnumbers are obtained. The negative values originate\nfrom the negative chirality (circulation ×polarity) of\nthe high skyrmion density qSkregion that can be seen\ninFig. 13.The maximum skyrmion number value Smaxas a\nfunction of external field for different MCA values is\nrepresented in Fig. 6. Topological quantity Smaxat-\ntains negative values in contrast to the cases where\nMCA is in-plane or perpendicular to the nanoele-\nment’s base. For Smax<0the respective MCA\nvalue range is Ku= 150−350kJ/m3. Calculation of\nthe skyrmion number at different external field val-\nues for the representative Ku= 150,250,350kJ/m3val-\nues are shown in Fig. 13 . Evident is the mani-\nfestation of micromagnetic configurations mainly hav-\ning negative skyrmion numbers for external fields\nranging from Hext= 5×105A/mto approximately\nHext=−2×105A/m. The minimum value of Sis ob-\nserved for the case of Ku= 150kJ/m3and is close to\nS=−0.4. For the three different Kuvalues shown\n9REFERENCES REFERENCES\nFigure 13: Sas a function of Hextfor\nKu= 150,250,350kJ/m3with MCA running parallel\nto [111] direction alongside with the micromagnetic\nconfiguration in top and bottom view (rotated with\nrespect to top view by 60 degrees) for representative\npoint P with S <0atKu= 350kJ/m3.\ninFig. 13 the crossover to negative Shappens at\nHext= 5×105A/m. Also the maximum antivortex\nstates are located at the vicinity of zero external mag-\nnetic field which significantly differentiate compared to\nthe previous studied cases.\nConclusions\nThe topological invariant known as skyrmion num-\nberShas been calculated for FePt triangular prism\nnanonelements with different directions and magnitude\nof MCA for varying magnitude of the external mag-\nnetic field Hext. The direction of the external field\nHextwas normal to nanoelement’s surface in all con-\nducted numerical simulations. Magnetization config-\nurations during reversal process were studied. It was\npossible to explore the formation of skyrmionic regions\nand qualitatively-quantitatively characterize them by\na variety of calculated properties such as the skyrmion\nnumber, the different contributions on the magnetic\nenergy such as demagnetization energy, exchange en-\nergy and uniaxial anisotropy energy and visualization\nof the micromagnetic configurations. Skyrmionic mag-\nneticconfigurationshavebeendetectedinhighsymme-\ntry positions with respect to the geometry of the tri-\nangular prism having skyrmion number greater than\none (S > 1) for the case where MCA was normal\nto nanoelement’s base. For MCA attaining values be-\ntweenKu= 200−500kJ/m3three distinct skyrmions\nare formed and persist for a range of external fields.\nIn conjunction with previous studies it is clear that\nmagnetic skyrmionscan be produced in a widerange of\nexternalfieldsjustbytuningMCA’smagnitudeanddi-\nrection, even in the absence of chiral interactions such\nas Dzyaloshinsky-Moriya. It is challenging to extent\nthe numerical simulations including thermal effects in\nthe form of Brownian term in LLG equation in order\nto investigate the formation of skyrmions at room tem-\nperature.Acknowledgments\nV.D. Stavrou would like to thank the State Scholar-\nship Foundation of Greece (IKY) for the financial sup-\nport under the scholarship grant (appl. no.14386). 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In\nthe vicinity of the Verwey transition, the exchange bias field is substantially enhanced because of a\nsharp increase in magnetocrystalline anisotropy constant from high-temperature cubic to low-temperature monoclinic structure. Moreover, with respect to the Fe\n3O4(40 nm)/MgO (001) thin\nfilm, the coercivity field of the CoO (5 nm)/Fe 3O4(40 nm)/MgO (001) bilayer is greatly increased\nfor all the temperature range, which would be due to the coupling between Co spins and Fe spinsacross the interface. Published by AIP Publishing. https://doi.org/10.1063/1.5023725\nI. INTRODUCTION\nExchange bias (EB) refers to a shift in the hysteresis\nloop along the magnetic field axis due to the interfaceexchange coupling between ferromagnetic (FM) and antifer-romagnetic (AFM) materials, which was first discovered by\nMeiklejohn and Bean in oxide-coated Co particles.\n1This\nphenomenon has been extensively studied because of thetechnological application in spintronic devices and magnetic\nrecording.\n2–4Recently, many investigations of EB in thin\nfilms were carried out for the case that FM layer is locatedon top of AFM oxide layer. These oxides present uniqueproperties, such as magnetism, superconductivity, metal-\ninsulator transitions, electron transfer, or ferroelectricity.\n5–12\nIn these systems, the exchange coupling affects the physical\nproperties of these oxides and in turn some parameters of theAFM oxides can be used to manipulate the variation of EB.\nA representative example is the strain controls the exchange\nbias in FM/AFM (ferroelectric) system.\n10–12It has been\nreported that the EB can be influenced by many factors in FM/\nAFM system, such as the FM magnetization MFM, the thick-\nness of FM layer tFMor AFM layer tAFM, and the anisotropy of\nAFM ( KAFM)o rF M( KFM).2–4,13,14The exchange bias field\n(HE) is inversely proportional to the MFMandtFM,a n dt h e\nKAFM is reported to affect the critical thickness of the AFM\nlayer.2,3,15,16Furthermore, the AFM or FM domain formation\nis also claimed to play a dominant role in EB. Mauri et al.17\nand Malozemoff18predicted that HE/(KAFMAAFM)1/2/\nMFMtFMwhen the domain wall formed in AFM layer; more-\nover, Ball et al.19,20found that the domain wall might occur\nin the FM layer in the Fe 3O4/NiO or Fe 3O4/CoO systems\nby polarized neutron reflectometry studies, and the HE\n/(KFMAFM)1/2/MFMtFMdepending on the domain wall formed\non the FM side of the interface is also proposed,3where AAFM\nandAFMare the exchange stiffness of AFM and FM layer,\nrespectively. Therefore, it is found that the HEcan be mediated\nby varying tFM,MFM,o rKFMfor different FM materials.2–4To\nsearch a FM material that exhibits a big change in KFMleadingto a large variation of the HEwithin the narrow temperature\nrange will be a very interesting issue.\nAs one of the oldest known oxide materials, magnetite\n(Fe3O4) keeps on attracting extensive attention in fundamen-\ntal science as well as for possible applications in spin-\ntronics21–25due to its rather unique and interesting set of\nelectrical and magnetic properties,26–29and the first-order\nmetal-insulator transition known as the Verwey transition\naround 124 K.30At the Verwey transition temperature ( TV),\nthe Fe 3O4undergoes a structural transition from spinel cubic\nto monoclinic structure with a sharp change in electrical and\nmagnetic properties.26,27This transition provides an external\ntuning capacity of the properties by varying the temperature.\nTherefore, it can be expected that the rapid change in KFMin\nthe vicinity of the TVwill result in the obvious variation of\nHE. The exchange bias effect with Fe 3O4has been reported\nby many groups, such as the Fe 3O4/CoO bilayers,31–33yet\nthe Verwey transition is not mentioned in their systems.31–41\nVenta et al.8reported the effect of the Verwey transition on\nthe exchange bias in Ni(Ni 80Fe20)/V2O3system, the Fe 3O4is\nconsidered to be formed with the reaction between Ni 80Fe20\nand V 2O3, the Fe 3O4film is extremely thin, whereas the\nVerwey transition is found to disappear for very thin film\n(<5 nm).42–46Their discussion about the exchange bias\naffected by the Verwey transition, to our knowledge, is still\ndebated. Therefore, it is very necessary to study the effect of\nthe Verwey transition on the exchange bias in FM/AFM sys-\ntem with thicker Fe 3O4layer. It has been reported that the\nVerwey transition is greatly influenced by the thickness of\nFe3O4thin films.42–46With decreasing the thickness, the TV\nand the transition become lower and broader, respectively, or\neven disappears for very thin film (such as 5 nm in Refs. 45\nand 46or thinner than 30 nm in Refs. 42–44), thus the\nchange in KFMaround TVbecomes very small for the thin\nFe3O4film. On the other hand, the HEis inversely propor-\ntional to the FM thickness tFM.2,3Considering these two\naspects, in order to observe a clear variation of the HEin the\nvicinity of TV, we chose the thicknesses of Fe 3O4and CoO\nas 40 nm and 5 nm, respectively, in our FM/AFM bilayer\nsystem.a)E-mail: xhliu@alum.imr.ac.cn\n0021-8979/2018/123(8)/083903/5/$30.00 Published by AIP Publishing. 123, 083903-1JOURNAL OF APPLIED PHYSICS 123, 083903 (2018)\nIn this work, we report the exchange bias effect tuned\nby the Verwey transition of Fe 3O4in CoO (5 nm)/Fe 3O4\n(40 nm)/MgO (001) bilayer. A sharp increase in KFMleads to\na rapid enhancement of the exchange bias field in the vicinityofT\nV. Furthermore, compared with the Fe 3O4(40 nm)/MgO\n(001) thin film, the coercivity of the CoO (5 nm)/Fe 3O4\n(40 nm)/MgO (001) is greatly enhanced for all the tempera-\nture range, which would be induced from the strong coupling\nbetween Co spins and Fe spins across the interface, the par-tial interface spins of CoO rotating with magnetic field dur-ing the hysteresis loop measurement.\nII. EXPERIMENTAL METHODS\nThe CoO (5 nm)/Fe 3O4(40 nm) and CoO (5 nm)/Fe 3O4\n(20 nm) bilayers and the 40 nm-thick Fe 3O4thin film were\ngrown on MgO (001) by using molecular beam epitaxy\n(MBE) in an ultrahigh vacuum system with a background\npressure of 1 /C210/C010mbar range. The 40 nm (or 20 nm)\nFe3O4was grown on clean MgO (001) using an iron flux of\n1A˚per minute in an oxygen background pressure of 1 /C210/C06\nmbar with a substrate temperature of 250/C14C,45and then, 5 nm\nCoO was in situ grown on Fe 3O4(40 nm)/MgO (001) using a\ncobalt flux of 1 A ˚per minute in an oxygen background pres-\nsure of 3 /C210/C07mbar with a substrate temperature of 260/C14C.\nThe growth temperature or the growth oxygen pressure in ourwork is different from that reported by Lind et al.\n47and Wolf\net al.48To determine the structural quality and the chemical\nstates, the films were analyzed in situ by using reflection high-\nenergy electron diffraction (RHEED), low-energy electrondiffraction (LEED), and X-ray photoemission spectroscopy\n(XPS). High-resolution X-ray diffraction (HR-XRD) wasemployed for further ex situ investigation of the structural\nquality and the microstructure of the films. The transport andmagnetic properties of the thin films were measured with astandard four probe technique using physical property mea-surement system (PPMS) and superconducting quantum inter-\nference device (SQUID), respectively.\nIII. RESULTS AND DISCUSSION\nFigure 1shows RHEED and LEED electron diffraction\npatterns of the clean substrate MgO (a) and (d), the 40 nm-thick Fe\n3O4film on MgO (001) (b) and (e), and the CoO\n(5 nm)/Fe 3O4(40 nm)/MgO (001) bilayer (c) and (f). Sharp\nRHEED streaks and the high contrast and sharp LEED spots[Figs. 1(b) and1(e)] indicate a flat and well ordered (001)\ncrystalline surface structure of the 40 nm Fe\n3O4film grown\non MgO (001). The presence of the (ffiffiffi ffi\n2p\n/C2ffiffi ffi\n2p\n)R45/C14surface\nreconstruction patterns both in the RHEED and in the LEEDimages provides another indication for the high structuralquality of the Fe\n3O4film.45The lattice parameter of Fe 3O4\n(8.397 A ˚) is nearly twice as that of CoO (4.267 A ˚),49with\ngrowing CoO (5 nm) on Fe 3O4(40 nm)/MgO (001), the spi-\nnel and reconstruction streaks of RHEED in Fig. 1(b) and\ncorresponding spots of LEED in Fig. 1(e) disappear. The\nsharp RHEED and LEED patterns of CoO (5 nm)/Fe 3O4\n(40 nm)/MgO (001) in Figs. 1(c)and1(f)become similar to\nthat of the MgO (001) [see Figs. 1(a) and1(d)]. To further\ncheck the chemical states of the CoO thin film, the CoO\nFIG. 1. RHEED and LEED electron diffraction patterns of the following: clean substrate MgO (001) (a) and (d); 40 nm-thick Fe 3O4grown on MgO (001) (b)\nand (e); 5 nm-thick CoO grown on Fe 3O4(40 nm)/MgO (001) (c) and (f).083903-2 Liu et al. J. Appl. Phys. 123, 083903 (2018)(5 nm)/Fe 3O4(40 nm)/MgO (001) bilayer was analyzed in\nsituby XPS shown in Figs. 2(a)–2(c) . The wide scan with\nbinding energy from 1200 eV to /C018 eV shows a clear and\ntypical XPS pattern of CoO.50The Co 2p core-level XPS\nspectrum shown in Fig. 2(b)represents the typical character-\nistic for Co2þwith a clear satellite feature at 786.3 eV and\n802.9 eV, marked as S1 and S2, respectively,50,51and the\nvalence band presents insulating behavior for the CoO [Fig.2(c)]. Furthermore, the long range h–2hhigh-resolution X-\nray diffraction showing only single phase of CoO (5 nm)/Fe\n3O4(40 nm) in Fig. 2(d)also demonstrates the high quality\nof the epitaxial thin film.\nThe resistivity ( q) as a function of temperature ( T)f o r\nthe Fe 3O4(40 nm)/MgO (001) thin film in Fig. 3(a)displays\na sharp Verwey transition with a clear hysteresis [The CoO(5 nm)/Fe\n3O4(40 nm)/MgO (001) bilayer is insulating, and\nwe could not measure the electrical properties.] Moreover,the zero-field and field cooling (ZFC-FC) magnetization(M) dependent on temperature for CoO (5 nm)/Fe\n3O4\n(40 nm)/MgO (001) bilayer in Fig. 3(b) shows a rapid\nchange in magnetization at the Verwey transition, similarto that reported in the previous work.\n52,53The Verwey tran-\nsition temperatures TV/C0andTVþare defined as the temper-\nature of the maximum slope of log [ q(T)] or M(T) curve\nfor the cooling down and warming up temperaturebranches, respectively [see the insets of Figs. 3(a) and\n3(b)]. The values of 114.5 K and 119.5 K for T\nV/C0andTVþ\n[the inset of Fig. 3(a)] and very sharp transition indicate the\nquite high quality of our thin film, as compared to the pre-vious work.\n42–45,54–56Furthermore, it is found that the TV/C0\nandTVþin Fig. 3(a) are consistent with those in Fig. 3(b),\nindicating no influence of CoO on the chemical composi-tion of the Fe\n3O4layer.Magnetic hysteresis loops [ M(H)] at different tempera-\ntures of CoO (5 nm)/Fe 3O4(40 nm)/MgO (001) are shown in\nFig. 4(a), in which each M(H) curve was measured in the\napplied field of 50 kOe after field cooling ( HCF) at 10 kOe ( H\nalong the [100] direction) from 300 K (higher than the Neel\nFIG. 2. XPS spectra of the following:\nwide scan with binding energy from\n1200 to /C018 eV (a), Co 2 pcore-level\n(b) and valence band (c), and long\nrange h–2hX-ray diffraction pattern\n(d) of CoO (5 nm)/Fe 3O4(40 nm)/MgO\n(001).\nFIG. 3. Resistivity as a function of temperature for the 40 nm-thick Fe 3O4\nfilm grown on MgO (001) (a) and ZFC-FC temperature-dependent magneti-\nzation curve of CoO (5 nm)/Fe 3O4(40 nm)/MgO (001) sample in applied\nfield of 200 Oe (b). Inset: temperature dependence of d(log q)/dT(a) and d M/\ndT(b) around the Verwey transition, TV/C0andTVþcorrespond to the two\npeaks.083903-3 Liu et al. J. Appl. Phys. 123, 083903 (2018)temperature of CoO).2,3Obvious exchange bias is noticed at\nlow temperatures from 50 to 119 K, whereas it becomes very\nsmall when T/C21120 K ( /C24TVþ). In this system, the HEand\nHCare defined as HE¼(HL/C0HR)/2 and HC¼(HLþHR)/2,\nrespectively; here, HLandHRare the points where the hys-\nteresis loop intersects the field axis. Figure 4(b) shows the\nvalues of the HEcalculated from Fig. 4(a) as a function of\ntemperature for CoO (5 nm)/Fe 3O4(40 nm)/MgO (001)\nbilayer. Clearly, the HEdecreases with a rising temperature\nand exhibits a sharp drop at TVþ. The HEis about 68 Oe at\n200 K and nearly disappears at 250 K, implying the blockingtemperature of CoO is about 250 K.\n31–33Similarly, the HE\n(T) curve for HCF¼2 kOe also exhibits a rapid jump at TVþ\nbut with slightly smaller HEvalues due to non-fully oriented\ninterface magnetic moments at this cooling field.2,3\nHowever, this sharp change in HEatTVis not observed for\nthe CoO (5 nm)/Fe 3O4(20 nm)/MgO (001) bilayer [see the\ninset (left) of Fig. 4(b)] because of the broadened Verwey\ntransition for the thinner Fe 3O4film, similar to that reported\nin the previous work.31–33For Fe 3O4, the low-temperature\nmonoclinic magnetocrystalline anisotropy constants are con-\nsiderably greater (about 10 times) than those of the high-temperature cubic structure,\n57–59and we thus observe a great\nenhancement of HCatT<TV, see Fig. 4(c).\nThe coercivity is related to the anisotropy constants and\nthe saturation magnetization of the FM materials and can beroughly expressed as H\nC/KFM/MFM,60at the same time,\nthe HE/(KFMAFM)1/2/MFMtFMwhen the domain wall\nformed on the FM side of the interface;3thus, the relation-\nship HC/HE2can be obtained, which is well expressed as\nan inset (right) of Fig. 4(b), indicating the domain wall\nformed in the FM layer proposed by Ball et al.19,20and the\ntunability of the exchange bias by the Verwey transition;moreover, Ijiri et al.\n61also found that the CoO AFM order-\ning is long-range and propagates coherently through the\nintervening Fe 3O4layer in Fe 3O4/CoO superlattices. van derZaag et al.32calculated the low-temperature unidirectional\nanisotropy constant KE(0)¼l0MFMHE(0)tFM,w h e r e KE(0)\nandHE(0)are the values at 0 K. Similarly, extrapolated our\nresults to 0 K we estimated the KE(0)of about 0.68 mJ/m2,\nwhich is smaller than that reported by van der Zaag et al.32\nFurthermore, compared with the pure Fe 3O4(40 nm)/MgO\n(001) thin film, the HCof CoO (5 nm)/Fe 3O4(40 nm)/MgO\n(001) bilayer is much larger for all the temperature range\n[see Fig. 4(c)], meaning the strong coupling between Co\nspins and the Fe spins across the interface of CoO/Fe 3O4\nbilayer, that partial interface spins of CoO rotate with mag-\nnetic field during the hysteresis loop measurement.2,3\nTherefore, both the obvious exchange bias and the enhance-\nment of coercivity are observed in our CoO (5 nm)/Fe 3O4\n(40 nm)/MgO (001) bilayer, see Figs. 4(b) and 4(c).\nMoreover, the HC(T)c u r v e si nF i g . 4(c) show the same\nVerwey transition temperature for Fe 3O4(40 nm)/MgO\n(001) and CoO (5 nm)/Fe 3O4(40 nm)/MgO (001), which is\nin agreement with that in Fig. 3. Finally, we have to point\nout that the saturation magnetization MFMalso changes at\nTV, but this DMFMis very small,62only around 1% [see the\ninset of Fig. 4(c)]; we thus can omit the effect of MFMon\nthe exchange bias in our system.\nIV. SUMMARY\nIn conclusion, we have investigated the exchange bias\ntuned by the Verwey transition of Fe 3O4in the CoO (5 nm)/\nFe3O4(40 nm)/MgO (001) bilayer. The HEis significantly\nenhanced because of a sharp increase in KFMfrom high tem-\nperature cubic to low temperature monoclinic structure at\nTV. Moreover, the coercivity of the bilayer is greatly\nincreased for all the temperature range as compared to thepure Fe\n3O4(40 nm)/MgO (001) thin film due to the partial\ninterface spins of CoO rotating with magnetic field during\nthe hysteresis loop measurement.\nFIG. 4. Magnetic hysteresis loops at different temperatures in applied field of 50 kOe after field cooling at 10 kOe from 300 K (a) and the exchange bias fiel d\n(HE) as a function of temperature from 50 to 200 K at cooling field of 10 kOe and 2 kOe (b), of CoO (5 nm)/Fe 3O4(40 nm)/MgO (001). Inset of (b): HCvs H E2\nfrom 100 to 200 K (right) and temperature dependence of the HEat cooling field of 10 kOe of CoO (5 nm)/Fe 3O4(20 nm)/MgO (001) (left); temperature depen-\ndence of coercivity field ( HC) for CoO (5 nm)/Fe 3O4(40 nm)/MgO (001) and Fe 3O4(40 nm)/MgO (001) from 50 to 200 K (c). Inset of (c): saturation magneti-\nzation ( MFM) as a function of temperature of CoO (5 nm)/Fe 3O4(40 nm)/MgO (001).083903-4 Liu et al. J. Appl. Phys. 123, 083903 (2018)ACKNOWLEDGMENTS\nWe thank L. H. Tjeng from the MPI CPfS for\nstimulating discussions. 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Phys. 123, 083903 (2018)"    },    {        "title": "1810.01158v1.Magnetocrystalline_anisotropy_and_exchange_probed_by_high_field_anomalous_Hall_effect_in_fully_compensated_half_metallic_Mn2RuxGa_thin_films.pdf",        "content": "arXiv:1810.01158v1  [cond-mat.mtrl-sci]  2 Oct 2018Magnetocrystalline anisotropy and exchange probed by high -field anomalous Hall\neffect in fully-compensated half-metallic Mn 2RuxGa thin films\nCiar´ an Fowley,1,∗Karsten Rode,2Yong-Chang Lau,2Naganivetha Thiyagarajah,2Davide\nBetto,2Kiril Borisov,2Gwena¨ el Atcheson,2Erik Kampert,3Zhaosheng Wang,3,†Ye Yuan,1\nShengqiang Zhou,1J¨ urgen Lindner,1Plamen Stamenov,2J.M.D. Coey,2and Alina Maria Deac1\n1Institute of Ion Beam Physics and Materials Research,\nHelmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstra ße 400, 01328 Dresden, Germany\n2AMBER and School of Physics, Trinity College Dublin, Dublin 2, Ireland\n3Hochfeld-Magnetlabor Dresden (HLD-EMFL), Helmholtz-Zent rum Dresden - Rossendorf,\nBautzner Landstraße 400, 01328 Dresden, Germany\n(Dated: October 3, 2018)\nMagnetotransport is investigated in thin films of the half-m etallic ferrimagnet Mn 2RuxGa in\npulsed magnetic fields of up to 58T. A non-vanishing Hall sign al is observed over a broad tem-\nperature range, spanning the compensation temperature (15 5K), where the net magnetic moment\nis strictly zero, the anomalous Hall conductivity is 6673 Ω−1m−1and the coercivity exceeds 9T.\nMolecular field modelling is used to determine the intra- and inter-sublattice exchange constants\nand from the spin-flop transition we infer the anisotropy of t he electrically active sublattice to be\n216kJm−3and predict the magnetic resonances frequencies. Exchange and anisotropy are compa-\nrable and hard-axis applied magnetic fields result in a tilti ng of the magnetic moments from their\ncollinear ground state. Our analysis is applicable to colli near ferrimagnetic half-metal systems.\nPhySH: Ferrimagnetism, Magnetotransport, Half-metals, Anomalous Hall effect, Magnetic anisotropy, Exchange\ninteraction\nThin films with ultra-high magnetic anisotropy fields\nexhibit magnetic resonances in the range of hundreds of\nGHz [1–3] which is promising for future telecommunica-\ntions applications. Spin-transfer driven nano-oscillators\n(STNOs) working on the principle of angular momentum\ntransferfromaspin-polarisedcurrenttoasmallmagnetic\nelement[4,5], haveachievedoutputpowersofseveral µW\nand frequency tuneabilities of ∼GHzmA−1[6, 7], useful\nfor wireless data transmission [8]. Output frequencies of\nSTNOs based on standard transition-metal based ferro-\nmagnets, such as CoFeB, or cubic Heulser alloys such as\nCo2Fe0.4Mn0.6Si are in the low GHz range [9–13].\nCertain Heusler alloys [14, 15] are a suitable choice\nfor achieving much higher output frequencies, aimed at\nenabling communication networks beyond 5G [16]. The\nMn3−xGa family contains two Mn sublattices which are\nantiferromagnetically coupled in a ferrimagnetic struc-\nture [14]. They have low net magnetization, Mnet, and\nhigh effective magnetic anisotropy, Keff, with anisotropy\nfields of µ0HK= 2Keff/Mnetexceeding 18T [17, 18],\nwhich results in resonance frequencies two orders of mag-\nnitude higher [1, 2] than Co-Fe-B.Furthermore, the mag-\nneticpropertiesoftheseferrimagneticalloyscanbetuned\neasily with composition [19–21]. Mn 3−xGa films have\nshown tuneable resonance frequencies between 200GHz\nto 360GHz by variation of the alloy stoichiometry and\nmagnetic anisotropy field [2].\nHere we focus on the fully-compensated half-metallic\nHeusler compound, Mn 2RuxGa (MRG) [20–25]. Filmsof MRG were first shown experimentally [20] and subse-\nquently confirmed by DFT calculations [25] to exhibit a\nspin gap at EF. The material crystallises in the cubic\nspace group, F¯43m. Mn on the 4 aand 4csites are anti-\nferromagnetically coupled, while those on the same sites\nare ferromagnetically coupled. The crystal structure is\nFIG.1: (a)CrystalstructureofMn 2RuxGa, themagneticmo-\nments of the Mn 4aand Mn 4care aligned antiparallel. (b) and\n(c), a two-sublattice macrospin model used to explain the ob -\nserved temperature and field dependences of electronic tran s-\nport in the presence and absence of an applied field µ0Hz,\nrespectively. Two key points of the model are: below (above)\nTcompthe moment of the Mn 4cis parallel (antiparallel) to\nMnet; and, in the absence of an applied field the sublattice\nmoments donot change their orientation uponcrossing Tcomp.2\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s32/s49/s48/s32/s75\n/s32/s49/s49/s48/s32/s75\n/s32/s49/s56/s48/s32/s75/s100\n/s120/s121/s47/s100 /s181\n/s48/s72\n/s122/s32/s40/s97/s46/s117/s46/s41\n/s181\n/s48/s72\n/s122/s32/s40/s84/s41/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s32/s49/s55/s53/s32/s75\n/s32/s49/s54/s53/s32/s75\n/s32/s49/s51/s53/s32/s75\n/s32/s49/s50/s53/s32/s75\n/s32/s49/s49/s48/s32/s75/s77/s82/s32/s40/s37/s41\n/s181\n/s48/s72\n/s122/s32/s40/s84/s41/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s45/s49/s48/s107/s48/s49/s48/s107/s50/s48/s107/s51/s48/s107/s52/s48/s107/s53/s48/s107/s54/s48/s107/s55/s48/s107/s120/s121/s32/s40/s79/s104/s109/s46/s109/s41/s45/s49\n/s181\n/s48/s72\n/s122/s32/s40/s84/s41/s32/s49/s55/s53/s32/s75\n/s32/s49/s54/s53/s32/s75\n/s32/s49/s51/s53/s32/s75\n/s32/s49/s50/s53/s32/s75\n/s32/s49/s49/s48/s32/s75\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s50/s52/s54/s56/s49/s48/s181\n/s48/s72\n/s99/s32/s40/s84/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48/s52/s53/s53/s48\n/s181\n/s48/s72\n/s115/s102/s32/s40/s84/s41/s45/s52/s53 /s45/s51/s48 /s45/s49/s53 /s48 /s49/s53 /s51/s48 /s52/s53/s45/s50/s48/s107/s48/s50/s48/s107/s52/s48/s107/s54/s48/s107/s56/s48/s107/s49/s48/s48/s107 /s120/s121/s32/s40/s79/s104/s109/s46/s109/s41/s45/s49\n/s48/s72\n/s122/s32/s40/s84/s41/s32/s50/s50/s48/s32/s75\n/s32/s49/s56/s48/s32/s75\n/s32/s49/s53/s48/s32/s75\n/s32/s49/s49/s48/s32/s75\n/s32/s56/s48/s32/s75\n/s32/s53/s48/s32/s75\n/s32/s49/s48/s32/s75\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s45/s49/s48/s107/s45/s56/s107/s45/s54/s107/s45/s52/s107/s45/s50/s107/s48/s50/s107/s52/s107/s54/s107/s56/s107/s49/s48/s107\n/s32/s47/s32 /s32/s32/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s115/s119/s101/s101/s112/s115/s32/s40/s48/s32/s84/s41\n/s32/s43/s32/s32/s53/s56/s32/s84/s32\n/s32/s43/s32/s54/s46/s53/s32/s84/s32/s120/s121/s32/s40/s79/s104/s109/s46/s109/s41/s45/s49\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s40/s102/s41/s40/s100/s41/s40/s99/s41/s40/s98/s41/s40/s97/s41\n/s40/s101/s41\nFIG. 2: (a) AHC loops up to 6 .5T for Mn 2Ru0.61Ga around the compensation temperature (155K). Loops are off set vertically\nfor clarity. (b) Magnetoresistance loops recorded at the sa me time as the data in (a). Loops are offset vertically for clar ity. (c)\nAHC loops up to 58T, where the spin-flop transition is indicat ed by the grey arrows. The linear slope is due to the ordinary\nHall effect. Loops are offset vertically for clarity. (d) Deri vative of the selected data in (c) clearly highlighting the s pin-flop.\n(e)µ0Hc(black circles) and µ0Hsf(red squares) as a function of temperature. The divergence o f the coercivity is expected at\nTcompsince with Mnet= 0 and Keff∝negationslash= 0. (f) Temperature dependence of the remanent Hall conduct ivity when saturated at\n10K in negative (solid line) and positive (dashed line) appl ied field. The black open (closed) circles record the remanen t Hall\nresistivity after the application of 6 .5T (58T).\nshown in figure 1 (a). The Ga is on the 4 bsites and Ru\noccupies a fraction of the 4 dsites [20]. We will discuss\nMn on the 4 aand 4csites by referring to the Mn 4aand\nMn4csublattices. By changing the Ru concentration, the\nmagnetic properties of the Mn 4csublattice are altered,\nwhile those of the Mn 4asublattice remain relatively con-\nstant [21]. Thin films grown on MgO have an out-of-\nplane magnetic easy axis due to biaxial strain induced by\nthe substrate during growth [23]. Unlike the uncompen-\nsated tetragonal D022Mn3−xGa family of alloys, MRG\nhas a compensation temperature, Tcomp, where there isno net magnetization [20, 21]. Nonetheless, there is non-\nvanishing tunnel magnetoresistance [22], spin Hall angle\n[23] and magneto-optical Kerr effect [24], which all arise\nfrom the Mn 4csublattice. The occupied electronic states\noriginating from the Mn 4asublattice lie below the spin\ngap [25].\nThe electrical transport on MRG reported to date [22,\n23] can be explained using the model shown in figure\n1 (b) and (c) where the direction of spin polarisation\nis governed by the direction of the Mn 4csublattice and\nnot Mn 4aorMnet. Here we make use of the dominant3\ninfluence of a single sublattice on the electron transport\nto study the magnetism of a compensated half-metal at\ncompensation, and evaluate the exchange and anisotropy\nenergies.\nWe measure magnetotransport, especially the anoma-\nlous Hall effect in the temperature range 10K to 300K\nin magnetic fields up to 58T. The anomalous Hall con-\nductivity (AHC) of a metallic ferromagnetic film, σxy, is\nproportional to the out-of-plane component of magneti-\nzation,Mz, which is defined as Mcosθwhereθis the an-\ngle between the z-axis and the magnetization, M[26]. In\nferrimagnets, however, the AHC will depend on the band\nstructure at the Fermi level, EF, so when the material is\nhalf-metallic, one expects σxy∝MslcosθMsl, whereMsl\nis the magnetization of the sublattice that dominates the\ntransport.\nMn2RuxGa layers of varying composition, x= 0.55,\n0.61 and 0.70, were deposited on MgO substrates in a\nfully automated Shamrocksputtering system. The thick-\nness of the films, ≈27nm, was determined by X-ray re-\nflectivity. Hallcrossesofwidth100 µmandlength900 µm\nwere patterned using direct-laser-write lithography, Ar+\nion milling and lift-off. The Hall bars were contacted\nwith Cr 5nm / Au 125nm pads.\nA Lakeshore Hall system was used to measure the\nlongitudinal ( ρxx) and transverse ( ρxy) resistivities from\n10K to 300K in out-of-plane fields up to 6 .5T. The\nAHC,σxy=ρxy/ρ2\nxx[27, 28], is obtained from the raw\ndata. In-plane, µ0Hx, and out-of-plane, µ0Hz, pulsed\nmagnetic fields of up to 58T were applied at the Dres-\ndenHighMagneticFieldLaboratoryatselectedtempera-\nturesbetween10Kand220K. WefocusonMn 2Ru0.61Ga\nwithTcomp≈155K. All three compositions were found\nto have compensation temperatures between 100K and\n300K, and exhibit similar properties.\nAHC loops versus µ0HzaroundTcompare shown in\nfigure 2 (a). At all temperatures, MRG exhibits strong\nperpendicular magnetic anisotropy. The reversal of the\nsign ofσxybetween 135K and 165K indicates a rever-\nsal of the spin polarisation at EFwith respect to the\napplied field direction, as expected on crossing Tcomp.\nThe coercivity, µ0Hc, varies from 3T to 6T between\n110K and 175K. The longitudinal magnetoresistance,\nρxx(H)/ρxx(0) shown in figure 2 (b) is small ( <1%), as\nexpected fora half-metal[29]. Pulsedfield measurements\ninfigure2(c)showthat, closeto Tcomp,µ0Hcexceeds9T\nand that MRG exhibits a spin-flop transition at higher\nfields, indicated in the figure by the grey arrows. The\nderivative of selected curves of σxyversus applied field,\nfigure 2 (d), show clearly the spin-flop field especially at\nlower temperatures. We note that the longitudinal mag-\nnetoresistance up to 58T also does not exceed 1% (not\nshown). The divergence in coercivity (black circles in\nfigure 2 (e)) is expected at Tcompbecause the anisotropy\nfield in uniaxial magnets is µ0HK= 2Keff/Mnet, where\nKeffis the effective anisotropyenergy and Mnetis the netmagnetization. The anisotropy field is an upper limit on\ncoercivity. The temperature dependence of the spin-flop\nfield,µ0Hsf, is also plotted in figure 2 (e) (red squares).\nThe solid (dashed) line in figure 2 (f), traces the\ntemperature dependence of σxywhen the sample is ini-\ntially saturated in a field of −6.5T (6.5T) at 10K\nand allowed to warm up in zero-applied magnetic field.\nThe spontaneous Hall conductivity, σxy, decreases from\n7859Ω−1m−1to5290Ω−1m−1anddoesnotchangesign\nfor either of the zero-field temperature scans. The rema-\nnent value of σxyafter the application of 6 .5T (58T) is\nplotted with open (solid) symbols. The combined data\nestablish that, in MRG films, neither the AHE nor the\nAHCareproportionalto Mnet. Theydepend onthemag-\nnetization of the sublattice that gives rise to σxy. While\nsimilar behaviour is well documented for the anomalous\nHall effect in rare-earth – transition-metal (RE-TM) fer-\nrimagnets, where both RE and TM elements contribute\nto the transport [30–32], in MRG both magnetic sub-\nlattices are composed of Mn which has been confirmed\nto have the same electronic configuration, 3 d5[21]. If\nboth sublattices contributed equally to the effect, the\nsum should fall to zero at Tcomp.\nWe refer to the model presented in figure 1 (b) and\n(c) to explain the behaviour shown in figure 2 (f). Fig-\nure 1 (b) shows the Mn 4aand Mn 4csublattice moments\nand the net magnetic moment in the case of an applied\nfield,µ0Hz, along the easy-axis of MRG. Below Tcomp,\nthe Mn 4cmoment (green arrow) outweighs that of Mn 4a\n(blue arrow), and Mnet(orange arrow) is parallel to the\nMn4csublattice. At Tcomp,Mnetis zero but the direc-\ntions of the sublattice moments have not changed with\nrespect to µ0Hz. AboveTcomp,µ0Hzcauses a reversal of\nMnet(provided it exceeds µ0Hc). Here, the Mn 4asublat-\ntice has a larger moment than Mn 4candMnetwill be in\nthe samedirection asthe Mn 4amoment. Due to the anti-\nferromagnetic alignment of both sublattices the moment\non Mn 4cis parallel (antiparallel) to µ0Hzbelow (above)\nTcomp.\nIn the absence of an applied field (figure 1 (c)), the\ndirection of Mnetwill reverse crossing Tcompdue to the\ndifferent temperature dependences of the sublattice mo-\nments. However, the sublattice moments only change in\nmagnitude, and not direction. The uniaxial anisotropy\nprovided by the slight substrate-induced distortion of the\ncubic cell [20] provides directional stability along the z-\naxis. Therefore, crossing Tcompin the absence of applied\nfield, we expect no change in sign of σxy, nor should\nit vanish. The Mn 4csublattice dominates the electron\ntransport and determines spin direction of the available\nstates at EF, while the Mn 4astates form the spin-gap.\nThe results of a molecular field model [33] based on\ntwo sublattices are presented in figure 3. The molecular\nfield,Hi, experienced by each sublattice is given by:\nHi\n4a=n4a−4aM4a+n4a−4cM4c+H(1)4\nHi\n4c=n4a−4cM4a+n4c−4cM4c+H (2)\nwheren4a−4aandn4c−4care the intra-layer exchange\nconstants and n4a−4cis the inter-layer exchange con-\nstant.M4aandM4care the magnetizations of the 4 a\nand 4csublattices. His the externally applied mag-\nnetic field. The moments within the Mn 4aand Mn 4csub-\nlattices are ferromagnetically coupled and hence n4a−4a\nandn4c−4care both positive. The two sublattices couple\nantiferromagnetically and therefore n4a−4cis negative.\nThe equations are solved numerically for both tempera-\nture and applied field dependences to obtain the projec-\ntion of both sublattice magnetizations along the z-axis,\nMz−α=Mαcosθα, whereα= 4a,4c. In the absence of\nan applied field, θ= 0, therefore Mz−αreduces simply\ntoMα.\nThe model parameters are given in table I. Based on\npreviousXMCDmeasurements[21]aswellasDFTcalcu-\nlations [25] we take values of547kAm−1and 585kAm−1\nfor the magnetizations on the 4 aand 4csublattice, re-\nspectively. The values of n4a−4a,n4c−4candn4a−4c\nare fit to reproduce Tcompand the Curie temperature,\nTC. The temperature dependences of Mz−4a(blue line),\nMz−4c(green line) and Mnet(orange line) with n4a−4a=\n1150,n4c−4c= 400 and n4a−4c=−485 are shown in fig-\nure 3 (a). In orderto numericallyobtain the temperature\ndependence in zero applied field a strong field of 60T is\nusedtosetthedirectionof Mnetandthenreducedtozero,\nso the sublattice moments reverse at Tcomp= 155K as in\nthe experiment. TCis625K. Mnetvariesfrom38kAm−1\nat 10K to a maximum of 97kAm−1at 512K, close to\nTC.\nFigure 3 (b) shows the measured AHC (circles), along\nwith|Mz−4c|(green line) from the molecular field model.\nIt can be seen clearly that σxyfollows the temperature\ndependence of Mz−4cbelowTcompand not Mnet. As\na further step, we plot |Mnet|from the molecular field\nmodel (orange line) with |σxy(T)−σxy−comp|(triangles).\nAsσxyis proportional only to M4cand at compensa-\nTABLE I: Initial parameters input to the molecular field\nmodel according to equations 1 and 2. M4a,M4candK4a,\nK4care the magnetizations and uniaxial anisotropies on the\n4a, 4csublattices. n4a−4aandn4c−4care the intralayer ex-\nchange constants. n4a−4cis the interlayer exchange constant.\nDerived parameters are outputs of the molecular field model.\nInitial parameters\nM4a(0K) 547kAm−1n4a−4a 1150\nM4c(0K) 585kAm−1n4c−4c 400\nK4a 0kJm−3n4a−4c -485\nK4c 216kJm−3\nDerived parameters\nMnet(10K) 38kAm−1TC 625K\nMnet(max.) 97kAm−1Tcomp 155K/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s56/s48/s46/s48/s49/s48/s48/s46/s48/s49/s50/s48/s46/s48/s49/s52/s48/s46/s48/s49/s54/s48/s46/s48/s49/s56\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s120/s121/s47/s77\n/s110/s101/s116\n/s40/s79/s104/s109/s46/s109/s46/s65/s47/s109/s41/s45/s49\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s120/s121/s47/s77\n/s122/s32/s40/s79/s104/s109/s46/s109/s46/s65/s47/s109/s41/s45/s49\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32\n/s120/s121/s47/s77\n/s122/s45/s52 /s99/s32 /s32/s76/s105/s110/s101/s32/s102/s105/s116/s32\n/s120/s121/s47/s77\n/s122/s45/s52 /s99\n/s32\n/s120/s121/s47/s77\n/s122/s45/s52 /s97/s32 /s32/s76/s105/s110/s101/s32/s102/s105/s116/s32\n/s120/s121/s47/s77\n/s122/s45/s52 /s97/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48 /s55/s48/s48/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s32/s77\n/s122 /s45/s52 /s97\n/s32/s77\n/s122 /s45/s52 /s99\n/s32/s124 /s77\n/s110/s101/s116/s124 /s77\n/s122/s32/s40/s77/s65/s47/s109/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48 /s55/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s32/s124 /s77\n/s122/s45/s52 /s99/s124\n/s32/s124 /s77\n/s110/s101/s116/s124/s124/s77\n/s122/s124/s32/s40/s77/s65/s47/s109/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s84\n/s99/s111/s109/s112\n/s48/s50/s107/s52/s107/s54/s107/s56/s107\n/s32\n/s120/s121\n/s32/s124\n/s120/s121/s40/s84 /s41/s32/s45/s32\n/s120/s121 /s45/s99/s111/s109 /s112/s124\n/s32\n/s120/s121/s32/s32/s40/s79/s104/s109/s46/s109/s41/s45/s49\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s52/s48/s46/s54/s32/s124 /s77\n/s122 /s45/s52 /s97/s124\n/s32/s124 /s77\n/s122 /s45/s52 /s99/s124/s124/s77\n/s122/s124/s32/s40/s77/s65/s47/s109/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s54/s107/s56/s107/s32\n/s120/s121\n/s120/s121/s32/s32/s40/s79/s104/s109/s46/s109/s41/s45/s49\n/s40/s99/s41/s40/s97/s41\n/s40/s98/s41\nFIG. 3: (a) Temperature dependence of Mz−4aandMz−4c\nandMnet. The data is obtained from numerical integration\nwith an applied field to set the direction of the net magneti-\nzation and then reduced to zero, therefore the magnetizatio n\nreverses at Tcomp.Tcompis 155K and the Curie temperature\nis 625K. (b) σxy,|σxy(T)−σxy−comp|,|Mz−4c|, and|Mnet|as\nafunction of temperature. Inset: σxyplotted with Mz−4aand\nMz−4cas a function of temperature, to show that σxydoes\nindeed follow Mz−4cand not Mz−4a. (c) Ratio of σxy/Mz−4c\nandσxy/Mz−4a(dotted lines) over the experimentally mea-\nsuredtemperature rangecomplete withlinear fits(solid lin es).\nThe ratio is almost constant with no significant linear back-\nground slope showing that σxy∝Mz−4c. The inset shows the\nclear divergence of σxy/MnetatTcomp.\ntionM4c=M4a, subtracting the value of σxyatTcomp\n(|σxy(T)−σxy−comp|) gives an approximate indication of5\n/s48 /s49/s53 /s51/s48 /s52/s53/s45/s54/s107/s45/s52/s107/s45/s50/s107/s48/s50/s107/s52/s107/s54/s107/s120/s121/s32/s40/s79/s104/s109/s46/s109/s41/s45/s49\n/s181\n/s48/s72\n/s105/s32/s40/s84/s41/s32/s47/s32 /s32/s181\n/s48/s72\n/s122\n/s32/s47/s32 /s32/s181\n/s48/s72\n/s120\n/s50/s50/s48/s32/s75/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s77\n/s122/s45 /s52 /s99/s32/s40/s77/s65/s47/s109/s41\nFIG. 4: Comparison of experimental data (solid lines) and\nmolecular fieldmodel (dashed lines) at 220K for fields applie d\nalongµ0Hzandµ0Hx.µ0Hsfis observed in both cases at\n26T.\nhowMnetbehaves with temperature. Even though this\nignorestheweak M4atemperaturedependence, thetrend\nofMnetfollows|σxy(T)−σxy−comp|, showingthat σxyis a\nreflection of M4cand notMnet. The inset in figure 3 (b)\nshows both Mz−4aandMz−4cwith the experimentally\nobtained σxy, and shows that σxymore closely follows\nMz−4c. Figure 3 (c) shows the ratio of σxytoMz−4c\n(green dotted line), Mz−4a(blue dotted line) and Mnet\n(inset). Linear fitting of σxy/Mz−4c(solid green line),\nandσxy/Mz−4a(solid blue line) show that σxy/Mz−4c\nremains constant over the measured temperature range,\nand is equal to 0 .0136Ω−1m−1A−1m, similar to what\nhas been reported for other itinerant ferromagnetic sys-\ntems [34, 35]. The linear slope for σxy/Mz−4aand the\ndivergence of σxy/Mnetshows that σxyreflects neither of\nthese two quantities.\nA recent study has shown via ab initio calculations\nthat this must be the case for a fully-compensated half-\nmetallic ferrimagnetic system [36] although previous re-\nports on bulk films found ρxy, and hence, σxyfalling to\nzero atTcomp[37].\nFor the evaluation of the magnetic anisotropy we use\nthe initial low-field change of σxyversusµ0Hxand ex-\ntrapolate to zero and obtain K4c(not shown). The val-\nues obtained vary from 100kJm−3to 250kJm−3over\nthe entire data range. We also calculate the anisotropy\ndirectly from the spin-flop transition Hsf=/radicalbig\n2HKHex\n4c,\nwhereHKis thesublattice anisotropyfield and Hex\n4cis the\nexchange field, the first term in Eqn. 2. The anisotropy\nfield,HK, is related to the sublattice anisotropy energy\nK4c.\nA comparison between the experiment and the model\nat 220K for both µ0Hzandµ0Hxis shown in figure\n4. The solid lines plot the experimentally obtained σxy,\nwhile the dashed lines plot Mz−4cfrom the model. The\nspin-flop field is observed in both cases at µ0Hz= 26T.\nFor the case of µ0Hx, it can first be seen that the 4 c\nmoment does not saturate along the field as one wouldexpect [18, 38]. It initially decreases but then returns\nto a saturated value in both the experimental data and\nthe model. This behaviour is due to the fact that in\nMRG the exchange and anisotropy energies are compa-\nrable and weak. If the exchange coupling is strong then\nthe net magnetic moment could be saturated along µ0Hx\nas both sublattices can remain antiparallel up to the\nanisotropy field µ0HK= 2(K4a+K4c)/(M4a+M4c) =\n2Keff/Mnet. If the exchange coupling is weak then both\nsublattice moments will tilt from their antiparallel align-\nment, breaking exchange, before the net magnetic mo-\nment can be saturated along µ0Hxat the appropriate\nsublattice anisotropy field µ0HK= 2Ksl/Msl, sl = 4a,4c.\nThe model and experiment disagree slightly on the\ntemperature dependence of HsfbelowTcomp. Bet-\nter agreement can be obtained by using much higher\nanisotropyenergiesofoppositesign: K4a=−1.5MJm−3\nandK4c= 1.7MJm−3. This has the effect of increasing\n(decreasing) Hsfabove (below) Tcomp. While this im-\nproves the match between σxyversusµ0HzbelowTcomp,\nit worsens the match of σxyversusµ0Hx, at all tem-\nperatures. This and the slight discrepancies between\nthe model and experiment when a low value of K4c\nis used (Fig. 4) indicate that additional anisotropies,\nlikely cubic, in MRG, as well as anti-symmetric exchange\n(Dzyaloshinskii-Moriya interaction) should be taken into\naccount.\nWe have shown that the uniaxial molecular field model\nreproduces the main characteristics of the experimental\ndata and we confirm the relationship σxy∝M4ccosθM4c.\nKnowing HKandHex\n4cwe can predict the frequencies\nof the anisotropy, fanis=γµ0HK, and the exchange,\nfexch=γµ0/radicalbig\n2HKHex\n4c=γµ0Hsf, magnetic resonance\nmodes, where γ= 28.02GHzT−1[39]. At 220K, µ0Hsf\n= 26T and µ0Hex\n4c=n4a−4cM4a= 294T, therefore\nµ0HK= 1.15T and the resonances are fanis= 32GHz\nandfexch= 729GHz.\nIn conclusion, σxyfor fully-compensated half-metallic\nferrimagnetic alloys follows the relevant sublattice mag-\nnetization, MslcosθMsl, and not MnetcosθMnet. High-\nfield magnetotransport and molecular field modelling al-\nlows the determination of the anisotropy and exchange\nconstants provided the half-metallic material is collinear.\nMn2RuxGa behaves magnetically as an antiferromagnet\nand electrically as a highly spin polarised ferromagnet; it\nis capable of operation in the THz regime and its trans-\nport behaviour is governed by the Mn 4csublattice. The\nimmediate, technologically relevant, implication of these\nresultsis that spin-transfertorqueeffects in compensated\nferrimagnetic half-metals will be governed by single sub-\nlattice.6\nACKNOWLEDGEMENTS\nThis project has received funding from the Euro-\npean Union’s Horizon 2020 research and innovation pro-\ngramme under grant agreement No 737038 (TRAN-\nSPIRE). This work is supported by the Helmholtz Young\nInvestigator Initiative Grant No. VH-N6-1048. 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H.,\nFelser, C., and Chadov, S., arxiv.org/abs/1710.04453\n(2017).\n[37] Stinshoff, R., Fecher, G., Chadov, S., Nayak, A., Balke,\nB., Ouardi, S., Nakamura, T., and Felser, C., AIP Ad-\nvances7(10), 105009 (2017).\n[38] Guo, V., Lu, B., Wu, X., Ju, G., Valcu, B., and Weller,\nD.,Journal of Applied Physics 99(8), 08E918 (2006).\n[39] Morrish, A. H., Physical Principles of Magnetism, Ch.\n10, pp 539–639. Wiley, New York (1965)."    },    {        "title": "1810.02560v3.Magnetic_anisotropy_and_spin_polarized_two_dimensional_electron_gas_in_the_van_der_Waals_ferromagnet_Cr__2_Ge__2_Te__6_.pdf",        "content": "Magnetic anisotropy and spin-polarized two-dimensional electron gas\nin the van der Waals ferromagnet Cr 2Ge2Te6\nJ. Zeisner,1, 2,\u0003A. Alfonsov,1,\u0003S. Selter,1, 2S. Aswartham,1M. P. Ghimire,3, 1, 4\nM. Richter,1, 5J. van den Brink,1B. B uchner,1, 2and V. Kataev1\n1Leibniz Institute for Solid State and Materials Research IFW Dresden, 01069 Dresden, Germany\n2Institute for Solid State and Materials Physics, TU Dresden, 01062 Dresden, Germany\n3Central Department of Physics, Tribhuvan University, Kirtipur, 44613, Kathmandu, Nepal\n4Condensed Matter Physics Research Center, Butwal-11, Rupandehi, Lumbini, Nepal\n5Dresden Center for Computational Materials Science (DCMS), TU Dresden, 01062 Dresden, Germany\n(Dated: April 10, 2019)\nWe report a comprehensive experimental investigation on the magnetic anisotropy in bulk single\ncrystals of Cr 2Ge2Te6, a quasi-two-dimensional ferromagnet belonging to the family of magnetic lay-\nered transition metal trichalcogenides that have attracted recently a big deal of interest with regard\nto the fundamental and applied aspects of two-dimensional magnetism. For this purpose electron\nspin resonance (ESR) and ferromagnetic resonance (FMR) measurements have been carried out\nover a wide frequency and temperature range. A gradual change in the angular dependence of the\nESR linewidth at temperatures above the ferromagnetic transition temperature Tcreveals the de-\nvelopment of two-dimensional spin correlations in the vicinity of Tcthereby proving the intrinsically\nlow-dimensional character of spin dynamics in Cr 2Ge2Te6. Angular and frequency dependent mea-\nsurements in the ferromagnetic phase clearly show an easy-axis type anisotropy of this compound.\nFurthermore, these experiments are compared with simulations based on a phenomenological ap-\nproach, which takes into account results of static magnetization measurements as well as high tem-\nperaturegfactors obtained from ESR spectroscopy in the paramagnetic phase. As a result the de-\ntermined magnetocrystalline anisotropy energy density (MAE) KUis (0:48\u00060:02)\u0002106erg/cm3.\nThis analysis is complemented by density functional calculations which yield the experimental MAE\nvalue for a particular value of the electronic correlation strength U. The analysis of the electronic\nstructure reveals that the low-lying conduction band carries almost completely spin-polarized, quasi-\nhomogeneous, two-dimensional states.\nI. INTRODUCTION\nLayered van der Waals crystals constitute a large\nclass of materials which is intensively studied in the\n\feld of current solid state research [1, 2]. Weak cou-\nplings between individual layers render these quasi-two-\ndimensional compounds optimal as a starting point for\nfabrication of two-dimensional (2D) atomic crystals [3]\nwhich show a plethora of electronic properties and, con-\nsequently, a high potential for technical applications (see,\ne.g., Refs. [1, 2, 4] and references therein). Recently,\nsome members of the family of layered transition metal\ntrichalcogenides (TMTC) came into focus of research due\nto their peculiar magnetic properties in combination with\nsemiconducting or insulating behavior [5{10]. In particu-\nlar, TMTC are currently considered as promising materi-\nals for future technological applications but also o\u000ber an\nextensive materials base for exploring fundamental mag-\nnetic properties of strongly correlated 2D electron sys-\ntems. Cr 2Ge2Te6belongs to this class of materials and\nit was \frst synthesized by V. Carteaux et al. [11]. The\ncarriers of magnetism in this material are Cr3+ions with\nspinS= 3=2 which are octahedrally coordinated by Te\nligands and form a 2D honeycomb-like magnetic lattice\nin theabplane. In the bulk crystal individual magnetic\n\u0003These authors contributed equally to this work.layers stack along the crystallographic caxis and are only\nweakly coupled by van der Waals forces. By virtue of in-\ntrinsic magnetic two-dimensionality Cr 2Ge2Te6has been\nproposed, e.g., as magnetic substrate for nanoelectronic\ndevices [12] or as a candidate for next generation memory\ndevices [13]. From the fundamental point of view the ob-\nservation of intrinsic ferromagnetism in atomically thin\n\flms of Cr 2Ge2Te6is intriguing [14]. Such phenomenon\nwas also observed in single layers of the related materials\nCrI3[15], and VSe 2[16].\nMotivated by the above-mentioned aspects, Cr 2Ge2Te6\nreceived considerable attention both experimentally [11,\n17{23] and theoretically [5, 24] during the last years.\nAiming at a detailed understanding of the magnetic prop-\nerties of this material, in bulk as well as in monolayer\nform, a quantitative characterization of the magnetic\nanisotropies represents a key task. Regarding the lat-\nter, \frst insights were recently obtained by X. Zhang et\nal.[17] for the case of bulk samples by studying ferro-\nmagnetic resonance (FMR) at frequencies below 20 GHz\nas a function of temperature with external magnetic \feld\napplied in the abplane. In the present work we report a\ncomprehensive investigation of the magnetic anisotropy\nof bulk Cr 2Ge2Te6single crystals by means of electron\nspin resonance (ESR) spectroscopy over a wide frequency\nand temperature range for several \feld orientations which\nallows a re\fned quantitative analysis of the magnetocrys-\ntalline anisotropy energy density (MAE). Furthermore,arXiv:1810.02560v3  [cond-mat.str-el]  9 Apr 20192\nwe compare experimentally obtained MAE with our re-\nsults from density functional (DF) calculations and \fnd\na good agreement for a particular strength of electronic\ncorrelations U. In addition, the two-dimensionality of\nspin dynamics in Cr 2Ge2Te6in the vicinity of the ferro-\nmagnetic transition temperature Tcis proven by angular\ndependent measurements of the ESR linewidth at various\ntemperatures.\nPrevious theoretical [24, 25] and experimental [25]\nwork suggests strong spin polarization of the electronic\nstates of Cr 2Ge2Te6at the valence band maximum\n(VBM) and the conduction band minimum (CBM). Since\nthese investigations propose di\u000berent relative polariza-\ntions at VBM and CBM, we use the obtained electronic\nstructure results for a further analysis.\nThe paper is organized as follows. Details of synthesis\nand characterization of the samples as well as further ex-\nperimental and calculation details are provided in Sec. II.\nExperimental and theoretical \fndings are presented and\ndiscussed in Sec. III. Main conclusions of this study are\ndrawn in Sec. IV. Appendices A and B contain additional\ninformation on the samples used for the measurements\nand further results obtained from low-frequency FMR,\nrespectively.\nII. SAMPLES AND METHODS\nSingle crystals of Cr 2Ge2Te6were grown by using the\nself \rux technique described by X. Zhang et al. [17]. A\nmixture of chromium granules (4N, MaTeck), germanium\nchips (5N, Sigma Aldrich) and tellurium lumps (5N, Alfa\nAesar) with molar ratios of Cr : Ge:Te = 10 : 13 : 77 were\ntaken in an alumina crucible which was then sealed in\na quartz ampule under a partial atmosphere of Ar (ap-\nprox. 300 mbar). Then, the quartz ampule was placed\nupright in a Mu\u000bel furnace (Nabertherm) and heated\nto 1000\u000eC for 24 h followed by cooling with a rate of\n2\u000eC/h to 450\u000eC. At this temperature excessive melt was\ncentrifuged. Shiny, plate-like crystals of up to 11 mm \u0002\n10 mm\u00020.2 mm were obtained.\nCrystals were characterized by powder X-ray di\u000brac-\ntion (PXRD) and energy dispersive X-ray spectroscopy\n(EDX). EDX was performed at an accelerating volt-\nage of 30 kV using a ZEISS EVO MA 10 scanning\nelectron microscope (SEM) with an energy dispersive\nX-ray analyzer. A mean elemental composition of\nCr : Ge : Te = (20.9 \u00060.2) % : (20.4\u00060.2) % : (58.7\u00060.1) %\nwas obtained by EDX. This is in good agree-\nment with the expected elemental composition of\nCr : Ge : Te = 20 % : 20 % : 60 % assuming a general un-\ncertainty of the method of approximately \u00062% for\neach element. In the range of this uncertainty, the low\nstandard deviation of the chemical composition over\nall measured spots indicates a homogeneous elemental\ndistribution. PXRD was performed on pulverized\ncrystals on a STOE STADI laboratory di\u000bractometer\nin transmission geometry with Cu K \u000b1radiation from\nFIG. 1. Powder X-ray di\u000braction pattern from Cu-K \u000b1ra-\ndiation (1.54059 \u0017A) of pulverized crystals of Cr 2Ge2Te6(red\ndots), calculated pattern from Rietveld analysis (black line),\ndi\u000berence between measured and calculated intensity (blue\nline) and expected Bragg positions for the Cr 2Ge2Te6struc-\nture (black bars).\na curved Ge(111) single crystal monochromator and\ndetected by a 6\u000e-linear position sensitive detector manu-\nfactured by STOE & Cie. Rietveld analysis was carried\nout on the measured X-ray pattern using the Jana2006\nsoftware package [26], the di\u000braction pattern with the\n\ftted model is shown in Fig. 1. Structural parameters\nand corresponding R-values are summarized in Table II\nin Appendix A. These results are in good agreement\nwith literature [11, 27].\nDC magnetization was measured as a function of tem-\nperature and magnetic \feld using a vibrational sam-\nple quantum interference device magnetometer (SQUID-\nVSM) from Quantum Design. All measurements were\ncarried out with a vibrational amplitude of 2 mm. The\nvalues obtained for magnetic moments were corrected due\nto deviation of the measured sample shape and size from\na point dipole. This correction follows the procedure de-\nscribed in Ref. [28] and further details are presented in\nAppendix A. For temperature dependent measurements,\nthe sample was demagnetized at 300 K by applying an\nexternal \feld of 7 T and reducing it to zero while period-\nically switching the polarity of the \feld. After that the\nsample was zero \feld cooled (ZFC) down to 1.8 K. The\nmeasurement itself was performed upon warming from\n1.8 K to 300 K with an external applied \feld of 0.35 T or\n3 T, respectively. For \feld sweep experiments the sam-\nple was demagnetized as described above and cooled to\nthe desired temperature of 4 K or 7 K, respectively. Dur-\ning the measurement the \feld was ramped from zero to\n7 T to obtain a virgin magnetization curve, followed by\na measurement of the full hysteresis curve by cycling the\n\feld from 7 T to -7 T and back.\nESR and FMR experiments were performed using3\nTABLE I. Comparison of atomic positions of Cr 2Ge2Te6as determined from PXRD and DF calculations.\nMethod Atom Wycko\u000b site x y z\nTe 18 f -0.0017(11) 0.3642(6) 0.0853(10)\nPXRD Cr 6 c 0 0 0.3330(30)\nGe 6 c 0 0 0.0588(14)\nTe 18 f -0.003 0.374 0.085\nGGA Cr 6 c 0 0 0.334\nGe 6 c 0 0 0.059\nTe 18 f -0.003 0.373 0.085\nGGA+U, Cr 6 c 0 0 0.334\nU= 2 eV Ge 6 c 0 0 0.058\nthree di\u000berent setups. For high-\feld/high-frequency ESR\n(HF-ESR) measurements a home made setup comprising\na superconducting magnet equipped with a variable tem-\nperature insert (Oxford Instruments), a vector network\nanalyzer (PNA-X from Keysight Technologies) for gener-\nation and detection of microwaves, and oversized waveg-\nuides were used. This setup allows measurements over a\nwide frequency (20 - 330 GHz), magnetic \feld (0 - 16 T),\nand temperature (1.8 - 300 K) range. All HF-ESR mea-\nsurements were performed in transmission mode using a\nFaraday con\fguration. Resonance \felds were determined\nfrom the position of the minimum in the transmission. In\naddition, a cantilever-based torque detected ESR setup\nwas employed. In this setup magnetic \felds up to 10 T\nand temperatures between 7 and 300 K are provided by\na magneto-optical cryostat (Oxford Instruments). Mi-\ncrowaves with frequencies of \u001863 GHz and\u0018125 GHz\nare generated by an ampli\fer/multiplier chain (AMC\nfrom Virginia Diodes Inc.) and are focused on the sample\nby means of the microwave optics. The sample is placed\non the tip of a piezoresistive cantilever (see Appendix A),\nwhose vibration amplitude and frequency changes are\nused to detect the FMR signal. Furthermore, a standard\nX-band ESR-spectrometer (EMX from Bruker) operat-\ning at a microwave frequency of 9.6 GHz and providing\nmagnetic \felds up to 0.9 T was used. It is equipped with\na He-gas \row cryostat (Oxford Instruments) and a go-\nniometer allowing temperature and angular dependent\nmeasurements between 4 and 300 K.\nThe DF calculations were carried out with the all-\nelectron full-potential local-orbital (FPLO) code [29,\n30], version 18.00, using the standard generalized-\ngradient approximation (GGA) in the parameteriza-\ntion of Perdew, Burke, and Ernzerhof (PBE-96) [31].\nThe GGA+ Ufunctional with atomic-limit (AL) double-\ncounting corrections [32] was applied to the Cr-3 dor-\nbitals in a part of the calculations. The AL \ravor of\ndouble-counting is in general to be preferred against the\naround-mean-\feld version since it ful\flls Hund's \frst\nrule [33]. The values of Uwere varied in the range\nfrom 0.5 eV to 4.0 eV, while the related exchange pa-\nrameterJwas \fxed to 0.6 eV. All k-space integrations\nwere carried out with the linear tetrahedron method\nand 12\u000212\u000212 subdivisions in the full Brillouin zone.The densities of states were evaluated with a re\fned\nmesh of 24\u000224\u000224 subdivisions. The experimental\nspace group R \u00163 (no. 148) and lattice parameters as ob-\ntained in the present experiments were used for all cal-\nculations. Starting from the experimentally determined\nvalues, the atomic positions were optimized by scalar-\nrelativistic GGA or GGA+ Ucalculations for each value\nofUuntil all residual forces were smaller than 1 meV/ \u0017A.\nAtomic positions as obtained from PXRD and two of\nthe DF calculations are compared in Table I. As can be\nseen from this table, the re\fned calculated atomic posi-\ntions agree with their experimental counterparts except\nfor the Te-ycoordinates, which di\u000ber by 0.01. This dif-\nference could be caused by omitting spin-orbit coupling\nin the optimization. For the determination of magnetic\nanisotropy energy, self-consistent calculations using the\noptimized structure were carried out in full relativistic\nmode for both of the considered orientations of the mag-\nnetization, [001] (perpendicular to the quasi-2D plane)\nand [100] (arbitrarily chosen in the plane).\nIII. RESULTS AND DISCUSSION\nA. Magnetization\nMagnetization Mmeasured as a function of external\nmagnetic \feld His shown in Fig. 2. Measurements were\nconducted with magnetic \feld aligned parallel and per-\npendicular to the caxis at temperatures of 4 and 7 K,\nrespectively. At both temperatures a similar behav-\nior is found: While raising the \feld from zero, M(H)\nrapidly increases until the onset of saturation at \felds\nHab\nsat\u00190:5 T andHc\nsat\u00190:3 T forH?candHkcat\n4 K, respectively. The faster saturation of the magneti-\nzation forHkc(Hc\nsat< Hab\nsat) evidences the easy-axis\ntype anisotropy of Cr 2Ge2Te6. Furthermore, all mea-\nsured magnetization curves reach an isotropic saturation\nvalue of about 3 \u0016Bper Cr ion which is in very good\nagreement with the value theoretically expected for Cr3+\nions withS= 3=2 and ag-factor of 2.\nThe ferromagnetic transition temperature Tcwas\ndetermined from measurements of the temperature-\ndependent magnetization in an external \feld of 0.1 T ap-4\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51\n/s32/s72 /s32/s124 /s124 /s32/s99\n/s32/s72 /s32 /s32/s99/s77 /s32/s40\n/s66/s32/s47/s32/s67/s114/s41\n/s48/s32/s72 /s32/s40/s84/s41/s84 /s32/s61/s32/s52/s32/s75/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51\n/s32/s72 /s32/s124/s124/s32/s99\n/s32/s72 /s32 /s32/s99/s84 /s32/s61/s32/s55/s32/s75/s77 /s32/s40\n/s66/s32/s47/s32/s67/s114/s41\n/s48 /s32/s72 /s32/s40/s84/s41\nFIG. 2. Magnetization as a function of external magnetic \feld\napplied parallel (red circles) and perpendicular (blue squares)\nto thecaxis at 4 K. Corresponding curves recorded at 7 K are\nshown in the inset.\nplied parallel to the caxis from which the static mag-\nnetic susceptibility \u001fwas calculated (Fig. 3). From the\nposition of the minimum in the \frst derivative of the\nsusceptibility @\u001f=@T , aTcof (66\u00061) K was obtained.\nThe uncertainty in Tcis caused by a \fnite width of the\nminimum, see Fig. 3. Our value of Tcis consistent with\nthe results of Ref. [11] where, depending on the method\nof determination, transition temperatures of 61 K and\n65 K were found. Similar values were reported for Tc\nin other recent studies on Cr 2Ge2Te6[19, 20, 23]. The\ninverse susceptibility is shown as a function of tempera-\nture in the inset of Fig. 3 together with a linear \ft ac-\ncording to the Curie-Weiss-law. This \ft yields a Curie-\nWeiss temperature \u0002 cwof (92\u00061) K and an e\u000bective\nmoment\u0016e\u000bof (4:04\u00060:01)\u0016B/Cr. The latter is consis-\ntent with the value 3.87 \u0016B/Cr theoretically expected for\na spin-only system with a g-factor of 2. The deviation\nbetweenTcand \u0002 cwis in agreement with previous stud-\nies [12, 19, 20] and can be attributed to the development\nof short-range correlations already above Tcas will be\ndiscussed in Sec. III B.\nFurthermore, results of temperature dependent mea-\nsurements in di\u000berent magnetic \felds are presented in\nFig. 4 for two orientations of the \feld with respect to the\ncrystal axes. At an external \feld of 0.35 T, i.e., below\nthe saturation \feld in the abplane at low temperatures,\nan anisotropic behavior of the magnetization is found.\nFor magnetic \felds applied perpendicular to the caxis a\nmaximum of the magnetization is observed around 40 K,\nwhereas magnetization increases monotonously with de-\ncreasing temperature for Hkc. A similar di\u000berence be-\ntween measurements with \felds perpendicular and paral-\nlel to the easy direction was found in other Cr-based 2D\nhoneycomb materials, for example CrCl 3[34] and CrI 3\n[35, 36]. This behavior might be caused either by antifer-\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s45/s52/s45/s50/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s32/s40/s101/s109/s117/s32/s47/s32/s109/s111/s108/s32/s79/s101/s41\n/s84 /s32/s40/s75/s41/s84\n/s67 /s32/s61/s32/s54/s54/s32/s75\n/s72 /s32/s124/s124/s32/s99\n/s45/s52/s45/s50/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56\n/s47 /s84 /s32/s40/s101/s109/s117/s32/s47/s32/s109/s111/s108/s32/s79/s101/s32/s75/s41\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s45 /s49 \n/s32/s40/s109/s111/s108/s32/s79/s101/s32/s47/s32/s101/s109/s117/s41\n/s84 /s32/s40/s75/s41/s67/s87 /s32/s61/s32/s57/s50/s32/s75\n/s101 /s102 /s102 /s32/s61/s32/s52/s46/s48/s52/s32/s181\n/s66/s47/s67/s114FIG. 3. Temperature dependence of the static magnetic sus-\nceptibility measured with an external \feld of 0.1 T oriented\nparallel to the caxis (left axis of the main panel). The right\naxis represents the \frst derivative @\u001f=@T , which shows a min-\nimum atTc. The inverse susceptibility is given together with\na linear \ft in the inset. For better visibility, error bars of \u001f\nand 1=\u001fare only shown for every 25thdata point.\nromagnetic couplings between the ferromagnetic layers, a\nmore complex ferromagnetic state in low magnetic \felds\nor a temperature dependence of the magnetic anisotropy\n[34{36]. On the other hand, for a \feld of 3 T, which\nis larger than the low-temperature saturation \felds for\nboth orientations in Cr 2Ge2Te6, an isotropic behavior\nwas found with the onset of ferromagnetic (FM) correla-\ntions being shifted to higher temperatures as the external\nmagnetic \feld enhances FM correlations.\nB. ESR and FMR\nThe major distinction between ESR and FMR is that\nthe former is the phenomenon of a resonant excitation of\nan ensemble of paramagnetic, possibly exchange coupled\nspins, whereas the latter is the resonance of the total\nmagnetization of the ferromagnetically ordered sample.\nSince Cr3+(3d3,S= 3=2) ions in Cr 2Ge2Te6have no\norbital momentum in \frst order. Thus, their spectro-\nscopicgfactor is generally expected to be very close to\nthe spin-only value of 2 and only slightly anisotropic due\nto the second-order spin-orbit coupling e\u000bects [37] and\ninteractions with the heavier Te-ligands, see discussion\nin Sec. III D. Therefore, an ESR signal in the paramag-\nnetic state of Cr 2Ge2Te6has to be almost isotropic with\nrespect to the orientation of the applied external \feld.\nIn contrast, an FMR signal of Cr 2Ge2Te6in the ordered\nstate may be appreciably anisotropic due to the magnetic\nshape anisotropy of the non-spherical sample as well as\ndue to the intrinsic magnetocrystalline anisotropy, which\nboth a\u000bect the resonance behavior of the magnetization5\n/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48 /s49/s50/s53 /s49/s53/s48 /s49/s55/s53/s48/s49/s50\n/s51/s50/s49\n/s32/s77 /s32 /s97/s116/s32/s48/s46/s51/s53/s32/s84/s58/s32 /s72 /s32/s124 /s124 /s32/s99\n/s32/s77 /s32 /s97/s116/s32/s48/s46/s51/s53/s32/s84/s58/s32 /s72 /s32 /s32/s99 /s32/s77 /s32/s40\n/s66/s32/s47/s32/s67/s114/s41\n/s84 /s32/s40/s75/s41/s45/s49/s53/s48/s45/s49/s48/s48/s45/s53/s48/s48/s53/s48/s49/s48/s48\n/s32/s69/s83/s82/s58/s32 /s72 /s32/s124 /s124 /s32/s99/s44/s32 /s32/s61/s32/s57/s46/s54/s32/s71/s72/s122\n/s32/s69/s83/s82/s58/s32 /s72 /s32 /s32/s99/s44/s32 /s32/s61/s32/s57/s46/s54/s32/s71/s72/s122\n/s48/s32/s40/s72\n/s114/s101/s115/s32/s45/s32 /s72\n/s114/s101/s115/s40/s103\n/s49/s50/s48/s75/s41/s41/s32/s40/s109/s84/s41\n/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48 /s49/s50/s53 /s49/s53/s48 /s49/s55/s53/s48/s50/s52\n/s55/s54/s53/s52/s51/s50/s49\n/s32/s77 /s32/s97/s116/s32/s51/s32/s84/s58/s32 /s72 /s32 /s32/s99\n/s32/s77 /s32/s97/s116/s32/s51/s32/s84/s58/s32 /s72 /s32 /s32/s99/s77 /s32/s40\n/s66/s32/s47/s32/s67/s114/s41\n/s84 /s32/s40/s75/s41/s45/s50/s48/s48/s45/s49/s53/s48/s45/s49/s48/s48/s45/s53/s48/s48/s53/s48/s49/s48/s48\n/s32/s69/s83/s82/s58/s32 /s72 /s32/s124 /s124 /s32/s99/s44/s32 /s32/s61/s32/s55/s53/s46/s53/s32/s71/s72/s122\n/s32/s69/s83/s82/s58/s32 /s72 /s32 /s32/s99/s44/s32 /s32/s61/s32/s56/s51/s46/s56/s32/s71/s72/s122\n/s48/s32/s40/s72\n/s114/s101/s115/s32/s45/s32 /s72\n/s114/s101/s115/s40/s103\n/s49/s50/s48/s75/s41/s41/s32/s40/s109/s84/s41/s40 /s97 /s41 /s40 /s98/s41 \nFIG. 4. Resonance shift \u000eH(right vertical scale) as a function of temperature at a microwave frequency \u0017of 9.6 GHz (a) and at\nhigher frequencies (b) for external magnetic \feld applied parallel and perpendicular to the caxis, respectively (open symbols)\ntogether with the T-dependence of the static magnetization M(T) (left vertical scale) for the respective \feld orientations (closed\nsymbols). For better visibility, error bars for static magnetization are shown only for every tenth data point. The zero-shift of\nan ideal paramagnet is indicated by a gray dashed line. The black dotted line on panel (b) denotes the shift at 4 K obtained\nfrom the analysis of the frequency dependent FMR data (see the text).\nof the sample depending on the particular direction of\nthe magnetic \feld [38{40]. This expectation has been in\nfact con\frmed by measurements of the frequency depen-\ndence of the position of the resonance signals for Hkc\nandH?c\feld geometries at two selected temperatures\nof 120 K and 4 K, i.e., well above and well below the FM\ntransition temperature Tc\u001866 K, respectively (Fig. 5).\nIn the paramagnetic state above Tcthe resonance \feld\nHresshows a linear dependence on the microwave fre-\nquency\u0017, as expected for a paramagnetic resonance:\n\u0017=g\u0016B\u00160Hres=h : (1)\nHere\u0016Bdenotes the Bohr magneton, \u00160is the vacuum\npermeability, his the Planck constant, and gis the (ef-\nfective)g-factor of the resonating spins. Indeed, only a\nsmall anisotropy between the two \feld orientations was\nobserved at 120 K and by \ftting of Eq. (1) to the data\nshown in Fig. 5 (a) the g-factorsgjj= 2:040\u00060:005 and\ng?= 2:023\u00060:005 were obtained for \feld applied par-\nallel and perpendicular to the caxis, respectively (the\nrespective \fts are plotted by solid lines in Fig. 5 (a)).\nAs will be discussed in the following in more detail, our\nESR study reveals the existence of short-range FM cor-\nrelations already above Tc. In low-dimensional spin sys-\ntems this can give rise to a frequency-dependent shift of\nthe resonance line [41]. Thus, the g-factors determined\nfrom Eq. (1) are, in fact, e\u000bective g-factors that contain\nboth, the ionic g-factor of the magnetic ions and the ad-\nditional contribution arising from low-dimensional spin\ncorrelations.\nAt 4 K, the frequency dependence exhibits a pro-nounced anisotropy as well as an orientation-dependent\nnon-zero intercept with the ordinate of the frequency-\n\feld diagram, see Fig. 5 (b). Such a behavior is typical\nfor FMR with magnetic \felds applied parallel or perpen-\ndicular to the magnetic anisotropy axis [40]. Thus, the\n\u0017(Hres) dependence of the FMR signal measured over a\nwide frequency range from 10 to 330 GHz gives an addi-\ntional proof for the identi\fcation of the caxis as the mag-\nnetic easy axis, in line with the static magnetization mea-\nsurements and previous observations in Refs. [11, 14, 17].\nNote that solid lines in Fig. 5 (b) represent the simula-\ntion results using a standard phenomenological theory of\nFMR, as will be discussed in detail in Sec. III C.\nIn order to gain detailed insights into the devel-\nopment of the magnetic anisotropy upon a transition\nfrom the high-temperature paramagnetic state into the\nlow-temperature ferromagnetically ordered state, the\nshift\u000eH(T) of the resonance line from its param-\nagnetic resonance position was measured as a func-\ntion of temperature at several selected excitation fre-\nquencies\u0017(Fig. 4). We de\fne the resonant shift\nas\u000eH(T) =Hres(T)\u0000Hres(120 K), where the ref-\nerence resonance \feld Hres(120 K) was calculated ac-\ncording to Eq. (1) using the g-factor values obtained\nfrom the frequency dependent measurements in the\nparamagnetic state at T= 120 K [Fig. 5 (a)].\nAs can be seen in Fig. 4, at all measured frequencies\nthe signal is shifted to lower \felds (resulting in a neg-\native shift) if the magnetic \feld is oriented along the c\naxis, which is the magnetic \\easy\" axis, while it is shifted\nto higher \felds (corresponding to a positive shift) for6\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49 /s49/s50/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48/s52/s48/s48\n/s50/s55/s49/s46/s49/s32/s71/s72/s122 \n/s84 /s32/s61/s32/s52/s32/s75\n/s32/s72 /s32/s124 /s124 /s32/s99\n/s32/s72 /s32 /s32/s99/s32/s40/s71/s72/s122/s41\n/s48/s32/s72\n/s114/s101/s115/s32/s40/s84/s41/s51/s50/s46/s52/s32/s71/s72/s122 /s50/s50/s48/s46/s56/s32/s71/s72/s122 \n/s49/s52/s48/s46/s53/s32/s71/s72/s122 \n/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s115/s105/s103/s110/s97/s108/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s56/s50/s46/s56/s32/s71/s72/s122 \n/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s48/s48/s50/s48/s52/s48/s54/s48/s32/s40/s71/s72/s122/s41\n/s48 /s32/s72\n/s114/s101 /s115 /s32/s40/s84/s41\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49 /s49/s50/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48/s52/s48/s48\n/s32/s72 /s32/s124 /s124 /s32/s99/s58/s32 /s103\n/s124/s124/s32/s61/s32/s50/s46/s48/s52/s48\n/s32/s72 /s32 /s32/s99/s58/s32 /s103 /s32/s61/s32/s50/s46/s48/s50/s51/s32/s40/s71/s72/s122/s41\n/s48/s32/s72\n/s114/s101/s115/s32/s40/s84/s41/s50/s55/s51/s46/s54/s32/s71/s72/s122 \n/s49/s57/s54/s46/s50/s32/s71/s72/s122 \n/s49/s54/s55/s46/s56/s32/s71/s72/s122 \n/s56/s50/s46/s53/s32/s71/s72/s122 /s84 /s32/s61/s32/s49/s50/s48/s32/s75\n/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s115/s105/s103/s110/s97/s108/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s40 /s98/s41 /s40 /s97 /s41 \nFIG. 5. Relation between the excitation frequency \u0017(left vertical scale) and the corresponding resonance \feld \u00160Hresat 120 K\n(a) and 4 K (b) (closed symbols) for external magnetic \feld applied parallel and perpendicular to the caxis. Representative\n\feld-sweep ESR spectra are shown for various \u0017(right vertical axis) and H?c. For comparison, all spectra are normalized and\nshifted vertically. Solid lines in (a) are \fts to the data using the linear frequency-\feld dependence of a paramagnet [Eq. (1)],\nwhereas solid lines in (b) denote simulations of resonance \felds in the FM phase according to Eq. (2). Inset on panel (b) shows\nthe low-frequency part of the \u0017(Hres) diagram on an enlarged scale.\nmagnetic \feld perpendicular to the caxis. Despite the\nqualitative similarity between measurements at di\u000berent\nfrequencies concerning the observed trends of the shift\n\u000eH(T), the measured \u000eH(T)k;?dependences at distinct\nfrequencies di\u000ber from each other regarding the details\nof the shape of the resulting \u000eH(T) curves. While the\nshift evolves rather smoothly for higher frequencies (open\nsymbols in Fig. 4 (b)), measurements at 9.6 GHz (open\nsymbols in Fig. 4 (a)) revealed a much abrupter change\nof\u000eH(T) below 80 K. A similar behavior was observed in\nthe temperature dependent magnetization measurements\nM(T)k;?at 0.35 and 3 T (full symbols in Fig. 4), respec-\ntively. Though in particular for strong \felds where the\nM(H) dependence at T\u001cTcis saturated (Fig. 2), the\nM(T)kandM(T)?curves are rather symmetric in shape,\nas expected, the resonance shifts are not symmetric with\nrespect to the abscissa as the functional form of the res-\nonance condition in the ferromagnetic case depends on\nthe orientation of external \feld relative to the magnetic\nanisotropy axis [38, 40].\nIt is worthwhile mentioning that the resonance lines\nare shifted from the paramagnetic position already be-\nlow 100 K, i.e. above Tc. In a classical ferromagnet,\na temperature-dependent resonance shift is caused by\ninternal magnetic \felds resulting from an increasingly\nstatic net magnetization which, in turn, arises from FM\ncouplings between the spins below the ordering temper-\nature, see, e.g., Ref. [39]. The fact that the \u000eH(T) de-\npendence sets in at higher temperatures suggests that\nthe FM correlations in Cr 2Ge2Te6on the typical ESR\ntimescale of the order of \u0017\u00001\u001810 ps develop at temper-atures signi\fcantly exceeding the Tc. Such a behavior is\nreminiscent of one-dimensional spin systems where shifts\nof the resonance positions have been investigated exper-\nimentally and theoretically [41{45] and were attributed\nto the reduced dimensionality as well as anisotropic mag-\nnetic interactions. Consequently, an e\u000bective reduction\nof the dimensionality in the spin system could be the\norigin of the observed temperature-dependent resonance\nshift. Indeed, clear-cut evidence for the two-dimensional\ncharacter of the spin correlations in Cr 2Ge2Te6is ob-\ntained from measurements of the angular dependence of\nthe linewidth \u0001 H(\u000b). Fig. 6 shows the angular depen-\ndent linewidth at three di\u000berent temperatures together\nwith \fts to the data (solid red lines) as detailed in the\nfollowing. At room temperature, upper panel in Fig. 6,\n\u0001H(\u000b) reveals a (cos2(\u000b) + 1)-dependence with \u000bbe-\ning the angle between external magnetic \feld and the\nabplane. This kind of angular dependence is typical\nfor spin systems in an uncorrelated, i.e. paramagnetic,\nphase in which no signi\fcant interactions between the\nspins exist in any direction and, as a consequence, the\nreduced dimensionality does not a\u000bect the spin dynamics\nre\rected in \u0001 H(\u000b) [46, 47]. Upon lowering the temper-\nature, spin correlations start to grow, leading to a grad-\nual change of the angular dependence. At 100 K, center\npanel in Fig. 6, a slight deformation of the (cos2(\u000b) + 1)-\ndependence is visible around Hkcwhich can be de-\nscribed by a small additional contribution to \u0001 H(\u000b)\nof the form (3 cos2(\u0019=2\u0000\u000b)\u00001)2. Finally, at 70 K,\nlower panel in Fig. 6, the angular dependence of the\nlinewidth reveals a pure (3 cos2(\u0019=2\u0000\u000b)\u00001)2-behavior7\n/s45/s50/s48 /s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s51/s48/s51/s53/s52/s48/s52/s53/s53/s48/s53/s53\n/s55/s48/s32/s75/s48 /s32/s72 /s32/s40/s109/s84/s41\n/s82/s111/s116/s97/s116/s105/s111/s110/s32/s97/s110/s103/s108/s101/s32 /s32/s40/s68/s101/s103/s41/s72 /s32/s124 /s124 /s32/s99/s72 /s32 /s32/s99/s49/s48/s48/s49/s48/s53/s72 /s32/s124 /s124 /s32/s99/s72 /s32 /s32/s99\n/s49/s48/s48/s32/s75/s48 /s32/s72 /s32/s40/s109/s84/s41\n/s72 /s32/s124 /s124 /s32/s99/s72 /s32 /s32/s99/s52/s48/s48/s52/s50/s48/s52/s52/s48/s52/s54/s48/s48 /s32/s72 /s32/s40/s109/s84/s41/s50/s57/s56/s32/s75\nFIG. 6. Angular dependence of the ESR linewidth obtained\nfrom measurements at 9.6 GHz and temperatures of 298 K\n(upper panel), 100 K (center panel), and 70 K (lower panel),\nrespectively. Solid red lines are \fts to the data as described\nin the main text.\nwithout (cos2(\u000b) + 1)-contribution. Note, that the max-\nimal linewidth occurs for the external \feld oriented per-\npendicular to the abplane, which requires in our de\fni-\ntion of the rotation angle a shift of \u000bby 90\u000ein the ar-\ngument of the trigonometric function. Such an angular\ndependence is typical for two-dimensional systems and is\ncaused by the increasing dominance of long-wavelength\n\ructuations (or, in reciprocal space, q\u00180 modes) in low-\ndimensional magnets, as discussed in Refs. [46{48]. As\nat 70 K no additional contributions to \u0001 H(\u000b) apart from\nthe one related to the low dimensionality was observed,\nwe conclude that the shift of the resonance line above Tc\nis not caused by inhomogeneities in the sample, for which\nwe also did not \fnd any signature in the structural char-\nacterizations of the crystals, see Appendix A. Thus, a\ncontinuous slowing down of the correlated FM dynamics\nof the Cr spins by approaching the ordering transition\nfrom above, as evidenced by the onset of a resonance\nshift and a gradual change in \u0001 H(\u000b), demonstrates the\nintrinsically two-dimensional nature of the magnetism inCr2Ge2Te6, even in bulk crystals. Moreover, the two-\ndimensionality of magnetism as inferred from the analysis\nof static magnetization in the immediate vicinity of Tc,\nsee Refs. [19, 20, 23], is found to be extended to much\nhigher temperatures in the dynamic regime probed by\nESR.\nIn addition to the frequency dependent studies of mag-\nnetic resonance signals in Cr 2Ge2Te6and the angular\ndependence of the linewidth at low frequencies above Tc\nwe investigated the angular dependence of the resonance\n\feldHres(\u0012) at 7 K, i.e., deep in the FM phase, and at\nfrequencies of 62.6 GHz and 125.2 GHz with a cantilever-\nbased torque detected HF-ESR/HF-FMR setup. In this\nexperiment the cantilever with the sample was rotated in\nthe external magnetic \feld from Hkc(\u0012= 90\u000e) toH?c\n(\u0012= 0\u000e) directions. The obtained variations of the reso-\nnance position as a function of the angle \u0012between the\nvector of external magnetic \feld and the crystallographic\nabplane and exemplary spectra are shown in Fig. 7.\nFor both studied frequencies a similar sinusoidal be-\nhavior ofHres(\u0012) was found: the maximal resonance \feld\nwas observed when the magnetic \feld was aligned in the\nabplane, while a minimal resonance position corresponds\nto the magnetic \feld oriented parallel to the caxis, which\nfurther supports a magnetic easy-axis scenario. More-\nover, as the shift \u000eHofHresfrom the paramagnetic value\nis governed by the magnetic anisotropy, angular depen-\ndent measurements allow a further independent determi-\nnation of the anisotropy constants. Thus, the precision\nof the obtained value for the MAE could be increased\nby a combined evaluation of Hres(\u0017) andHres(\u0012) data.\nFor simulation of experimental data the same approach\nas for the frequency dependent measurements was em-\nployed, as described in Sec. III C. Numerical results are\nshown as solid lines in Fig. 7 (a) together with the mea-\nsuredHres(\u0012) dependences.\nWe note that the general qualitative behavior of the\nangular dependence of Hresis consistent with the one\nfound in the low-frequency FMR/ESR experiments at\n\u0017= 9:6 GHz at temperatures between 120 and 50 K (see\nFig. 16 (a) in Appendix B). Unfortunately, with further\nlowering the temperature the FMR signal at 9.6 GHz cor-\nresponding to magnetic \felds of about 0.35 T became\nmore and more structured and \fnally acquired a very dis-\ntorted shape which prevents a reliable analysis of the data\n(see Fig. 16 (b) in Appendix B). Most probably magnetic\ndomains in the crystal are responsible for the observed\ndistorted lineshapes at lower frequencies since magnetic\n\felds required for full polarization of the magnetization\nin the sample are larger than the external \feld applied\nin this low-frequency magnetic resonance experiment (cf.\nFig. 2). In contrast, measurements at much higher fre-\nquencies\u0017corresponding to the \felds above the satu-\nration \feld yielded well de\fned single or, in the case of\nsome cantilever measurements, double line spectra with-\nout any distortions down to the lowest measured temper-\nature of 4 K (Figs. 5 and 7). Therefore, only HF-FMR\ndata were used as the input for the simulations presented8\n/s45/s56/s48 /s45/s54/s48 /s45/s52/s48 /s45/s50/s48 /s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s50/s46/s48/s48/s50/s46/s50/s53/s52/s46/s48/s48/s52/s46/s50/s53/s52/s46/s53/s48/s40/s99/s41 /s40/s98/s41\n/s32 /s61/s32/s49/s50/s53/s46/s49/s53/s50/s32/s71/s72/s122\n/s32/s115/s105/s109/s117/s108/s97/s116/s105/s111/s110\n/s32 /s61/s32/s54/s50/s46/s53/s55/s54/s32/s71/s72/s122\n/s32/s115/s105/s109/s117/s108/s97/s116/s105/s111/s110/s48/s32/s72\n/s114/s101/s115/s32/s40/s84/s41\n/s65/s110/s103/s108/s101/s32/s40/s111\n/s41/s84 /s32/s61/s32/s55/s32/s75\n/s72 /s32 /s32/s99/s40/s97/s41\n/s49/s46/s54 /s49/s46/s56 /s50/s46/s48 /s50/s46/s50 /s50/s46/s52/s57/s48/s176/s55/s53/s176/s54/s48/s176/s52/s53/s176/s51/s48/s176/s49/s53/s176/s67/s97/s110/s116/s105/s108/s101/s118/s101/s114/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s115/s104/s105/s102/s116/s32/s110/s111/s114/s109/s46/s32/s40/s97/s114/s98/s46/s32/s117/s46/s41\n/s48/s32/s72 /s32/s40/s84/s41/s61/s32/s54/s50/s46/s53/s55/s54/s32/s71/s72/s122/s44/s32/s32/s32 /s84 /s32/s61/s32/s55/s32/s75\n/s48/s176\n/s51/s46/s56 /s52/s46/s48 /s52/s46/s50 /s52/s46/s52 /s52/s46/s54 /s52/s46/s56/s67/s97/s110/s116/s105/s108/s101/s118/s101/s114/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s115/s104/s105/s102/s116/s32/s110/s111/s114/s109/s46/s32/s40/s97/s114/s98/s46/s32/s117/s46/s41\n/s48/s32/s72 /s32/s40/s84/s41/s61/s32/s49/s50/s53/s46/s49/s53/s50/s32/s71/s72/s122/s44/s32/s32/s32 /s84 /s32/s61/s32/s55/s32/s75\n/s48/s176\n/s49/s53/s176\n/s51/s48/s176\n/s52/s53/s176\n/s54/s48/s176\n/s55/s53/s176\n/s57/s48/s176\nFIG. 7. Angular dependence of the resonance \feld Hres(\u0012) obtained from torque detected FMR at 7 K and excitation frequencies\n\u0017of 62.6 GHz and 125.2 GHz (a). Solid lines are results of the simulations according to Eq. (2). Representative spectra obtained\nfrom measuring the shift of the cantilever frequency at \u0017= 62:6 and 125.2 GHz are shown in (b) and (c), respectively. For\ncomparison, all spectra are normalized and shifted vertically.\nin the following section as the employed model is strictly\napplicable only for the case of the fully saturated mag-\nnetization.\nC. Simulations and determination of the magnetic\nanisotropies\nSimulation of the low-temperature frequency and an-\ngular dependences of the resonance \feld Hres[Fig. 5 (b)\nand Fig. 7 (a)] with a phenomenological model of fer-\nromagnetic resonance yields reliable quantitative infor-\nmation about the magnetic anisotropies of Cr 2Ge2Te6.\nWithin this well established approach (see, e.g., Refs. [39,\n40, 49]), the resonance frequency \u0017resis given by the fol-\nlowing equation:\n\u00172\nres=g2\u00162\nB\nh2M2ssin2\u0012\u0012@2F\n@\u00122@2F\n@'2\u0000\u0010@2F\n@\u0012@'\u00112\u0013\n:(2)\nHere,gdenotes the g-factor and Msrepresents the sat-\nuration magnetization of the sample, Fis a free energy\ndensity, and \u0012and'are the spherical coordinates of the\nmagnetization vector M(Ms;';\u0012). In order to calcu-\nlate the resonance position for a given orientation of the\nsample in the external magnetic \feld and for a given\nmicrowave frequency Eq. (2) has to be evaluated at the\nequilibrium position ( '0;\u00120) ofM. The latter is found\nvia minimization of Fwith respect to \u0012and'under the\nspeci\fc experimental conditions. In the present case, a\ncomplete description of our frequency and angular de-\npendent HF-FMR data could be achieved taking into ac-\ncount a uniaxial magnetic anisotropy as well as shape\nanisotropies of the respective samples used for the mea-surements. The free energy density then reads\nF=\u0000H\u0001M\u0000KUcos2(\u0012)+\n2\u0019M2\ns(Nxsin2(\u0012) cos2(\u001e)+\nNysin2(\u0012) sin2(\u001e) +Nzcos2(\u0012)):(3)\nHere, the \frst term describes the Zeeman energy den-\nsity of the sample in the external \feld H. The second\nterm represents the uniaxial anisotropy characterized by\nthe energy density KU. The last term denotes the shape\nanisotropy with Nx,Ny,Nzbeing demagnetization fac-\ntors [50, 51]. The speci\fc values for the demagnetization\nfactors of the two samples used for determination of KU\nare given in Table III of Appendix A together with the\nsample dimensions.\nAs can be seen in Figs. 5 (b) and 7 (a) it is possible\nto describe our HF-FMR results very well employing the\nmodel presented above. It should be noted that the \felds\nused in the HF-FMR experiments are larger than the sat-\nuration \felds of the magnetization Hab\nsatandHc\nsatdeter-\nmined from the M(H) curves in Sec. III A. This justi\fes\nthe use of this model which is applicable in the satu-\nrated regime of a ferromagnet. We emphasize that the\nsimulated curves for all data sets were obtained using\nthe same values for KUandg. For the latter, we used\ntheg-factors determined from HF-ESR data at 120 K as\nstarting values for the simulations (see Sec. III B). These\nvalues were re\fned in order to achieve the best agree-\nment between the simulated curves and the low temper-\nature experimental data yielding at the end an isotropic\ng-factor of 2:040\u00060:005, which is identical with the mea-\nsuredgjj-factor.\nFor a complete description of all frequency and an-\ngular dependent HF-FMR data it was merely necessary\nto correctly account for the magnetic shape anisotropy\nwhich is determined by the particular geometry of the9\nstudied crystals. This was realized by calculating the\nproper demagnetization factors based on the measured\ndimensions of the crystals [50, 51], see Appendix A. As a\nresult a value for KUof (0:48\u00060:02)\u0002106erg/cm3was\nobtained which corresponds to an anisotropy energy of\n(0:250\u00060:005) meV per unit cell Cr 6Ge6Te18, compris-\ning three formula units of Cr 2Ge2Te6. The value of KUis\npositive, which ultimately proves the magnetic easy axis\nanisotropy of Cr 2Ge2Te6, suggested by the magnetom-\netry data, and other qualitative experimental observa-\ntions. The magnetic anisotropy easy axis is collinear with\nthecaxis, i.e., it is directed perpendicular to Cr 2Ge2Te6\nlayers. It is worth to mention that the obtained KUvalue\nis larger than the one previously reported in Ref. [17].\nThe di\u000berence is likely to be ascribed to the in\ruence of\nthe shape anisotropy which was not taken into account\nin Ref. [17] and a saturation magnetization of about\n2.23\u0016B/Cr reported by these authors. This value is lower\nthan the theoretically expected one of 3 \u0016B/Cr which has\nbeen con\frmed by our measurements, see Sec. III A. In-\ndeed, simulations of our data without shape-anisotropy\nterm in Eq. (3) yield a KUof (0:29\u00060:03)\u0002106erg/cm3\nwhich is closer to the value reported in Ref. [17] for low\ntemperatures. However, for matching the simulation to\nthe data it was necessary to adjust the g-factors for both\nsamples individually. Moreover, the agreement between\nexperimental data and simulations was worse as in the\ncase of a proper treatment of the sample shape. There-\nfore, it is crucial to consider the shape anisotropy in order\nto obtain a correct value for the intrinsic uniaxial MAE\nwhich is, in turn, essential for the understanding of the\nsalient details of the magnetic properties of Cr 2Ge2Te6,\nin particular the origin of the easy axis type anisotropy.\nD. DF calculations\nThe scalar-relativistic electronic structure of the title\ncompound has been discussed in some detail already two\ndecades ago in Ref. [52]. In particular, a gap of 0.06 eV\nwas found in the mentioned calculation, separating the\noccupied majority-spin Cr-3 d-t2gstates from the other,\nunoccupied Cr-3 dstates and ensuring an integer total\nspin moment of 3 \u0016Bper Cr atom. The present scalar-\nrelativistic GGA results con\frm this picture of a ferro-\nmagnetic band insulator with a somewhat larger gap of\n0.28 eV, Fig. 8. However, if spin-orbit coupling is taken\ninto account by switching to the full relativistic mode,\nthe gap is almost closed, Fig. 9. The remaining narrow\ngap of 0.03 eV is consistent with the experimental trans-\nport gap of 0.007 eV, measured below the ferromagnetic\nordering temperature [11].\nThe unusual reduction of the gap by spin-orbit cou-\npling can be understood by analyzing the orbital charac-\nter of the states in the vicinity of the Fermi level, see Figs.\n9 and 10. The top of the valence band is dominated by\nTe-5p3=2states, while the bottom of the conduction band\nhosts a hybrid of Cr-3 d, Te-5p, and Ge-4pstates (the lat-\nFIG. 8. Density of states (DOS) of Cr 2Ge2Te6obtained by a\nscalar-relativistic GGA calculation in a ferromagnetic state.\nArrows indicate the majority (up-arrow) and minority (down-\narrow) spin channels. The lines limiting the empty areas de-\nnote the total spin-projected DOS and the \flled areas denote\nthe Cr-3dspin-projected DOS.\nFIG. 9. Density of states (DOS) of Cr 2Ge2Te6obtained by a\nfull relativistic GGA calculation carried out in a ferromagnetic\nstate with moment orientation along [100], i.e., within the\neasy plane. Thick lines denote the total DOS, for both spin\nchannels added up, thin lines denote the Te-5 p3=2projected\nDOS, and dotted lines denote the Te-5 p1=2projected DOS.\nter are not shown) with contributions to the total density\nof states of 36%, 21%, and 27%, respectively. Being the\nheaviest element of the investigated compound, Te ex-\nperiences the largest spin-orbit coupling. Its 5 pstates\nare split by about 0.8 eV, both in the free atom and in\nthe pseudo-atom used for the construction of the valence\nbasis for the title compound. Due to this splitting, the\ncenter of gravity of the Te-5 p3=2states moves upward in\nenergy by about 0.2 eV against the other atomic states,\ncompared with the case without spin-orbit coupling. In\nthe bulk compound, this is visible by a related shift of10\nFIG. 10. Density of states (DOS) of Cr 2Ge2Te6obtained by\na full relativistic GGA calculation carried out in a ferromag-\nnetic state with moment orientation along [100]. Upper panel:\nsame data as in Fig. 9 at an expanded energy-scale. Note the\ndominance of Te-5 p3=2states at the top of the valence band\nand the asymmetric shape of the gap with a two-dimensional\nvan-Hove singularity at the bottom of the conduction band.\nLower panel: Spin-resolved presentation showing the total\nDOS (thick lines) and the Cr-3 dprojected DOS. Note the\nprevalence of minority-spin states at the bottom of the con-\nduction band.\nFIG. 11. Density of states (DOS) of Cr 2Ge2Te6obtained by\na full relativistic GGA+ Ucalculation carried out in a fer-\nromagnetic state with moment orientation along [001], i.e.,\nalong the easy axis. The lines limiting the empty areas de-\nnote the total spin-projected DOS and the \flled areas denote\nthe Cr-3dspin-projected DOS.\nband weights, see Fig. 9: Below -3 eV, Te-5 p1=2and Te-\n5p3=2states contribute almost the same weight to the\nDOS, while Te-5 p3=2states clearly dominate at higher\nenergies including the region close to the gap.\nAn interesting peculiarity is observed at the bottom\nof the conduction band, which is of almost pure (99%)\nminority-spin character, see Fig. 10. Essential majority\nspin population sets in only about 100 meV above the\nFIG. 12. Upper panel: Calculated MAE of Cr 2Ge2Te6as\nobtained by GGA (full circle) and GGA+ Uapproaches for\ndi\u000berent values of U(diamonds connected by guiding lines).\nThe MAE depends on Umonotonously for all considered val-\nues up toU= 5 eV. A horizontal line denotes the experi-\nmental value obtained in this work. Data with open symbols\ndenote the total magnetic anisotropy energy, i.e., the sum of\ncalculated MAE of this work and intrinsic dipolar anisotropy\nenergy of 0.067 meV/f.u. taken from Ref. [53]. Note, that the\nsign de\fnition of MAE in Ref. [53] di\u000bers from our de\fnition.\nLower panel: Calculated gap size (same meaning of the sym-\nbols as in the upper graph). Note, J= 0.6 eV in all GGA+ U\ncalculations. Thus, the GGA result is not the same as the\nlimitU!0 of the GGA+ Udata. The position of the GGA\nresults on the abscissa is arbitrary.\nconduction band minimum. The strong spin polarization\nat CBM is in qualitative agreement with results of earlier\nGGA+Ucalculations taking into account van der Waals\ninteractions [24] and also with similar recent calculations\nfor a few-layer Cr 2Ge2Te6system [25]. Di\u000berent from\nthose scalar-relativistic calculations we \fnd that the top\nof the valence band is strongly spin-mixed by the strong\nspin-orbit interaction within the Te-5 pshell and that the\nCBM hosts minority-spin bands.\nWe note that the steep van-Hove singularity at the con-\nduction band bottom indicates a quasi two-dimensional\nband. As mentioned above, this band is composed of or-\nbitals from all constituents of the system. Thus, it can\nbe considered to host quasi-homogeneous states. Occu-\npation of these states with charge carriers will result in an\nalmost completely spin-polarized, quasi two-dimensional,\nand quasi-homogeneous electron gas.\nWhile the narrow experimental transport gap is rea-\nsonably well described by the full relativistic GGA data,\nthe magnetocrystalline anisotropy (MA) turns out to be\nof easy-plane type in this approach, at variance with the\neasy-axis MA found in experiment, see Fig. 12. For this\nreason, further calculations were performed relying on\nthe GGA+ Umethod. Depending on the speci\fc value11\nofU, the occupied and the unoccupied Cr-3 dbands are\nshifted further away from each other, compare Figs. 8\nand 11. This changes the relative positions of spin-\npolarized Cr-3 dstates and spin-orbit split Te-5 pstates.\nAs a result, the MA strongly depends on the chosen value\nofU. At about U\u00192 eV, a reasonable value for Cr-\n3dstates in a narrow-gap system, the calculated MAE\nmeets the measured value. However, the calculated gap\nof about 0.15 eV at U= 2 eV is more than an order of\nmagnitude larger than the experimental transport gap.\nA similarly strong dependence \u000eMAE=\u000eU of about\n3\u000210\u00004/f.u. (see Fig. 12) was reported for the title com-\npound by C. Gong et al. recently (cf. Extended Data\nFigure 8 of Ref. [14]). Those authors found a transi-\ntion from easy-plane to easy-axis MA at U= 0.2 eV, to\nbe compared with our value of 1.75 eV. This o\u000bset can\nbe understood by the fact that the authors of Ref. [14]\nused in their GGA+ Ucalculation only the Slater integral\nF0=U, while we included two further Slater integrals\nviaJ= (F2+F4)/14 which, \frst, tends to reduce the\ne\u000bect ofUand, second, introduces anisotropic contribu-\ntions to the correlation.\nAn additional contribution to the intrinsic magnetic\nanisotropy energy is due to the magnetic dipole inter-\naction. The latter is known to be responsible for the\nmagnetic shape anisotropy. Less known is that it also\naccounts for an intrinsic (bulk) contribution [54] which\nis usually very small [55]. In layered materials, it may\nhowever provide larger contributions as has been demon-\nstrated for the title compound recently [53]. We included\nthe additional term in Fig. 10 as a constant shift, as it\nonly depends on the size of the atomic moments and on\nthe structure, both being not very sensitive to the value\nofU.\nIt should be noted, that the polarized quasi two-\ndimensional band at the conduction band bottom is ro-\nbust with respect to changing the direction of magneti-\nzation and also with respect to the application of Uup\nto 2 eV. For this value of Uand easy-axis orientation\nof the magnetic moment, its distance from the lowest\nmajority-spin conduction band is reduced to 60 meV as\ncompared to 100 meV in the GGA calculation with mag-\nnetic moment oriented along [100]. If Uis increased be-\nyond 2 eV, the minority Cr-3 dstates are shifted further\ntoward higher energies. At U\u00192:3 eV, the character of\nCBM switches to majority spin. While the CBM is still\nalmost completely spin polarized (with the exception of\nthe switching value of U), it is no longer quasi-2D.\nIV. CONCLUSIONS\nIn conclusion, the spin dynamics and magnetic\nanisotropy of the bulk form of the 2D van der Waals\nferromagnet Cr 2Ge2Te6was investigated in detail by\nmeans of high-\feld ESR and FMR spectroscopies, mag-\nnetization measurements, and density functional calcu-\nlations. Magnetic resonance data evidence a gradualonset of the ferromagnetic correlations at temperatures\nwell above the magnetic phase transition which con-\ntinuously grow and slow down by approaching the Tc.\nSuch a pronounced correlated spin dynamics appears to\nbe a characteristic feature of the two-dimensional mag-\nnetism of Cr 2Ge2Te6. Both static and dynamic experi-\nmental methods directly evidence the easy-axis-type na-\nture of the total magnetic anisotropy, including magne-\ntocrystalline and shape anisotropy, in line with previ-\nous studies[11, 17, 19, 20]. Simulations of the HF-FMR\nexperiments as a function of microwave frequency and\nangle of the magnetic \feld with respect to the sample\nyield a MAE KUof (0:48\u00060:02)\u0002106erg/cm3, explic-\nitly taking into account the shape of individual crys-\ntals. Thus, the obtained value represents an accurate\nmagnitude of the intrinsic magnetocrystalline anisotropy\nenergy density as compared to the value reported in\nRef. [17] where the shape anisotropy apparently was not\ntaken into account. Comparing the MAE obtained from\nphenomenological simulations of experimental data with\nDF calculations employing the GGA+ Umethod, we \fnd\na good agreement with the theoretical MAE for U=\n2 eV. However, the gap calculated at this particular U\nvalue di\u000bers from experimental observation. Analysis of\nthe electronic structure indicates the presence of quasi\ntwo-dimensional states at the bottom of the conduction\nband. Upon electron doping [25], these states may host\na quasi-homogeneous, almost completely spin-polarized\nelectron gas.\nACKNOWLEDGMENTS\nThis research was supported by the Deutsche\nForschungsgemeinschaft (DFG) via grants AL 1771/4-1\nand KA1694/8-1. S.A. and S.S. acknowledge \fnancial\nsupport from GRK-1621 graduate academy of the DFG.\nM.P.G. thanks the Alexander von Humboldt Foundation\nfor \fnancial support through the Georg Forster Research\nFellowship Program. M.P.G. and M.R. are grateful for\ntechnical assistance by Ulrike Nitzsche.\nAppendix A: Details on samples used in this study\nDetailed information about structural characterization\nof the synthesized Cr 2Ge2Te6crystals by means of PXRD\nat room temperature are given in Tab. II. The homogene-\nity of the crystals was con\frmed by scanning electron\nmicroscopy (SEM) imaging using a high de\fnition back\nscattered electron detector (HDBSD). SEM images using\na secondary electron (SE) detector and a back scattered\nelectron (BSE) detector are shown in Fig. 13. Using a\nSE detector, the contrast in the image is given by di\u000ber-\nences in height and topology while using a BSE detector\nresults in a contrast given by di\u000berences in chemical com-\nposition. The homogeneous color and lack of contrast of\nthe SEM(BSE) image (Fig. 13(b)) over a wide area indi-12\nFIG. 13. SEM images of a Cr 2Ge2Te6single crystal using the height contrast SE mode (a) and the chemical contrast BSE\nmode (b).\ncates a homogeneous chemical composition of the sample\nand its purity. Nevertheless, some spots of di\u000berent con-\ntrast can also be seen on the sample surface. If compared\nto the SEM(SE) image (Fig. 13(a)), those spots are also\nshowing a height contrast. They originate from dust and\ndirt particles on the sample surface and are not obtained\ndue to a secondary phase in the sample.\nDC magnetization measurements were carried out on\na 4 mm\u00023 mm\u00022\u0016m crystal. For the correction of\nthe sample magnetic moment an equivalent surface area\nsquare \flm sample with a side length of 3.3 mm and thick-\nness of 2\u0016m was assumed. The correction factors for this\nsample were obtained based on the experimental data on\na squared nickel \flm presented in Ref. [28]. To extract\ncorrection factors for the assumed side length of 3.3 mm\nthe data on the squared nickel \flm for the same vibra-\ntion amplitude (2 mm) was \ftted by exponential func-\ntions yielding a good agreement between \ft and experi-\nTABLE II. Details of structural parameters and R-values of\nRietveld analysis.\nParameter Cr 2Ge2Te6\nWavelength ( \u0017A) 1.54059\n\u0012Range (\u000e) 10.00 - 101.99\nStep Size (\u000e) 0.01\nTemperature (K) 293\nSpace Group R\u00163 H (No. 148)\na(\u0017A) 6.828(3)\nc(\u0017A) 20.572(12)\nUisotropic :\nUCr 0.041(9)\nUGe 0.016(6)\nUTe 0.029(2)\nRp 0.0476\nRwp 0.0615\nRf 0.0902\nGoodness-Of-Fit 1.24mental data (for Hkc (horizontal): \u001f2= 7.7\u000110\u00006; forH\n?c (vertical): \u001f2= 8.0\u000110\u00007). From these functions the\nfactors of 1.157 ( \u00060.031) forHkc and 1.024 (\u00060.011)\nforH?c for the overestimation of the measured mo-\nment by the experimental setup were extracted. The\nmeasured moments were then divided by those factors\nto account for the non point-like dipole sample shape.\nAs an example for the e\u000bect of this correction, Fig. 14\nshows the uncorrected and corrected magnetization as\nfunction of \feld at 4 K close to and in the saturation\nmagnetization regime of Cr 2Ge2Te6. It is clearly visi-\nble that without this correction two di\u000berent values of\nsaturation magnetization for the di\u000berent \feld directions\nare obtained. After correction, the di\u000berence in satura-\n/s49 /s50 /s51 /s52 /s53 /s54 /s55/s50/s46/s55/s50/s46/s56/s50/s46/s57/s51/s46/s48/s51/s46/s49/s51/s46/s50/s51/s46/s51/s51/s46/s52/s77 /s32/s40\n/s66/s32/s47/s32/s67/s114/s41\n/s48/s72 /s32/s40/s84/s41/s32/s72 /s32/s124/s124/s32/s99/s32/s117/s110/s99/s111/s114/s114/s101/s99/s116/s101/s100\n/s32/s72/s32 /s124/s124/s32/s99/s32/s99/s111/s114/s114/s101/s99/s116/s101/s100\n/s32/s72 /s32 /s32/s99/s32/s117/s110/s99/s111/s114/s114/s101/s99/s116/s101/s100\n/s32/s72/s32 /s32/s99/s32/s99/s111/s114/s114/s101/s99/s116/s101/s100\nFIG. 14. Example for the e\u000bect of magnetization correction\nfor a \feld-dependent measurement of the magnetization at\n4 K forHkc(red circles) and H?c(blue squares), respec-\ntively. The as-measured data are shown as full symbols, while\nopen symbols represent the corrected data.13\ntion magnetization for the di\u000berent \feld directions van-\nishes. The strongly di\u000berent values of saturation magne-\ntization for the uncorrected case would indicate either a\nstrongly anisotropic g-factor or an unusual interplay with\na secondary magnetic phase. We can rule out both sce-\nnarios for the Cr 2Ge2Te6sample by our ESR and struc-\ntural (PXRD and SEM-EDX) investigations. In fact, the\nisotropic saturation magnetization after the correction is\nin good agreement with the nearly isotropic g-factors ob-\ntained by ESR. Furthermore, the correction leads to a\nsaturation moment close to 3 \u0016B/Cr, which is the ex-\npected moment for Cr3+withS= 3/2 andg\u00182 and is\nin good agreement with literature [11, 12, 19, 23]. We\nnote that the use of the correction factors yield the the-\noretically expected results which can be understood as a\npositive self-consistency check of the applied correction\nmethod.\nCantilever-based FMR measurements were carried out\non a small crystal (CL sample) mounted on the can-\ntilever as shown in Fig. 15. Compared to this sample,\nthe crystal employed in the main part of the X-band\n(9.6 GHz) and HF-ESR measurements (HF sample) was\nmuch larger and was chosen in order to increase the sig-\nnal intensity (note that the ESR intensity is proportional\nto the number of spins in the sample). For an evaluation\nof demagnetization factors, lengths of the sample edges\nwere measured from which the dimensions l,w, andhof\nan equivalent cuboid with volume l\u0002w\u0002hwere deter-\nmined. The resulting dimensions are given in Table III\nfor both samples together with the respective demagneti-\nzation factors calculated based on the equivalent-ellipsoid\nmethod [50, 51]. Uncertainty in measurements of the\nsample dimensions caused uncertainties in the calculated\ndemagnetization factors which did not exceed 9%. Note\nthat deviations of the real sample shape from the ap-\nproximated cuboid are small compared to the di\u000berences\nbetween the lateral dimensions landwand the sample\nthicknessh.\nTABLE III. Dimensions and demagnetization factors for the\nsamples used in the FMR study.\nl(\u0016m)w(\u0016m)h(\u0016m)NxNyNz\nCL sample 84 64 10 0.1643 0.0523 0.7834\nHF sample 1250 1030 35 0.0430 0.0017 0.9553\nAppendix B: Additional data obtained from X-band\nESR measurements\nIn addition to the experimental data presented in\nSec. III B of the main text angular dependence of the\nresonance \feld Hres(\u0012) was studied at a microwave fre-\nquency of 9.6 GHz at several selected temperatures. Res-\nonance shifts \u000eHobtained from these measurements\nare summarized in Fig. 16 (a). At 120 K only a weak\nanisotropy, as re\rected in the angular dependence of the\nFIG. 15. Cr 2Ge2Te6sample used for the torque detected\nFMR study presented in this work as mounted on a piezo-\ncantilever.\nresonance shift, was observed. Upon decrease of tem-\nperature the anisotropy increases due to the growth of\nFM correlations. Below 50 K such an analysis of the\nanisotropy is hampered by a strong distortion of the line,\nas is illustrated by exemplary spectra in Fig. 16 (b). As\ndiscussed in the main text, such a distortion is most prob-\nably related to magnetic domains likely to be still present\nat small \felds used in this experiment. Consequently, the\nresonance shift cannot be determined unambiguously at\nlow frequencies below 50 K.14\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48 /s55/s48/s48 /s56/s48/s48 /s57/s48/s48/s56/s48/s32/s75/s54/s48/s32/s75/s53/s48/s32/s75/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s115/s105/s103/s110/s97/s108/s32/s97/s116/s32/s100/s101/s116/s101/s99/s116/s111/s114/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s48/s32/s72 /s32/s40/s109/s84/s41/s52/s32/s75\n/s72 /s32 /s32/s99 /s44/s32 /s32/s61/s32/s57/s46/s54/s32/s71/s72/s122/s45/s50/s48 /s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s45/s49/s52/s48/s45/s49/s50/s48/s45/s49/s48/s48/s45/s56/s48/s45/s54/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s32/s49/s50/s48/s32/s75\n/s32/s49/s48/s48/s32/s75\n/s32/s56/s48/s32/s75\n/s32/s55/s48/s32/s75\n/s32/s54/s48/s32/s75\n/s32/s53/s48/s32/s75/s48/s32/s40/s72\n/s114/s101/s115/s32/s45/s32 /s72\n/s114/s101/s115/s40/s103/s61/s50/s41/s41/s32/s40/s109/s84/s41\n/s65/s110/s103/s108/s101/s32/s40/s176/s41/s72 /s32 /s32/s99/s72 /s32 /s32/s99/s32/s61/s32/s57/s46/s54/s32/s71/s72/s122\n/s40/s98/s41/s40/s97/s41\nFIG. 16. 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Magnetic defects have been modeled as local variations in the material parame-\nters, such as the exchange sti\u000bness, saturation magnetization, magnetocrystalline anisotropy and\nDzyaloshinskii-Moriya constant. We observe both pinning (potential well) and scattering (potential\nbarrier) traps when tuning either a local increase or a local reduction for each one of these magnetic\nproperties. It is found that the skyrmion-defect aspect ratio is a crucial parameter to build traps for\nskyrmions. In particular, the e\u000eciency of the trap is compromised if the defect size is smaller than\nthe skyrmion size, because they interact weakly. On the other hand, if the defect size is larger than\nthe skyrmion diameter, the skyrmion-defect interaction becomes evident. Thus, the strength of the\nskyrmion-defect interaction can be tuned by the modi\fcation of the magnetic properties within a\nregion with suitable size. Furthermore, the basic physics behind the mechanisms for pinning and for\nscattering is discussed. In particular, we discover that skyrmions move towards the magnetic region\nwhich tends to maximize its diameter; it enables the magnetic system to minimize its energy. Thus,\nwe are able to explain why skyrmions are either attracted or repelled by a region with modi\fed\nmagnetic properties. Results here presented are of utmost signi\fcance for the development and\nrealization of future spintronic devices, in which skyrmions will work as information carriers.\nI. INTRODUCTION\nTopologically protected objects known as skyrmions\nwere \frst introduced by Skyrme in the context of particle\nphysics [1, 2]. From then on, the concept of these topo-\nlogical excitations has been extended to the condensed\nmatter physics, where skyrmions can be found in liquid\ncrystals [3, 4], Bose-Einstein condensates [5{7], supercon-\nductors [8, 9] and nanoscaled magnetic thin \flms [10{15].\nNanomagnets are suitable systems to investigate ex-\notic magnetic structures, such as vortices, skyrmions and\ndomain walls. These quasiparticles are not only rele-\nvant to fundamental micromagnetism, but also enable\nus to engineer spintronic devices [16, 17]. In this work\nwe focus on skyrmions, the information carriers that\nwill probably form the basis of the next generation in\ndata storage and logic devices [18, 19]. Initially, mag-\nnetic skyrmions has been experimentally observed at low\ntemperature and under external magnetic \felds [20{26].\nNowadays, they have been stabilized at room temper-\nature without the aid of a magnetic \feld [27{33]. Usu-\nally skyrmions arise in magnetically ordered systems with\nDzyaloshinskii-Moriya (DM) couplings [34{38], which\ncan be either intrinsic in chiral magnets (for example,\ncompounds of transition metals with the noncentrosym-\nmetric B20-type structure) or induced in magnetic multi-\nlayer systems with broken inversion symmetry and strong\nspin-orbit coupling (for example, in Co/Pt multilayers).\nIndeed, skyrmions can be stabilized in the absence of DM\ninteractions [15, 39, 40], but it generates giant skyrmions.\n\u0003Author to whom correspondence should be addressed. Electronic\nmail: danilotoscano@\fsica.ufjf.brIn these magnetic systems, the skyrmion sizes are typi-\ncally of the same order of magnitude than the vortex\nsizes (from 100 nm to 1 \u0016m). On the other hand, in\nmagnetic systems in which DM interactions are signi\f-\ncant, the skyrmion diameters are of the order of tens of\nnanometers. Unlike vortices which have a free chirality\n(clockwise or anticlockwise rotation), skyrmions have a\n\fxed rotation sense which is speci\fed by the sign of the\nDM interaction parameter. Importantly, several works\nrefer to this structural property of the skyrmion as helic-\nity instead of chirality [15, 41]. Furthermore, skyrmions\ncan occur in two types [19]: Bloch's skyrmion (vortex-\ntype con\fguration) and N\u0013 eel's skyrmion (hedgehog-type\ncon\fguration). Thus, magnetically ordered systems with\nDM interactions can exhibit either a single skyrmion or\na close-packed lattice of identical skyrmions.\nDue to the peculiar characteristics of the skyrmions,\nnanoscale sizes, topological protection, and low current\ndensities needed to drive them, there is a lot of inter-\nest in replacing domain walls with skyrmions in spin-\ntronic technologies. The overview as well as a sketch\nof skyrmionic racetrack is presented in Ref. [42]. The\nstability of skyrmions has been extensively investigated\nand there are many ways of writing and deleting them in\nmagnetic nanostructures [14, 27, 41, 43{53]. Skyrmions\ncan be manipulated by spin-polarized currents applied in-\nplane as well as perpendicular to the plane of the nanos-\ntructure [14, 15, 54{60]. Unlike domain walls which are\nrestricted to the unidirectional movement along the nan-\notrack, skyrmions can not be driven by spin-polarized\ncurrents without being moved away from the longest axis\nof the nanotrack. As a result of the skyrmion Hall ef-\nfect [15, 61], skyrmions can be accumulated or even an-\nnihilated at the edges of the nanowire. The reason whyarXiv:1810.03754v2  [physics.comp-ph]  26 Oct 20182\nthe skyrmions move away from the nanotrack longest axis\nwhen applying a spin-polarized current can also be un-\nderstood through Thiele approach [62, 63], which takes\ninto account the Magnus force [56, 64]. From the tech-\nnological point of view, this is an issue to the develop-\nment of spintronic devices based on skyrmion transport\nand various strategies have been planned to suppress this\nundesirable phenomenon. An experimental challenge is\nto \fnd a special class of ferromagnetic materials which\npresents the same value for the Gilbert damping and non-\nadiabatic spin-transfer torque parameters, i.e., \u000b=\f.\nIn this case, there is no skyrmion Hall e\u000bect, as ob-\nserved in micromagnetic simulations [14, 43]. Instead of\nusing electrical currents to control the skyrmion trans-\nport, spin waves [65, 66] as well as temperature gra-\ndients [67, 68] can be used to drive skyrmions in nan-\notracks. Another possibility for controlling the skyrmion\nmotion along the nanotrack central axis by using a spin-\npolarized current is to change the magnetic medium, that\nis, when replacing the ferromagnet with the antiferro-\nmagnet [69, 70]. Furthermore, it was reported that the\nskyrmions were signi\fcantly in\ruenced by defects in the\nmagnetic medium. The inhomogeneity of a material can\nbe intrinsic (impurities) or induced (intentionally incor-\nporated imperfections). Generally, non-magnetic defects,\nwhich distort locally the nanomagnet geometry, generate\nthe skyrmion pinning in the neighborhood of a hole, the\nregion with missing magnetic moments [71, 72]. Chap-\npert et al. [73] pioneered the located modi\fcation of the\nmagnetic properties by employing the ion irradiation in\nmagnetic thin \flms and multilayers, for a review see\nRef. [74]. As a result, magnetic defects can be inten-\ntionally incorporated in nanomagnets in order to create\ntraps for vortex [75, 76], domain walls [77{79] as well\nas skyrmions [80]. References [78, 79] showed that it\nis possible to use focused ion beam irradiation to mod-\nify the saturation magnetization at the interface of non-\nmagnetic/ferromagnetic multilayers. We believe that the\nintermixing of a non-magnetic metal and Permalloy, in-\nduced by the Ga+ion beam at the interface of the mul-\ntilayer, is able to vary not only the saturation magneti-\nzation but also other material parameters in the irradi-\nated area. In particular, the exchange sti\u000bness constant,\nsince the exchange energy represents the main term in\nthe total energy of a ferromagnet, see [74]. Once DM\ninteractions are induced in ferromagnetic/heavy metal\nmultilayer systems with broken inversion symmetry and\nstrong spin-orbit coupling (for example, in Co/Pt multi-\nlayers), we believe the modi\fcation of the DM interaction\nstrength can be achieved by varying of layer thickness in\na selected region, see Refs. [81, 82].\nThe in\ruence of a triangular shaped defect character-\nized by a local increase of the easy axis anisotropy has\nbeen extensively investigated using micromagnetic simu-\nlations [14, 19, 49], and it was observed that skyrmions\nare repelled by this magnetic defect. It is known since\n1998 that the simplest e\u000bect induced by the irradia-\ntion of He+ion in a Co/Pt multilayer is to reduce theperpendicular anisotropy of the irradiated area [73, 80].\nNot surprisingly, Fook et al. [83] observed the con\fne-\nment of skyrmions along of a reduced anisotropy groove\nstrategically located on the axis of the nanotrack. Thus,\nskyrmions are attracted by a region of reduced perpen-\ndicular anisotropy and can be driven by an in-plane spin-\npolarized current without being drift from the direction\nof the electron \row. A similar strategy using potential\nbarriers instead of potential wells has been proposed re-\ncently [84]. In particular, the top and bottom edges of\nthe nanotrack were considered made of a material with a\nhigher perpendicular anisotropy. Another kind of mag-\nnetic defect, consisting in local variations of the exchange\nconstant, has been investigated in Refs. [85, 86]. In their\nmicromagnetic simulations, the authors observed pinning\ncenter as well as obstacle for magnetic skyrmions. In pre-\nvious works, our team has modeled magnetic defects as\nlocated variations on the exchange sti\u000bness constant and\nit was observed pinning and scattering traps for both\nthe vortex core [87] and the domain wall [88]. In sum-\nmary, we have observed that a local decrease of the ex-\nchange constant work as a pinning center, whereas a local\nincrease of the exchange constant work as a scattering\ncenter for the quasiparticle in soft ferromagnetic materi-\nals, such as Permalloy. In this work, similar mechanisms\nof pinning (attractive interaction) and scattering (repul-\nsive interaction) have also been observed for skyrmions\nin ferromagnetic nanowires with perpendicular magnetic\nanisotropy, provided that the skyrmion size is smaller\nthan defect size. Very recently, a study using atomistic\nspin simulation investigated the interaction of skyrmions\nwith di\u000berent kinds of atomic-scale defects [89]. The au-\nthors considered magnetic moments arranged on a trian-\ngular lattice, and magnetic defects having a hexagonal\ngeometry with typical size of the order of a few nanome-\nters (from 1 to 5 nm). Mechanisms to pin skyrmions in\na Co monolayer on Pt substrate were observed for di\u000ber-\nent kinds of magnetic defects: either a local reduction of\nthe exchange constant, a local increase of the strength of\nthe DM interaction, or a local reduction of the magnetic\nanisotropy.\nAs we have reviewed, there are a number of works in\nthe literature that have investigated the skyrmion-defect\ninteraction. However, these works have considered de-\nfects which are relatively small. Besides ensuring a poor\nthermal stability around the trap, the incorporation of\nsmall defects in magnetic nanostructures can be a chal-\nlenge for current experimental techniques. Moreover, the\nmechanisms behind of skyrmion pinning reported in these\nworks are very di\u000ecult to understand. Up to now, no\nwork explains why skyrmions are either attracted or re-\npelled by a region modi\fed magnetically. Besides com-\nplementing the previous studies by including the e\u000bect\nof the dipolar coupling, our results indicate that it is\nalso possible to obtain skyrmion traps through located\nvariations in the saturation magnetization. Our work\nsystem consists in Co/Pt multilayer nanowires hosting\na single skyrmion with the hedgehog con\fguration. In3\nsuch ferromagnetic nanotracks, it was predicted theo-\nretically [90] and was observed experimentally [27{33]\na high thermal stability. The main goal in this paper\nis to investigate how skyrmions are in\ruenced by a lo-\ncated variation on the intrinsic parameters of the ferro-\nmagnetic nanotrack, including exchange sti\u000bness, satu-\nration magnetization, magnetocrystalline anisotropy and\nDzyaloshinskii-Moriya constant. For this purpose, mag-\nnetic defects were strategically located on the nanotrack\naxis and in the neighborhood of the skyrmion. In order\nto demonstrate the Physics of the skyrmion-defect inter-\naction, we do not consider magnetic system under the\nin\ruence of an external agent, such as magnetic \felds\nor spin-polarized currents. Our methodology consists in\ncomputing energy di\u000berences by the Hamiltonian of the\nsystem. Thus, we obtain both well and barrier potentials,\nwhich identify, as quickly as possible, the kind of the in-\nteraction that we are dealing with, either attractive or re-\npulsive. Predictions of our results are veri\fed solving the\nLandau-Lifshitz-Gilbert equation without any external\nagent. Here, we answer the following question, magnetic\ndefects in nanowires: traps for skyrmions in spintronic\ntechnologies?\nII. MODELING AND SIMULATION DETAILS\nTo describe the magnetic system we have considered\nexchange, Dzyaloshinskii-Moriya, perpendicular mag-\nnetic anisotropy and dipole-dipole interactions, included\nin the following Hamiltonian:\nH=\u0000X\n<i;j>Jij[ ^mi\u0001^mj] +\n+X\n<i;j>Dijh\n^dij\u0001( ^mi\u0002^mj)i\n+\n\u0000X\niKi[ ^mi\u0001^n]2+\n+X\ni;jMij\u0014^mi\u0001^mj\u00003( ^mi\u0001^rij)( ^mj\u0001^rij)\n(rij=a)3\u0015\n(1)\nwhere ^mk\u0011(mx\nk;my\nk;mz\nk) is a dimensionless vector, cor-\nresponding to the magnetic moment located at the site\nkof the lattice. Once Jij>0, the \frst term in Eq. (1)\ndescribes the ferromagnetic coupling. Due to the short\nrange of the exchange interaction, the summation is over\nthe nearest magnetic moment pairs <i;j > . The second\nterm in Eq. (1) represents the Dzyaloshinskii-Moriya in-\nteractions, where the versor ^dijdepends on the type of\nmagnetic system considered. Dzyaloshinskii-Moriya in-\nteractions originate from inversion asymmetry and large\nspin-orbit coupling. In bulk materials with a lack of\nspatial inversion symmetry (B20 structures), the ver-\nsor of the Dzyaloshinskii-Moriya interaction is given by\n^dij= ^uij, where ^uijis unit vector joining the sites iandjin the same layer. In this case, Bloch skyrmions (vortex-\ntype con\fguration) can be stabilized in the framework of\na classical Heisenberg spin model with Dzyaloshinskii-\nMoriya interaction. On the other hand, for a mag-\nnetic multilayer system containing the interface between\na magnetic ultrathin layer and a strong spin-orbit cou-\npling adjacent layer, the versor of the Dzyaloshinskii-\nMoriya interaction is given by ^dij= ^uij\u0002^z, where\n^zis a versor perpendicular to the multilayer surface.\nThese magnetic systems favor the nucleation of N\u0013 eel\nskyrmions (hedgehog-type con\fguration). The constant\nof the Dzyaloshinskii-Moriya interaction Dijcan be ei-\nther positive or negative, thus it de\fnes the chirality\nof the magnetic structure. For a vortex-like skyrmion,\nDij<0 produces a structure with clockwise rotation,\nwhereasDij>0 produces a structure with anticlock-\nwise rotation. For a hedgehog-like skyrmion, Dij<0\nproduces a structure with inward radial \row, whereas\nDij>0 produces a structure with outward radial \row.\nThe third term in Eq. (1) describes the uniaxial mag-\nnetocrystalline anisotropy, since Ki>0 and ^n= ^zis a\nversor perpendicular to the magnetic layer surface. The\nlast term in Eq. (1) represents the dipolar coupling;\ndue to the long range of this interaction we consider\nall the dipole-dipole interactions. The strength of the\nmagnetic interactions, Jij; Dij; Ki; Mijhave the same\ndimension (energy unity) and they assume di\u000berent val-\nues depending on the local variation of the material pa-\nrameters: exchange sti\u000bness constant A, Dzyaloshinskii-\nMoriya constant D, magnetocrystalline anisotropy con-\nstantKand saturation magnetization constant MS. In\nour simulations we use the micromagnetic approach, in\nwhich the work cell has an e\u000bective magnetic moment\n~ mk= (MSVcell) ^mk. For the case in which the magnetic\nsystem is discretized into cubic cells Vcell=a3, the pos-\nsible values for the strength of the magnetic interactions\nare as follow:\nJij= 2a8\n<\n:A\nA0\nA00(2)\nDij=a28\n<\n:D\nD0\nD00(3)\nKi=a3\u001a\nK\nK00 (4)\nMij=\u0012\u00160a3\n4\u0019\u00138\n<\n:MSMS\nMSM00\nS\nM00\nSM00\nS(5)\nwhereA;D;K;M Sare parameters of the host material.\nA00;D00;K00;M00\nSare the parameters of the guest material.4\nA0;D0are parameters which describe interactions at the\ninterface between two ferromagnetic materials, see \fgures\n3(a) and 4. The geometric mean is adopted for the inter-\nface parameters: A0=p\nA\u0001A00andD0=p\nD\u0001D00in\norder to allow the magnetic parameters to vary gradually.\nIn the simulations we chose the typical parameters for\nCo/Pt multilayers, the values are as follow: exchange\nsti\u000bness constant A= 1:5\u000210\u000011J/m, Dzyaloshinskii-\nMoriya constant D= 4:0\u000210\u00003J/m2, magnetocrys-\ntalline anisotropy constant K= 1:2\u0002106J/m3and sat-\nuration magnetization constant MS= 5:8\u0002105A/m. In\norder to choose a suitable size for the work cell we esti-\nmate the characteristic lengths that are relevant to the\nproblem: the exchange length \u0015=r2A\n\u00160M2\nS\u00198:42 nm,\nthe wall width parameter \u0001 =r\nA\nK\u00193:54 nm and the\nlength associated with the Dzyaloshinskii-Moriya inter-\naction\u0018=2A\nD\u00197:50 nm, see Ref. [45]. The size of the\nwork cell used in the simulations was Vcell= 2\u00022\u00022nm3.\nOnce the side of the cell is smaller than the smallest char-\nacteristic lengths, a= 2 nm<\u0001, the chosen unit cell of\n2\u00022\u00022 nm3is accurate enough for the current study.\nFor the geometric parameters of the planar nanowires\nwe have considered the length l= 500 nm, the width\nw= 60 nm and the thickness t= 2 nm, di\u000bering one to\nanother only in the parameters of the magnetic defect:\nthe local variation of the magnetic property into a region\nof areaS. Guest material parameters, A00;D00;K00;M00\nS\nare regarded as tuning parameters to obtain traps for\nthe skyrmion. We have considered magnetic defects with\ndi\u000berent areas, where S ranging from 4 to 2116 nm2, con-\ntaining either a local reduction or a local increase of the\nmagnetic properties. It is worth mentioning that local\nvariations of the magnetic properties were considered in-\ndividually. Thus, we have studied four possible sources of\nmagnetic defects: Type A, TypeD, TypeKand Type\nMS. TypeAmagnetic defects are those characterized\nonly by local variations in the exchange sti\u000bness con-\nstant (other parameters of the magnetic material were\nunchanged in the region of the defect), Type Dmagnetic\ndefects are those characterized only by local variations in\nthe Dzyaloshinskii-Moriya constant, and so on.\nThe magnetization dynamics is governed by the\nLandau-Lifshitz-Gilbert (LLG) equation, whose discrete\nversion can be written as following:\nd^mi\nd\u001c=\u00001\n1 +\u000b2h\n^mi\u0002~bi+\u000b^mi\u0002( ^mi\u0002~bi)i\n(6)\nwhere~biis the dimensionless e\u000bective \feld at the lattice\nsitei, containing individual contributions from the ex-\nchange, Dzyaloshinskii-Moriya, anisotropy, dipolar, and\nZeeman \felds. The Gilbert damping parameter is \fxed\nat\u000b= 0:3, which corresponds to a typical value for\nCo/Pt multilayers. The connection between the time and\nits dimensionless corresponding is given by d\u001c=\u0017 dt,where\u0017=\u0012\u0015\na\u00132\n\r\u00160MS, being\r= 1:76\u00021011(T.s)\u00001\nthe electron gyromagnetic ratio. The LLG equation was\nintegrated by using a fourth-order predictor-corrector\nscheme with time step \u0001 \u001c= 0:01.\nIn order to obtain the remanent of the planar nanowire\nwith a single skyrmion we have chosen as initial condition\nan analytical solution in which a N\u0013 eel skyrmion is placed\nexactly at the geometric center of the nanowire. If no ex-\nternal agent (magnetic \feld or current) were present, the\nintegration of the LLG equation, Eq. (6), leads the mag-\nnetic system to the minimum energy con\fguration. This\nmakes possible the adjustment not only of the skyrmion\nradius but also its out-of-plane magnetization. The equi-\nlibrium con\fguration obtained in this way has been used\nas initial con\fguration in other simulations where a mag-\nnetic defect was inserted into the nanowire. To calculate\nthe interaction energy U(s) between the skyrmion and\nthe magnetic defect as a function of the center-to-center\nseparation s, we have \fxed the skyrmion at the center\nof the nanowire and only varied the defect position along\nthe nanowire axis, see Fig. (4). For each separation s, the\ntotal energy E(s) of the system was calculated using the\nEq. (1), and the interaction energy has been estimated\nusing the following expression:\nU(s) =E(s)\u0000E(s!1 ) (7)\nAfter computing the interaction energy between the\nskyrmion and the magnetic defect, we investigate how\nmagnetic defects a\u000bect the dynamics of the skyrmion\nwhen integrating the LLG equation, Eq. (6), with zero\nexternal agent.\nIII. RESULTS AND DISCUSSION\nA. Nanotracks made of a single magnetic material\nWe start our study obtaining the con\fguration of an\nisolated skyrmion in a planar nanowire. After the ini-\ntial con\fguration was relaxed by the integration of the\nLLG equation we obtain skyrmions with di\u000berent sizes,\nsee Fig. (1). The skyrmion diameter is de\fned as the\ncircle diameter at which the perpendicular magnetiza-\ntion changes of sign ( mz= 0). When comparing your\nresults with those of a previous study [14], one can verify\nthat they agreed with each other. If the magnetic nanos-\ntructure is large enough to host the skyrmion without\nit being deformed, one can note that the skyrmion di-\nameter does not depend on the dimensions of the nanos-\ntructure. In this regime, the skyrmion diameter only de-\npends on the parameters of the magnetic medium. The\ndependence of the skyrmion diameter and the magnetic\nproperties of the medium is shown in Fig. (2). Our\nresults for the skyrmion diameter as a function of the\ninteraction strength of Dzyaloshinskii-Moriya and per-\npendicular anisotropy are consistent with those reported5\nin Ref. [49]. In our study we \ft exponential behaviors\nfor all magnetic parameters. The skyrmion diameter de-\ncreases exponentially with the strength of the parameters\nexchange sti\u000bness and perpendicular anisotropy, whereas\nit increases exponentially with the strength of the pa-\nrameters Dzyaloshinskii-Moriya and saturation magneti-\nzation. The dependence of the skyrmion size on magnetic\nproperties of the medium can be understood by compe-\ntition between exchange, Dzyaloshinskii-Moriya, perpen-\ndicular anisotropy and dipolar energies. We observe the\nferromagnetic state for large values of AandK, thus,\nthe skyrmion diameter decreases gradually until it dis-\nappears. Therefore, the ferromagnetic alignment is fa-\nvored by increasing of AandK, so that the skyrmion\ndiameter decreases by increasing these parameters. We\nobserve the ferromagnetic state for small values of D, so\nthat the skyrmion diameter enlarges by increasing the D\nparameter strength. Obviously, the skyrmion shape is\ndistorted when its diameter is larger than the width of\nthe nanowire. To explain the dependence of the skyrmion\nsize on the saturation magnetization, we observe that the\nstrength of dipole-dipole interactions is directly propor-\ntional to the square of the MS. Once we are dealing with\nultrathin \flms, the dipolar coupling originates the shape\nanisotropy that work as an easy-plane anisotropy [45].\nThus, we have the exactly opposite situation of an easy-\naxis anisotropy, so that the skyrmion diameter increases\nby increasing the MSparameter.\nB. Nanotracks made of two magnetic materials\nSo far we have reviewed that the balance of the mag-\nnetic interactions de\fnes the skyrmion size. From now\non, we consider a planar nanowire made of two magnetic\nmedia, as shown in Fig. 3(a). The magnetic properties of\ntwo media are very close, being the magnetic parameters\nof the medium 2 about \u000610% of magnetic parameters of\nthe medium 1. The skyrmion is located exactly at the\ninterface between two magnetic media. Using this ini-\ntial condition, we numerically calculated the relaxed mi-\ncromagnetic state of 60-nm-wide nanowires in zero \feld\nfor di\u000berent values of the magnetic parameters of the\nmedium 2. Results of these simulations show that the\nskyrmion is stabilized either in the medium 1 or in the\nmedium 2, see Fig. 3. From the results shown in \fg-\nures (c), (e), (g) and (i), one can see that the magnetic\nsystem decreases its energy by moving the skyrmion to\nthe medium 2, which is the region of AandKreduced\norDandMSincreased. On the other hand, one can see\nthat the medium 2 is avoided in \fgures (b), (d), (f) and\n(h). In these simulations, the medium 2 is the region of\nAandKincreased or DandMSreduced. To understand\nthe two distinct behaviors, that is, the skyrmion is either\nattracted or repelled by the medium 2, we remember the\nreader of our previous results about the balance of the\nmagnetic interactions, which de\fnes the skyrmion size.\nIn Fig. (2), we emphasized that the skyrmion diameter\n17\n(a)D=3.0×10−3J/m2;K=0.9×106J/m3\n(b)D=4.0×10−3J/m2;K=1.3×106J/m3\n(c)D=3.0×10−3J/m2;K=0.8×106J/m3\n(d)D=4.0×10−3J/m2;K=1.2×106J/m3\n(e)D=3.0×10−3J/m2;K=0.7×106J/m3\n(f)D=4.0×10−3J/m2;K=1.1×106J/m3\nFigura 1.1– Raio do skyrmionFIG. 1. (Color online). Equilibrium con\fgurations of a single\nskyrmion in the planar nanowire 60 nm wide as a function\nof the magnetic properties of the medium. Figures (a) to (f)\nwere obtained using A= 1:5\u000210\u000011J/m andMS= 5:8\u0002105\nA/m. In (a) the skyrmion diameter is d= 9:61 nm. In (b)\nthe skyrmion diameter is d= 10:01 nm. In (c) the skyrmion\ndiameter is d= 15:02nm. In (d) the skyrmion diameter is d=\n13:21 nm. In (e) the skyrmion diameter is d= 22:02 nm. In\n(f) the skyrmion diameter is d= 17:62 nm. Unless otherwise\nstated, the skyrmion of \fgure (d) was used in most of our\nsimulations.\nis reduced by increasing of the exchange and perpendic-\nular anisotropy parameters, whereas it is increased by\nincreasing of the Dzyaloshinskii-Moriya and saturation\nmagnetization constants. Thus, we can summarize the\nresults of the Fig. (3) saying that the skyrmion prefers\nthe magnetic region which enlarges its diameter, because\nit not only minimizes the system energy, but also ensures\nthe skyrmion survival. In other words, the system energy\nincreases as the skyrmion diameter decreases. Once the\nskyrmion can disappear collapsing to the ferromagnetic\nstate, the skyrmion avoids the magnetic region which\ntends to shrink its size.\nC. Nanotracks with a single magnetic defect\nNow we investigate the possibility of building traps for\nmagnetic skyrmions. As shown in Fig. (4), a trap con-\nsists in a magnetic defect incorporated into the nanos-\ntructure which hosts the skyrmion. Our study of the\ndefect-skyrmion interaction revealed two kinds of traps.\nIn a pinning trap, the skyrmion core moves towards the6\n18Skyrmion diameter, d (nm)\n810121416182022\n \nExchange stiffness constant, A (10-11 J / m)1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9D = 4.0 x 10-3 J / m2; K = 1.2 x 105 J / m3; M s = 5.8 x 105 A / m\n(a)\nSkyrmion diameter, d (nm)\n5101520253035\n \nDzyaloshinskii-Moriya constant, D (10-3 J / m2)3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0A = 1.5 x 10-11 J / m; K = 1.2 x 105 J / m3; M s = 5.8 x 105 A / m\n(b)Skyrmion diameter, d (nm)\n51015202530\n \nPerpendicular anisotropy constant, K (106 J / m3)0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35A = 1.5 x 10-11 J / m; D = 4.0 x 10-3 J / m2; M s = 5.8 x 105 A / m\n(c)\nSkyrmion diameter, d (nm)\n101214161820\n \nSaturation magnetization constant, M s (105 A / m)4.5 5.0 5.5 6.0 6.5 7.0 7.5A = 1.5 x 10-11 J / m; D = 4.0 x 10-3 J / m2; K = 1.2 x 105 J / m3\n(d)\nFigura 1.1– global variation - skyrmion diameter - Crop 20 15 10 49\nFIG. 2. (Color online). Skyrmion diameter as a function of\nthe magnetic properties of the medium: (a) exchange sti\u000b-\nness constant, (b) Dzyaloshinskii-Moriya constant, (c) per-\npendicular anisotropy constant and (d) saturation magne-\ntization constant. We \ft exponential behaviors. In (a)\nd(A) =d0;A+d1;Ae\u0000(A=A0), whered0;A= 6:415 nm,\nd1;A= 320:685 nm and A0= 0:387\u000210\u000011J/m. In (b)\nd(D) =d0;D+d1;De(D=D 0), whered0;D=\u00001:646 nm,\nd1;D= 0:245 nm and D0= 0:976\u000210\u00003J/m2. In (c)\nd(K) =d0;K+d1;Ke\u0000(K=K 0), whered0;K= 2:934 nm,\nd1;K= 736:457 nm and K0= 0:281\u0002106J/m3. In (d)\nd(Ms) =d0;Ms+d1;Mse(Ms=Ms0), whered0;Ms= 5:260 nm,\nd1;Ms= 0:739 nm and Ms0= 2:444\u0002105A/m.\nmagnetic defect, indicating an e\u000bective attractive poten-\ntial of interaction between the skyrmion and the mag-\nnetic defect (potential well). In a scattering trap, the\nskyrmion core moves away from the magnetic defect,\nindicating an e\u000bective repulsive potential of interaction\nbetween the skyrmion and the magnetic defect (poten-\ntial barrier). Figures (5), (6), (7), (8) show that both\npinning and scattering traps can be individually origi-\nnated by the local variation of A;D;K;M S, respectively.\nFrom these \fgures, one can note that potential wells and\npotential barriers arise naturally as the defect size in-\ncreases. Thus, magnetic defects large enough relative to\nthe skyrmion size can be used as traps, either to pin or to\nscatter skyrmions. One can see that the interaction po-\ntential changes its shape for some critical size of the mag-\nnetic defect. When the defects are small, the strength of\nthe skyrmion-defect interaction is very weak. Besides,\nin these cases, the interaction depends on the skyrmion-\ndefect separation. Therefore, the skyrmion-defect aspect\nratio (that is, the ratio of sizes) is a crucial parameter to\ndesign traps for skyrmions. In particular, the e\u000eciency\nof the trap is compromised if the defect size is smaller\nthan the skyrmion size.\n19\nMedium 1: A, D, K and MsMedium 2: A'', D'', K'' and Ms''Interface: A' and D'\n(a)Initial condition: the skyrmion is located at the interface between two magnetic media\n(b)A��>A;D��=D;K��=K;M��\nS=M S\n (c)A��<A;D��=D;K��=K;M��\nS=M S\n(d)A��=A;D��<D;K��=K;M��\nS=M S\n (e)A��=A;D��>D;K��=K;M��\nS=M S\n(f)A��=A;D��=D;K��>K;M��\nS=M S\n (g)A��=A;D��=D;K��<K;M��\nS=M S\n(h)A��=A;D��=D;K��=K;M��\nS<M S\n (i)A��=A;D��=D;K��=K;M��\nS>M S\nFigura 1.2– Skyrmion at an interface Crop 20 15 13 12FIG. 3. (Color online). (a) Schematic view shows a piece\nthe magnetic nanowire, which contains the skyrmion located\nat the interface between two magnetic media. At the left\nside of the nanowire is the medium 1 characterized by the\nmagnetic parameters: A;D;K;M Sand at the right side of\nthe nanowire is the medium 2 characterized by the magnetic\nparameters: A00;D00;K00;M00\nS. Using this con\fguration as ini-\ntial condition, we integrate the LLG equation and obtain the\n\fnal con\fgurations shown in \fgures (b) to (i), which show\nthat the skyrmion choose a medium. In \fgures (b) and (c)\nthe medium 2 di\u000bers from medium 1 only in the exchange\nsti\u000bness constant, A00= 1:1AandA00= 0:9A, respectively.\nIn \fgures (d) and (e) the medium 2 di\u000bers from medium 1\nonly in the Dzyaloshinskii-Moriya constant, D00= 0:9Dand\nD00= 1:1D, respectively. In \fgures (f) and (g) the medium\n2 di\u000bers from medium 1 only in the perpendicular anisotropy\nconstant,K00= 1:1KandK00= 0:9K, respectively. In \fg-\nures (h) and (i) the medium 2 di\u000bers from medium 1 only\nin the saturation magnetization constant, M00\nS= 0:9MSand\nM00\nS= 1:1MS, respectively. Figures (b), (d), (f) and (h) show\nthat the skyrmion moves to the medium 1, thus, it is repelled\nby the medium 2. On the other hand, \fgures (c), (e), (g) and\n(i) show that the skyrmion moves to the medium 2, thus, it\nis attracted by the medium 2.7\nFIG. 4. (Color online). Schematic view shows a piece the\nmagnetic nanowire, which contains the skyrmion and the\nmagnetic defect. The magnetic defect consists in the local\nvariation of the magnetic properties in the area represented\nby the blue spots. The majority of the wire's magnetic mo-\nments is going out of the plane of this \fgure except at the\ncore of the skyrmion, where they are pointing in the oppo-\nsite direction (red region). The center-to-center separation\nbetween the skyrmion and the magnetic defect is s= 60 nm.\nThe magnetic defect has an area of 196 nm2.\nIn order to systematically understand the skyrmion-\ndefect interaction, we de\fne U0=U(s= 0) which mea-\nsures the strength of the interaction, that is, the maxi-\nmum value or the minimum value of U. IfU0>0, then\nthe maximum value U0represents the height of the poten-\ntial barrier, whereas if U0<0, then the minimum value\njU0jrepresents the depth of the potential well. As shown\nin Figs. (9), (10) and (11), the strength of the skyrmion-\ndefect interaction depends on the local variations of the\nmagnetic property as well as on the skyrmion-defect as-\npect ratio. In particular, from \fgures (10) and (11), one\ncan see that the strength of the interaction increases not\nonly by increasing of the area of the magnetic defect,\nbut also by increasing of the skyrmion size. Thus, the\nstrength of the skyrmion-defect interaction can be tuned\nby the modi\fcation of the magnetic properties within a\nregion with suitable size. It is notably that the e\u000eciency\nof the trap is improved as the skyrmion size becomes\nmuch smaller than the defect size.\nIn Table (I) we summarize four distinct ways to build\ntraps for pinning and scattering magnetic skyrmions. Al-\nthough the discussions have been held with the N\u0013 eel\nskyrmion, we would like to emphasize that the qualita-\ntive results of the Table (I) remain unchanged for Bloch\nskyrmions; it has been checked through micromagnetic\nsimulations. Moreover, we have veri\fed that the re-\nsults remain essentially the same for both chirality of\nthe skyrmion ( D> 0 andD< 0).\nNow we verify the predictions of the \fgure (5) through\nmicromagnetic simulations, see \fgures (12) and (13). By\nconsidering a magnetic defect with a suitable size, we\nshow that the skyrmion can be either attracted by a re-\nducedAregion or repelled by an increased Aregion. Al-\nthough the results presented in \fgures (12) and (13) are\nfor the case of Type Amagnetic defects, we have observed\nsimilar results (both pinning and scattering traps) for\nother types of magnetic defects which were approached\nin this work, see Table (I).\n25Interaction energy, U (x10-21 J)\n−350−300−250−200−150−100−50050\n \nSkyrmion-magnetic defect separation, s (nm)−60−40−20 0 20 40 602116 nm²\n1444 nm²\n900 nm²\n484 nm²\n324 nm²\n196 nm²\n100 nm²\n36 nm²\n4 nm²S\nA'' = 0.7 A\n(a)Interaction energy, U (x10-21 J)\n−50050100150200250300350\n \nSkyrmion-magnetic defect separation, s (nm)−60−40−20 0 20 40 602116 nm²\n1444 nm²\n900 nm²\n484 nm²\n324 nm²\n196 nm²\n100 nm²\n36 nm²\n4 nm²S\nA'' = 1.3 A\n(b)\nFigura 1.8– AFIG. 5. (Color online). Local variations in the exchange sti\u000b-\nness constant: Type Amagnetic defects. Figures (a) and (b)\nshow the interaction energy between the skyrmion and the\nmagnetic defect as a function of the center-to-center separa-\ntion. In Fig. (a) is shown a local reduction of 30% in A,\nwhereas in Fig. (b) is shown a local increase of 30% in A.\nThe behavior is shown for di\u000berent areas of a type Amag-\nnetic defect.\nTABLE I. Two types of traps for magnetic skyrmions can\nbe originated in located variations of the magnetic properties\nwhen tuning either a local increase ( X00> X ) or a local\nreduction ( X00< X ), whereXcan beA;D;K orMS. A\npinning trap corresponds to a potential well for the skyrmion,\nwhereas a scattering trap corresponds to a potential barrier.\nMagnetic PropertyPinning\nTrapScattering\nTrap\nexchange sti\u000bness A00<AA00>A\nDzyaloshinskii-Moriya constant D00> DD00< D\nperpendicular anisotropy K00< KK00> K\nsaturation magnetization M00\nS> M SM00\nS<M S\nIt is also interesting to discuss the in\ruence of the de-8\n26Interaction energy, U (x10-21 J)\n−50050100150200250300350\n \nSkyrmion-magnetic defect separation, s (nm)−60−40−20 0 20 40 602116 nm²\n1444 nm²\n900 nm²\n484 nm²\n324 nm²\n196 nm²\n100 nm²\n36 nm²\n4 nm²S\nD'' = 0.7 D\n(a)Interaction energy, U (x10-21 J)\n−350−300−250−200−150−100−50050\n \nSkyrmion-magnetic defect separation, s (nm)−60−40−20 0 20 40 602116 nm²\n1444 nm²\n900 nm²\n484 nm²\n324 nm²\n196 nm²\n100 nm²\n36 nm²\n4 nm²S\nD'' = 1.3 D\n(b)\nFigura 1.9– DM\nFIG. 6. (Color online). Local variations in the constant of the\nDzyaloshinskii-Moriya interaction: Type Dmagnetic defects.\nFigures (a) and (b) show the interaction energy between the\nskyrmion and the magnetic defect as a function of the center-\nto-center separation. In Fig. (a) is shown a local reduction\nof 30% inD, whereas in Fig. (b) is shown a local increase of\n30% inD. The behavior is shown for di\u000berent areas of a type\nDmagnetic defect.\nfect size on the magnitude of the skyrmion-defect inter-\naction. As shown in \fgures (5), (6), (7) and (8), the\ninteraction potential can present a double-well potential\n(or double-barrier potential) instead of a single poten-\ntial well (or a single potential barrier). Particularly, this\nsituation occurs whenever the defect is smaller than the\nskyrmion and the interaction between them is extremely\nweak. Besides, micromagnetic simulations show that the\nskyrmion pinning due to a double-well potential gener-\nates a pinning center which does not coincide with the\ndefect center.\nFrom the technological point of view, the reader can\nwonder whether magnetic defects considered in this work\nwould work as traps for skyrmions in a nanotrack at room\ntemperature? By analyzing our results for skyrmion-\n27Interaction energy, U (x10-21 J)\n−200−150−100−50050\n \nSkyrmion-magnetic defect separation, s (nm)−60−40−20 0 20 40 602116 nm²\n1444 nm²\n900 nm²\n484 nm²\n324 nm²\n196 nm²\n100 nm²\n36 nm²\n4 nm²S\nK'' = 0.70 K\n(a)Interaction energy, U (x10-21 J)\n−50050100150200\n \nSkyrmion-magnetic defect separation, s (nm)−60−40−20 0 20 40 602116 nm²\n1444 nm²\n900 nm²\n484 nm²\n324 nm²\n196 nm²\n100 nm²\n36 nm²\n4 nm²S\nK'' = 1.3 K\n(b)\nFigura 1.10– KFIG. 7. (Color online). Local Variations in the perpendicular\nmagnetic anisotropy: Type Kmagnetic defects. Figures (a)\nand (b) show the interaction energy between the skyrmion\nand the magnetic defect as a function of the center-to-center\nseparation. In Fig. (a) is shown a local reduction of 30%\ninK, whereas in Fig. (b) is shown a local increase of 30%\ninK. The behavior is shown for di\u000berent areas of a type K\nmagnetic defect.\ndefect interaction energy, one can see that the heights\nof the potential barriers as well as the depths of the po-\ntential wells depend strongly on the type of defect be-\ning considered, on the strength of the modi\fed magnetic\nproperty, and on the defect size (skyrmion-defect aspect\nratio). Thus, characteristics of trap can be experimen-\ntally controlled. For instance, the skyrmion-defect in-\nteraction strength can be tuned in one or more order\nof magnitude larger than the thermal energy, that is,\nEthermal\u001810\u000021[J]\u0018kBTroom.\nIt is worth mentioning that the region with modi-\n\fed magnetic properties can be designed to present a\nsingle or a combination of tuned magnetic parameters.\nThe balance of the magnetic defect parameters will be\nresponsible for adjusting a proper trap, either to cap-9\n28Interaction energy, U (x10-21 J)\n−10010203040506070\n \nSkyrmion-magnetic defect separation, s (nm)−60−40−20 0 20 40 602116 nm²\n1444 nm²\n900 nm²\n484 nm²\n324 nm²\n196 nm²\n100 nm²\n36 nm²\n4 nm²S\nMs'' = 0.7 M s\n(a)Interaction energy, U (x10-21 J)\n−90−80−70−60−50−40−30−20−10010\n \nSkyrmion-magnetic defect separation, s (nm)−60−40−20 0 20 40 602116 nm²\n1444 nm²\n900 nm²\n484 nm²\n324 nm²\n196 nm²\n100 nm²\n36 nm²\n4 nm²S\nMs'' = 1.3 M s\n(b)\nFigura 1.11– Ms\nFIG. 8. (Color online). Local Variations in the saturation\nmagnetization: Type MSmagnetic defects. Figures (a) and\n(b) show the interaction energy between the skyrmion and\nthe magnetic defect as a function of the center-to-center sep-\naration. In Fig. (a) is shown a local reduction of 30% in\nMS, whereas in Fig. (b) is shown a local increase of 30% in\nMS. The behavior is shown for di\u000berent areas of a type MS\nmagnetic defect.\nture or to blockade the skyrmions. Thus, the kind of\nthe trap depends on the suitable choice which mate-\nrial parameters will be modi\fed within the selected area.\nMaybe, the a\u000bected region by ion beam irradiation con-\ntains more than one modi\fed magnetic property. How-\never, the magnitude of the a\u000bected magnetic parameters\nwill not be the same; they can di\u000ber even in the order\nof magnitude. From Figs. (5), (6), (7) and (8), one can\nsee that the individual contribution of the magnetic pa-\nrameterA;D; andKare of the same magnitude order\n(\u0018102Ethermal ), whereas the MSparameter modi\f-\ncation results in the weakest skyrmion-defect interaction\n(\u001810Ethermal ). Our results could guide the design of ex-\nperimental works that intend to investigate the realistic\nincorporation of such magnetic defects into nanomagnets.\n18Interaction energy, U 0 (x10-21 J)\n−1,500−1,000−50005001,0001,500\nMagnetic modification, A'' /  A−0.5 0.0 0.5 1.0 1.5 2.0 2.5S = 196 nm²\nS = 900 nm²Scattering increases\nPinning increases\n(a)TypeAmagnetic defects\nInteraction energy, U 0 (x10-21 J)\n−1,000−50005001,000\nMagnetic modification, D'' /  D−0.5 0.0 0.5 1.0 1.5 2.0 2.5S = 196 nm²\nS = 900 nm²Scattering increases\nPinning increases\n(b)TypeDmagnetic defectsInteraction energy, U 0 (x10-21 J)\n−800−600−400−2000200400600800\nMagnetic modification, K'' /  K−0.5 0.0 0.5 1.0 1.5 2.0 2.5S = 196 nm²\nS = 900 nm²Scattering increases\nPinning increases\n(c)TypeKmagnetic defects\nInteraction energy, U 0 (x10-21 J)\n−400−300−200−1000100200\nMagnetic modification, M s'' / M s−0.5 0.0 0.5 1.0 1.5 2.0 2.5S = 196 nm²\nS = 900 nm²Scattering increases\nPinning increases\n(d)TypeM Smagnetic defects\nFigura 1.1– Modification - Uo X Crop 20 15 10 49FIG. 9. (Color online). Interaction energy U0as a function\nof a local variation in the magnetic properties. The behav-\nior is shown for di\u000berent types of magnetic defects. Linear\nand quadratic dependencies of the skyrmion-defect interac-\ntion with the located magnetic modi\fcations do not surprise,\nonce the same dependencies are presents in the strength of\nthe magnetic interactions, it can be con\frmed in Eqs. (2),\n(3), (4) and (5).\n23Interaction energy, U 0 (x10-21 J)\n−800−600−400−20002004006008001,000\n \nArea of the magnetic defect, S (nm²)−200 0 200 400 600 800 1,000A'' = 2.0 A\nA'' = 1.5 A\nA'' = 1.3 A\nA'' = 0.7 A\nA'' = 0.5 A\nA'' = 0.3 A\n(a)TypeAmagnetic defects\nInteraction energy, U 0 (x10-21 J)\n−1,000−800−600−400−2000200400600800\n \nArea of the magnetic defect, S (nm²)−200 0 200 400 600 800 1,000D'' = 2.0 D\nD'' = 1.5 D\nD'' = 1.3 D\nD'' = 0.7 D\nD'' = 0.5 D\nD'' = 0.3 D\n(b)TypeDmagnetic defectsInteraction energy, U 0 (x10-21 J)\n−600−400−2000200400600800\n \nArea of the magnetic defect, S (nm²)−200 0 200 400 600 800 1,000K'' = 2.0 K\nK'' = 1.5 K\nK'' = 1.3 K\nK'' = 0.7 K\nK'' = 0.5 K\nK'' = 0.3 K\n(c)TypeKmagnetic defects\nInteraction energy, U 0 (x10-21 J)\n−400−300−200−1000100200\n \nArea of the magnetic defect, S (nm²)−200 0 200 400 600 800 1,000Ms'' = 2.0 Ms\nMs'' = 1.5 Ms\nMs'' = 1.3 Ms\nMs'' = 0.7 Ms\nMs'' = 0.5 Ms\nMs'' = 0.3 Ms\n(d)TypeM Smagnetic defects\nFigura 1.6– Crop 20 15 10 49\nFIG. 10. (Color online). Interaction energy U0=U(s= 0)\nas a function of the defect area S. The behavior is shown for\ndi\u000berent types of magnetic defects. Exponential behaviors\nhave been \ftted for all types of defects. The missing points\nin the curves for defects of small areas represent the situa-\ntion in which the graph of the interaction potential does not\ncorrespond neither a potential well nor a potential barrier.\nAs has been mentioned previously, the traps do not work for\nmagnetic defect of small areas.10\n24Interaction energy, U 0 (x10-21 J)\n−600−400−2000200400600\n \nArea of the magnetic defect, S (nm²)0 800 1,600 2,400 3,200 4,000d = 17.62 nm; A'' = 1.5 A\nd = 13.21 nm; A'' = 1.5 A\nd = 10.01 nm; A'' = 1.5 Ad = 17.62 nm; A'' = 0.5 A\nd = 13.21 nm; A'' = 0.5 A\nd = 10.01 nm; A'' = 0.5 A\n(a)TypeAmagnetic defects\nInteraction energy, U 0 (x10-21 J)\n−800−600−400−2000200400600800\n \nArea of the magnetic defect, S (nm²)0 800 1,600 2,400 3,200 4,000d = 17.62 nm; D'' = 1.5 D\nd = 13.21 nm; D'' = 1.5 D\nd = 10.01 nm; D'' = 1.5 Dd = 17.62 nm; D'' = 0.5 D\nd = 13.21 nm; D'' = 0.5 D\nd = 10.01 nm; D'' = 0.5 D\n(b)TypeDmagnetic defectsInteraction energy, U 0 (x10-21 J)\n−600−400−2000200400600\n \nArea of the magnetic defect, S (nm²)0 800 1,600 2,400 3,200 4,000d = 17.62 nm; K'' = 1.5 K\nd = 13.21 nm; K'' = 1.5 K\nd = 10.01 nm; K'' = 1.5 Kd = 17.62 nm; K'' = 0.5 K\nd = 13.21 nm; K'' = 0.5 K\nd = 10.01 nm; K'' = 0.5 K\n(c)TypeKmagnetic defects\nInteraction energy, U 0 (x10-21 J)\n−250−200−150−100−50050100150\n \nArea of the magnetic defect, S (nm²)0 800 1,600 2,400 3,200 4,000d = 17.62 nm; Ms'' = 1.5 Ms\nd = 13.21 nm; Ms'' = 1.5 Ms\nd = 10.01 nm; Ms'' = 1.5 Msd = 17.62 nm; Ms'' = 0.5 Ms\nd = 13.21 nm; Ms'' = 0.5 Ms\nd = 10.01 nm; Ms'' = 0.5 Ms\n(d)TypeM Smagnetic defects\nFigura 1.7– Crop 20 15 10 49\nFIG. 11. (Color online). Interaction energy U0as a function of\nthe defect area Sand the skyrmion diameter d. The behavior\nis shown for di\u000berent types of magnetic defects. The missing\npoints in the curves for defects of small areas represent the\nsituation in which the graph of the interaction potential does\nnot correspond neither a potential well nor a potential barrier.\nAs has been mentioned previously, the traps do not work for\nmagnetic defect of small areas.\n177\n(a)t=0\n(b)t=210 ps\n(c)t=438 ps\nFigura 7.1– Successive snapshots show the skyrmion being attracted by a pinning trap, which is cha-\nracterized by a local reduction in the exchange stiffness constant A��=0.7A (a local re-\nduction of 30 % in A) into an area of 196 nm2. (a) Initial configuration at t = 0, where the\ncenter-to-center separation between the skyrmion and the magnetic defect is 20 nm. (b)\nConfiguration after 210 ps. (c) Configuration after 438 ps, where the skyrmion is captured\nby the pinning trap.\nFIG. 12. (Color online). E\u000bect of a pinning trap. Successive\nsnapshots show the skyrmion being attracted by a trap, which\nis characterized by a local reduction in the exchange sti\u000bness\nconstantA00= 0:7A(a local reduction of 30 % in A) into\nan area of 196 nm2. (a) Initial con\fguration at t = 0, where\nthe center-to-center separation between the skyrmion and the\nmagnetic defect is 20 nm. (b) Con\fguration after 210 ps. (c)\nCon\fguration after 438 ps, where the skyrmion is captured\nby the pinning trap.\n178\n(a)t=0\n(b)t=210 ps\n(c)t=438 ps\nFigura 7.2– Successive snapshots show the skyrmion being repulsed by a scattering trap, which is\ncharacterized by a local increase in the exchange stiffness constant A��=1.3A (a local\nincrease of 30 % in A) into an area of 196 nm2. (a) Initial configuration at t = 0, where\nthe center-to-center separation between the skyrmion and the magnetic defect is 14 nm.\n(b) Configuration after 210 ps. (c) Configuration after 438 ps.FIG. 13. (Color online). E\u000bect of a scattering trap. Succes-\nsive snapshots show the skyrmion being repelled by a trap,\nwhich is characterized by a local increase in the exchange sti\u000b-\nness constant A00= 1:3A(a local increase of 30 % in A) into\nan area of 196 nm2. (a) Initial con\fguration at t = 0, where\nthe center-to-center separation between the skyrmion and the\nmagnetic defect is 14 nm. (b) Con\fguration after 210 ps. (c)\nCon\fguration after 438 ps.\nIV. CONCLUSION\nGenerally, the interaction between a skyrmion and a\nmagnetic defect is di\u000ecult of being observed because of\ntwo very simple reasons: (1) Skyrmions are topologically\nprotected entities. Thus, these quasiparticles have a kind\nof shield, which hinders the skyrmion interaction with\ninhomogeneities of the magnetic medium. Ultimately,\ndue to the topological protection of the skyrmion, the\nskyrmion-defect interaction is weak by nature. (2) Up\nto now, most of the theoretical and computational works\nhas considered magnetic defects with very small sizes.\nBesides masking the skyrmion-defect interaction, the in-\ncorporation of small defects in nanomagnets can be a\nchallenge for current experimental techniques.\nIn this work, we show that the skyrmion-defect inter-\naction becomes evident as the defect size increases. Both\nattractive and repulsive interactions have been observed\nwhenever the defect size was larger than the skyrmion\nsize. Therefore, traps for magnetic skyrmions can be cre-\nated through magnetic defects with sizes large enough.\nIndeed, we conclude that the skyrmion-defect aspect ra-\ntio is a crucial parameter to build traps for skyrmions,\nwhich can be captured or blocked around a magnetic\ndefect intentionally incorporated into the nanomagnet.\nMagnetic defects have been modeled as located varia-\ntions on the material parameters, such as A;D;K;MS.\nThrough micromagnetic simulations we show that any of11\nthese material parameters can be used to create both pin-\nning and scattering traps by tuning either a local increase\nor a local decrease of a given magnetic property. In or-\nder to provide a background for experimental studies, we\nhave varied the parameters of the magnetic defects. In\ncontrast to previous studies on the skyrmion-defect inter-\naction, we consider the e\u000bect of the full dipolar coupling.\nAll the possibilities for a skyrmion trap were investigated\nand they are summarized in Table (I). Furthermore, we\nunderstand the basic physics behind the mechanisms for\npinning and for scattering magnetic skyrmions, that is,\nskyrmions move towards the magnetic region which tends\nto maximize its diameter. It enables the magnetic system\nto minimize its energy. Thus, we are able to explain why\nskyrmions are either attracted or repelled by a region\nwith modi\fed magnetic properties.\nWe believe that it is possible to engineer magnetic de-\nfects working as skyrmion traps, and we are hoping for\nthis paper encourages future experimental works to in-\nvestigate the skyrmion-defect interaction as well as to\nverify our predictions. As we emphasized, the magnetic\ndefect size must be large enough relative to the skyrmionsize, somehow, it facilitates the realistic incorporation of\nsuch magnetic defects into nanomagnets.\nUnderstanding and controlling the static properties\nand dynamics of skyrmions in magnetic nanostructures\nis of utmost signi\fcance for the development and real-\nization of future spintronic devices. Although the re-\nsults presented here are for a very simple distribution of\nmagnetic defects into a ferromagnetic nanotrack, we be-\nlieve their consequences can be planned and extended.\nFor instance, a nanotrack containing a proper distribu-\ntion of magnetic defects can be designed to solve the\nissue of transporting skyrmions in a usual ferromagnetic\nnanowire.\nACKNOWLEDGMENTS\nThis work was partially supported by CAPES, CNPq,\nFAPEMIG and FINEP (Brazilian Agencies). 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Rep. 4, 6784 (2014)."    },    {        "title": "1810.05435v1.Room_temperature_biaxial_magnetic_anisotropy_in_La0_67Sr0_33MnO3_thin_films_on_SrTiO3_buffered_MgO__001__substrates_for_spintronic_applications.pdf",        "content": "1\n \n \nRoom temperature \nbiaxial magnetic anisotropy in \nLa\n0.67\nSr\n0.33\nMnO\n3\n \nthin \nfilm\ns\n \non SrTiO\n3\n \nbuffered MgO (001) substrates\n \nfor spintronic \napplications\n \nSandeep Kumar Chaluvadi,\n1\n \nFernando Ajejas,\n2,6\n \nPasquale Orgiani,\n3,4\n \nOlivier Rousseau,\n1 \nGiovanni Vinai,\n3\n \nAleksandr Yu Petrov,\n3\n \nPiero Torelli,\n3\n \nAlain Pautrat,\n5\n \nJulio Camarero,\n2,6\n \nPaolo Perna,\n2 \nand Laurence Mechin\n1\n,\na\n)\n \n1\nNormandie Univ, UNICAEN, ENSICAEN, CNRS, GREYC, 14000 Caen, France\n \n2\nIMDEA\n-\nNanociencia, Campus de Cantoblanco, 28049 Madrid, Spain\n \n3\nCNR\n-\nIOM TASC National Laboratory, Area Science Park\n-\n \nBasovizza, 34149 Trieste, Italy\n \n4\nCNR\n-\nSPIN, UOS Salerno, 84084 Fisciano (SA), Italy\n \n5\nNormandie Univ, UNICAEN, ENSICAEN, CNRS, CRISMAT, 1405\n0 Caen, France\n \n6\nUniversidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain\n \nSpintronics exploit\ns\n \nthe magnetoresistance effects to store or sense the magnetic information. Since the \nmagnetoresistance strictly depends on the magnetic anisotropy of the system, it is fundamental to \nset a \ndefined\n \nanisotropy to the system. Here, we investigate by means of \nvectorial Magneto\n-\nOptical Kerr Magnetometry (v\n-\nMOKE)\n,\n \nhalf\n-\nmetallic \nLa\n0.67\nSr\n0.33\nMnO\n3\n \n(LSMO) thin films that exhibit at room temperature pure \nbiaxial \nmagnet\nic \nanisotropy \nif grown onto MgO (001) substrate with a thin SrTiO\n3\n \n(STO) buffer. In this way, we can \navoid unwanted uniaxial\n \nmagnetic anisotropy contribution\ns that may be detrimental for specific application\ns. \nThe\n \ndetailed study of the \nangular\n \nevolution of \nthe magnetization reversal pathways, critical fields (\ncoercivity and switching\n) \nallows for \ndisclosin\ng the \norigin of the magnetic anisotropy\n, which\n \nis \nmagnetocrystalline\n \nin nature and \nshows four\n-\nfold\n \nsymmetry \nat any \ntemperature\n. \n \n \nHalf\n-\nmetallic perovskite oxides \npromise\n \ngreat advantages over conventional spintronics metallic materials \nfor applications such as magnetic sensors, magnetic random access memory (MRAM), magnetic tunnel junctions \n(MTJs)\n \nand\n \ndomain wall race\n-\ntrack memories\n.\n1\n \nPerovskite oxides\n, in general, appear to be a new contender for many \nnovel applications that were considered traditionally beyond its range\n.\n2\n \nThe conduction mechanism in these materials, \nin fact, strongly depends on the interplay between orbital and spin\n \ndegrees of freedom\n3\n \nth\nat \nmay be exploited to add \nmultiferroic\n \nor ferroelectric functionalities\n.\n4\n–\n6\n \nThe \ncomplexity\n \nof such mechanism\n, however, can determine \nentangled \nmagneto transport response becoming in \nsome\n \ncase\ns\n \nundesired. For example, it has been seen that a switchable \nanisotropic magnetoresistance (AMR) response in manganites may be hidd\nen by the colossal magnetoresistance \n(CMR) if the magnetic anisotropy of the system is not accurately designed\n.\n7\n \nIrrespective of the applications, one of \n                                        \n        \n \na\n)\n \nAuthor to whom correspondence should be addressed: \nlaurence.mechin@ensicaen.fr\n 2\n \n \nthe key properties that \nneed\n \nto be considered for a ferromagneti\nc sample is \nthe\n \nmagnetic anisotropy that dictates the \nmagnetization reversals pathways\n \nas well as the MR output\n.\n8\n \nWhile\n \na \ndefined\n \nuniaxial\n \nanisotropy\n \nis essential for \nmagnetic field sensors based on \nAMR\n,\n7\n \na\n \nbiaxial\n \nanisotropy\n, which\n \nprovides \nfour \nstable magnetization states\n,\n \nhas\n \ncapability to encode more information (four binary bits: “00”, “01”, “10”, “11”), \nand \ncan be used in memory and logic \ndevices\n.\n9,10\n \nHowever, \nespecially in \nhalf\n-\nmetallic\n \nperovskite compounds with four\n-\nfold \nmagnetocrystalline\n \nanisotropy \nin bulk\n, \nit results\n \ndifficult\n \nto avoid \neither \nstrain \n(\ninduced by \nthe \nsubstrate\n)\n \nor \nshape\n \nuniaxial anisotropy in thin films\n.\n11,12\n \n \nAmong manganite\ns\n, \nLa\n0.67\nSr\n0.33\nMnO\n3\n \n(LSMO) \nhas arisen special interest \nfor its peculiar properties \nsuch \nas \nnearly \n100%\n \nspin \npolarization\n \nand\n \nroom temperature \n(RT) ferromagneti\nsm\n \nwith Curie temperature\n \nof about\n \nT\nc\n~370 \nK\n,\n \nhence\n,\n \nenabling RT spintronic applications\n. \nThe magnetotransport properties of these compounds in thin film\ns are \nstrongly affected by \nexternal perturbations\n \nsuch as substrate\n-\ninduced \nstrain\n.\n13\n \nIn general, in \ntensile (compressive) \nstrained film\ns\n, the electron occupancy in ‘\ne\ng\n’ \ndoublet \nfavors\n \nin\n-\nplane (out\n-\nof\n-\nplane) \nx\n2\n-\ny\n2\n \n(\n3z\n2\n-\nr\n2\n) \norbitals\n14\n \ndetermining\n \nin\n-\nplane (out\n-\nof\n-\nplane) \nmagnetization \neasy axis\n.\n15\n \nIn addition, tensile (compressive) strain reduces (enhances) the Tc \nwith respect to the bulk value\n.\n16,17\n \nT\nhe\nrefore, the\n \nchoice of the su\nbstrate for the LSMO growth is extremely important. \nThe most \ncommonly \nused\n \nsingle crystal substrate is\n \nSrTiO\n3\n \n(STO) \n(a\nSTO\n=0.3905 nm) \nwith (001) crystallographic \norientation\n \ns\nince LSMO \n(a\nLSMO\n= 0.387 nm) \ngrown epitaxially on \nSTO \nexhibits low \ntensile \nstrain (0.82%) and \nwith \ngood structural and morphological properties.\n18\n \nHowever, i\nn the\n \nlow thickness \nregime\n,\n \na sizeable uniaxial (two\n-\nfold) \nanisotropy contribution\n19\n \ndue to surface steps and terraces originated \nfrom\n \nthe \nmis\n-\ncut\n \nangle of the substrate\n7,11,20,21\n \ngenerally \nhide the \nbiaxial \n(\nfour\n-\nfold\n) \nmagnetocrystalline\n \nanisotropy \nof the LSMO film\n.\n \nIn the \nhigher thic\nkness regime, \nLSMO on STO (001) \nusually show a competition between (\nbiaxial\n) bulk\n12\n \nand shape or strain \n(uniaxial) magnetic \nanisotropy\n.\n22\n \nAnother single crystal oxide that can be used as \na \nsubstrate\n \nfor \nthe growth of \nLSMO \nfilms \nis MgO with \n(001) crystallographic orientation. However, \nthe \nlattice mismatch between LSMO and MgO (a\nMgO\n= 0.4212)\n \nresults in \nlarge\n \ntensile strain\n \n(\n~\n8%)\n.\n \n \nThis generally \ndegrades the film \nmagneto\n-\ntransport\n \nand morphological \nproperties \nthat \nlimited its use\n. \n \n \nOne of the utmost challenges is to control the magnetic anisotropy of the system via \nsubstrate\n-\ninduced\n \nstrain \nand surface engineering. \nI\nn this letter, \nto \ndefine\n \nmagnetic anisotropy \nsymmetry in\n \nLSMO thin films\n,\n \ni\nnstead of \ngrowing \nit \ndirectly \non \ncubic \nSTO \n(001)\n \ncrystals\n;\n \nwe \ndeposited \n50 nm \nLSMO \nfilm \non \ncubic \nMgO (001) substrate\n \nand employed \n12 nm \nSTO as a buffer layer\n. \nThe\n \ninclusion of 12 nm \nSTO\n \nbuffer layer \non MgO substrate not only helps in\n 3\n \n \nincorporating the \nstructural and interfacial \ndefects\n \nin \nit\n \nbut also minimizes the misfit strain and acts as a template for \ndepositing good quality epitaxial LSMO film\n. \nThe angular dependent magnetic anisotropy symmetry in our STO \nbuffered LSMO films shows dominant biaxial anisotropy\n \neven\n \nat RT, enhanced by an ord\ner of magnitude at 40 K.\n \n \nThe \nSTO (001) buffered \nLSMO thin films \nwe\nre\n \nepitaxially grown on MgO (001) substrate by pulsed laser \ndeposition (PLD) \nfrom \ncommercial \nstoichiometric targets \nby \nusing KrF excimer laser of wavelength 248 nm\n. High \nenergetic laser pul\nses (1.4 \n–\n \n1.7 J/cm\n2\n) were used for transferring correct elemental ratio of heavy elements for growing \nstoichiometric LSMO and STO films.\n23,24\n \nThe deposition was made at 0.35\n \nmbar oxygen \npressure\n \nwhile maintaining \nthe substrate temperature at 720\n \n°C. After deposition, the \nsamples \nwere cooled down to \nRT\n \nat 10\n \n°C per minute \nin \n7x10\n2\n \nmbar oxygen pressure. \nT\nhe thickness\nes\n \nof LSMO and STO layer\ns\n \nwere set at \n50 and 12 nm\n,\n \nrespectively. \nThe \nstructural\n, morphology, \nmagnetic and \nelectrical \ntransport measurements \nwere done by PANalytical X’Pert four\n-\ncircle \nX\n-\nRay Diffraction (XRD)\n \nin \nlow and high\n-\nresolution modes\n, Atomic Force Microscopy (AFM), \nSuperconducting \nQuantum Interference D\nevice (SQUID)\n, \nand four\n-\nprobe technique, \nrespectively\n. Angular dependent \ni\nn\n-\nplane \nm\nagnetization reversal process, \ncoercivity\n,\n \nand magnetic anisotropy measurements were performed at 300 K and 40 \nK by using vectorial Magneto\n-\nOptical Kerr (v\n-\nMOKE) magnetometry.\n25\n \n \nThe overall optimal structural \nand compositional qualit\nies\n \nof the film \nare\n \nconfirmed by the transport and \nmorphological properties. In fact, resistivity and magnetization vs. temperature measurements (see Fig\n.\n \nS1 in the \nSupporting Information) demonstrate low residual resistivity, Metal\n-\nInsulator transition temperature ‘T\nMI\n’ and Curie \ntemperature ‘T\nC\n’ close to the bulk values, i.e. >420 and 362 K, respectively. \nWhereas, t\nhe \nT\nMI\n \nof LSMO film\n, when\n \ndirectly grown\n \non MgO\n \n(001)\n \nshowed\n \nreduced \nvalue (T\nMI\n \n~325 K\n)\n \n(\nsee \nFig. S2\n \nSupporting Information\n)\n. \nTherefore, \nSTO buffer layer \nhelps to \nreduce structural defects in LSMO layer and \nimproves the \nfilm properties.\n \nThe film surface \nprobed by AFM reveals \nan RMS\n \n(root mean sq\nuare) roughness of about 0.35 nm (\nvery \nsmooth\n, \ni.e. \nless than one unit \ncell).\n \nThe \nepitaxial structure of the \nSTO and \nLSMO film\ns \ndemonstrated by the \nθ\n-\n2θ \nXRD spectrum \nthat \nshows only \n(00\nl\n) peaks, indicating the preferential c\n-\naxis orientation along the [00\nl\n] substrate crystallographic direction (Fig. \n1\n(a)). \nThe \ncalculated \n‘c’ \nlattice parameter\ns\n \nof \nSTO and \nLSMO films \nare 0.3903 and \n0.387\n3\n \nnm, \nrespectively, that indirectly \nindicates the optimal oxygen composition in the film.\n26\n \nIn order to determine t\nhe in\n-\nplane epitaxial relationship \nbetween the LSMO film and MgO (001) substrate\n,\n \nazimuthal\n \nϕ\n-\nscans performed around the (0\n-\n24) MgO and (0\n-\n13) \nLSMO asymmetric reflections, as shown in Fig. 1(b). \nThe \npeaks \nwith \nseparation of 90 degrees\n \nobserved for both MgO 4\n \n \nsubstrate and LSMO film demonstrat\nes\n \nthe \nfour\n-\nfold symmetry \nwith the [100] plane of the fi\nlm parallel to [100] plane \nof the substrate. \n \n \n \n \nFIG. 1: (a) Theta\n-\n2 theta XRD scan of 50 nm thick LSMO film on STO buffered MgO (001) substrate, (b) asymmetrical \nPhi scans around (0\n-\n13) peak of LSMO (001) film and (0\n-\n24) peak of MgO (001) substrate \nshows 90° separation cube\n-\non\n-\ncube epitaxy.\n \nIn order t\no investigate in details the LSMO film structure, we have performed XRD reciprocal space map (RSM) \naround the asymmetric (0\n-\n13) LSMO and (0\n-\n24) MgO reflections. \nInterestingly, in low\n-\nresolution mode\n \n(Fig\n. 2(a))\n, a \nsingle diffraction peak is evident thus inferring a cubic structure of the film (i.e.\n,\n \nidentical in\n-\nplane lattice parameters, \na = b\n). \nHowever, in \nhigh\n-\nresolution mode (i.e.\n,\n \nusing a Ge (220) double\n-\nbounce monochromator), \nthe \nRSM around\n \nthe\n \n(0\n-\n13\n) LSMO \ncrystallographic \nreflection\n \nshows \na double\n-\npeaks structure\n \nalong the Qx in\n-\nplane direction \n(\ninset of \nFig.\n \n2\n(a)\n)\n. Such a feature indicates the presence of two slightly \ndifferent in\n-\nplane lattice parameters (i.e. a \n≠\n \nb) thus \nsupporting \na\n \npossible \northorhombic\n \nor monoclinic\n \nstructure for the LSMO film.\n \nThe average in\n-\nplane lattice parameter \n5\n \n \nof LSMO is 0.388 nm, which is very close to the relaxed pseudocubic lattice value. \nFig. 2(b) shows the omega scans \nmeasured around the LSMO (002) peak at differe\nnt \nϕ\n \nangles (0°, 45° and 90°). At \nϕ\n \n= 0° and 90°, the presence of \ndouble\n-\npeak along the LSMO out\n-\nof\n-\nplane direction indicates, as \na \nfirst\n \napproximation, the presence of two LSMO \ndomains with the c\n-\naxis direction tilted of ~0.4° with respect to the [001]\n-\nMg\nO direction. However, rocking curve \ntaken at \nϕ\n \n= 45°, does not show any double\n-\npeak structure. \nNote that in phi\n-\nscan in Fig.1b, we do have a 6° broad \npeak, which might “cover\n-\nup” the two peaks expected for a rhombohedral structure, and the observed 0.4° sp\nlitting of \nthe rocking curves in\n \nFig.2b might be correlated to the actual “rhombohedral” arrangement of the LSMO unit cells. \nTherefore, from the phi scans (α=β=90°) and omega scans (γ≠90°), we deduce that \nthe most probable scenario is based \non a 4\n-\nmonoclin\nic domain structure\n.\n \n \n \n \nFIG.\n \n2\n: (Color online) (a) Low\n-\nresolution RSM scans of (0\n-\n13) LSMO and (0\n-\n24) MgO peaks shows that the LSMO film is \nfully relaxed. \nThe inset\n \npresents the high\n-\nresolution RSM scan of (0\n-\n13) LSMO that shows two peaks, marked with arrows.\n \n(b)\n \nOmega scans\n \nmeasured\n \naround LSMO (002) \npeak \nat \ndifferent \nphi \nangles i.e.,\n \n0\n°\n, 45\n°\n \nand 90\n°.\n \n \n6\n \n \nThe magnetic \nanisotropy \nof the \nfilm \nalso \nreveals a four\n-\nfold \nsymmetry. This w\nas \ninvestigated\n \nat RT in details \nby \nacquiring\n \ni\nn\n-\nplane\n \nKerr hysteresis loops \nby \nv\n-\nMOKE \nin\n \nthe full angular range\n \n(i.e., \n0°\n-\n360°) \nwhile \nkeeping fixed \nthe external magnetic field direction. At\n \nθ=0°\n, the applied external magnetic field is alig\nned parallel to the [100] \ncrystallographic axis of MgO (001) substrate. \nBy exploiting the \nsimultaneous acquisition of two in\n-\nplane \nmagnetization \ncomponents\n,\n \ni.e., parallel \n(M\n||\n(H))\n \nand\n \nperpendicular\n \n(M\n\n(H)), we \ncan \ndeduce the \nsymmetry \nof\n \nmagnetic \nanisotropy \npresent in \nthe film.\n7\n,\n11\n,\n27\n \nIn fact, by \ninspecting the cha\nnge of sign in the\n \nM\n\n(H),\n \nwe \ncan accurately locate \nthe easy \naxis (e.a.) \nand hard ax\ni\ns\n \n(h.a.)\n \ndirections.\n \nFig. 3 \npresents \nthe normalized Kerr hysteresis loops\n \nof the LSMO \n(001) film \nacquired at\n \nand around \ne.a. and h.a. i.e., \nθ=45°, 90°,\n \nand \n±9°\n.\n \nSquare loop with sharp irreversible transitions \nin the \nM\n||\n(H) \nis found at \nθ=45°, \nwhereas the M\n\n(H) \ncomponent\n \nis almost zero\n.\n \nThis indicates that \nthe magnetization is \naligned \nin the film plane \neither parallel or anti\n-\nparallel\n,\n \nwhich corresponds to \na \nmagneti\nzation reversal proceeding by \nnucleation and \nfurther \npropagation of magnetic domains (\ncharacteristic of an \ne.a. \nbehavior\n). When the field is applied \naway from \nthe e\n.a., smoother \nreversible \ntransition\ns\n \nare found indicating that the reversal starts by rotation followed by \npropagation of domains. In particular, \nwhen \nthe magnetic field is applied \naround the e.a., i.e.\n \nfrom \nθ=36°\n \nto \n54°\n, \nthe \nmagnetization switches with\n \njust \none irreversible transition\n \nand fr\nom\n \nthe\n \nM\n\n(H)\n \ncomponent\n,\n \nwe could observe that \nthere is a\n \nsignificant change of sign\n. This \nmeans that the magnetization rotates within the film plane in\n \nclockwise to \nanticlockwise\n \ndirection \nor vice\n-\nversa\n \nwith respect to the applied field angle. \nThe \nremanence of\n \nM\n||\n(H)\n \nat\n \nthe \ne\n.a\n.\n \nis \nM\nR,||\n\nM\nS\n \nand the coercive field is about 1.8 mT\n.\n \n \nT\nhe M(H) loops taken at \nθ=81°\n \nand \nθ=99°\n \nshows that the\n \nmagnetization reversal takes place in \nthree \nsteps, \nwhich are marked \nwith arrows in \nM\n||\n(H)\n \nand circles\n \nin\n \nM\n\n(H)\n \n(\nFig.\n \n3\n \n(top right))\n, respectively. \nSweeping the filed \nfrom positive to negative values, the reversal occurs by a first reversible magnetization rotation, followed by \ntwo \nirreversible transitions \nthat \ntake\n \nplace by nucleation and propagation of two consecutive \n90° domain walls\n.\n28\n \nA\nt \nh.a.\n \ni.e.,\n \nθ\n=90°,\n \nas the field strength decreases\n, \nthe\n \nM\n||\n(H) l\noop show rotation of the magnetization followed by sharp \nirreversible transition and final rotation towards the applied field direction. \nThe \nreversal proceeds\n \nthus\n \nwith one (two) \nirreversible transition, related to nucleation and propagation \nof 180° (90°) domain walls, when the field is applied \nclose to one of the two \ne.a. \n(\nh.a.\n) orientations of magnetization.\n \nThese are the typical signatures of a four\n-\nfold magnetic \nsymmetry.\n27\n \nThe \nexperimental \ndata have been properly reproduced in the whole angular range with a modified \ncoherent rotation Stoner\n-\nWohlfarth model\n11,27\n \nby using \nexclusively \na single magnetic ani\nsotropy term with four\n-\nfold 7\n \n \nsymmetry \nwhose strength was extracted \nfrom\n \nthe experimental \ncurves at the hard axis (H\nK\nb\n=\n5 \nmT). \nTo note\n,\n \nsince \nthis \nmodel is based on coherent rotation, it fails in reproducing the experimental data in the regions in which nucle\nation \nand propagation of domains mechanisms dominate\n \n(i.e., close \nto e\n.a.)\n. However, it provides a qualitative \nand \nquantitative estimation of \nthe relevant anisotropy contributions involved\n \n(at the h.a. region)\n.\n \n \n \nFIG.\n \n3\n: (Color online) Normalized magnetic\n \nhysteresis cycles of 50 nm thick LSMO  film grown on STO buffered MgO \n(001) measured by v\n-\nMOKE at 300 K at and around easy (e.a.) (left) and hard (h.a.) (right) axis. The corresponding applied \nexternal magnetic field angles (θ) with respect to\n \nthe\n \ncrystal\nlographic\n \naxis are\n \nspecified in the figures. The \nM\n||\n(H)\n \n(black) \nand \nM\n\n(H) \n(red) loops acquired simultaneously \nare\n \nshown. The arrow (circles) in the top right panel indicates the double \ntransition, which is the signature of biaxial anisotropy.\n \nThe cyan and \ngreen solid lines correspond to simulation results of \nStoner\n-\nWohlfarth \nmodel.\n \nT\nhe\n \nfour\n-\nfold \nsymmetry of the \nmagnetic anisotropy becomes more evident \nby plotting the \nangular dependent\n \nremanence \nand critical fields \nin the \nfull angular \nrange\n \n(\nFig. 4\n)\n.\n \nThe normalized remanence \nmagnetization \nplots \n(\nM\nR,||\n/M\nS \nand\n \nM\nR\n,\n\n/M\nS\n) \nextracted \nfrom\n \nthe experimental\n \nM\n||\n(H) \nand\n \nM\n\n(H) \nloops\n \na\nt the applied field \nµ\no\nH=0\n \nas a \nfunction of angle ‘θ’ \nare \nshown in\n \nFig. 4(a). \n \nBoth \nthe magnetization components show repeating\n \nfeatures with the \nperiodicity of 90°. \nIn addition\n,\n \nM\nR\n,\n\n \nchanges its sign for every 45° \ni.e., \nwhen\never\n \nit crosses the characteristic axes. \nThe \npolar plot of \nM\nR,\n||\n/M\nS,\n \nand\n \nM\nR\n,\n\n/M\nS\n \nare presented in panels c\n-\nd of Fig. 4 revealing the four\n-\nfold symmetry. In particular, \n8\n \n \nthe \nM\nR,||\n/M\nS\n \nresembles a butterfly structure with the highest and lowest values pointing towards e.a. and h.a. of film, \ni.e., \nalong\n \nwith\n \n[110] and the [010] crystallogr\naphic directions, whereas the \nM\nR\n,\n\n/M\nS\n \nshows \nfour lobe\n \nsymmetrical \nshape with positive and negative values are depicted in solid and open circles.\n \n \nFig. 4\n(b)\n \nshows the angular dependence of the critical fields \ni.e., coercive\n \n(\nH\nC\n) and switching (\nH\nS\n) \nfields\n,\n \nextracted \nfrom\n \nM\n||\n(H) \nand\n \nM\n\n(H) \nloops\n. \nThe \nvalue of \n‘\nH\nC\n’ (H\nS\n)\n \nis higher \n(lower) \nat \ne\n.a.\n \ni.e.,\n \n[\n110\n]\n \nand \ndecreases\n \n(increases) \nas it\n \napproaches towards \nh.a.\n \ni.e.,\n \n[01\n0]\n. \nThe H\nC\n \nand H\nS\n \ncoincide at and around \ne.a. \ncorresponds to \none\n \nirreversible transition \n(grey shaded area) \nleading to 180° domain walls\n \nreversals\n. \nAs \nthe\n \nfield is applied\n \naway from \nthe \ne\n.a.\n \n(\ni.e., approaching the h.a., \nwhite region\n)\n, \nH\nS\n \nincreases\n \nand reaches the maximum \nexactly \nat the \nh.a.\n \nIn \nthe white \nshaded \nregion\ns\n, the magnetization reversal takes place with two irreversible transitions \nthat\n \nrelate\n \nto \nthe \nnucleation \nand propagation of two consecutive 90° domain walls. The polar plots of the H\nC\n \nand H\nS\n \nare presented in \nFig. 4\n(e,\n \nf)\n \nshow \nsymmetrical four \nlobe\ns\n \nand asteroid shape\n, respectively,\n \nwith 90\n° \nperiodicity\n. \nFrom \nth\nese\n \nangular dependent \nanalys\ne\ns\n, it is evident that\n \nt\nhe \nangles between two adjacent \ne.a. and h.a. \nare orthogonal to each other\n \nand the minima \nof \n \nM\nR,\n||\n  \nat the consecutive h.a. \nhave \nidentical v\nalues\n, which experimentally prove that no uniaxial magnetic anisotropy \ncontribution \nexists\n.\n27\n \nThese features \ndemonstrate \nthe four\n-\nfold symmetry of the magnetic anisotropy in the \nLSMO/STO/MgO\n \n(001) s\ntructure, which is \nsimilar\n \nto the \nmagnetocrystalline\n \nanisotropy of the \nbulk \nLSMO \nwith easy \naxes aligned \ntowards 45°\n \n[\n110\n]\n.\n29\n \nThe analysis of both magnetization components allows \ntherefore \nfor disclosing with \nhigh accuracy the symmetry of the magn\netic anisotrop\ny. \nTo note that\n,\n \neven \nthough \nthe authors in the Ref.\n12\n \nclaim to \nhave biaxial anisotropy\n \nin LSMO/STO film\n, the difference between the magnetization remanence states at two \nconsecutive h.a\n. reveals an additional uniaxial contribution, as explained \nby the authors \nin \nthe Ref.\n \n19\n \nand\n \n27\n.\n 9\n \n \n \nFI\nG. 4\n: (Color online) Angular evolution at 300 K of the magnetic properties of 50 nm thick LSMO film grown on STO \nbuffered MgO (001) substrate. (a) Normalized remanence magnetization \nM\nR,||\n/M\ns\n \n(black) and M\nR,\n\n/M\ns\n \n(red), \n \n(b) critical \nfields (coercivity ‘H\nC\n’ (blu\ne) and switching ‘H\nS\n’ (magenta)) as a function of the \napplied\n \nfield angle ‘\n\n’ shows well\n-\ndefined \n90° periodicity i.e., a pure bi\n-\naxial anisotropy. The grey shaded regions in (b) indicate the system exhibits only one \nirreversible transition, whereas, in the\n \nwhite regions, the system exhibits two consecutive irreversible transitions. (c\n-\nf) Polar \nplots of \nM\nR,\n||\n/M\ns,\n \nM\nR,\n\n/M\ns,\n \nH\nC\n \nand H\nS\n \nrespectively.\n \nPositive and negative values in (d) \nrepresent\n \nby\n \nsolid and open circles.\n \n \nIn order to \ndisclose \nthe \nnature of the four\n-\nfold magnetic anisotropy in our systems, we performed \ntemperature \ndependent studies. \nT\nhe normalized Kerr hysteresis loops \n(Fig. 5(a)\n) \nmeasured \nat 40 K \nat different in\n-\nplane \nmagnetic \nfield \ndirections\n \npresent similar features and symmetry \nwith respect \nto the RT case\n.\n \nT\nhe \ne\n.a. and h.a. are\n \npresent at 45\n° \n[\n110\n]\n \nand 90° \n[\n01\n0\n]\n. \nNear\n \nh.a.\n \ni.e., at \nθ\n=70°\n \nand 80°, two irreversible transitions are observed owing to nucleation and \npropagation of two consecutive 90° domain walls. \nIn this case, the data have been reproduced by using again a single \nmagnetic anisotropy term with four\n-\nfold anisotropy and \nthe \nanisotropy field \nis \none order of magnitude larger than the \none found at RT (H\nK\nb\n=40 mT). \nT\nemperature\n-\ndependent coercive fields \n(Fi\ng. \n5(b)\n) \nin the angular range of \n0°\n-\n180°\n \nhas \n90\n° periodicity \nas the RT \nbehavior\n \nbut \nwith\n \na tremendous increase \nof the \ncoercive fields of about \none order of \nmagnitud\ne\n.\n \nTherefore, a\ns the temperature decreases from 300 K to 40 K, the \nsignatures \nof biaxial anisotropy \nbecome \nmore evident\n. The cause of anisotropy is due to magnetocrystalline anisotropy of LSMO, which is usually dominant \nat low temperatures\n.\n30\n \n \n10\n \n \n \n \nFIG. 5: (Color online) (a) Normalized magnetic hysteresis loops of 50 nm thick LSMO film grown on STO buffered MgO \n(001) around the first half quadrant measured at 40 K, and (b) \nAngular dependent coercive field ‘H\nC\n’ at 40 and 300 K. The \ncyan and green soli\nd lines in (a) corresponds to simulation results of \nStoner\n-\nWohlfarth\n \nmodel.  \n \n \nIn summary, we have fabricated \nhigh\n-\nquality\n \nepitaxial LSMO thin films on STO buffered MgO (001) \nsubstrate by PLD technique and studied \nin\n-\nplane \nmagnetic anisotropy properties at\n \n300 and 40 K. We demonstrated \nthat the use of STO buffer layer improved the quality of LSMO film by accommodating structural defects. The LSMO \nfilm \nha\ns\n \na \n4\n-\ndomain monoclinic structure with domains present along [100] and [010] directions. The magnetic \nanisotropy \nalso showed biaxial anisotropy with the e.a. \ndirection\n \nis aligned towards 45\n°\n \nor [110], where the tilted \n11\n \n \ndomains are absent and the h.a. is aligned\n \nalong \nthe tilted monoclinic domains i.e., \n0\n°\n \nor [100]\n.\n \nAs temperature \ndecreases, the strength of the biaxial anisotropy increases, being one order of magnitude larger at 40 K, revealing its \nmagnetocrystalline origin. \nUnlike the LSMO films directly grown o\nn STO (001) substrates that show anomalies in \nmagnetic anisotropy, films that were grown \nonto\n \nSTO buffered MgO (001) substrates displays well\n-\ndefined biaxial \nanisotropy, that can be useful in four bits logic devices based on purely anisotropic magnetoresis\ntance \nresponse\n.\n \n \nSee supplementary \ninformation \nfor morphology, magnetic and electrical transport properties of LSMO/STO/MgO\n \n(001)\n \nfilm.\n \nSKC \nacknowledge\n \nProgramme International de Coopération Scientifique (PICS) du CNRS under grant no. \n6161 and SIMEM Doctor\nal School\n \nat Universite de Caen Normandie\n \nfor financial support. IMDEA\n-\nNanociencia \nacknowledges support from the ‘Severo Ochoa’ Program for Centres of Excellence in R&D (MINECO, Grant SEV\n-\n2016\n-\n0686). FA, \nJC,\n \nand PP acknowledge the support of Spanish MINECO\n \nProjects No. FIS2015\n-\n67287\n-\nP and \nFIS2016\n-\n78591\n-\nC3\n-\n1\n-\nR, and from the Comunidad de Madrid through Project N\nANOFRONTMAG CM. This project \nhas\n \nreceived funding from the European Union’s Horizon 2020 research and innovation programme under grant \nagreements No. \n737116 \n(byAxon) and N\no. 654360 NFFA\n-\nEurope.\n \n \n \n 12\n \n \n1\n \nS.S.P. Parkin, M. Hayashi,\n \nand L. Thomas, Science\n \n320\n, 190 (2008).\n \n2\n \nJ. Heber, Nature \n459\n, 28 (2009).\n \n3\n \nE. Dagotto, Science\n \n309\n, 257 (2005).\n \n4\n \nF.A. Cuellar, Y.H. Liu, J. Salafranca, N. Nemes, E. Iborra, G. Sanchez\n-\nSantolino, M. Varela, \nM.G. Hernandez, J.W. Freeland, M. Zhernenkov, M.R. Fitzsimmons, S. Okamoto, S.J. Pennycook, \nM. Bibes, A. 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We show that multiple nodal lines coexist in momentum space,\naccompanied by the electron and hole pockets that touch each other linearly at the nodal lines.\nInclusion of spin-orbit coupling enriches the magnetic and electronic properties of h-InC. Spin-orbit\ncoupling leads to an easy-plane type magnetocrystalline anisotropy, and the nodal lines can be tuned\ninto topological nodal points, contingent upon the magnetization direction. Symmetry analysis and a\ntight-binding model are provided to explain the nodal structure of the bands. Our \fndings suggest\nh-InC as a new venue for supporting carbon-based magnetism and exotic band topology in two\ndimensions.\nI. INTRODUCTION\nA striking consequence of symmetry and topology in\nthe electronic structure of materials is the occurrence of\na symmetry-protected topological state, such as topo-\nlogical insulators1,2. Notably, the protected degener-\ncies of electronic energy bands by nontrivial band topol-\nogy are of particular interest, leading to the discovery\nof condensed-matter realizations of fundamental parti-\ncles, such as Majorana fermions3{5, Weyl fermions6,7,\nand Dirac fermions8,9. Topological materials can also\nrealize more exotic fermions beyond the conventional\nform of elementary particles10{12, such as nodal line\nsemimetals13{15. Nodal line semimetals are a class of\ntopological semimetals that are characterized by hosting\none-dimensional lines of nodes formed from the conduc-\ntion and valance bands. There are multiple mechanisms\nthat topologically protect nodal lines associated with dif-\nferent symmetries, including inversion14,16, mirror17, and\nnonsymmorphic glide-mirror18,19symmetries. More re-\ncently, topological semimetals without time-reversal sym-\nmetry hosted in magnetic materials have spurred a deal\nof community interest in \fnding exotic phenomena, such\nas tunable nodal points20, enhanced anomalous quantum\nHall e\u000bect21,22.\nOn the other hand, the recent development of synthe-\nsis and characterization techniques for two-dimensional\n(2D) materials23{30has led to the exciting discovery of a\nroom-temperature ferromagnet in two dimensions31{34.\nWhereas magnetism is one of the oldest phenomena\nobserved in materials, its realization in a 2D system\nhas remained elusive; it is only recently that convinc-\ning evidence of ferromagnetism has been observed in\n2D materials, which include thin \flms of chromium ger-\nmanium telluride Cr 2Ge2Te635and chromium triiodide\nCrI336. The discovery of these 2D magnetic materials\nhas motivated numerous experimental and theoretical\nstudies37{42, which have potential implications for future\na1\na2(a) (b)\nC Inb1\nb2ΓK\nM3.79 Å\nxy\nzkxky\nkzFIG. 1. (a) Atomic structure of h-InC in a planar 2D hon-\neycomb lattice. The primitive unit cell is indicated by black\ndashed lines. (b) High-symmetry kpoints in the 2D hexago-\nnal Brillouin zone (BZ).\nspintronic device applications43{46and fundamental sci-\nence47{58.\nApart from these e\u000borts, there have been ongoing stud-\nies regarding metal-free magnetism using the localize pz\norbitals of carbon allotropes. According to Lieb's theo-\nrem59, the ground electronic state of a bipartite lattice\nis accompanied by the electron spin of s=/divides.alt0NA−NB/divides.alt0/slash.left2,\nwhereNAandNBare the number of sites in the Aand\nBsublattices, respectively. The crucial requirement for a\ncarbon-based bipartite lattice to induce ferromagnetism\nis thus to unbalance the number of the pzorbitals be-\ntween the two sublattices. This condition enforces two\nelectrons to occupy a single \u0019-state near the Fermi level,\npaying an extra Coulomb repulsion energy U, and ne-\ncessitating spin polarization depending on the strength\nofUwith respect to the hopping strength tof the elec-\ntrons59{61. Considered as possible realizations of Lieb's\ncondition, diverse carbon allotropes have been investi-\ngated based on a variety of schemes62{87.\nIn this paper we employ the design principle for the bi-\npartite magnetism dictated from Lieb's theorem to \fndarXiv:1810.08563v2  [cond-mat.mtrl-sci]  25 Jun 20192\na new ferromagnetic nodal-line material in two dimen-\nsions. We suggest that a single layer of hexagonal in-\ndium carbide ( h-InC), shown in Fig. 1, should ful\fll the\nLieb's condition. Using \frst-principles calculations, we\nshow that h-InC is a stable ferromagnetic metal, which\nhosts symmetry-protected twofold-degenerate nodal lines\nin momentum space, referred to as Weyl nodal lines.\nBased on the ferromagnetic phase of h-InC, we pro-\nvide \frst-principles calculations that establish h-InC as\na promising platform for realizing exotic fermiology and\npotential band topology. In the absence of spin-orbit\ncoupling (SOC), the perfectly planar structure of h-InC\nallows for the existence of Weyl nodal lines protected\nby mirror symmetry. We \fnd that di\u000berent types of\nnodal lines, respectively referred to as type-I and type-\nII nodal lines10,88{91, coexist near the Fermi level. The\nWeyl nodal lines are accompanied with the electron and\nhole pockets that touch each other linearly at the nodal\nline. SOC can gap out the Weyl nodal line, or tune to a\npair of nodal points, depending on magnetization orien-\ntation. The nodal points are topological, characterized\nby the\u0019Berry phase. Considered as a 2D descendant\nof a three-dimensional Weyl node, which is described by\na massless two-band Weyl-like Hamiltonian, the nodal\npoint is often referred to as a 2D Weyl node. We pro-\nvide symmetry analysis that o\u000bers the rationales for the\npresence and protection of the nodal structure, which is\nfurther supported by our a tight-binding theory. The\nnew insight from this work could lead to the discovery of\na new 2D material that allows for tunable band topology\nvia the magnetization.\nII. METHODS\nIn the present study, we performed \frst-principles cal-\nculations based on density functional theory (DFT) as\nimplemented in the Vienna ab initio simulation package\n(VASP )92within the projector-augmented-plane-wave\n(PAW) method93,94. The exchange-correlation energy\nwas calculated within the Perdew-Burke-Ernzerholf-type\ngeneralized gradient approximation95. The 48 ×48 of\nk-grid were sampled from the \frst Brillouin zone (BZ)\nusing the gamma-centered Monkhorst-Pack scheme96.\nThe plane wave basis was constructed within the en-\nergy cuto\u000b of 550 eV. The two-dimensional (2D) crystal\nstructure was approximated by placing a vacuum space\nof 20 \u0017A along the out-of-plane z-direction. The con-\nvergence thresholds for the energy and the Hellmann-\nFeynman force were set to 10−6eV and 0:01 eV/ \u0017A, re-\nspectively. The phonon calculations were carried out\nby using the Phonopy package97. The spin-polarized\n(spin-unpolarized) phononic energy spectra were calcu-\nlated within the force criterion of 10−5eV/\u0017A using a\n4×4×1 (6×6×1) supercell and the 6 ×6×1 Monkhorst-Pack\nsampling of k-points. The cohesive energy of h-InC\nwas calculated using Ecoh=(EInC−EC−EIn)/slash.leftNatoms ,\nwhereEInC,ECandEInare the total energies of h-InC,\n(b)\nΓ K M ΓFrequency [THz]020\n10Non-magnetic Ferromagnetic(a)\n-6\n-7\n-8Non-magnetic\nFerromagnetic\nΔE = 69 meVEnergy [eV]\n3.4 3.6 3.8 4.0 4.2 4.43.7 3.74 3.78 3.82 3.86 3.9-7.92\n-7.96\n-8.00\n-8.04\nLattice constant  [Å]\nΓ K M ΓFIG. 2. (a) Total energy of h-InC evaluated as a function of\nthe unit cell area. (Inset) Magni\fed view of the energy curves\nnear the energy minima, highlighted by a green box. (b)\nPhononic energy spectra calculated under the ferromagnetic\n(left) and non-magnetic (right) spin con\fgurations.\nC, and In, respectively. Natoms =2 is the number of\natoms per unit cell. The magnetic anisotropy energy\nEMAE ofh-InC was calculated based on EMAE (\u0012;\u001e)=\n~E(\u0012;\u001e)−~E(0;\u0019/slash.left2), where ~E(\u0012;\u001e)is the total energy\nevaluated by constraining the direction of the magnetic\nmoments along ^n=(sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012)within\nthe magnetic force theorem method98,99implemented in\nVASP100. To topologically characterize the 2D Weyl\npoints, the Berry phases were calculated via the wannier-\nization of DFT bands using the Wannier90101code.\nIII. RESULTS AND DISCUSSION\nA. Atomic structure and structural stability of\nh-InC\nWe begin our study by delineating the crystal structure\nofh-InC. Geometry optimization leads to the planar 2D\nhoneycomb structure shown in Fig. 1(a) with the primi-\ntive unit cell comprising one InC formula unit. The space\ngroup of h-InC isP6m2 (# 187). The generating point\ngroup ofP6m2 is isomorphic to D1\n3h, which contains mir-\nror symmetry Mz(My) about the x-y(x-z) plane. As\nshown in detail later, Mz(My) plays an important role to\nprotect nodal lines (nodal points) of the electronic energy\nbands. The lattice constant is calculated as a=/divides.alt0a1/divides.alt0=/divides.alt0a2/divides.alt0\n= 3.79 \u0017A. Note that the nearest-neighboring C atoms are\nseparated by 3.79 \u0017A, which is much greater than the case3\nof graphene (1.42 \u0017A). This large spacing between carbons\nhelps electrons localized at the C pzorbitals, thereby pro-\nmoting spin-polarization.\nWe expect that the h-InC lattice is thermodynami-\ncally and kinetically stable. The reason for this follows.\nFirst, the cohesive energy of h-InC is calculated as -\n3.91 eV per atom. This value is comparable to those\nof known 2D materials, such as silicene (-3.96 eV), ger-\nmanene (-2.6 eV)102, as well as existing monolayer tran-\nsition metal dichalcogenides, including MoTe 2and WSe 2\n(-4∼-6 eV)103{107. Moreover, the h-InC lattice forms a\nlocal minimum in atomic con\fguration space as shown in\nthe total energy curve in Fig. 4(a), which supports that\nh-InC is thermodynamically stable. In addition to the\nthermodynamic stability, the kinetic stability of h-InC\nis also expected from our phononic energy calculations.\nFigure 2(b) shows the phononic energy spectra calculated\nwith and without spin-polarization. Both results produce\nno negative mode, irrespective of spin-polarization, which\nestablish the structural stability of h-InC.\nB. Magnetism of h-InC\nHaving discussed the atomic structure of h-InC, we\nnow turn to its magnetic properties. We \frst evidence\nthat h-InC has the ferromagnetic ground state. After\nmany trials to minimize the total energy of the system\nwith di\u000berent spin con\fgurations, including the coplanar\n120○antiferromagnetic con\fguration shown in Fig. 3(b),\nwe \fnd that the ferromagnetic spin con\fguration in\nFig. 3(a) results in the lowest total energy. In detail, the\ntotal energy per unit cell of the ferromagnetic state is cal-\nculated as 69 meV lower per carbon atom than the non-\nmagnetic state [See Fig. 2(a)]. With respect to the copla-\nnar 120○antiferromagnetic spin con\fguration shown in\nthe right panel of Fig. 3(a), the energy of the ferromag-\nnetic state is 34 meV lower per carbon atom. As we\nwill show below, spins are localized at carbon atoms that\nform a triangular sublattice, in which geometric frustra-\ntion forbids the colinear 180○antiferromagnetic spin con-\n\fguration to become a ground spin state.\nWe explain the ferromagnetic ground state of h-InC as\nbeing due to the unbalanced pzelectrons. As an element\nin group XIII (main group III) of the periodic table, In\nprovides one less valence electrons than C per unit cell.\nIn this respect, h-InC can be viewed as In-substituted\ngraphene, in which one sublattice of graphene is substi-\ntuted with In atoms. The In-substitution plays a role to\nremove half of the pzelectrons of graphene, which neces-\nsitates the spin-polarization ful\flling Lieb's magnetiza-\ntion condition. Indeed, the net spin moment of the fer-\nromagnetic ground state is calculated as 0.8 \u0016Bper unit\ncell, which is close to 1 \u0016Bthat we expected from Lieb's\ntheorem. Moreover, the magnetization density, shown\nin Fig. 3(c), shows that a large amount of spins are lo-\ncalized at the carbon sites, which indicate that carbon\natoms mainly contribute to the spin moment. The cross-sectional views of the magnetization density in Fig. 3(d)\nfurther reveal that the magnetization density originates\nfrom the electrons in the carbon pzorbitals, featuring\nthepzorbital characters. Therefore, carbon-based mag-\nnetism is considered to be at the heart of the magnetism\nofh-InC.\nIn the presence of SOC, h-InC exhibits a weak mag-\nnetocrystalline anisotropy of an easy-plane type. Fig-\nure 3(e) shows the magnetic anisotropy energy EMAE of\nh-InC evaluated as a function of the azimuth angle \u0012mea-\nsured from the out-of-plane axis and the horizontal az-\nimuth angle \u001emeasured on the basal plane [See inset of\nFig. 3(f)]. The result shows that EMAE has the minimum\nat\u0012=\u0019/slash.left2 with a negligible dependence on \u001e, thus lead-\ning to the formation of (0001) easy plane and in-plane\nisotropic. The (out-of-plane) hard axis energy is amount\nto 0.4 meV, as shown in Fig. 3(f). The in-plane isotropy\ncould make the ferromagnetic phase susceptible to ther-\nmal \ructuation108and correspondingly di\u000ecult to realize\na 2D magnet at \fnite temperature using h-InC.\nC. Electronic structure of h-InC\n1. Electronic band structure and PDOS without SOC\nHaving predicted the existence of a new 2D magnetic\nmaterial, we turn to the characterization of its electronic\nstructure. It is readily noticed that h-InC is a spin-\npolarized metal from its electronic energy band struc-\nture presented in Fig. 4(a). The Fermi level ( E=0) in-\ntersects with the spin-polarized energy bands. Notably,\nthe largest spin-splitting occurs between the C pzbands\nthat correspond to the energy bands containing the two\nhighest occupied energy levels at the \u0000 point, depicted\nby dashed lines in Figs. 4(a) and 4(b). The projected\ndensity of states (PDOS) in the right panel of Fig. 4(a)\nfurther reveal that these energy bands mainly consist of\nthe Cpzorbitals, giving rise to the strong PDOS peaks\natE=−0:430 eV from the spin-up band and at E=1:161\neV from the spin-down band. As expected, these C pz\nbands have relatively a narrow bandwidth within ∼1.8\neV, re\recting the hindered electron hopping between the\nCpzorbitals due to the interstitial In atoms.\nThe PDOS further reveal that all the other energy\nbands except the C pzbands near the Fermi level consist\nof thes,px, andpyorbitals from both C and In atoms.\nIn detail, the bonding states of the sorbitals are located\ndeep inside the occupied region of the energy bands near\nE=-10 eV, whereas the anti-bonding states of the s-\norbitals are placed above the Fermi level near E=1.43\neV and 1.9 eV at the \u0000 point. In addition, the strong\nPDOS peaks appear right below the Fermi level, con-\ntributed from the pxandpyorbitals of both C and In as\nwell as with a sizable contribution from the In and C s\norbitals. This indicates that \u001bbonds are formed between\nthesp2-type orbitals of In and C. Hereafter we refer to\nthese bands as sp2 bands. Note that the sp2 (pz) bands4\nxy\nzxy\nzxy\nz(a) (b)   (c)   \n(f)\nxyz\nΘ\nφ \n014\n3\n2\n0 60 120 180 240 300 3600 30 60 90\nφ [degree]Θ [degree]EMAE [10-4 eV](e)\nxyz\n5\n4\n3\n2\n1\n0EMAE [10-4 eV] Θ ( φ = 0 )\nφ ( Θ = 90)(d)\n0310-3 e/ Å3\n10-410-3\n10-2\n10-310-4\nS2zS1zFM (ground state)\nS2S1\nFIG. 3. (a) Ferromagnetic spin con\fguration aligned along the in-plane y-direction. The corresponding unit cells are represented\nby green-dashed rhombuses. (b) Coplanar 120○spin con\fguration. (c) Magnetization density isosurface at n↑(r)−n↓(r)=\n0:001e/\u0017A3. (d) Cross-sectional views of the magnetization density plotted along the S1 (top panel) and S2 (bottom panel)\ngreen-dashed lines introduced in (c). (e) Uniaxial magnetocrystalline anisotropic energy EMAE drawn in polar coordinates\n(r;\u0012;\u001e ), wherer=EMAE(\u0012;\u001e). (f) Uniaxial magnetocrystalline anisotropic energy curves EMAE=EMAE(\u0012;\u001e)evaluated as a\nfunction of \u0012at\u001e=0 (blue) and as a function of \u001eat\u0012=\u0019/slash.left2 (green), respectively. Inset illustrates the parametrization of spin\norientation in terms of the azimuth angle \u0012measured from the out-of-plane axis and the horizontal azimuth angle \u001emeasured\non the basal plane. The red arrow represents spin orientation ^n=(sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012)in real space.\nare symmetric (anti-symmetric) under the mirror opera-\ntionMz, as they are formed from the s,px, andpy(pz)\norbitals.\nNode (mz;sz)(mz;sz)nodal line\nA (+,−)(+,+)Type-II\nB (+,−)(−,−)Type-I\nC (+,+)(−,+)Type-I\nD (+,−)(−,−)Type-I\nE (+,+)(−,+)Type-I\nF (+,+)(−,−)Type-I\nG (+,−)(−,+)Type-I\nH (+,+)(−,−)Type-I\nI(+,−)(−,+)Type-I\nTABLE I. Mirror ( mz=±1) and spin ( sz=±1) eigenvalues for\nthe energy bands without SOC that form the corresponding\nnodes de\fned in Fig. 4(b). The second (third) column lists\nthe symmetry eigenvalues for the bands that are located on\nthe left upper (lower) side of the node in Fig. 4(b). The type-I\nand type-II classi\fcation of the Weyl nodal lines is listed in\nthe fourth column.\nThe energy bands with di\u000berent mirror eigenvalues\n(mz=±1) can cross each other without opening a band\ngap as shown in Figs. 4(a) and 4(b). A mirror +band\n(mz=+1, solid line) intersects with a mirror −band\n(mz=−1, dashed line). Similarly, the energy bands withdi\u000berent spin states ( sz=±1) can also overlap each other\nwithout opening a band gap; a spin-up band ( sz=+1, col-\nored by red) crosses a spin-down band ( sz=−1, colored\nby blue). Thereby, the mirror and spin-rotational SU(2)\nsymmetries independently protect the nodal structure of\nthe energy bands, and all the band crossings in Fig. 4(c)\nare explained based on the mirror and spin eigenval-\nues. For example, protected by mirror symmetry Mz, the\nband crossings at B, C, D, and E in Fig. 4(c) are formed\nfrom the energy bands with the same spin state. Table I\nlists the mirror and spin eigenvalues for the bands that\nform the nodal points that are introduced in Fig. 4(b).\n2. Weyl nodal lines and Fermi surface topology\nBeing preserved independently without SOC, the mir-\nrorMzand spin-rotational SU(2)symmetries are global\nsymmetries in momentum space in the sense that they\nare respected at any kpoint of the entire BZ. Therefore,\nthe occurrence of band crossings that we found from the\nband structure signals the existence of one-dimensional\nnodal lines in the 2D BZ. A close inspection has found\nmultiple nodal lines, such as shown in Fig. 4(d). No-\ntably, the nodal line that contains the band crossing A\nin Fig. 4(c) is of a unique type, referred to as a type-II\nnodal line formed from the energy bands that have the5\n-1Energy [eV](b)1\n0\n-2A\nB\nD\nECGF\nH\nIC↑ C↓ In↑ In↓\nPDOS [a.u] Γ K M\nMΓ\nΓ4\n2\n0\n-2\n-4Energy [eV]\n(c)(a)\nTotal \nDOS \ns\npx  , py\npz\n0.4\n0.2\n0\n-0.2[eV]\n-0.4\nkxkyE\nType-II\nType-IType-II\nType-I\nFIG. 4. (a) Electronic energy band structure and projected density of states (PDOS) of h-InC. (Left panel) Spin-polarized\nelectronic energy band structure. The Fermi level is set to zero. The energy bands are calculated without SOC along the\nhigh-symmetry k-points of the hexagonal BZ shown in Fig. 1(b). The spin-up (majority spin) and spin-down (minority spin)\nbands are colored by red and blue, respectively. The mirror eigenbands with mz= +1 andmz=-1 are illustrated with solid\nand dashed lines, respectively. (Right panel) PDOS of h-InC. The same energy scale is used between the band structure and\nPDOS. (b) Magni\fed view of the gray box in the left panel of (a). A band crossing is highlighted by a green circle. (c) Weyl\nnodal lines in energy-momentum space. Figure plots only the nodal lines within the energy range /divides.alt0E/divides.alt0<0:4 eV. The color scheme\nis used to represent the energy of the Weyl nodal lines. The node that is labeled by A (B) in (b) is a member of the type-II\n(type-I) Weyl nodal line in (c).\nsame band velocities along the nodal lines88,90. Note that\nthepz(sp2) bands have a positive (negative) band cur-\nvature at \u0000. While conventional type-I nodal lines are\nformed dominantly from overlapping one pzband and\nonesp2 band, a type-II nodal line coexists, formed from\nthe overlapping two sp2 bands in the energy range from\n0.345 eV to 0.348 eV.\nWe \fnd a contrasting behavior between the type-I and\ntype-II nodal lines, featured in the Fermi surface geom-\netry. In the type-I case, an alternating chain of electron\nand hole pockets appears such that the position of the\nnodal line is placed inside the chain; in contrast, in the\ntype-II case, a chain of electron and hole pockets occurs\navoiding the position of the nodal line. Note that in both\ncases, the electron and hole pockets exist and touch each\nother linearly at the nodal line. Figures 5(a) and 5(d)illustrate the Fermi surface diagrams at E=−0:238 eV\nandE=0:346 eV, where type-I and type-II nodal lines\noccur, respectively. They demonstrate that the nodal\nline that is represented by a dashed line in Fig. 5(b) [(e)]\nis located at the interior (exterior) part of the area sur-\nrounded by the Fermi surfaces that are represented by\nsolid contours. Note that the Fermi surfaces are warped\nhexagonally around the \u0000 point. The hexagonal warping\nis more prominent in the type-I nodal line in Fig. 5(b)\nthan the type-II nodal line in Fig. 5(e), which is due to\nthe type-II nodal line residing closer to the \u0000 point.\nWe propose the contrasting geometry of the Fermi sur-\nfaces as a generic feature of the type-I and type-II nodal\nlines in two dimensions, originated from the hexagonal\nwarping and the di\u000berent signs of the band velocities.\nTo demonstrate this, we plot the energy bands in mo-6\n(a) (b) (c)\n(d) (e) (f)↓↑\nE = -0.238 eVΓO\n00\nky\nkx\nky\nkx5 -5-55\n0\n  kx [10 -3π/a]ΓRSTky [10 -3  ]\n10\n105\n5 1515\nE = 0.346 eV\n↓↑ qop\n-346\n-345-347-348l m\nn\n-150\n-200\n-250\n-300\n-350\nOΓ P Q Γ Γ\nRΓ SΓ TΓE [meV]\nE [meV]ky [10 -2  π/a]\n  kx [10 -2π/a]\nΓΓ\nΓΓΓΓ\nP\nQkx kx kx\nkx kx kxky kyπ/a\nFIG. 5. Contrasting Fermi surfaces of type-I and type-II nodal lines in h-InC. (a) Fermi surfaces at E= -0.238 eV, which\nintersect with a type-I nodal line. The orange hexagon illustrates the BZ. The Fermi surfaces for the spin-up and spin-down\nbands are colored by red and blue, respectively. The momenta of the nodal line are indicated by the dashed line. (b) Magni\fed\nblack-boxed region from (a). Only the Fermi surfaces (contours) that intersect with the type-I nodal line are highlighted. (c)\nMagni\fed black-boxed regions from (b) (top panels) and the energy bands depicted along the green line from the corresponding\ntop panel (bottom panels). (d) Fermi surfaces at E= 0.346 eV, which intersect with the type-II nodal line. (e) Magni\fed\nblack-boxed region from (d). Only the Fermi surfaces (contours) that intersect with the type-II nodal line are highlighted. (f)\nMagni\fed black-boxed regions from (e) (top panels) and the energy bands depicted along the green line from the corresponding\ntop panel (bottom panels).\nmentum space along the \u0000 −A, \u0000−B, and \u0000 −Clines in\nFig. 5(c). We \frst notice that the energy disperses along\nthe nodal line due to the hexagonal warping. The \fg-\nures show that the band crossing on the \u0000 −A(\u0000−C)\nline is shifted down (up) in energy with respect to the\nFermi energy ( E=−0:238), forming an electron (hole)\npocket, whereas the band crossing point occurs at the\nFermi energy on \u0000 −B. Thus, from \u0000 −Ato \u0000−Cthrough\n\u0000−B, the electron pocket is reduced to the crossing point,\nand evolves to the hole pocket. Meanwhile, the di\u000berent\nsigns of the band velocity necessitates the occurrence of\nthe crossing points in the inner area of the two Fermi\nsurfaces. Similarly, from \u0000 −Dto \u0000 −Fthrough \u0000 −E\nfor the Fermi surfaces at E=0:346 eV, where the type-\nII nodal line occurs [See Fig. 5(e).], an electron pocket\nevolves to a hole pocket via a band crossing point. Note\nthat the weaker warping of the energy bands results in\na weaker energy dispersion along the type-II nodal line\nwith respect to the type-I nodal line. Moreover, unlike\nthe type-I case, in the type-II case, the same sign of the\nband velocities necessitate the occurrence of the type-II\ncrossing points in the outer area of the two Fermi surfaces\nas illustrated in Fig. 5(f). These contrasting patterns ofthe Fermi surfaces are intrinsic to the very de\fnition of\nthe type-I/type-II classi\fcation, done based the band ve-\nlocities, thus characterizing the types of the nodal lines\nin two dimensions. We expect that the characteristic fea-\nture should be observable in h-InC as in the case of topo-\nlogical nodal lines in three dimensions109{115and type-II\nWeyl nodes in topological semimetals116{118.\n3. E\u000bect of spin-orbit coupling\nIn general, SOC couples the spin and orbital sectors\nof the Bloch states, such that the nodal lines that we\nfound without SOC can be annihilated. The nodal lines\nonly survive when they are protected by a global sym-\nmetry that is respected in the entire BZ even in the pres-\nence of SOC. An example is the mirror symmetry with\nrespect to the basal plane Mz, which is preserved only\nwhen spins are colinearly aligned along the out-of-plane\nz-direction when including SOC. In this particular case of\nspin con\fguration, the nodal lines can survive when they\nare formed from the energy bands with distinct mirror\neigenvalues. Inclusion of SOC necessitates the de\fnition7\nNode (mz;sz)\u0016z(mz;sz)\u0016zSOC gap(eV) Nodal line\nA (+,−)−(+,+)+ − Type-II\nB (+,−)−(−,−)+ − Type-I\nC (+,+)+(−,+)− − Type-I\nD (+,−)−(−,−)+ − Type-I\nE (+,+)+(−,+)− − Type-I\nF (+,+)+(−,−)+ 0.11 −\nG (+,−)−(−,+)− 0.06 −\nH (+,+)+(−,−)+ 0.07 −\nI(+,−)−(−,+)− 0.05 −\nTABLE II. Mirror and spin eigenvalues. The mirror mz=±1\nand spinsz=±1 eigenvalues are evaluated from the non-\nSOC calculations for the corresponding nodes that were found\nwithout SOC. The mirror eigenvalues \u0016z=±iare evaluated\nfrom the SOC calculations with the \fxed spin orientation\nalong the out-of-plane z-direction. The third (\ffth) column\nlists the mirror eigenvalues of the band that appears on the\nleft top (left bottom) side of a crossing or an anti-crossing in\nFig. 6(a).\nof a new mirror quantum number \u0016z=mzsz=±icapable\nof protecting the band crossings. Some nodal lines indeed\nsurvive in our \frst-principles calculations calculated by\nconstraining the spin orientation along the out-of-plane\nzdirection as shown in Fig. 6(a). While the nodal lines\nthat contain the crossings A, B, C, D, or E survive [See\nFig. 4(b) and Fig. 6(a)], the other nodal lines that contain\nF, G, H, or I are gaped out by including SOC. These re-\nsults are in good agreement with the calculated mirror\neigenvalues of the corresponding band, listed in Table II.\nIn contrast to the out-of-plane spin-polarization, when\nspins are aligned uniaxially along an in-plane direction,\nwhich is the energetically most favored, the nodal lines\nare expected to disappear, except for the case where\nspins are all aligned along y-direction. In this particular\ncase, theMymirror-symmetry is preserved and the nodal\nlines that we found without SOC are tuned to 2D Weyl\npoints, residing along the Mymirror symmetric \u0000 −M1\nline (ky=0). The DFT band structure in Fig. 6(b), which\nis calculated by constraining spins along the y-direction,\nexhibits the Weyl points at B 1, C1, and H 1along the\nMy-invariantM1−\u0000 line. In contrast, all the nodes in\nthe \u0000 −M2line, where Mzis broken, open a gap. We\nnote that the Weyl points are topological in the sense\nthat they carry the Berry phase \u0019. We con\frmed that\nthe Weyl point B 1gives rise to the Berry phase \u0019when\ncalculated along the closed path of radius 0.0038 \u0017A−1that\nencircles B 1.\n4. Tight-binding theory of h-InC\nFinally, we construct a tight-binding theory that de-\nscribes the sandp-orbitals of C and In of h-InC. These\norbitals are responsible for the electronic structure of h-\nInC near the Fermi energy. The underlying interactions\nthat give rise to the low-energy band structure obtained\n(a)6\n4\n2\n0\n-2\n-4\n6\n4\n2\n0\n-2\n-4SOC\nΓ KM1(M2)Γ M1(M2) Γ0\n-1\n-2DA\nB\nC\nEG\nHF\n(b)\nEnergy [eV] Energy [eV]\nΓ KM1Γ M2ΓK\nM1M2\n0\n-1\n-2SOC\nM2 M1 ΓEnergy [eV]A1 A2\nB1\nI1B2\nC1H1\nE1 E2H2\nI2C2G1 G2(c)\nF1 F2\nD1D2SOC SOCSOC\nkxky\nkzI\nWeyl point Weyl point\nWeyl pointFIG. 6. SOC band structures of h-InC with di\u000berent spin\norientations. (a) Spins are constrained to the out-of-plane z\ndirection. The gray-boxed region is magni\fed on the right\npanel. (b) Spins are constrained to the in-plane y-direction.\nThe gray-boxed region is magni\fed on the bottom panel. The\nred-dashed circles (blue-dashed boxes) indicate the crossing\n(anti-crossing) points. (c) High-symmetry kpoints of the 2D\nBZ. The sky-blue dashed line at ky=0 isMyinvariant, where\nthe Weyl points appear. The location of the Weyl points\n(labeled by B 1) is marked with a Weyl cone.\nfrom the \frst-principles calculations can essentially be as-\ncribed to 1) the kinetic energy term H0, contributed from\nboth the nearest and next-nearest neighbor hopping of\nthe valence electrons, 2) on-site energy term Hon−sitefor\nboth the electrons that are bound to each sublattice, 3)\nthe e\u000bective Zeeman term HZeeman , which is constructed\nto discern the sites and the spins, and 4) the SOC term\nHSOC, summarizing\nH=H0+Hon−site+HZeeman +HSOC; (1)8\nwhere\nH0=/summation.disp\nj;j′;l;l′/summation.disp\n/uni27E8r;r′/uni27E9tll′\n1^ c†\njl\u000b(r)^ cj′l′\u000b(r′)\n+/summation.disp\nj;l;l′/summation.disp\n/uni27EAr;r′/uni27EBt2;j^ c†\njl\u000b(r)^ cjl′\u000b(r′);\nHon−site=/summation.disp\n\u000b/summation.disp\nj;l;r\u0001l\nj^ c†\njl\u000b(r)^ cjl\u000b(r);\nHZeeman =−/summation.disp\n\u000b;\f/summation.disp\nj;l;r(Bl\nj⋅s)\u000b\f^ c†\njl\u000b(r)^ cjl\f(r);\nHSOC =/summation.disp\n\u000b;\f;ri\u0015s\u000b\f⋅^c†\nj\u000b(r)×^cj′\f(r):\nHere,landjdescribe the orbital and sublattice in-\ndices, respectively. \u000band\frepresent spin states, respec-\ntively.t1andt2are hopping parameters for the near-\nest and the next nearest neighbor hopping, respectively.\nHZeeman describes the site-dependent Zeeman splitting\nforpzbands and sp2 bands. In the SOC Hamiltonian\nHSOC,Lj=i\"jklis the orbital angular momentum in\nthep-orbital basis. ^cj\u000b(r)describes the p-orbitals for\nspin\u000band sublattice j.\nEnergy [eV]4\n2\n0\n-2\n-4\nΓ K M ΓDFT(SOC) TB(SOC)\nFIG. 7. DFT and tight-binding (TB) electronic energy band\nstructures with SOC. The DFT (TB) bands are colored by\ngray (red). The spin orientation is set to the out-of-plane z-\ndirection. The following set of parameters are used for the\ntight-binding band structure: \u0001s\nC= -5.69 eV, \u0001s\nIn= -0.99\neV, \u0001px;y\nC;In= 3.31 eV, \u0001pz\nC= -0.62 eV , \u0001pz\nIn= 2.62 eV , Vss\u001b\n= -1.5 eV, Vsp\u001b= 2.4 eV , Vpp\u001b= 2.8 eV,Vpp\u0019= -1.0 eV,\nBpz\nC= 0.8 eV,Bpz\nIn= 0 eV,Bpx;y\nC= 0.19 eV, Bpx;y\nIn= 0 eV,\ntpz\n1= 1.0 eV,t2;C= 0.1 eV,t2;In= 0.01 eV, \u0015= 0.034 eV.Figure 7 shows the calculated tight-binding bands com-\npared with the DFT bands, where the e\u000bective Zeeman\n\feld is applied along the out-of-plane z-direction. We\n\fnd that the next nearest-neighbor hopping between pz-\norbitals is indispensable for reproducing hexagonal warp-\ning of the Fermi surface. The hexagonal warping results\nin the dispersion of energy along the nodal lines, allow-\ning for the formation of the electron and hole pockets.\nThe energy bands nicely reproduce the essential features\nof the \frst-principles results, especially the nodal struc-\nture. The symmetry analysis is also in good agreement\nwith the \frst-principles results, which supports that our\nsymmetry analysis is valid.\nIV. CONCLUSION\nIn summary, we have performed \frst-principles cal-\nculations to investigate the atomic, magnetic, and elec-\ntronic structures of monolayer hexagonal indium carbide\n(h-InC), which realizes a stable ferromagnetic nodal-line\nmetal. Without spin-orbit coupling, we found that dou-\nbly degenerate nodal lines are formed from spin-polarized\nbands. 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Lett. 117,\n077202 (2016)."    },    {        "title": "1811.03469v1.Single_Layer_Ferromagnetic_and_Piezoelectric_CoAsS_with_Pentagonal_Structure.pdf",        "content": "Single-Layer Ferromagnetic and Piezoelectric CoAsS with Pentagonal Structure\nLei Liu and Houlong L. Zhuang\u0003\nSchool for Engineering of Matter Transport and Energy,\nArizona State University, Tempe, AZ 85287, USA\n(Dated: November 9, 2018)\nSingle-layer pentagonal materials are an emerging family of two-dimensional (2D) materials that\ncould exhibit novel properties due to the building blocks being pentagons instead of hexagons as in\nnumerous 2D materials. Based on our recently predicted single-layer pentagonal CoS 2that is an\nantiferromagnetic (AFM) semiconductor, we replace two S atoms by As atoms in a unit cell to form\nsingle-layer pentagonal CoAsS. The resulting single-layer material is dynamically stable according\nto the phonon calculations. We \fnd two drastic changes in the properties of single-layer pentagonal\nCoAsS in comparison with those of CoS 2. First, we \fnd a magnetic transition from the AFM to FM\nordering. We understand that the transition is caused by the lower electronegativity of As atoms,\nleading to the weakened bridging roles on the superexchange interactions between Co ions. Single-\nlayer pentagonal CoAsS also shows signi\fcantly stronger magnetocrystalline anisotropy energy due\nto stronger spin-orbit coupling. We additionally perform Monte Carlo simulations to calculate the\nCurie temperature of single-layer pentagonal CoAsS and the predicted Curie temperature is 103\nK. Second, we \fnd that single-layer pentagonal CoAsS exhibits piezoelectricity, which is absent\nin single-layer pentagonal CoS 2due to its center of symmetry. The computed piezoelectric coe\u000e-\ncients are also sizable. The rare coexistence of FM ordering and piezoelectricity makes single-layer\npentagonal CoAsS a promising multifunctional 2D material.\nI. INTRODUCTION\nMultifunctional materials have been immensely stud-\nied due to their combinations of two or more chemical\nand physical properties that can be used and tailored for\nspeci\fc applications.1{3For example, various biomedi-\ncal applications of gold nanostructures have been real-\nized ranging from diagnostics to medical imaging mainly\ndue to the functional properties such as localized surface\nplasmon resonance.2Multifunctional properties are also\ndesirable for 2D materials. For instance, 2D oxide thin\n\flms that possess magnetism, ferroelectricity, and ther-\nmoelectricity simultaneously, have shown great potential\nin electronic device applications.4{6\nAmong a number of functional properties, magnetism\nis certainly an important one that makes a 2D mate-\nrial attractive and promising for spintronic applications.\nA number of 2D materials have been predicted to ex-\nhibit ferromagnetic (FM) or antiferromagnetic (AFM)\norderings.7{9Some of these predictions such as single-\nlayer CrI 3and Fe 3GeTe 2have also been con\frmed in\nrecent experiments.10{12\nAs another critical functional property for a semicon-\nducting 2D material, piezoelectricity describes a phys-\nical phenomenon that couples mechanical strains and\nelectric \felds. Reed et al. \frst performed theoretical\nstudies of piezoelectricity on several 2D transition-metal\ndichalcogenides.13,14A recent experiment on single-layer\nMoS 2has con\frmed the theoretical predictions.15\nBut few of the above examples of 2D materials have\nshown the coexistence of magnetism and piezoelectricity.\nThe former property often involves a transition-metal el-\nement being a component of a 2D material and the latter\nrequires the absence of a center of symmetry. Searching\nfor multifunctional 2D materials with both magnetism\n(a) (b)\nabcbCoAs\nSFIG. 1. (a) Top and (b) side views of a 2 \u00022\u00021 supercell\nof single-layer CoAsS with pentagonal structure. A type 2\npentagon encloses the cyan shaded area shown in (a).\nand piezoelectricity is therefore critical to expand the\nfunctional applications of 2D materials.\nTo discover novel 2D materials with both magnetism\nand piezoelectricity, we adopt the strategy of explor-\ning structure-properties relationships by modifying the\nbuilding blocks of a 2D material from hexagons to pen-\ntagons. In particular, we used pentagonal geometries in\nconjunction with density functional theory (DFT) calcu-\nlations to predict novel 2D materials.16,17Furthermore,\nthrough datamining the Materials Project,18we identi-\n\fed a series of pyrites structures with embedded type\n2 pentagons,19one of which is illustrated in Fig.1(a).\nTessellating this type of pentagons in a plane forms the\nCairo tessellation, a ubiquitous pattern seen on the Cairo\nstreet.20We recently studied the electric structure and\nmagnetic properties of single-layer CoS 2with the Cairo\ntessellation. Although this single-layer pentagonal ma-arXiv:1811.03469v1  [cond-mat.mtrl-sci]  8 Nov 20182\nterial exhibits the AFM ordering, the structure remains\nto have an inversion center, prohibiting the occurrence\nof the piezoelectric e\u000bect. We therefore aim to design\na 2D multifunctional material based on single-layer pen-\ntagonal CoS 2by modifying the structure to achieve the\ncombination of magnetism and piezoelectricity.\nTo break the inversion symmetry of single-layer CoS 2,\nwe partially replacing the S atoms bridging the Co atoms\nby As atoms to form single-layer CoAsS. We expect the\nsubstitution will introduce the piezoelectricity. Figure 1\nshows the top and side views of a 2 \u00022\u00021 supercell\nof single-layer CoAsS. Each unit cell consists of the same\nnumber (2) of Co, As, and S atoms. Similar to single-\nlayer pentagonal CoS 2, single-layer pentagonal CoAsS\nhas a bulk counterpart, which is a type of sulfarsenide\nminerals that include NiAsS and FeAsS.21Bulk CoAsS\nis also an example of arsenopyrite-like structure with a\ngeneral chemical formula ABS (A= Fe, Co, Ni; B=\nAs, Sb).22Although bulk CoAsS was successfully syn-\nthesized in the 1970s23and recently studied using DFT\ncalculations,22single-layer CoAsS has not yet been ob-\ntained or computationally characterized. A DFT + U\nstudy of single-layer CoAsS therefore will stimulate in-\nterest in this new 2D material.\nII. METHODS\nWe use the Vienna Ab-initio Simulation Package\n(VASP, version 5.4.4)24to perform all the DFT calcu-\nlations. We also use the Perdew-Burke-Ernzerhof (PBE)\nfunctional for approximating the exchange-correlation\ninteractions.25Furthermore, we use the standard PBE\nversion of the potential datasets for Co, As, and S gen-\nerated based on the projector-augmented wave (PAW)\nmethod.26,27The potential datasets treat the 3 d8and\n4s1electrons of Co atoms, the 4 s2and 4p3electrons of\nAs atoms, and the 3 s2and 3p4electrons of S atoms as va-\nlence electrons. The plane waves with the cut-o\u000b kinetic\nenergy of 550 eV are used to approximate the electron\nwave functions. We adopt a \u0000-centered 12 \u000212\u00021\nMonkhorst-Pack k-point grid to sample the kpoints in\nthe reciprocal space.28We keep using the same e\u000bec-\ntiveUparameter ( Ue\u000b= 3.32 eV) with the Dudarev\nmethod29to treat the dorbitals of Co atoms. This pa-\nrameter has been shown to lead to similar energy di\u000ber-\nences between the AFM and FM structures of single-\nlayer pentagonal CoS 2using the PBE+ Uand HSE06\nmethods.19The vacuum spacing of the surface slab is set\nto 18.0 \u0017A to avoid image interactions between neighbour-\ning nanosheets of CoAsS. During the VASP calculations,\nthe in-plane lattice constants along with atomic coordi-\nnates are fully optimized until the residual forces reach a\nthreshold value of 0.01 eV/ \u0017A.III. RESULTS AND DISCUSSION\nWe follow the same procedure applied to single-layer\npentagonal CoS 2to determine the ground-state magnetic\nordering of single-layer pentagonal CoAsS.19Namely, we\nassume three di\u000berent initial magnetic moments (5, 3,\nand 1\u0016B, respectively) for Co ions to form high-spin\n(HS), intermediate-spin (IS), and low-spin (LS) states,\nfor each of which we account for the AFM and FM ar-\nrangements of the two Co ions in a unit cell. We also\nperform non-spin polarized DFT calculations and the\nresulting state is denoted as non-magnetic (NM). Ta-\nble I summarizes the energies of single-layer pentagonal\nCoAsS adopting the seven di\u000berent states. As can be\nseen, the NM state exhibits the highest energy. More\nimportantly, all the FM states are relaxed into the same\nenergy and in-plane lattice constants, independent of the\ninitially speculated spin states. By contrast, there are\ntwo di\u000berent AFM states. The HS-AFM and IS-AFM\nstates have the same energy, whereas the LS-AFM state\nexhibits a much higher energy. But both energies are\nhigher than the energies of FM states. We therefore con-\nclude that the FM state is the ground state of single-layer\npentagonal CoAsS, and HS/IS-AFM state is the second\nlowest-energy state. We henceforth focus on single-layer\npentagonal CoAsS with the FM ordering.\nWe also see from Table I that each Co ion has an in-\nteger magnetic moment of 2 \u0016B, showing localized mag-\nnetism in contrast to itinerant magnetism exhibited by\ne.g., single-layer Fe 3GeTe 2.30Table I also lists the in-\nplane lattice constants of single-layer pentagonal CoAsS.\nAs expected, the in-plane lattice constants increase to\n5.75\u0017A and 5.69 \u0017A from 5.34 \u0017A and 5.43 \u0017A, respectively,\nTABLE I. Energies \u0001 E(in meV per formula unit) and mag-\nnetic moments m(in\u0016Bper Co ion), in-plane lattice constants\naandb(in\u0017A), and band gaps Eg(in eV) of single-layer CoAsS\nwith di\u000berent spin states. The energies are calculated using\nthe energy of the LS-FM state as the reference. HS, IS, LS,\nAFM, FM, and NM stand for high spin, intermediate spin, low\nspin, antiferromagnetic, ferromagnetic, and non-magnetic, re-\nspectively. All these results are obtained from the PBE + U\n(Ue\u000b= 3.32 eV) calculations.\nSpin state \u0001 E m a b E g\nHS-AFM 45 0.00 5.69 5.52 0.32\nHS-FM 0 2.00 5.75 5.69 0.34a,0.64b\nIS-AFM 45 0.00 5.69 5.52 0.32\nIS-FM 0 2.00 5.75 5.69 0.32a,0.64b\nLS-AFM 149 0.00 5.65 5.61 0.09\nLS-FM 0 2.00 5.75 5.69 0.34a,0.64b\nNM 228 0.00 5.64 5.64 0.48\naSpin-up component\nbSpin-down component3\n0100200300400\nГXM YM ГFrequency (cm-1)ГXYM\nFIG. 2. Computed phonon spectrum of single-layer pentago-\nnal CoAsS using 4 \u00024\u00021 supercells.\n-4-3-2-101234Energy (eV)\nГXM YMГ ГXM YMГ(a) (b)\nГXYMCo:3d\nAs:4pS:3p\nFIG. 3. (a) Spin-up and (b) spin-down orbital-resolved band\nstructures of single-layer pentagonal CoAsS calculated with\nthe PBE + U(Ue\u000b= 3.32 eV) method. The inset of (a)\nshows the high- symmetry k-point path.\nas in single-layer pentagonal CoS 231due to the larger\nradius of As atoms.\nTo examine the e\u000bects on the dynamical stability\ncaused by partially replacing S by As atoms, we calculate\nthe phonon spectrum of single-layer pentagonal CoAsS.\nFigure 2 shows the computed phonon spectrum using the\nforce constants of 4 \u00024\u00021 supercells. We observe\nthat, similar to single-layer pentagonal CoS 231, single-\nlayer pentagonal CoAsS is dynamically stable ( i.e., no\nimaginary frequencies) except that the maximum phonon\nfrequencies at di\u000berent kpoints is reduced because of the\nheavier As atoms. For example, at the \u0000 point the max-\nimum vibrational frequency is 381 cm\u00001for CoAsS, in\ncontrast to 455 cm\u00001for CoS 2at the same kpoint.\nFigure 3 shows the orbital-resolved band structures\nfor the spin-up and spin-down electrons, revealing that\nsingle-layer pentagonal CoAsS is a semiconductor with\nthe spin-up and spin-down bandgaps of 0.32 and 0.64 eV,\nrespectively. The spin-up bandgap is an indirect bandgap\nwith the conduction band minimum (CBM) at the M\npoint and the valence band minimum (VBN) between\nthe \u0000 andYpoints, whereas the spin-down bandgap is a\ndirect one with both the CBM and VBM at the Mpoint.\nThree types of orbitals (3 dorbitals of Co; 4 pand 3por-\nCo\nSAsFIG. 4. Charge density di\u000berence of single-layer pentagonal\nCoAsS. Green and red isosurfaces refer to charge depletion\nand accumulation, respectively. The isosurface value is 0.007\ne/a3\n0(a0: Bohr radius).\nbitals of As and S, respectively.) dominate the bands near\nthe bandgaps. In particular, the CBMs at the Mpoints\nof the spin-up and spin-down bandgaps are dominated\nfrom the 4porbitals of As. For the spin-up band struc-\nture, 4porbitals of As and 3 porbitals of S are mixed to\nform the VBM. For the spin-down band structure, more\nvisible contributions to the VBM result from the mixed\n3dorbitals of Co and 3 porbitals of S. These mixed or-\nbitals contribute to the exchange interactions between Co\nions leading to the FM ordering.\nWe also apply the HSE06 hybrid density functional to\ncalculate the bandgaps of the optimized structure from\nthe PBE + U(Ue\u000b= 3.32 eV) method. We \fnd that\nthe HSE06 functional results in the same conclusion that\nsingle-layer pentagonal CoAsS is a semiconductor and\nthe FM structure is more stable than the AFM structure\nby 35.85 meV per formula unit. This energy di\u000berence is\nalso similar to that (44.91 meV per formula unit) using\nthe PBE + U(Ue\u000b= 3.32 eV) method. Furthermore,\nthe spin-up and spin-down bandgaps using the HSE06\nfunctional are 0.29 and 1.64 eV, respectively.\nTo understand the mechanism of the transition from\nsingle-layer pentagonal, AFM CoS 2to FM CoAsS, we\nnote that the Co-S-Co superexchange interactions in\nsingle-layer pentagonal, AFM CoS 2are rather weak and\nthe calculated exchange integral is 3.01 meV leading to\nthe low N\u0013eel temperature. Unlike the Co-Co exchange in-\nteractions in single-layer pentagonal CoS 2bridged only\nby the S ions, the Co-Co exchange interactions in single-\nlayer pentagonal CoAsS are bridged by both As and S\nions. This is re\rected by the orbital mixing in the orbital-\nresolved band structure shown in Fig.3.4\nThe electrons surrounding the As and S ions in single-\nlayer pentagonal CoAsS play the roles of bridging the ex-\nchange interactions between the Co ions. The magnitude\nof electron density around an As or S ion therefore qual-\nitatively determines the \\bridge width\" describing the\nstrength of exchange interactions. Intuitively, the larger\n\\bridge width\" corresponds to stronger exchange inter-\nactions. To compare the \\bridge width\" of As and S ions,\nFig.4 displays a plot of the charge density di\u000berence \u0001 \u001a,\nwhich is calculated as the di\u000berence of the charge den-\nsity of single-layer pentagonal CoAsS with reference to\nthe total charge density of isolated Co, As, and S atoms.\nThe \u0001\u001aplot shows that electron accumulates more signif-\nicantly near the S ions than near the As ions, consistent\nwith the higher electronegativity of S (2.5) over As (2.0)\non the Pauling scale.32The narrower \\bridge width\" re-\nsults in the weakened Co-As-Co superexchange interac-\ntions that would otherwise enhance the AFM ordering.\nConsequently, the spins in Co ions prefer an alignment\nin the FM manner, competing with the AFM alignment\nowing to the Co-S-Co superexchange interactions.\nTo further demonstrate the importance of electroneg-\nativity in a\u000becting the magnetic ordering of single-layer\npentagonal CoAsS, we perform an energy calculation on\nan imaginary 2D material CoOS. Namely, the As atoms\nare replaced by more electronegative O atoms (3.5 on\nthe Pauling scale).32As a result of this replacement, the\nenergy of CoOS with the AFM ordering is lower than\nthat with the FM ordering by 86.44 meV per formula\nunit. As a reference, for single-layer pentagonal CoS 2,\nthe corresponding energy di\u000berence is merely 11.99 meV\nper formula unit.31We therefore conclude that the elec-\ntronegativity of the atoms bridging the Co ions plays a\ncritical role in a\u000becting the magnetic ordering. As such,\none may use the electronegativity of bridging ions as a\nmetric to assess the exchange coupling strength in 2D\nmagnets.\nHaving understood the mechanism of magnetic tran-\nsition, we set to evaluate the magnetocrystalline\nanisotropy energy (MAE) of single-layer pentagonal\nCoAsS. To calculate the MAE, we consider the spin-\norbit coupling (SOC) in the calculations.33We set the\nparallel spin vectors of the two Co ions initially along\ntheaorbaxis and gradually change the angle of these\ntwo vectors with reference to one of these two axes in\ntheabandac, andbcplanes, respectively. The MAE is\ntherefore de\fned as the energy at di\u000berent rotation an-\ngles subtracting the minimum energy in each of the three\nplanes. Figure 5 shows the calculated MAE as a function\nof rotation angle in the ab,ac, andbcplanes. In the ab\nplane, the minimum-energy spin orientation is along the\nbaxis, and the highest MAE (135 \u0016eV/Co ion) occurs\nwhen the spin vectors are along the aaxis. In the ac\nplane, the minimum-energy spin orientation is along the\ndirection that has an angle of 10\u000eabout theaaxis. The\ncorresponding largest MAE is (365 \u0016eV/Co ion). In the\nbcplane, the MAEs show the largest energy variation,\nfrom the energy minimum along the baxis to the en-\n03 06090120150180210240270300330360\nRotation angle (o)0100200300400500 MAE (µeV/Co ion)ab\nac\nbcFIG. 5. Magnetocrystalline anisotropy energy of single-layer\npentagonal CoAsS in the ab,ac, andbcplanes. The energy\nminimum is set to zero for each plane.\n0 50 100 150 200 250 300\nTemperature (K)00.20.40.60.81Normalized magnetizationm = (1-T/103)0.45\nFIG. 6. Normalized magnetization m(represented by blue\ntriangles) as a function of temperature Tin single-layer fer-\nromagnetic, pentagonal CoAsS obtained from Monte Carlo\nsimulations. The magnetization is \ftted to the equation:\nm(T) =\u0010\n1\u0000T\nTC\u0011\f\n, plotted as a solid blue line.\nergy maximum (460 \u0016eV/Co ion) along the caxis. This\nmaximum MAE is much higher than that (153 \u0016eV/Co\nion) of single-layer pentagonal CoS 2due to the presence\nof heavier As ions associated with stronger SOC. Strong\nmagnetocrystalline anisotropy is a necessary condition\nfor single-layer pentagonal CoAsS to exhibit long-range\nmagnetic ordering.34,35\nWe next employ the VAMPIRE package36to compute\nthe Curie temperature of single-layer pentagonal CoAsS\nvia Monte Carlo (MC) simulations. The simulations are\nbased on the Heisenberg model with the exchange Hamil-\ntonian for the exchange interaction of a system36\nH =\u00001\n2X\ni6=jJijSiSj; (1)5\nTABLE II. In-plane elastic constants C11(N/m),C22(N/m), and C12(N/m), and piezoelectric coe\u000ecients e11(10\u000010C/m),\ne12(10\u000010C/m),d11(pm/V),d12(pm/V), and d26(pm/V).\nC11 C12 C66 C22 e11 e12 e26 d11 d12 d26\n75.44 6.17 26.94 51.53 -2.47 1.38 1.09 -3.52 3.10 2.02\nwhereJijis the exchange integral between adjacent spins\nof Co ions, and SiandSjare the spin moment ( i.e.1.0\n\u0016B) of thei-th andj-th Co ions, respectively. Follow-\ning the convention, positive exchange integral Jijdenotes\nFM ordering. We consider only the nearest-neighbour\ninteractions between Co ions and calculate the exchange\nintegral from the energy di\u000berence between the AFM and\nFM con\fgurations as\nJ=EAFM\u0000FFM\n8; (2)\nwhereEAFM andEFMare the energies of a unit cell of\nsingle-layer pentagonal CoAsS with the same structure\nand di\u000berent magnetic con\fgurations (AFM and FM, re-\nspectively). We obtain a larger J(12.77 meV) than that\n(3.01 meV) of single-layer pentagonal CoS 2.19\nTo perform the MC calculations, we construct a su-\npercell of single-layer CoAsS with the in-plane size of 25\nnm\u000225 nm, su\u000eciently large to minimize the statis-\ntical noise caused by the \fnite-size e\u000bects. We apply\nthe periodic boundary conditions in both the aandbdi-\nrections. Figure 6 displays temperature-dependent nor-\nmalized magnetization using 104equilibration and 104\naveraging steps. We \ft the results to the equation36\nm(T) =\u0010\n1\u0000T\nTC\u0011\f\n; (3)\nand we obtain an estimated Curie temperature TCof\n103K and the critical exponent \fof 0.45. Although the\nTCis still below room temperature, it appears higher\nthan the N\u0013 eel temperature ( \u001820K19) of single-layer\nCoS 2due to the enhanced exchange interactions.\nIn addition to the magnetic transition from single-layer\nAFM CoS 2to FM CoAsS, another important di\u000berence\nbetween these two single-layer materials is their symme-\ntries. We use the Phonopy package37to perform a sym-\nmetry analysis and to identify the space groups of the\nsurface slabs of single-layer pentagonal CoS 2and CoAsS.\nThe space group numbers of these two single-layer ma-\nterials are 14( P21=c) and 7(Pc), respectively. A sym-\nmetry change is likely to cause novel properties such as\nthe piezoelectricity to occur in 2D materials like MoS 2.38\nDue to the presence of the inversion symmetry, single-\nlayer pentagonal CoS 2shows no piezoelectric e\u000bect. We\nalso con\frm this in our calculations. By contrast, the in-\nversion center is absent in single-layer pentagonal CoAsS,\nas can be seen in Fig.1(a).\nThe strength of piezoelectricity is quanti\fed by piezo-\nelectric coe\u000ecients. To calculate the piezoelectric coe\u000e-cients of single-layer pentagonal CoAsS, we \frst compute\nthe elastic constants with a symmetry-general method.39\nThe surface slab is a monoclinic structure. It is there-\nfore associated with 13 independent elastic constants.40\nFor a 2D material, we only need to focus on the elas-\ntic constants (a four-rank tensor) that contains no index\nof 3 (thezcomponent). As a result, there are four inde-\npendent elastic constants of interest: C1111,C1122,C1212,\nandC2222, which are written as C11,C12,C66, andC22in\nthe Voigt notation.41Table II shows the four computed\nelastic constants. We convert the dimension from N/m2\nto a common dimension N/m for 2D materials by multi-\nplying the obtained elastic constants from VASP calcu-\nlations with the zlength (18.0 \u0017A) of the surface slab. By\ndoing this, we remove the dependence of the computed\nelastic constants on the vacuum spacing. The same unit\ntransformation is also applied to the piezoelectric coe\u000e-\ncients (see below).\nWe next calculate the piezoelectric coe\u000ecients eijkand\ndimnof single-layer pentagonal CoAsS. These two sets\nof piezoelectric coe\u000ecients are related via the following\nequation using the Einstein notation:38,42,43\neijk=dimnCmnjk: (4)\nDue again to the 2D nature, only the components with\nindices 1 and/or 2 are of interest. Moreover, because of\nthe symmetry of the surface slab, only three coe\u000ecients\ne111,e122, ande212are independent. These three coe\u000e-\ncients are calculated as\ne111=d111C1111+d122C2211+d112C1211+d121C2111;(5)\ne122=d111C1122+d122C2222+d112C1222+d121C2122;(6)\nand\ne212=d211C1112+d222C2212+d212C1212+d221C2112;(7)\nrespectively. Reducing Eqs. 5, 6, and 7 with two-index\nnotations gives rise to d11,d12, andd26in the fol-\nlowing equations that are previously used for other 2D\nmaterials:38,42,43\nd11=e11C22\u0000e12C12\nC11C22\u0000C2\n12; (8)\nd12=e12C11\u0000e11C12\nC11C22\u0000C2\n12; (9)6\nand\nd26=e26\n2C66: (10)\nTo obtaineijk, we calculate the polarizations from both\nelectronic and ionic contributions using density func-\ntional perturbation theory.44Table II shows that the cal-\nculated piezoelectric coe\u000ecients are sizable. For exam-\nple, the computed d11(-3.52 pm/V) is signi\fcantly larger\nthan that of a variety of single-layer materials such as\nMoS 2(1.50 pm/V) and WS 2(1.93 pm/V) with the 2 H\nstructure.45The coe\u000ecients d11ande11of single-layer\nCoAsS also exceed all the engineered piezoelectric co-\ne\u000ecients of atom doped graphene.14Such a signi\fcant\npiezoelectric e\u000bect endows single-layer pentagonal CoAsS\nwith an additional functional property besides the exist-\ning FM ordering. It might lead to potential applications\nin sensors, energy conversion and electronics.15,46IV. CONCLUSIONS\nIn summary, we have predicted a single-layer ternary\ncompound CoAsS with pentagonal structure using DFT\n+Ucalculations. In comparison with single-layer pen-\ntagonal CoS 2with the AFM ordering, single-layer pen-\ntagonal CoAsS exhibits the FM ordering and signi\fcantly\nstronger MAEs. We suggest that electronegativity plays\nan important role in leading to the magnetic transition\nfrom the AFM to the FM ordering. In addition to\nthe FM ordering, we \fnd single-layer pentagonal CoS 2\npossesses piezoelectricity with sizable piezoelectric coef-\n\fcients. Our prediction shows that this novel single-layer\npentagonal material may be useful for a variety of appli-\ncations owing to its multifunctional properties.\nACKNOWLEDGMENTS\nWe thank the start-up funds from Arizona State Uni-\nversity (ASU). This research used computational re-\nsources of the Agave Research Computer Cluster of ASU\nand the Texas Advanced Computing Center under Con-\ntracts No.TG-DMR170070.\n\u0003zhuanghl@asu.edu\n1H. Sasabe and J. Kido, Chemistry of Materials 23, 621\n(2010).\n2C. M. Cobley, J. Chen, E. C. Cho, L. V. Wang, and Y. Xia,\nChemical Society Reviews 40, 44 (2011).\n3C. Cha, S. R. Shin, N. Annabi, M. R. Dokmeci, and\nA. Khademhosseini, ACS nano 7, 2891 (2013).\n4M. Dawber, K. 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Nandadasa,1, 2Yang-Ki Hong,3and Seong-Gon Kim1, 2,\u0003\n1Department of Physics and Astronomy, Mississippi State University, Mississippi State, MS 39762, USA\n2Center for Computational Sciences, Mississippi State University, Mississippi State, MS 39762, USA\n3Department of Electrical and Computer Engineering and MINT Center,\nThe University of Alabama, Tuscaloosa, AL 35487, USA\n(Dated: November 21, 2018)\nThe site preference and magnetic properties of Zn, Sn and Zn-Sn substituted M-type strontium\nhexaferrite (SrFe 12O19) have been investigated using \frst-principles total energy calculations based\non density functional theory. The site occupancy of substituted atoms were estimated by calculating\nthe substitution energies of di\u000berent con\fgurations. The distribution of di\u000berent con\fgurations\nduring the annealing process at high temperature was determined using the formation probabilities\nof con\fgurations to calculate magnetic properties of substituted strontium hexaferrite. We found\nthat the magnetization and magnetocrystalline anisotropy are closely related to the distributions\nof Zn-Sn ions on the \fve Fe sites. Our calculation show that in SrFe 11:5Zn0:5O19, Zn atoms prefer\nto occupy 4 f1, 12k, and 2asites with occupation probability of 78%, 19% and 3%, respectively,\nwhile in SrFe 11:5SnO 19, Sn atoms occupy the 12 kand 4f2sites with occupation probability of 54%\nand 46%, respectively. We also found that in SrFe 11Zn0:5Sn0:5O19, (Zn,Sn) atom pairs prefer to\noccupy the (4 f1, 4f2), (4f1, 12k) and (12k, 12k) sites with occupation probability of 82%, 8% and\n6%, respectively. Our calculation shows that the increase of magnetization and the reduction of\nmagnetic anisotropy in Zn-Sn substituted M-type strontium hexaferrite as observed experimentally\nis due to the occupation of (Zn,Sn) pairs at the (4 f1, 4f2) sites.\nI. INTRODUCTION\nHexagonal strontium hexaferrite SrFe 12O19(SFO),\nalong with other M-type hexaferrites XFe12O19(X=\nSr, Ba, Pb), has large saturation magnetization, high\ncoercivity, and excellent chemical stability [1{3]. Sev-\neral experimental studies of substituted hexaferrite have\nbeen performed where magnetic properties are tailored\nto \ft speci\fc applications by the partial substitution\nof Fe in its crystallographic sites by divalent, triva-\nlent, tetravalent, and divalent-tetravalent combination of\nmetal atoms. The substitution of rare-earth elements\nsuch as Pr [4, 5], La [6, 7], Sm [8], and Nd [9] in M-type\nhexaferrite has shown to enhance the coercivity without\nmuch reduction in magnetization. Rare earth elements\nincrease the spin-orbit coupling which in turn strengthen\nthe magnetocrystalline anisotropy, hence the coercivity.\nHowever, reducing the use of rare-earth elements is highly\ndesired for economical reasons. The substitution of Al\nleads to enhancement in coercivity [10{12]. Wang et al.\n[13] have reported coercivity values as high as 17.6 kOe\nfor Al-substituted strontium hexaferrite. Even higher\nvalues of coercivity (21.3 kOe) are possible by the dou-\nble substitution of Ca and Al atoms [14], which is even\nhigher than the coercivity of Nb-based magnets. The\nsubstitution of Fe by the trivalent metals such as Al,\nGa, and Cr leads to an increase in coercivity and magne-\ntocrystalline anisotropy (MAE) with reduction in magne-\n\u0003Corresponding author: sk162@msstate.edutization [12]. In particular, the substitution of divalent-\ntetravalent pairs such as Zn-Nb [15], Zn-Sn [16{19], and\nSn-Mg [2] result in signi\fcant enhancement in saturation\nmagnetization with rapid reduction in coercivity.\nTheoretical studies on pure and substituted M-type\nhexaferrite have also been performed. Fang et al. in-\nvestigated the electronic structure of strontium hexafer-\nrite using density-functional theory (DFT) [20]. Park\net al. have calculated the exchange interaction of stron-\ntium hexaferrite from the di\u000berences of the total energy\nof di\u000berent collinear spin con\fgurations [21]. Magnetism\nin La-substituted strontium hexaferrite has been studied\nusing DFT [22, 23]. Zn-Sn substituted strontium hexa-\nferrite has been studied by Liyanage et al. [24]. The site\noccupancy and magnetic properties of Al, In, and Ga-\nsubstituted strontium hexaferrite has been investigated\nby Dixit et al. [10, 25].\nIn this work, we used \frst-principles total-energy cal-\nculations to investigate the relationship between the site\noccupation and magnetic properties of substituted stron-\ntium hexaferrite, SrFe 12\u0000xMxO19withM= Zn or Sn\nand SrFe 12\u00002x(ZnSn)xO19withx= 0:5. The Boltzmann\ndistribution function was used to determine the forma-\ntion probabilities of various con\fgurations at a typical\nannealing temperature (1000 K) of strontium hexaferrite,\nwhich was further used to compute the weighted average\nof various magnetic properties. We show that our model\npredicts an increase of saturation magnetization as well\nas a decrease in magnetic anisotropy energy (MAE) of\nSrFe 11(ZnSn) 0:5O19compared to the pure M-type SFO\nin good agreement with the experimental observations\n[19, 26].arXiv:1811.04101v2  [cond-mat.mtrl-sci]  20 Nov 20182\nFIG. 1. (a) One double formula unit cell of SrFe 12O19. Two\nlarge gray spheres are Sr atoms and small pink spheres are\nO atoms. Colored spheres enclosed by polyhedra formed by\nO atoms represent Fe3+ions in di\u000berent inequivalent sites:\n2a(cyan), 2b(orange), 4 f1(green), 4f2(purple), and 12 k\n(blue). (b) A schematic diagram of the lowest-energy spin\ncon\fguration of Fe3+ions of SrFe 12O19. The arrows represent\nthe local magnetic moment at each atomic site.\nII. METHODS\nSFO belongs to space group P63=mmc (No. 194) and\nhas a hexagonal magnetoplumbite crystal structure [27]\nwhose double formula unit cell containing 64 atoms is\nshown in Fig. 1. The unit cell structure of SFO con-\nsists of ten oxygen layers and the Fe+3ions occupy \fve\ncrystallographically inequivalent sites: three octahedral\nsites (2a, 12k, and 4f2), one tetrahedral site (4 f1), and\none trigonal bipyramid site (2 b) as indicated by the co-\nordination polyhedra in Fig. 1(a). As a ferrimagnetic\nmaterial, SFO has 16 Fe+3ions with spins in the ma-\njority direction (2 a, 2b, and 12ksites) and 8 Fe+3ions\nwith spins in the minority direction (4 f1and 4f2sites) as\nshown in Fig. 1(b). First-principles total-energy calcula-\ntions were performed to determine the site preference of\nZn and Sn atoms in M-type Sr-hexaferrite. Total ener-\ngies and forces were calculated using density-functional\ntheory (DFT) with projector augmented wave (PAW) po-\ntentials as implemented in VASP [28, 29]. The exchange\ncorrelation e\u000bect was described using the Perdew-Burke-\nErnzerhof (PBE) within generalized gradient approxima-\ntion (GGA) [30]. All calculations were spin polarized ac-\ncording to the ground state ferrimagnetic ordering of Fe\nspins [20, 31]. Electronic wave functions were expanded\nin a plane-wave basis with an energy cuto\u000b of 520 eV.\nReciprocal space was sampled with a 7 \u00027\u00021 Monkhorst-Pack mesh [32] with a Fermi-level smearing of 0.2 eV ap-\nplied through the Methfessel-Paxton method [33] for re-\nlaxations and the tetrahedron method [34] for static cal-\nculations. Full geometrical optimization was performed\nto relax the positions of ions, cell shape, and cell volume\nuntil the largest force component on any ion was less than\n0.01 eV/ \u0017A. Since the on-site Coulomb interactions are\nparticularly strong for localized delectrons, we employed\nGGA+U method in the simpli\fed rotationally invariant\napproach in order to avoid the self-interaction error in lo-\ncalized electron state of Fe-3 d[35]. Based on our previous\nstudy [24], Ue\u000bfor Fe atoms was set to 3.7 eV. The Ue\u000b\nfor Zn (3d104s2), Sn (5s25p2), Sr (5s2), and O (2 s22p4)\nwere set to zero. Drawings in Fig. 1 are produced using\nVESTA code [36].\nThe magnetic properties of SFO can be modi\fed by\nsubstitution of foreign atoms for Fe. There are \fve crys-\ntallographic inequivalent Fe sites in SFO. When foreign\natoms are substituted in a SFO unit cell, there can be\nseveral energetically di\u000berent con\fgurations. Due to the\nferrimagnetic nature of SFO, the magnetization of sub-\nstituted SFO strongly depends on site preference of the\nsubstituted atoms. In order to investigate the e\u000bect of\nsubstitution on the magnetic properties, it is imperative\nto understand site preference of substituted atoms. The\nsite preference of the substituted atom can be determined\nby calculating the substitution energy. The substitution\nenergyEsub[i] for con\fguration iat 0 K is given by\nEsub[i] =ESFXO [i]\u0000ESFO\u0000X\n\u000bn\u000b\u000f\u000b (1)\nwhereESFXO [i] is the total energy per unit cell of sub-\nstituted SFO in con\fguration i, whileESFOis the total\nenergy per unit cell of pure SFO and \u000f\u000bis the total en-\nergy per atom for element \u000b(\u000b= Zn, Sn and Fe) in its\nmost stable crystal structure. Zn belongs to space group\nP63=mmc (No. 194) has a hexagonal crystal structure\nwhile Sn belongs to Fd\u00163m(No. 227) with cubic crystal\nsystem.\u000fZn,\u000fSnand\u000fFewere found to be \u00000:789 eV,\n\u00003:835 eV and\u00008:461 eV, respectively. n\u000bis the num-\nber of atoms of type \u000badded or removed; if one atom is\nadded then n\u000b= +1 while n\u000b=\u00001 when one atom is\nremoved.\nThe magnetocrystalline anisotropy energy, Ea, was\nalso calculated. Eain the present case, is de\fned as\nthe di\u000berence between the two total energies where the\nspin quantization axes are aligned along two di\u000berent di-\nrections [37]:\nEa=E(100)\u0000E(001) (2)\nwhere,E(100) is the total energy with spin quantization\naxis in the magnetically hard plane and E(001) is the total\nenergy with spin quantization axis in the magnetically\neasy axis. The total energies in Eq. (2) are computed\nby the non-self-consistent calculations, where the spin\ndensities are kept constant [38].3\nThe uniaxial magnetic anisotropy constant, K1, can be\ncomputed as [39, 40]:\nK1=Ea\nVsin2\u0012(3)\nwhereVis the equilibrium volume of the unit cell, and \u0012\nis the angle between the two spin quantization axis ori-\nentations (90\u000ein the present case). The anisotropy \feld,\nHa, which is related to the coercivity can be expressed\nas [41]:\nHa=2K1\nMs(4)\nwhereK1is the magnetocrystalline anisotropy constant\nandMsis the saturation magnetization.\nWhen the separation between Esubof di\u000berent con\fg-\nurations is not too big compared to the thermal energy\nduring the synthesis of these hexaferrites at a high an-\nnealing temperature ( &1000 K), we can expect the site\npreference of substituted atoms to change at such an el-\nevated temperature. This change in the site occupation\npreference can be modeled using Maxwell-Boltzmann dis-\ntribution. The site occupation probability or the forma-\ntion probability Pi(T) of con\fguration iat temperature\nTis given by\nPi(T) =giexp(\u0000\u0001Gi=kBT)P\njgjexp(\u0000\u0001Gj=kBT)(5)\n\u0001Gi= \u0001Ei+P\u0001Vi\u0000T\u0001Si (6)\n\u0001Si=kBln(gi)\u0000kBln(g0) (7)\nwhere \u0001Gi, \u0001Ei, \u0001Vi, and \u0001Siare the change in free en-\nergy, substitution energy, unit cell volume and entropy of\nthe con\fguration irelative to the ground state con\fgura-\ntion.P,kBandgiare the pressure, Boltzmann constant\nand multiplicity of con\fguration i.g0is the multiplicity\nof the ground state con\fguration. In our earlier work,\nwe considered \u0001 Sito be the same for all con\fgurations\n[42]. Eq. (7) improves the model by explicit calculation\nof change in entropy with respect to the most stable con-\n\fguration [43].\nTherefore, when the formation probability of higher\nenergy con\fgurations become non-negligible at the an-\nnealing temperature, it can be concluded that in a sample\nof substituted SFO there will not be a single con\fguration\nbut a distribution of several con\fgurations. Any physi-\ncal quantity of a sample of SFO will then be a weighted\naverage of respective quantity in di\u000berent con\fgurations:\nhQi=X\niP1000K (i)\u0001Qi (8)\nwhereP1000K (i) andQiare the formation probability at\n1000 K and a physical quantity Qof the con\fguration\ni. During the annealing process when the material ismaintained at high temperature, substituted atoms have\nsu\u000ecient energy to overcome the local energy barriers\nand acquire energetically the various con\fgurations that\nhave su\u000ecient formation probability. We note that the\nsubstituted SFO considered in the present work loses its\nmagnetic properties at a typical annealing temperature\n(1000 K or higher) that is near or above its Curie temper-\nature. When the material is cooled down to a low tem-\nperature below the critical temperature, it regains the\nmagnetic properties while the con\fgurations are locked in\nthose with higher substitution energy due to energy bar-\nriers between them. Consequently, the weighted average\ncalculated by Eq. (8) is the material's low temperature\nproperty even though P1000K is used for computation.\nIII. RESULTS AND DISCUSSION\nIn this work, two types of substitutions have been stud-\nied. In the \frst case one Zn or Sn atom was substituted\nin a SFO unit cell. In the second case, a pair of Zn and\nSn atoms were substituted in a SFO unit cell. Although\nthere are 24 Fe atoms in SFO unit cell, the application of\ncrystallographic symmetry operations shows that many\nof these Fe sites are equivalent and leaves only \fve in-\nequivalent sites. Therefore, one foreign atom can be sub-\nstituted in \fve di\u000berent ways, giving rise to \fve di\u000berent\ncon\fgurations. We label these inequivalent con\fgura-\ntions using the crystallographic name of the Fe site: [2 a],\n[2b], [4f1], [4f2], and [12k].Esubcorresponding to the\nsubstitution of one Zn at di\u000berent inequivalent Fe sites\nis given in Table I. The lowest Esubwas found when a\nZn atom was substituted at 4 f1site, followed by the [2 a]\nand [12k] cases. For the case of a single Sn atom substi-\ntution (Table II) the lowest Esubwas found to be for the\nsubstitution at the 4 f2site, and the second lowest Esub\nwas corresponding to the substitution at the 12 ksite. A\nhigh multiplicity of the 12 ksite and low Esubindicate\nthat at higher temperatures the 12 ksite is very likely to\nbe occupied. In the second case, where a pair of Zn and\nSn atoms were substituted in the SFO unit cell, there can\nbe 5\u00025 = 25 di\u000berent con\fgurations. Esubcorrespond-\ning to all these con\fgurations are listed in Table III. The\nlowestEsubwas found to be the con\fguration where Zn\ngoes to the 4 f1while Sn occupies the 4 f2site.\nFig. 2 shows the variation of site occupation probabil-\nity with temperature for the inequivalent con\fgurations\nof SrFe 12\u0000xZnxO19withx= 0:5. In low temperature\nrange the 4 f1site is most likely to be occupied. How-\never, as temperature rises this probability falls while the\nsite occupation probability of 12 krises. This is due to\nsmall substitution energy di\u000berence between 4 f1and 12k\nsites and higher multiplicity of the 12 ksite. Similar be-\nhavior of site occupancy was seen in the case of single\nSn atom substitution as shown in Fig. 3. In the elevated\ntemperature regime, the site occupancy of the 4 f2falls,\nwhile that of the 12 ksite rises. In fact, at the anneal-\ning temperature of SFO (1000 K), the occupation prob-4\nTABLE I. Physical properties of inequivalent con\fgurations of SrFe 12\u0000xZnxO19withx= 0:5: multiplicity ( g), substitution\nenergy (Esub), volume of the unit cell ( V), total magnetic moment ( mtot), saturation magnetization ( \u001bs), magnetocrystalline\nanisotropy energy ( Ea), uniaxial magnetic anisotropy constant ( K1), anisotropy \feld ( Ha), and the formation probability at\n1000 K (P1000K ). All values are for a double formula unit cell containing 64 atoms.\ncon\fgg E sub(eV)V(\u0017A3)mtot(\u0016B)\u001bs(emu/g) Ea(meV)K1(kJ/m3)Ha(kOe)P1000K\n[4f1] 4 -1.44 706.61 44 115.01 0.83 188.20 6.37 0.794\n[2a] 2 -1.27 705.15 34 88.74 0.90 204.49 8.95 0.025\n[12k] 12 -1.13 706.61 34 88.70 0.80 181.39 7.96 0.180\n[2b] 2 -0.86 708.08 36 94.13 0.63 142.55 5.91 0.000\n[4f2] 4 -0.63 708.14 44 115.15 0.85 192.31 6.52 0.000\nTABLE II. Physical properties of inequivalent con\fgurations of SrFe 12\u0000xSnxO19withx= 0:5: multiplicity ( g), substitution\nenergy (Esub), volume of the unit cell ( V), total magnetic moment ( mtot), saturation magnetization ( \u001bs), magnetocrystalline\nanisotropy energy ( Ea), uniaxial magnetic anisotropy constant ( K1), anisotropy \feld ( Ha), and the formation probability at\n1000 K (P1000K ). All values are for a double formula unit cell containing 64 atoms.\ncon\fgg E sub(eV)V(\u0017A3)mtot(\u0016B)\u001bs(emu/g) Ea(meV)K1(kJ/m3)Ha(kOe)P1000K\n[4f2] 4 -2.42 716.94 44 112.37 0.66 147.49 5.06 0.462\n[12k] 12 -2.25 716.65 36 91.95 0.83 185.56 7.77 0.538\n[2b] 2 -1.78 712.67 34 86.74 0.66 148.38 6.55 0.000\n[4f1] 4 -1.51 715.41 44 112.38 0.18 40.31 1.38 0.000\n[2a] 2 -1.51 716.45 35 90.65 0.90 201.27 8.55 0.000\nTABLE III. Physical properties of inequivalent con\fgurations of SrFe 12\u00002x(ZnSn) xO19withx= 0:5: multiplicity ( g), substitu-\ntion energy ( Esub), volume of the unit cell ( V), total magnetic moment ( mtot), saturation magnetization ( \u001bs), magnetocrystalline\nanisotropy energy ( Ea), uniaxial magnetic anisotropy constant ( K1), anisotropy \feld ( Ha), and the formation probability at\n1000 K (P1000K ). All values are for a double formula unit cell containing 64 atoms.\ncon\fg g E sub(eV)V(\u0017A3)mtot(\u0016B)\u001bs(emu/g) Ea(meV)K1(kJ/m3)Ha(kOe)P1000K\n[4f1;4f2] 16 -5.326 717.32 50 127.13 0.70 156.35 4.72 0.819\n[2b;4f2] 8 -5.124 719.68 40 101.97 0.58 129.12 4.88 0.020\n[4f1;2a] 8 -5.065 716.49 40 101.68 0.82 183.36 6.91 0.010\n[2a;4f2] 8 -4.991 714.67 40 101.72 0.82 183.83 6.91 0.004\n[4f1;12k] 48 -4.940 717.54 40 101.72 0.61 136.21 5.14 0.084\n[12k;12k] 132 -4.736 715.33 30 76.17 0.77 172.46 8.67 0.060\n[2a;12k] 24 -4.697 715.08 30 76.28 0.83 185.97 9.33 0.001\n[12k;4f2] 48 -4.663 717.04 40 101.58 0.63 140.77 5.32 0.003\n[4f1;2b] 8 -4.650 715.16 40 101.72 0.49 109.78 4.13 0.000\n[12k;2a] 24 -4.366 718.08 30 76.17 0.71 158.42 7.99 0.000\n[12k;2b] 24 -4.207 715.13 30 76.15 0.49 109.78 5.52 0.000\n[2a;4f1] 8 -4.206 714.79 40 101.68 0.90 210.73 7.59 0.000\n[2a;2a] 2 -4.156 720.35 30 76.26 0.85 189.05 9.56 0.000\n[2b;12k] 24 -4.119 721.64 30 76.39 0.58 128.77 6.51 0.000\n[4f2;2b] 8 -4.082 716.07 40 101.73 0.53 118.59 4.47 0.000\n[4f1;4f1] 12 -3.946 720.60 50 127.17 0.78 173.43 5.26 0.000\n[12k;4f1] 48 -3.932 717.21 40 101.59 0.84 187.65 7.09 0.000\n[4f2;4f2] 12 -3.876 721.34 50 127.18 0.72 159.92 4.85 0.000\n[2a;2b] 4 -3.726 717.56 30 76.58 0.79 176.39 8.83 0.000\n[2b;2a] 4 -3.706 717.50 30 76.58 0.79 175.79 8.85 0.000\n[4f2;12k] 48 -3.586 723.09 40 101.86 0.75 166.12 6.31 0.000\n[4f2;2a] 8 -3.493 724.60 40 101.73 1.12 246.98 9.41 0.000\n[2b;4f1] 8 -3.297 722.28 40 101.83 0.68 150.84 5.73 0.000\n[2b;2b] 2 -3.086 690.64 30 76.36 0.41 95.11 4.60 0.000\n[4f2;4f1] 16 -3.073 723.67 50 127.21 0.78 172.69 5.26 0.0005\nFIG. 2. Temperature dependence of the formation prob-\nability of di\u000berent con\fgurations of SrFe 11:5Zn0:5O19. The\ncon\fgurations with negligible probability are not shown. The\nvertical dotted line indicates the annealing temperature of\n1000 K.\nability of the 12 ksite (54%) becomes higher than that\nof the 4f2site (46%), which is the most energetically\nstable substitution site at 0 K. For the Zn-Sn substitu-\ntion (SrFe 11Zn0:5Sn0:5O19), the formation probability of\n[4f1;4f2] is high in the low as well as high temperature\nrange. Other important con\fgurations with signi\fcant\nformation probability at high temperature (1000 K) are\n[4f1;12k] and [12k;12k] (Fig. 4). Substituted Zn atoms\nare most likely to occupy 4 f1and 12ksites, while Sn\natoms occupy 4 f2and 12ksites.\nThe total magnetic moment mtotupon the substitution\nof a single Zn and Sn atom at various Fe site of SFO is\ngiven in Table I and Table II. It is evident that the substi-\ntution at the minority spin sites (4 f1and 4f2) enhances\nthe net magnetic moment, while moment is reduced for\nthe substitution at the majority spin sites (12 k, 2a, and\n2b). Similar changes in the total magnetic moment were\nalso noticed for (SrFe 11Zn0:5Sn0:5O19substitution (Ta-\nble III). The saturation magnetization \u001bsvalues are also\nlisted.\nEa,K1, andHaof various con\fgurations are presented\nin Table I, II and III. In these cases, substitutions at\n2asite seems to enhance the anisotropy values. Ta-\nble III also that the reduction of magnetic coercivity\nin SrFe 11(ZnSn) 0:5O19as experimentally observed [16]\nis mainly due to the occupation of (Zn,Sn) pair at the\n(4f1, 4f2) sites. Although the [2 b;4f2] con\fguration, the\none with second lowest substitution energy at 0 K, has\nmuch lower K1andHavalues, its multiplicity is much\nlower compared to other low-energy con\fgurations and\nits formation probability grows merely to 2% (compared\nto 82% for the [4 f1;4f2] con\fguration) even at 1000 K.\nFIG. 3. Temperature dependence of the formation prob-\nability of di\u000berent con\fgurations of SrFe 11:5Sn0:5O19. The\ncon\fgurations with negligible probability are not shown. The\nvertical dotted line indicates the annealing temperature of\n1000 K.\nFIG. 4. Temperature dependence of the formation probabil-\nity of di\u000berent con\fgurations of SrFe 11:0Zn0:5Sn0:5O19. The\ncon\fgurations with negligible probability are not shown. The\nvertical dotted line indicates the annealing temperature of\n1000 K.\nThis is a di\u000berent behavior from the related systems\nBaFe 12\u0000x(Zr0:5Zn0:5)xO19and LaZnFe 11O19, where the\n2bsite plays the main role in the reduction of anisotropy\n[44, 45].\nTable IV shows the weighted average of physical prop-\nerties of pure and substituted (Zn, Sn, Zn-Sn) stron-\ntium hexaferrite based on the formation probabilities6\nTABLE IV. Weighted averages of physical properties of pure and substituted (Zn, Sn, Zn-Sn) strontium hexaferrite: volume\nof the unit cell ( V), total magnetic moment ( mtot), saturation magnetization ( \u001bs), magnetocrystalline anisotropy energy ( Ea),\nuniaxial magnetic anisotropy constant ( K1), and anisotropy \feld ( Ha).\nmaterial hVi(\u0017A3)hmtoti(\u0016B)h\u001bsi(emu/g) Ea(meV) K1(kJ/m3)Ha(kOe)\nSrFe 12O19 706.83 39.0 110.19 0.85 193.00 7.35\nSrFe 11:5Zn0:5O19 706.83 41.7 109.22 0.83 187.39 6.75\nSrFe 11:5Sn0:5O19 718.61 39.7 101.34 0.75 168.05 6.52\nSrFe 11(ZnSn) 0:5O19 717.08 47.2 120.01 0.69 153.87 4.99\nTABLE V. Comparison of calculated and experimental magnetic properties of SrFe 12\u00002x(ZnSn) xO19withx= 0 andx= 0:5:\nsaturation magnetization ( \u001bs), uniaxial magnetic anisotropy constant ( K1), and anisotropy \feld ( Ha). The calculated values\nare for 0 K while the experimental values are measured at the room temperature. Relative di\u000berence w.r.t. pure SFO ( x= 0)\nvalues are given in parentheses.\n\u001bs(emu/g) K1(kJ/m3) Ha(kOe)\nx= 0 x= 0:5 x= 0 x= 0:5 x= 0 x= 0:5\nExp. [16, 18] 69.9 75.1 (+7 :4%) 280.9 210.7 ( \u000025:0%) 18.7 12.5 ( \u000033:0%)\nCalc. [This work] 110.2 120.0 (+8 :2%) 193.0 153.9 ( \u000020:0%) 7.4 5.0 ( \u000032:0%)\nCalc. [24] 113.1 127.2 (+12 :5%) 190.0 100.0 ( \u000047:4%) 7.1 3.0 ( \u000057:0%)\nat the annealing temperature. In all three cases of\nsubstituted SFO, the mtotwas estimated to be greater\nthan that of pure SFO. Biggest increase was found in\nthe case of Zn-Sn pair substituted SFO. A reduction in\nthe values of \u001bsand anisotropy of SrFe 11:5Zn0:5O19and\nSrFe 11:5Sn0:5O19is in agreement with previous experi-\nmental studies on substituted barium hexaferrite [46, 47].\nThe saturation magnetization ( \u001bs) of SrFe 11:5Zn0:5O19\nand SrFe 11Zn0:5Sn0:5O19were higher than that of pure\nSFO. In Table V we compare the magnetic properties of\nZn-Sn pair substituted SFO in this work with previously\nreported computed and experimental data. Results from\nthe present work show an increase of 8.2% in the value of\n\u001bs, while Ghasemi et al. [18] experimentally observed a\nsimilar increase of 7.4%. Quantities related to magnetic\nanisotropy viz. K1, andHashowed reduction compared\nto pure SFO. SrFe 11Zn0:5Sn0:5O19showed a reduction of\n20% and 32% in K1andHa, respectively, while exper-\nimentally Fang et al. observed a reduction of 25% and\n33% inK1andHavalues [16]. Table V indicates that our\npresent results are in better agreement with experimen-\ntal results compared to the previous computed results\nthat was obtained by considering the ground state con-\n\fgurations at 0 K only. Thus, our model based on the\nformation probability at the annealing temperature suc-\ncessfully explains the reduction in anisotropy of Zn-Sn\nsubstituted SFO as well as the increase in magnetization\nvalues.\nIV. CONCLUSION\nUsing \frst-principles total energy calculations based\non density functional theory, we calculated substitutionenergies of Zn, Sn, and Zn-Sn pair substituted strontium\nhexaferrite. These energy values were used to determine\nsite preferences of the substituted atoms at 0 K as well\nas at a high annealing temperature. The site occupa-\ntion probabilities or the formation probabilities of dif-\nferent con\fgurations were then used to estimate mag-\nnetic properties of substituted SFO. We found that in\nSrFe 11:5Zn0:5O19, Zn atoms prefer to occupy 4 f1, 12k,\nand 2asites with occupation probability of 78%, 19%\nand 3%, respectively, while in SrFe 11:5SnO 19, Sn atoms\noccupy the 12 kand 4f2sites with occupation probability\nof 54% and 46%, respectively. We further showed that in\nSrFe 11Zn0:5Sn0:5O19, the pair of (Zn,Sn) atoms prefers\nto occupy the (4 f1, 4f2), (4f1, 12k) and (12k, 12k) sites\nwith occupation probability of 82%, 8% and 6%, respec-\ntively. The results from our model based on the for-\nmation probability were found to be in good agreement\nwith recent experimental observations showing enhance-\nment of magnetization and the reduction in anisotropy\nfor Zn-Sn substituted strontium hexaferrites.\nV. ACKNOWLEDGMENT\nThis research was supported by the Creative Materials\nDiscovery Program through the National Research Foun-\ndation of Korea (NRF) funded by the Ministry of Sci-\nence, ICT, and Future Planning (2016M3D1A1919181).\nIt was also partly supported by the Center for Com-\nputational Sciences (CCS) at Mississippi State Univer-\nsity. YKH's work was partially supported by National\nScience Foundation (NSF) under Award Number CMMI\n1463301. 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Dimitrov 5, 6, 7  \n \n1 Institut Quantique, Université de Sherbrooke, J1K 2R1, QC, Canada.   \n2 Regroupement québécois sur les matériaux de pointe , Département de physique, Université de Sherbrooke , J1K \n2R1, QC, Canada.   \n3 Canadian Institute for Advanced Research, Toronto,  Ontario M5G 1Z8, Canada. \n4 Département de chimie, Université de Sherbrooke, J 1K 2R1, QC, Canada.   \n5 Institute of Solid State Physics, Bulgarian Academ y of Science, Sofia 1184, Bulgaria. \n \n6 Institute of Optical Materials and Technologies, Bu lgarian Academy of Sciences, Sofia 1113, Bulgaria. \n \n7 Department of Electrophysics, National Chiao Tung U niversity, Hsinchu 30010, Taiwan. \n \n \nAbstract \nIt is known that orthorhombic RMnO 3 multiferroics (R = magnetic rare earth) with low s ymmetry exhibit \na large rotating magnetocaloric effect because of t heir strong magnetocrystalline anisotropy. In this paper, we \ndemonstrate that the hexagonal ErMnO 3 single crystals also unveils a giant rotating magn etocaloric effect that can \nbe obtained by spinning them in constant magnetic f ields around their a or b axes. When the ErMnO 3 crystal is \nrotated with the magnetic field initially parallel to the c-axis, the resulting entropy change reaches  maximum values \nof 7, 17 and 20 J/kg K under a constant magnetic fi eld of 2, 5 and 7 T, respectively. These values are  comparable \nor even larger than those shown by some of the best  orthorhombic phases. More interestingly, the gener ated \nanisotropic thermal effect is about three times lar ger than that exhibited by the hexagonal HoMnO 3 single crystal. \nThe enhancement of the rotating magnetocaloric effe ct in the hexagonal ErMnO 3 compound arises from the unique \nfeatures of Er 3+  magnetic sublattice. In fact, the Er 3+  magnetic moments located at 2a sites experience a first-order \nmetamagnetic transition close to 3 K along the c-ax is resulting in a peaked magnetocaloric effect over  a narrower \ntemperature range. In contrast, the “paramagnetic” behaviour of Er 3+  magnetic moments within the ab-plane, \nproduces a larger magnetocaloric effect over a wide r temperature range. Therefore, the magnetocaloric effect \nanisotropy is maximized between the c and the ab-di rections, leading to a giant rotating magnetocalori c effect.  \n*Mohamed.balli@usherbrooke.ca 2 \n I. Introduction \nBased on the well-known magnetocaloric effect (MCE) , magnetic cooling is a trending technology that \ncontinues to attract a worldwide interest due to it s potential high thermodynamic efficiency as well a s its eco-\nfriendly character [1-8]. The implementation of thi s emergent technology in our daily life would enabl e to fully \nsuppress the harmful synthetic refrigerants such as  fluorinated fluids, usually present in standard re frigerators and \nair-conditioners [1-8], thus allowing to meet the r equirements of several treaties that were universal ly adopted by \nthe international community aiming to reduce the ut ilization of CFCs, HCFCs and HFCs gases and, green house \ngases (GHG) emissions [1]. However, the search for advanced magnetocaloric materials with outstanding thermal, \nchemical and mechanical properties is necessary for  the transfer of magnetic refrigeration technology towards the \nmarket place. In this context, a large MCE has been  pointed out in a wide variety of magnetocaloric ma terials \nincluding both intermetallics and oxides [1-8]. Som e of them such as LaFe 13-xSi x compounds, Fe 2P-type materials \nand ABO 3-based oxides have been successfully tested in room -temperature devices unveiling the bright future of  \nmagnetic refrigeration [1]. On the other hand, magn etic materials with excellent caloric effects at th e temperature \nrange below 30 K are of great importance in several  low-temperature applications such as space industr y, scientific \nfacilities and, hydrogen and helium liquefiers. Par ticularly, the production and utilization of liquif ied hydrogen is \nexpected to markedly increase in the forthcoming ye ars as an alternative source of energy [8]. To deal  with the \nincreasing demand for efficient solid state-refrige rants operating at very low temperatures, worldwide  research \nworks have been conducted leading to quite interest ing systems such as DyGdAl 5O12  garnets [9, 10], RAl 2 \nintermetallics (R = rare earth) [5], RMn 2O5 [11, 12] multiferroics and RMO 3 oxides (M = transition metal) [13-\n27]. \nParticularly, the RMnO 3 compounds are still attracting a growing interest,  because of their excellent \nmagnetocaloric properties as well as their potentia l implementation in spintronic devices [13-27, 28-4 0]. Also, \nthese oxides unveil degrees of freedom other than m agnetization such as electric polarization, charge and volume \nchange enabling additional potential caloric effect s. Their structural and physical properties strongl y depend on \nthe rare-earth size [28-40]. Usually when the ionic  radius of the rare-earth is lower than that of Dy,  RMnO 3 \ncrystallizes in a hexagonal structure with P63cm sp ace group while for larger ionic radius (r R > R Dy ), the crystalline \nstructure transforms into an orthorhombic symmetry with Pbnm space group. On the other hand, the ferro electricity \norigin as well as the strength of the magnetoelectr ic coupling in the RMnO 3 compounds highly depend on their \ncrystallographic structure. Hence, the hexagonal RM nO 3 show a weak magnetoelectric coupling since the \nferroelectricity arises from structural deformation s. In contrast, the appearance of electric polariza tion in 3 \n orthorhombic RMnO 3 originates from the magnetic frustration, leading accordingly to a strong coupling between \nmagnetic and electric ordering parameters [28-40]. More recently, it was reported that the gigantic \nmagnetocrystalline anisotropy particularly shown by  orthorhombic RMnO 3 compounds leads to a giant rotating \nmagnetocaloric effect (RMCE) that can be simply gen erated by spinning their crystals in constant magne tic fields, \ninstead of the standard magnetization-demagnetizati on process [27]. Such effect would enable more effi cient and \ncompact magnetic cooling machines with simplified d esigns.  \nIn this paper we mainly focus on the magnetic and m agnetocaloric properties of the hexagonal ErMnO 3 \nsingle crystal (h-ErMnO 3), aiming to get more insight about its real potent ial in low-temperature magnetic cooling. \nIn h-ErMnO 3, the competition between different magnetic exchan ge interactions involving the Mn and Er \nsublattices gives rise to the appearance of several  magnetic phase transitions over a temperature rang e going from \n1000 K down to the liquid helium temperature [41-50 ]. Therefore, a paraelectric to ferroelectric trans ition is \nobserved at about 833 K, while the Mn magnetic mome nts order antiferromagnetically close to 80 K. The Er \nmagnetic moments usually undergo ordering transitio ns below 10 K. It is worth noting that the Mn magne tic \nmoments are strongly frustrated in the-ab plane wit hin a triangular structure, making marginal their c ontribution \nto the total magnetization even under intense magne tic fields [41-50]. For this purpose, our present s tudy of the \nmagnetocaloric properties of h-ErMnO 3 single crystals is motivated mainly by the large a nisotropy expected in the \ntemperature range around 10 K, where the ordering s tate of Er magnetic moments can be easily manipulat ed under \nrelatively low driving magnetic fields. Moreover, t he choice of single crystals instead of polycrystal line samples \nwill allow us to avoid grain boundary effects and s tructural and stoichiometric inhomogeneity. Additio nally, the \nsingle crystals approach enables us to explore the strength of the MCE anisotropy in h-ErMnO 3 and accordingly, \nthe resulting rotating magnetocaloric effect. \n \nII. Experimental  \nh-ErMnO 3 single crystals were grown using the high temperat ure solution method with the help of \nPbF 2/PbO/B 2O3 flux [51]. The ErMnO 3 polycrystalline samples were prepared by the stand ard solid-state reaction \nmethod using stoichiometric quantities of Er 2O3 and MnO 2 precursors annealed at 1120 °C for 24 h. The flux was \nthen mixed with ErMnO 3 polycrystals in a 7:1 proportion and subjected to heat treatment in a platinum crucible at \n1250 °C for 48 h. The growth process was achieved b y cooling slowly the final product down to 1000 °C at a rate \nof 5 °C/h [51]. The symmetry of the crystals and th eir crystalline quality were checked using Raman sc attering. 4 \n Micro-Raman spectra were collected for two polarize r configurations [b(cc)b and c(ab)c] using a Labram -800 \nspectrometer equipped with a microscope, a He-Ne la ser and a nitrogen-cooled charge coupled device (CC D) \ndetector. c and b represent the incident and scatte red propagation directions, while (cc) and (ab) are  the polarizer \norientations for the incident and scattered beams. Careful investigations of Raman-active modes report ed in Figure \n1 unveils that like the reference material YMnO 3, h-ErMnO 3 crystals form in a high-quality hexagonal structur e \nwith P63cm  space group (Fig. 1 right) where the Er 3+  ions occupy two inequivalent lattice sites, one wi th C 3v \nsymmetry defined as site 2a, and the other one with  C 3 symmetry defined as site 2b. The magnetization \nmeasurements were carried out using a superconducti ng quantum interference device (SQUID) magnetometer  \nfrom Quantum Design (MPMS XL7). Aiming to minimize the demagnetization effect, before each measurement  \nwe ensure that the target crystallographic directio n (ab or c) is along the largest dimension of the s elected h-\nErMnO 3 crystal (roughly 0.5 mm*0.5 mm*5 mm). \nIII. Results and discussion  \nIn Fig.2-a, we report the 0.1 T-zero-field cooled ( ZFC) and field-cooled (FC) magnetization as a funct ion \nof temperature measured following the c-axis and th e ab-plane. As can be seen for both directions, the  \nthermomagnetic curves exhibit a nearly reversible b ehaviour unveiling the absence of thermal hystereti c losses. \nAs the temperature decreases, the magnetization is markedly enhanced at very low temperatures, which i s mainly \ncaused by the ordering of Er 3+ magnetic moments. The linear fitting of the 0.1 T- inverse susceptibility (1/ χ) shown \nin the inset of Figure 2-a using the Curie-Weiss la w χ-1=(T-Tθ)/C (C is the Curie-Weiss constant) leads to \nparamagnetic Curie temperatures (T ϴ) of about -3 and -63 K for the ab and c-orientatio ns, respectively. The strong \nnegative value of T ϴ reflects the presence of dominant antiferromagneti c interactions in the hexagonal ErMnO 3 \nalong the c-axis. In contrast, its weak negative va lue for the field applied along the ab-plane indica tes a weak \nantiferromagnetic ordering and/or a paramagnetic be havior of Er 3+  magnetic moments. The deduced effective \nmagnetic moment from the linear interpolation of 1/ χ at high temperatures was found to be 10.14 µ B for the c-axis \nand 10.39 µ B for the ab-plane, being in perfect accord with the  theoretical value given by \u0001\u0002\u0003\u0003 =\n\u0005(\u0001\u0002\u0003\u0003 (\u0007\b \t))\u000b+ (\u0001 \u0002\u0003\u0003  (\u000e\u000f \u0010\t))\u000b= 10.75\u0001 \u0016: here, the effective magnetic moments of Er 3+  and Mn 3+  ions [52] \nwere taken as 9,58 µ B and 4,889 µ B, respectively. \n \nIt is worth noting that with increasing magnetic fi elds parallel to the c-axis, a sudden change in the  \nmagnetization at about 2 K can be clearly seen from  the thermomagnetic curves reported in Figure 2-b. In addition, 5 \n this transition zone slightly moves to high tempera tures under magnetic fields above 1 T. The observed  behavior \ncontrasts markedly to that shown by the magnetizati on along the ab-plane. This can be mainly explained  by the \nfact that the Er 3+  magnetic moments undergo a field-induced metamagne tic transition from the antiferromagnetic \nto the ferromagnetic state. On the other hand, the anomaly at ̴ 80 K associated with the antiferromagnetic orderin g \nof Mn 3+  magnetic moments is not clearly visible from the r eported thermomagnetic curves in Figure 2-a. The \nabsence of Mn 3+  critical point can be essentially attributed to th e frustrated triangular magnetic structure of the \nMn 3+  magnetic moments within the ab-plane and the domin ance of the magnetic moment of Er 3+ . In the \nmultiferroic h-ErMnO 3, the Mn 3+  ions organize in layers of triangular sub-units st acked along the c-axis (Fig. 1), \nwhile their related moments form an angle of 120 ° with one another, leading to a negligible net magne tization \n[50]. Consequently, the magnetic features involving  the Mn 3+  sub-lattice are masked by the large magnetization \nof the Er 3+  sub-lattice. Figure 3-a displays the magnetization  as a function of magnetic field along the ab-plane  and \nthe c-axis at 2 K. As can be seen, the h-ErMnO 3 single crystal unveils a large magnetocrystalline anisotropy which \nis a common feature of frustrated RMnO 3 manganites [13-40]. However, in comparison to ortho rhombic RMnO 3 \nphases [13-40], this magnetic anisotropy remains le ss pronounced because of the high crystallographic symmetry. \nAccording to data of Figures 2-a and 3-a, the magne tization behavior clearly indicates that the ab-pla ne \ncorresponds to the easy-orientation, while the c-ax is looks like a hard-direction. Such behavior is in triguing since \nthe preferred orientation of Er 3+  magnetic moments is along the c-axis. In this case , the Mn 3+  moments create a \nmolecular magnetic field in this direction [45, 47] , which is also the case of the hexagonal DyMnO 3 single crystal. \nIn this context, it is worth to underline the work by Skumryev et al  [44] in which the magnetic anisotropy of \nhexagonal RMnO 3 manganites was rather attributed to the quadrupolar  charge distribution of the 4f shells of the \nrare-earth elements. \n \nAlong the “easy-plane” ab, the magnetization increa ses with magnetic field following a paramagnetic \ntrend without any steplike transitions, to finally reach a large value under high fields. The magnetiz ation at 7 T \napplied within the ab-plane is about 160 Am 2/kg (7.74 µ B/f.u.), which is more than 85% of the Er 3+  free magnetic \nmoment (9 µ B) [52] and 60 % of the total magnetization in the c ase where the magnetic moments of both, Er 3+  (9 \nµB) and Mn 3+  (4 µ B) [52] sublattices are fully aligned in the directi on of the magnetic field. Keeping in mind the \nmarginal contribution of the Mn sublattice to the t otal magnetization, this result reveals that the Er3+  magnetic \nmoments can be completely aligned when applying suf ficiently high magnetic fields along the ab-plane. In \ncontrast, the h-ErMnO 3 crystal shows distinguishing features along the c-a xis. It behaves like a ferromagnetic 6 \n material under very low magnetic fields (< 0.1 T). This “ferromagnetic” volume vanishes at about 2.8 K  (Fig. 3-\nb). Meier et al [41] attributed this pronounced res ponse under small magnetic fields to the formation of a \nferrimagnetic multidomain state of Er 3+  magnetic moments. However, as this remnant magneti zation is negligible, \nthe observed polarization is most probably associat ed with the appearance of a weak ferromagnetism due  to spin \ncanting, instead of a normal ferromagnetism or ferr imagnetism [47]. For magnetic fields higher than 0. 1 T applied \nalong the c-axis (Fig. 3-b), the magnetization incr eases almost linearly followed by a steplike enhanc ement after \noverpassing a critical field of roughly 0.8 T. For magnetic fields larger than 1 T (Fig. 3-a), the mag netization \ncontinues to increase slightly without any tendency  to saturate even under high magnetic fields. It re aches 40 \nAm 2/kg under a magnetic field of 7 T, being only 21.5 % of that shown by the Er 3+  free ion. On the other hand, \nthe h-ErMnO 3 crystal shows a large magnetic anisotropy when com pared to other hexagonal RMnO 3 phases such \nas HoMnO 3 [16] and DyMnO 3 [26] single crystals. When the magnetic field is c hanged from the easy-direction to \nthe hard-direction, the magnetization at 7 T is red uced by 75 % for h-ErMnO 3 and only 46 % for h-HoMnO 3 [16].  \n \nAs explained above, it is highly challenging to man ipulate the magnetic order of Mn 3+  sublattice even \nunder intense magnetic fields, since the Mn 3+  magnetic moments are strongly coupled in the ab-pl ane. This low \nmagnetic response of the Mn 3+  sublattice implies that the magnetocaloric propert ies of h-ErMnO 3 single crystals \naround the magnetic ordering point are involving ma inly the Er 3+  sublattice. The MCE was initially calculated in \nterm of the isothermal entropy change by numericall y integrating the reported magnetic isotherms in Fi gure 4, \nusing the well-known Maxwell relation [1-7] given by:  \n∆\u0018 (\u0019, 0 − \u001c )=\u001d\u001e\u001f \n\u001f! \"\n\u0016#$\u001c%\u0016\n&     (1) \nIn the absence of hysteretic effects [53], Eq. (1) enables us a fast and accurate characterization of \nmagnetocaloric materials. It is worth noting that t he utilization of Maxwell equation and magnetizatio n data to \nevaluate the entropy change was a subject of contro versy in the past [1, 53-55]. However, this mainly concerned \nmaterials that present large hysteretic effects res ulting in phase-separated states which is not the c ase of the here \nstudied compound. In fact, the h-ErMnO 3 crystal unveils almost zero hysteresis along the a b and c-directions. On \nthe other hand, magnetization data were successfull y used in the past to calculate the entropy change in similar \nmaterials (RMnO 3) showing both first and second order magnetic phas e transitions [13-27].  \nFigure 5-a, b presents the deduced - ΔS as a function of temperature under some represent ative magnetic \nfields applied along the ab-plane and the c-axis. B ecause of the magnetic frustration, the entropy cha nge presents 7 \n a large anisotropy. Following the magnetization tre nd, giant values for the entropy change can be obta ined along \nthe ab-plane. In magnetic fields changing from 0 to  2 T, 0 to 5 T, and 0 to 7 T, - ΔS reaches maximum values of \nabout 12.4, 20.5, and 22.7 J/kg K close to the orde ring temperature of Er 3+  magnetic moments ( ̴ 10 K), respectively. \nAlthough the h-ErMnO 3 crystal contains a rare-earth element with low mag netic moment as compared to Dy 3+  and \nHo 3+  ions (10 µ B) [52], its MCE in terms of - ΔS largely exceeds that shown by other hexagonal RMn O 3 manganites \nsuch as h-DyMnO 3 and h-HoMnO 3 compounds [16, 26]. In similar magnetic fields app lied within the same plane \nat similar temperature range, the maximum values of  –ΔS are evaluated to be 3.86, 13.9 and 18.7 J/kg K fo r h-\nHoMnO 3 [16] and, 8, 15 and 19 J/kg K for h-DyMnO 3 [26], respectively. The marked difference between e ntropy \nchanges shown by h-RMnO 3 (R = Er, Ho, Dy) crystals are mainly attributed to  the distinguished magnetic features \nof their rare-earth elements. Such features are usu ally driven by the 4f-4f and 3d-4f magnetic interac tions that \nsignificantly differ from one h-RMnO 3 to the other. For more details, we refer the inter ested reader to Refs. 27-40.  \nBecause of the strong magnetocrystalline anisotropy , the entropy change following the c-axis was found  \nto be much lower, with respect to the ab-plane. Whe n changing the applied magnetic field along the c-d irection \nfrom 0 to 2 T, 0 to 5 T and 0 to 7 T, - ΔS reaches maximum values of about 6, 9.5 and 11.4 J /kg K, respectively. \nThese values compare well with those obtained for h -HoMnO 3 [16] and h-DyMnO 3 [26] under relatively high \nmagnetic fields applied along the same crystallogra phic axis. However, to get more insight about the m echanism \nbehind the magnetocaloric effect along the c-axis, the contribution of the metamagnetic-like transitio n was also \ninvestigated. Its associated entropy change can be well evaluated by using the Clausius-Clapeyron equa tion given \nby [53, 56]:   \n \n∆\u0018'\u0002() = −∆\u000e*\u0016 +\n*! = −∆\u000e \u001e*!,\n*\u0016 \"-.\n    (2) \nwhere B C and T T are the critical field and the transition temperat ure, respectively. ΔM is the magnetization jump. \nIn contrast to the Maxwell relation, the above equa tion enables us to directly link ΔS with the sudden change in \nthe magnetization. The critical field which is usua lly defined as the inflection point shown by isothe rms within the \nmetamagnetic zone is reported in Figure 6-a as a fu nction of temperature. When increasing temperature,  it evolves \nalmost linearly with a positive rate of about 0.4 T /K. The temperature dependence of - ΔSmeta  is plotted in Figure \n6-b. As shown, the MCE at very low temperatures ori ginates mainly from the Er 3+  magnetic moments on the 2a \nsites that experience a field-induced magnetic tran sition from an antiferromagnetic state toward a fer romagnetic \norder [41, 45, 47], while those on the 4b sites con tribute for little. For example, the application of  a magnetic field 8 \n of 2 T parallel to the c-axis at 1.9 K results in a  full entropy change of 5.33 J/kg K, while the cont ribution of the \nmetamagnetic region is evaluated to be 5 J/kg K. Ev en though the magnetic field is increased by 5 T (f rom 2 to 7 \nT) the entropy change is enhanced only by 2 J/kg K which can be attributed to the ordering of the magn etic \nmoments on the 4b sites. These findings also underl ine the strong coupling between the Mn 3+  and the Er 3+  moments \non the 4b sites that forbids their easy orientation  or ordering in the presence of external magnetic f ields, giving \nthen rise to moderate levels of MCEs. However, as c an be seen in Figure 6-b the MCE associated with th e \ntransitional zone weakens while the temperature inc reases which is due to the disordering of the magne tic moments \nat the 2a sites. At high temperatures, the MCE is e xpected to originate from both 4b and 2a magnetic m oments. \nThe driving forces behind the MCE in h-ErMnO 3 along the c-axis can be well understood when looki ng \ncarefully at the complex magnetic structure of the Er sublattice. According to early works [45], it wa s found that \nthe 120 ° antiferromagnetic ordering of the Mn 3+  magnetic moments at the Néel temperature (80 K) tr iggers the \nmagnetic order of the Er 3+  magnetic moments occupying the 4b sites, while tho se located at the 2a sites remain \ndisordered. This is mainly due to the strong coupli ng between the Mn 3+  (3d 4) and Er 3+ (4f 11 ) 4b magnetic properties. \nBy using inelastic neutron scattering, terahertz an d far infrared spectroscopies as well as some theor etical models, \nChaix et al [45] have unveiled a distinct hybridiza tion between the magnetically coupled Mn and Er (4b ) via a \ncrystalline field transition. In contrast, there is  no coupling between the Er 3+ (2a) and Mn 3+  sublattices [45]. Because \nof the mean field created by the Mn magnetic moment s, it was found that the Er magnetic moments at the  4b sites \nare antiferromagnetically polarized along the c-axi s [45]. According to Meir et al [41], a ferromagnet ic ordering \nof Er 3+  magnetic moments at the 2a site occurs below 10 K following the c-axis. These new 2a established \nexchange interactions result in the reorientation o f both Er 3+  (4b) and Mn 3+  magnetic moments at about 7 and 2.5 \nK, respectively. Meir et al [41] also claimed that at very low temperatures the Er 3+  (4b) and Er 3+  (2b) sites maintain \na ferrimagnetic ground state. However, the reported  magnetic structure in Ref .41 fails to explain the  multiple \nmagnetic phase transitions experienced by h-ErMnO 3 under intense magnetic fields [47]. In a recent re port by \nSong et al [47], it was found that in addition to t he metamagnetic transition that occurs in an extern al magnetic \nfield close to 1 T, h-ErMnO 3 exhibits other magnetic phase transitions when sub jected to 12 and 28 T [47]. \nAccording to these reported data [47], it seems tha t the Er 3+  magnetic moments located at 2a sites are rather \nantiferromagnetically ordered along the c-axis. Fur thermore, under the effect of an external magnetic field of about \n1 T, they present a metamagnetic like transition to  reach a ferromagnetic state (Fig.3), leading accor dingly to high \nlevels of MCE. The Er 3+  magnetic moments on the 4b sites exhibit a spin re orientation under 12 T before being 9 \n fully oriented parallel to the c-axis at 28 T [47].  In contrast to the magnetic moments on the 2a site s, such spins \nordering is accompanied by a small change in the ma gnetization giving rise to negligible thermal effec ts.  \nThe strong coupling between the Mn and 4f (4b) magn etic sublattices seems to be supported by Raman \nscattering data. Figure 7 presents two phonon frequ ency shifts as a function of temperature for h-ErMn O 3 and \nYMnO 3 compounds. The first panel (a) is related to the p honon involving the Mn and the basal oxygen ions (O 2,3) \noscillating in the ab-plane (out-of-phase). For bot h materials, the frequency shift for this mode foll ows the typical \nanharmonic phonon temperature dependence (solid lin e) until an extra hardening occurs below 80 K (T N). This \nparticular atomic motion tends to deform the Mn 3+  magnetic arrangement in the ab-plane resulting in an additional \ncontribution to the constant force trough spin-phon on interaction [42]. This extra hardening has been observed in \nall the hexagonal manganites [57]. The second panel  (b) represents the temperature dependence of the p honon \nrelated to the apical oxygen ions (O1,2) oscillatin g along the c-axis. This mode involves atomic motio ns that are \nusually independent of the Mn 3+  spin frustration when the rare-earth element has n o magnetic moment [57] such \nas YMnO 3. Otherwise, a phonon extra hardening is observed b elow T N as can be seen for the h-ErMnO 3 crystal. \nThis underlines an anisotropic superexchange magnet ic interaction between the Mn 3+  planes and the rare-earth \nions through the apical oxygen ions (see Fig. 1 wit h orange shaded line). This type of magnetic intera ctions was \nfound to be responsible for the appearance of magne toelectric effects in the HoMnO 3 manganite [58]. \nIt is worth noting that the theoretical limit that can be reached by the entropy change in h-ErMnO 3 is \nabout ΔSLimit  = R*Ln(2J+1) = 81 J/kg K, with R is the universal gas constant and J is the angular quantum \nmomentum. The latter can be determined (effective v alue) from the magnetization saturation (g*J*µ B = 13 µ B) \nassuming the Landé factor g equal to 2. On the othe r hand, the obtained entropy change in the field va riation of 7 \nT applied within the ab-plane mostly arising from t he Er sublattice, is only 27 % of ΔSLimit  unveiling the hidden \nmagnetocaloric potential of h-ErMnO 3 single crystals. To attain the theoretical limit, it is necessary to orient the \nmagnetic moments of both Er 3+  (9 µ B) and Mn 3+ (4 µ B) from a perfect paramagnetic to a complete ordered  state. \nThis can be for example carried out under intense m agnetic fields. However, in the case of h-ErMnO 3, this seems \nto be highly challenging from a practical point of view since unusual magnetic fields far above 50 T a re required \n47 . In a recent work by Song et al [47], it was shown  that a magnetic field of 50 T is necessary to full y align the \nEr 3+  magnetic moments, while the triangular frustration  of Mn 3+  magnetic moments (negligible magnetization) \nremains stronger even under the application of this  “colossal” magnetic field. In order to break up th e 120 ° \ncoupling and allowing the Mn sublattice to contribu te to the MCE, new routes other than applying large  magnetic 10 \n fields must be explored. For example, the manipulat ion of the 4f-3d exchange interactions through chem ical doping \nand/or external excitations such as pressure and el ectric field would be interesting ways to investiga te.   \nThe refrigerant capacity (RC) [59] is an important “practical” parameter that enables us to probe the \napplicability of magnetocaloric materials. This is because the RC takes into consideration both the en tropy change \nmagnitude and the working temperature range. It is usually given by  \n/0 = \u001d1\u0018 (\u0019)$\u0019 !2\n!+ (3) \nwhere T C and T H are the cold and hot temperatures corresponding to  the half maximum of ΔS (T). It is worth \nnoting that in certain cases the RC cannot be used as a reliable parameter for the characterization of  magnetocaloric \nmaterials. This usually concerns some specific mate rials that exhibit a weak entropy change over a ver y large \ntemperature range, giving rise to unreasonable larg e values for RC. However, this is not the case of t he present \nstudied crystal that exhibits a giant entropy chang e over a wide working temperature window. In fact, sufficiently \nhigh MCE levels are necessary to achieve an efficie nt refrigeration process via the enhancement of hea t exchanges \nbetween the magnetocaloric refrigerant and the carr ier fluid in cooling devices. In this case, the RC could be used \nas a good figure of merit to identify the potential  of magnetocaloric materials in practical applicati ons [60]. For \nexample, it was more recently shown by Niknia et al  [60] that the RC linearly scales with the exergeti c cooling \npower of an AMR-magnetic refrigerator. For the h-Er MnO 3 compound, the obtained RC in the magnetic field \nchange of 7 T applied within the ab-plane is more t han 440 J/kg. This value is comparable to that of t he best \northorhombic RMnO 3 such as o-DyMnO 3 [27] and, much larger than those early reported fo r hexagonal RMnO 3 \nphases such as h-HoMnO 3 and h-DyMnO 3 single crystals [16, 26, 27]. Along the c-axis, th e h-ErMnO 3 reveals a \nmoderate RC of only about 14 % (64 J/kg) of that ob served for a field along the ab-plane (for 7 T). On  the other \nhand, in a similar magnetic field, the exhibited RC  by h-DyMnO 3 and h-HoMnO 3 single crystals along their c-axis \n[16, 26] is about three times larger if compared to  that of h-ErMnO 3 along the same axis. Such marked differences \ncould be mainly attributed to the specific magnetic  features of h-RMnO 3 (R = Er, Ho. Dy) crystals as explained \nabove. In addition, the h-ErMnO 3 exhibits a narrow magnetocaloric operating tempera ture range along the c-axis, \nmarkedly contrasting with the ab-plane as shown in Figure 8-a. Under 7 T, the difference T H-TC is lower than 7 K \nfollowing the c-axis, while it is more than 26 K al ong the ab-plane. The limited operating magnetocalo ric window \nalong the c-axis is mainly attributed to the first order character of the AF-F transition taking place  at temperatures \nbelow 4 K (see Figs. 2 and 3). In fact, the nature of the magnetic transitions along the ab and c-dire ctions were 11 \n checked using the Banerjee criterion also known as Arrott plots [61] (Fig. 8-c, d). This approach cons ists in the \nanalysis of the slope of M2 versus H/M isotherms. Their negative slope (Fig. 8 -c) along the c-axis (S-like shape) \nindicates that the phase transition associated with  the ordering of Er 3+  magnetic moments located at 2a sites (at \nvery low temperatures) is first-order in nature. Al so, the recently reported specific heat measurement s by Song et \nal. [47] for h-ErMnO 3 along the c-axis present pronounced spikes for mag netic fields higher or equal to 1 T being \na signature of first order phase transitions. On th e other hand, a rapid change in the magnetization a long the c-axis \nafter overpassing a critical field (see Fig3-b), wh ile the latter increases with increasing temperatur e (Fig.6-a), is \nanother indication for the occurrence of a first or der magnetic transition along the c-axis [53]. In c ontrast, the \npositive slope of M2 versus H/M along the ab-plane (Fig. 8-d) indicates  that the “paramagnetic” ordering of Er 3+  \nsublattice is of second order.   \nIt is known that the delivered cooling power by a m agnetocaloric device is mainly determined by the \nmagnitude of the entropy change. However, the adiab atic temperature change ΔTad  is extremely important since it \nstrongly influences the strength of heat exchanges and accordingly the temperature span between its ho t and cold \nsources. As the h-ErMnO 3 crystal exhibits a second magnetic order transition  along the ab-plane, its adiabatic \ntemperature change was deduced by using the followi ng equation [1, 14]: \n∆\u0019)* = −!\n34∆\u0018        (4) \nwhere the specific heat (C P) values were taken from Refs. 47 and 49. It is wor th noting that in certain cases where \nthe specific heat depends strongly on the magnetic field, Eq. (4) is may be not suitable for the calcu lation of the \nadiabatic temperature change ΔTad . However, according to a recently reported work by  Song et al [47], it can be \nclearly seen that for h-ErMnO 3, the specific heat and particularly the ratio (Cp /T) at temperatures far above the \nordering point of Er +3  magnetic moments, depend slightly on the magnetic field, which enables us to reasonably \nevaluate the adiabatic temperature change in this t emperature range using Eq (4). For practical reason s, the h-\nErMnO 3 adiabatic temperature change is evaluated close to  20 K that is the liquid hydrogen temperature. Arou nd \nthis temperature, the ratio Cp/T is about 1.35 J/kg  K 2 whatever the value of the applied magnetic field [ 47]. The \nfield dependence of ΔTad  along the ab-plane is reported in Figure 8-b. For magnetic field variations of 2, 5 and 7 \nT within the ab-plane, the ΔTad  at 19 K reaches values of about 1.5, 7.5 and 12 K,  respectively. However, the ΔTad  \nshown by h-ErMnO 3 crystals is expected to be much larger at low temp eratures since the MCE usually maximizes 12 \n around the magnetic ordering point. A detailed stud y of ΔTad  in h-ErMnO 3 crystals using specific heat \nmeasurements under magnetic fields will be reported  in the future.  \nMore recently, the magnetocaloric properties of h-E rMnO 3 polycrystalline samples were investigated by \nSattibabu et al  [19] and Vinod et al [20]. In both works, the repor ted entropy change is much lower than for our \nsingle crystals. Under a magnetic field variation f rom 0 to 5 T, the maximum ΔS that can be reached by using h-\nErMnO 3 polycrystals is only 14 J/kg K instead about 20.5 J/kg K shown by h-ErMnO 3 single crystals along the \nab-plane (see Fig.5-a.). This result points out the  negative impact of the magnetocrystalline anisotro py usually \nexhibited by non-cubic materials such as h-ErMnO 3 on the polycrystalline magnetocaloric performance [62]. As \ncan be clearly seen in Figure 3-a, the h-ErMnO 3 compound presents a large magnetic anisotropy. Bec ause of the \nrandom orientation of grains in polycrystalline sam ples, the obtained magnetization and accordingly th e MCE \nunder a given magnetic field are rather equivalent to the mean value of those associated to the main c rystallographic \naxes (hard, intermediate and easy-axes). In the cas e of our crystals, the mean value of its entropy ch anges along \nthe ab and c-orientations (15 J/kg K for 5 T) is pr actically similar to the reported entropy change fo r polycrystalline \nsamples [19, 20]. Therefore, in magnetocaloric rege nerators, it is necessary to orient the easy-axis o f \npolycrystalline refrigerants parallel to the magnet ic field direction. In this context, textured magne tocaloric \npowders under magnetic fields would be a promising solution [63]. It would enable us to maximize the M CE in \nthe regenerators, increasing then the efficiency of  magnetic cooling systems. \nAs demonstrated above, the h-ErMnO 3 single crystal generates a giant conventional MCE that can be \nobtained by the standard magnetization-demagnetizat ion process (field variation), within the ab-plane.  However, \nthe large anisotropy shown by the MCE between the a b and c-orientations indicates that additional MCEs  can be \nobtained by spinning the h-ErMnO 3 crystals around their a or b-axes in constant magn etic fields. Such process is \nof great interest from a practical point of view, s ince the resulting RMCE would enable the implementa tion of \nmore compact and efficient rotary magnetic refriger ators with simplified designs [11, 27]. It is worth  noting that \nthe difficulty associated with the growth of crysta ls remains a serious obstacle to their utilization in RMCE-based \ncooling devices. However, this drawback can be avoi ded by using, for example, textured polycrystalline  powders \nof these crystals under magnetic fields, which is m ore interesting from both practical and economical points of \nview. In this way, textured powders of anisotropic magnetic compounds using for example the described process \nin Ref. 63 would also enables us to generate large RMCEs without the need to single crystals.  13 \n The isothermal entropy change arising from the rota tion of h-ErMnO 3 single crystals within the bc or ac \nplanes was evaluated based on the entropy changes r eported above. Considering initially the magnetic f ield parallel \nto the c-axis, the rotation of h-ErMnO 3 around the b (or a)-axis by an angle of 90 ° resul ts in a change of the total \nentropy by [1, 14-16, 24, 27] \n1\u0018 5-)6 =  ΔS (H//ab)  −  ΔS (H//c)     (5) \nwith ΔS (H//ab) and ΔS (H//c) are the entropy changes associated with th e magnetization of h-ErMnO 3 along the \nab-plane and the c-axis, respectively. The temperat ure dependence of ΔSR,c-ab for several constant magnetic fields \nis plotted in Figure 9-a. The associated adiabatic temperature change ΔTR,c-ab  that can be calculated by using Eq. \n(4) is reported in Figure 9-b as a function of magn etic field at 19 K. As shown, the h-ErMnO 3 compound provides \na giant RMCE over the temperature range around 10 K . Surprisingly, the displayed RMCE is about three t imes \nlarger than that shown by h-HoMnO 3 crystals [16] in a similar working temperature ran ge (see Fig. 10-a). The \nrotation of h-ErMnO 3 crystal around the a (or b) -axis in constant magn etic fields of 2, 5, and 7 T leads to maximum \nentropy changes of 7, 17 and 20 J/kg K, respectivel y. Also, the generated ΔSR,c-ab  is even larger than that found in \nthe vast majority of orthorhombic RMn 2O5 and RMnO 3 phases, usually known for their gigantic \nmagnetocrystalline anisotropy [27] such as TbMnO 3 (18 J/kg K for 7 T) [13, 14, 24], o-DyMnO 3 (18 J/kg K for 7 \nT ) [18], HoMn 2O5 (12.3 J/kg K for 7 T) [11] and TbMn 2O5 (13.35 J/kg K for 7 T) [12]. On the other hand, in  a \nconstant magnetic field of 7 T initially parallel t o the c-axis, the adiabatic temperature change resu lting from the \nrotation of h-ErMnO 3 crystals at 19 K between their c and a (or b)-axes  is about 12 K. Because of the reversible \ncharacter of the adiabatic temperature change (abse nce of hysteresis), the rotation of h-ErMnO 3 crystals in the \nopposite direction at ̴31 K which is closer to the critical (boiling) poin t of hydrogen (33 K), enables us to reduce \ntheir temperature down to about 20 K which is the t emperature of fully liquid state of hydrogen. This generated \nnegative thermal effect could be then used for exam ple to liquify gaseous hydrogen [8] as proposed in Ref.11.\n   \nThe giant RMCE observed in h-ErMnO 3 single crystals is mainly attributed to their uniq ue magnetic \nfeatures when compared to other hexagonal RMnO 3 phases [27]. As shown in figure 3-b, the first-ord er \nmetamagnetic transition (involving 2a sites) giving  rise to a large magnetocaloric effect along the c- axis, occurs at \nvery low temperatures (around 3 K). Consequently, t he MCE along this direction is peaked around 3 K an d rapidly \ndeclines above this temperature (Fig. 8-a) due to t he weakness of AF-F transition. In contrast, becaus e of the \n“paramagnetic”-like behaviour of Er 3+  magnetic moments along the ab-direction, the MCE p eak position increases 14 \n significantly with magnetic field change. When the latter is varied from 2 to 7 T, the MCE maximum is shifted \nfrom 3.5 to 9.5 K for H//ab and, only from 2.7 to 3 .7 K for H//C (Fig. 5). Consequently, the MCE aniso tropy is \nmaximized between the c and ab-orientations particu larly under relatively high magnetic fields, leadin g then to a \ngiant RMCE. This markedly contrasts to other hexago nal RMnO 3 single crystals [16, 26, 27]. For example, under \na magnetic field changing from 0 to 7 T at 11 K, th e entropy change shown by h-ErMnO 3 along the ab-plane is \nabout 10 times larger than that along the c-axis. I n similar conditions of temperature and magnetic fi eld, the ΔS \nshown by h-HoMnO 3 along the ab-plane is only 1.4 times larger if com pared with its equivalent following the c-\naxis [16].  \nThe complexity as well as the role of the magnetocr ystalline anisotropy in the determination of the RM CE \nin h-ErMnO 3 crystals was also investigated. For this purpose, the coherent rotational (CR) model [11-12, 14, 24, \n52] was used to calculate the entropy change ΔSMCA  associated with their rotation between the c and a b directions. \nThis particularly allows us to quickly figure out w hether the RMCE arises directly from the magnetocry stalline \nanisotropy, while informing us on its complexity. I n Figure 10-b, we present the calculated ΔSMCA  as a function \nof angle ß at 8 and 15 K for a constant magnetic fi eld of 7 T. β represents the angle between the magnetic field \ndirection and the easy-axis (a or b). When the h-Er MnO 3 crystal is rotated from β = 90 ° to β = 0 ° under 7 T, the \nassociated ΔSMCA  is 16.5 J/kg K and 8 J/kg K at 8 K and 15 K, respe ctively. The corresponding experimental \nvalues are 16 and 16.3 J/kg K, respectively. These findings demonstrate that just below the ordering p oint of Er 3+  \nmagnetic moments along the ab-plane ( ̴ 11 K, under 7 T), the experimental and calculated values are in fair \nagreement. This underlines the fact that the shown RMCE at this temperature is mainly arising from the  \nmagnetocrystalline anisotropy. In contrast, a marke d deviation is observed between the calculated and \nexperimental rotating entropy changes at 15 K. Such  difference is probably due to the fact that the CR  model does \nnot include thermal fluctuations contributions usua lly taking place at high temperatures. At temperatu res below 8 \nK, it is highly challenging to identify the mechani sm behind the RMCE because of the complexity of the  \nmagnetocrystalline anisotropy. In this temperature range, the magnetization presents a pronounced jump  after \ncrossing a critical magnetic field making more diff icult the determination of anisotropy constants by using the CR \nmodel, pointing out to the limitations of the latte r. \nThe last point in this paper concerns the efficienc y of h-ErMnO 3 single crystals in functional \nmagnetocaloric devices. For this purpose, the sugge sted parameter in Refs. 64 and 65 and given by Ƞ = |Q/W| is \nutilized as a figure of merit. Q is the thermal ene rgy generated by the magnetocaloric material close to the transition 15 \n temperature under external magnetic fields, while W  represents the magnetic work that is required to d rive this \nisothermal heat. Following the reported procedure i n Moya et al [64], both Q and W can be determined f rom \nentropy change curves and magnetic isotherms, respe ctively. Preliminary investigations reveal that all  the \nquantities Ƞ, Q, and W depend strongly on the crystal orientation , magnetic field and initial temperature. For \nexample, when the used magnetic field varies from 0  to 2 T within the ab-plane at ̴3.5 K, a magnetic work of 131 \nJ/kg is needed to produce an isothermal heat of 43. 4 J/kg, leading then to an efficiency of about 0.33 . For a \nmagnetic field of 7 T, the required magnetic work a nd the generated heat (at 11 K) markedly increases to reach \n683 J/kg and 247 J/kg, respectively. The associated  efficiency is slightly increased to 0.36 which can  be more \nprobably explained by the saturation of magnetizati on along the ab-direction. On the other hand, it wa s observed \nthat under relatively low magnetic fields reachable  by permanent magnets, the magnetocaloric response of h-\nErMnO 3 is significantly efficient along the c-direction. For a driving magnetic field of 2 T applied along t he c-axis \nat ̴ 2.75 K, it was found to be 0.55 (W = 28 J/kg, Q = 15.52 J/kg), instead only 0.33 for the ab-plane (at  ̴3.5 K). It \nis worth noting that the Ƞ parameter is used as a guide to identify the effic iency of magnetocaloric materials only \nin isothermal processes and not within a complete t hermodynamic cycle. In fact, the thermodynamic effi ciency of \nmagnetocaloric devices depends on many other parame ters such as the temperature gradient that a materi al can \nachieve after several AMR-cycles [1], thermal losse s and additional works caused by eddy currents, mag netic field \nand energy recovery via the compensation of magneti c forces [66].  \nIV. Conclusions  \nIn this work, we have mainly focused on the magneti c and magnetocaloric properties of h-ErMnO 3 single \ncrystals. The latter reveal a giant conventional ma gnetocaloric effect that can be generated by applyi ng magnetic \nfields along the ab-plane. In the magnetic field ch ange of 7 T applied within the ab-plane, the result ing isothermal \nentropy change presents a maximum value of 22.7 J/k g K close to 10 K. In contrast, the partial polariz ation of Er 3+  \nmagnetic moments (located at 2a sites) along the c- axis via a field-induced magnetic transition leads to a less-\nimportant MCE that is peaked within a narrow temper ature range with a maximum of 11.4 J/kg K (7 T) clo se to 3 \nK. Additionally, the distinct features of Er 3+  magnetic moments along the c-axis and the ab-plane  lead to a strong \nanisotropy of the MCE around 10 K. Consequently, th e h-ErMnO 3 compound presents a giant RMCE that is several \ntimes larger than that showed by other hexagonal RM nO 3 phases. The rotation of h-ErMnO 3 single crystals in a \nconstant magnetic field of 7 T within the ac or bc- planes enables to generate a maximum entropy change  of 20 \nJ/kg K. These findings open the way for the impleme ntation of h-ErMnO 3 compounds as refrigerants in cryo-16 \n magnetocaloric liquefiers. 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Defay, V. Heine and N. D. Mathur, Nat. Phys. 11, 202 (2015) 20 \n [65] E. Defay, S. Crossley, S.  Kar-Narayan, X. Moy a & N. D. Mathur, Adv. Mater. 25, 3337 (2013). \n[66] M. Balli, C. Mahmed, P. Bonhote, and O. Sari, IEEE Trans. Magn. 47, 3383 (2011). \n \nAcknowledgments  \nThe authors thank M. Castonguay, S. Pelletier and B . Rivard for technical support. We acknowledge the financial \nsupport from NSERC (Canada), FQRNT (Québec), CFI, C IFAR, Canada First Research Excellence Fund (Apogée  \nCanada) and the Université de Sherbrooke. \nFigure captions \nFigure 1: (left) Micro-Raman spectra at 10 K for h-ErMnO 3. Those of YMnO 3 considered as a reference for \nhexagonal RMnO 3 phases are also reported for comparison. The backs cattering and light polarization \nconfigurations b(cc)b and c(ab)c, allow the observa tion of A 1 and E 2 phonon symmetries, respectively. (right) \nCrystallographic structure of h-ErMnO 3. R1 and R 2 correspond to 2a and 4b sites, respectively. \nFigure 2:  (a) Temperature dependence of ZFC and FC magnetiza tions of h-ErMnO 3 under a magnetic field of 0.1 \nT along the ab-plane and the c-axis. (b) Thermomagn etic curves of h-ErMnO 3 under magnetic fields going from \n0.2 T up to 4 T (H // c). \nFigure 3:  (a) Isothermal magnetization curves of the single crystal h-ErMnO 3 at 2 K along the ab-plane and the c-\naxis. (b) Collected magnetic isotherms along the c- axis at very low temperatures and, under low magnet ic fields. \nFigure 4:  Isothermal magnetization curves of the single crys tal h-ErMnO 3 for (a) H// ab, (b) H// c and (c) H//c. \nFigure 5:  Temperature dependence of the isothermal entropy c hange in h-ErMnO 3 under different magnetic fields \nfor (a) H// ab and (b) H// c. \nFigure 6: (a) Temperature dependence of the critical field B C (along the c-axis) corresponding to the metamagnet ic \ntransition taking place in the single crystal h-ErM nO 3. (b) The associated entropy change deduced from Cl ausius–\nClapeyron equation. \nFigure 7: Temperature dependence of the Raman-active phonons E2(5) (a) and A 1(9) (b) for both h-ErMnO 3 \nand YMnO 3 single crystals. (Right) Description of ion relati ve motions. 21 \n Figure 8  (a) Comparison between the h-ErMnO 3 entropy changes related to H// ab  and H// c for 7 T. (b) Adiabatic \ntemperature change of h-ErMnO 3 as a function of magnetic field at 19 K applied wi thin the ab-plane. (c) and (d) \nArrott plots of h-ErMnO 3 close to the ordering point of Er 3+  magnetic moments along the c-axis and the ab-plane , \nrespectively.  \nFigure 9:  (a) Entropy changes related to the rotation of h-E rMnO 3 from the c to the a (or b)  direction by 90 ° in \ndifferent constant magnetic fields, with magnetic f ield initially parallel to the c-axis. (b) Associated adiabatic \ntemperature change as a function of magnetic field at 19 K. \nFigure 10:  (a) Temperature dependence of the rotating entropy  change in the hexagonal ErMnO 3 and HoMnO 3 \n[16] single crystals for 7 T. (b)The calculated rot ating entropy change from the coherent rotational m odel for h-\nErMnO 3. The magnetic field is initially parallel to the c -axis. β is the angle between the easy direction (a or b ax es) \nand the magnetic field.  \n \n \n \n \n \n \n \n \n \n \n \n 22 \n  \nFigure 1 \n23 \n  \nFigure 2 \n \n \n \n24 \n  \nFigure 3 \n25 \n  \nFigure 4 \n26 \n  \nFigure 5 \n \n \n \n \n \n \n \n \n27 \n  \nFigure 6 \n \n \n \n \n \n \n28 \n  \nFigure 7  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n29 \n  \nFigure 8 \n \n \n \n30 \n  \nFigure 9 \n \n \n \n \n \n \n \n31 \n  \nFigure 10 \n"    },    {        "title": "1811.12100v1.Magnetocrystalline_Anisotropy_of_Fe_based__L1_0__Alloys__Validity_of_Approximate_Methods_to_Treat_the_Spin_Orbit_Interaction.pdf",        "content": "Magnetocrystalline Anisotropy of Fe-based L10Alloys: Validity of Approximate\nMethods to Treat the Spin-Orbit Interaction\nM. Blanco-Rey,1, 2,\u0003J.I. Cerd\u0013 a,3and A. Arnau4, 1, 2\n1Departamento de F\u0013 \u0010sica de Materiales, Facultad de Qu\u0013 \u0010mica, Universidad del\nPa\u0013 \u0010s Vasco UPV/EHU, Apartado 1072, 20080 Donostia-San Sebasti\u0013 an, Spain\n2Donostia International Physics Center, Paseo Manuel de Lardiz\u0013 abal 4, 20018 Donostia-San Sebasti\u0013 an, Spain\n3Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain\n4Centro de F\u0013 \u0010sica de Materiales CFM/MPC (CSIC-UPV/EHU),\nPaseo Manuel de Lardiz\u0013 abal 5, 20018 Donostia-San Sebasti\u0013 an, Spain\n(Dated: November 30, 2018)\nFirst-principles calculations are used to gauge di\u000berent levels of approximation to calculate the\nmagnetocrystalline anisotropy energies (MAE) of \fve L10FeMe alloys (Me=Co, Cu, Pd, Pt, Au).\nWe \fnd that a second-order perturbation (2PT) treatment of the spin-orbit interaction (SOI) breaks\ndown for the alloys containing heavier ions, while it provides a very accurate description of the MAE\nbehaviour of FeCo, FeCu, and FePd. Moreover, the robustness of the 2PT approximation is such\nthat in these cases it accounts for the MAE of highly-non-neutral alloys and, thus, it can be used to\npredict their performance when dopants are present or when they are subject to applied gate bias,\nwhich are typical conditions in working magnetoelectric devices. We also observe that switching of\nthe easy axis direction can be induced in some of these alloys by addition or removal of, at least,\none electron per cell. In all cases, the details of the bandstructure are responsible for the \fnally\nobserved MAE value and, therefore, suggest a limited predicting power of models based on the\nexpected orbital moment values and bandwidths. Finally, we have con\frmed the importance of\nvarious calculation parameters to obtain converged MAE values, in particular, those related to the\naccuracy of the Fermi level determination.\nI. INTRODUCTION\nIn a model system of interacting magnetic moments\nvarious contributions can be identi\fed that lead to the\nobservation of the magnetic anisotropy, i.e. the exis-\ntence of a preferential magnetization direction in the sys-\ntem. These terms are the classical dipole-dipole inter-\naction, resulting in the so-called shape anisotropy, and\nquantum-mechanical ones, as anisotropic exchange1and\nthe magnetocrystalline anisotropy (MCA). The latter, of\nrelativistic character, has its origin in the electronic spin-\norbit interaction (SOI).\nThe historical development of non-volatile memories\nbased in the property of magnetorresistivity has been\nclosely related to the ability of balancing out those com-\npeting contributions. The key achievement was the per-\npendicular magnetic anisotropy in thin \flm heterostruc-\ntures, since the out-of-plane MCA at the interface tends\nto dominate in-plane shape anisotropy. The e\u000ecient spin\norientation control in the ferromagnetic (FM) electrodes\nof magnetic tunneling junctions is still a technological\nchallenge. The use of external magnetic \felds for this\npurpose is evolving towards voltage control of spintronic\ndevices2,3and spin transfer torques induced by spin-\npolarized currents4. These advances motivate e\u000ecient,\nrobust and accurate modelling of the physics of the MCA\nfor di\u000berent materials/interfaces under external stimuli,\nsuch as external \felds or strain.\nThe interplay of MCA and dimensionality is con-\nsidered as a promising route for the development of\nspintronics. Indeed, the MCA plays a central role in\nthe magnetic properties of two-dimensional materials,since it can prevent thermal \ructuations from destroy-\ning long-range magnetic order5. In solids with cubic\nsymmetry, the MCA is an e\u000bect of fourth order in the\nSOI strength, but a tetragonal distortion can enable\na second-order MCA. This idea is behind the materi-\nals used in (or proposed for) some of the aforemen-\ntioned devices. For example, multiferroics allow for\nstrain-mediated magneto-electric coupling6,7and, more\nrecently, tetragonal Heusler alloys have attracted atten-\ntion for combining high MCA and half-metallicity8,9. We\nput the focus of this work on transition-metal alloys.\nThey are versatile, since their structure and magnetic\nproperties can be tailored by varying the stoichiome-\ntry, as it is the case of the strongly magnetostrictive\n\\galfenol\" (Fe 1\u0000xGax)10,11, Fe 1\u0000xCox12{14, and Cu-Ni\n\flms15. Several theoretical studies have shown that a\nbias voltage could signi\fcantly a\u000bect the MCA16,17and,\nin fact, an electric-\feld-induced MCA switching has been\nrealized in Fe 30Co70alloy \flms18.\nDensity Functional Theory (DFT) calculations that\ninclude SOI provide information on the magnetocrys-\ntalline anisotropy energy (MAE). Nonetheless, its small\nmagnitude, often in the sub-meV range, imposes strin-\ngent convergence to the DFT calculations at the ex-\npense of high computational demands. In practice,\nfully relativistic calculations that include SOI are car-\nried out by \frst computing self-consistently the Hamil-\ntonian of the system including the scalar relativistic\nterm and next adding the spin-orbit contribution. The\nlatter may be treated self-consistently (SCF) or non-\nself-consistently19,20(NSCF) within the so-called force\ntheorem21to obtain the MAE. NSCF is often consideredarXiv:1811.12100v1  [cond-mat.mtrl-sci]  29 Nov 20182\nFIG. 1. Structure of the L10tetragonal unit cell and carte-\nsian axes that provide a good match between the MLWFs and\ntheY2morbital functions.\nto produce a good estimate. Other electronic structure\ncalculations use the second-order perturbation theory\n(2PT) to calculate MAEs and orbital moments. These\nalternative approaches rely on the knowledge of the spin-\norbit e\u000bects on atomic orbitals22{24and have been for-\nmally formulated in the literature with Green's functions\nunder di\u000berent \ravours25{28.\nIn this work, we address various methodological as-\npects of the application of DFT to the study of the MCA\narising from spin-orbit e\u000bects in the bandstructure of ex-\ntended systems using FeMe (Me=Co, Cu, Pd, Pt, and\nAu) alloys with L10structure as case studies (see Fig. 1).\nMethodologically, our aim is two-fold: (i) focusing on the\nphysics of the problem, we take the SCF MAEs as refer-\nence values and examine the performance of the NSCF\nand 2PT approximations, and (ii) on the computational\nside, we analyze the convergence criteria required to get\nan accurate MAE with each method. The chosen model\nsystems allow to establish the applicability of these ap-\nproximations in terms of the SOI strength (larger as we\nmove downwards in the periodic table) and of the charac-\nter of the hybrid bands ( d\u0000dfor FeCo, FePd, and FePt,\nandd\u0000sfor FeCu and FeAu).\nOur calculations show that the 2PT description of the\nMAE of the lighter alloys is very reliable even under\nconditions of strong bias voltage, while it breaks down\nfor heavier elements. In general, the use of SCF hardly\nchanges the NSCF MAEs, as long as the values are care-\nfully converged. Regarding the nature of the MCA in\nthis family of alloys, we \fnd its physical origin in the\navailability of empty Fe electron states, although the\nwhole valence bandstructure contributes to its \fnal mag-\nnitude. The results presented here point to perturbative\napproaches as a feasible route to modelling the MCA\nand related properties of magnetic metals, speci\fcally of\ntetragonal alloys, in working conditions under electro-static \felds. The general character of these approaches\nsuggests that they can be applied to lower-dimensional\nsystems featuring dispersive bands.\nThe paper is organized as follows: section II describes\nthe three methods used here to compute the MAE,\nnamely SCF (section II A), NSCF (section II B) and, in\nmore detail, two di\u000berent implementations of a 2PT for-\nmula in DFT codes (section II C). Details of the DFT\ncalculation parameters for the Fe-based alloys are pre-\nsented in section III. The results for the charge neutral\nand non-neutral cases are shown in sections IV A and\nIV B, respectively. Finally, conclusions are drawn in sec-\ntion V.\nII. THEORETICAL BACKGROUND\nThe spin-orbit interaction (SOI) Hamiltonian is gener-\nally written as a sum over one-electron operators:\nHSO=X\ni\u0018ili\u0001si (1)\nwhere liandsiare the orbital and spin momentum op-\nerators, respectively, acting on the i-th electron in the\nsystem and \u0018iis a constant that accounts for the SOI\nstrength. In practice, as most of the relevant electronic\nand magnetic properties of solids derived from SOI orig-\ninate from valence electrons, only outer-shell and semi-\ncore electrons are considered in our \frst principles calcu-\nlations. Since \u0018iis proportional to the radial derivative of\nthe potential, it increases with the atomic number. Fur-\nthermore, it is often a good approximation to take the\nsame value for all the electrons within the same l-shell.\nIn the next subsections we describe the three methods\nconsidered in this work to evaluate the MAEs from \frst-\nprinciples, all of them including SOI at di\u000berent levels\nof approximation. In particular, we will examine under\nwhich conditions a second-order perturbative approach,\nwhere\u0018iacts as the perturbation constant, breaks down.\nA. Self-consistent MAE (SCF)\nOur reference ab initio MAE value is obtained by\nsubstracting the total energies Etotbetween two fully-\nrelativistic self-consistent calculations, which include\nSOI, for two di\u000berent orientations of the magnetization,\nMAE =Ex\ntot\u0000Ez\ntot (2)\nwhere the spins are aligned along the OX andOZdi-\nrections shown in the L10unit cell model of Fig. 1. The\nmain shortcoming of this method is its computational\ncost, since Eq. 2 implies substracting two large numbers,\nwhich requires demanding convergence criteria. In fact,\nthe obtention of well-converged MAEs from Eq. 2 is cru-\ncial in this work (see details in the next section), since\nthey are used as a benchmark for the approximations\ndescribed in the next subsections.3\nB. Non-self-consistent MAE (NSCF)\nA scalar relativistic ground state (GS) is converged in a\nspin-polarized calculation without SOI. The so-obtained\ncharge density is used to initialize a fully-relativistic cal-\nculation (i.e. non-collinear) by turning it into a block-\ndiagonal charge density matrix. Then, the spin-orbit\nHSOterm calculated for a given magnetization axis is\nadded to the scalar-relativistic hamiltonian H0and new\neigenvalues are calculated by diagonalization without fur-\nther self-consistent cycles. We denote the resulting total\nenergy change \u0001 Ex;z\ntotand the corresponding charge den-\nsity change \u0001 \u001ax;z. The MAE is approximated as the\ndi\u000berence in the band energies, Ex;z\nband, between the two\norientations of the magnetization, bearing in mind that\nthe Fermi levels are in general di\u000berent for the two ori-\nentations, as they are computed independently,\nMAE'\u0001Ex\ntot\u0000\u0001Ez\ntot'Ex\nband\u0000Ez\nband\n=NkX\nkNbX\nn[fx(\u000fx\nkn)\u000fx\nkn\u0000fz(\u000fz\nkn)\u000fz\nkn] (3)\nHere, the sum runs over one-electron eigenvalues \u000fx;z\nkn,\ncalculated with the spins aligned along the OXandOZ\ndirections, respectively, and integrated over the entire\n\frst Brillouin Zone (1BZ). Nbbands are considered (in-\ndexn) and a discrete grid of Nkpoints (index k) is used\nto sample the 1BZ. fx;zare the Fermi-Dirac distribu-\ntion functions, which depend on the magnetization axes\nthrough the Fermi energy, while the \fnite electronic tem-\nperaturekTacts as a smearing parameter. The approx-\nimation is based on the fact that \u0001 Ex\ntotand \u0001Ez\ntotare\ncorrect to order (\u0001 \u001ax)2and (\u0001\u001az)2, respectively. The\nmethod is thus sometimes called \\second variation\"20or\n\\force theorem\"19,21. Eq. 3 is correct to order \u0001 \u001ax;z\nwhile the (\u0001 \u001ax;z)2-order corrections have a small e\u000bect,\nsince there are cancellations from the two magnetiza-\ntion directions19. If one further assumes that the self-\nconsistency cycles introduce negligible modi\fcations in\nthe charge density matrix and in the exchange and cor-\nrelation potential, then Eq. 2 and Eq. 3 should provide\nvery similar MAE values, although with a considerable\nreduction in the computational cost in the latter case.\nC. Second-order perturbative MAE (2PT)\nA widely used alternative approximation treats the\nSOI as a second-order perturbation (2PT) to the many-\nbody GS,j\t(0)i. The general expression for the 2PT\nenergy correction is\n\u0001E(2)\nSO=X\ni6=0h\t(0)jHSOj\t(i)ih\t(i)jHSOj\t(0)i\nE0\u0000Ei(4)\nwhere the sum runs over excited states. In a many-body\nlanguage the GS j\t(0)iis formed by occupation of thelowest-lying one-electron Kohn-Sham eigenstates up to\nthe Fermi level. Each excited state j\t(i)iis then con-\nstructed by creating electron-hole ( e\u0000h) pairs using the\nunoccupied Kohn-Sham eigenstates. Thus, the E0\u0000Ei\nterm in the denominator is simply the energy di\u000berence\nbetween the occupied and unoccupied eigenvalues asso-\nciated to the particular e\u0000hexcitation22{25. The per-\nturbative expansion may then be written in terms of the\nGS Kohn-Sham eigenstates jkn\u001bias follows:\n\u0001E(2)\nSO=1\nNkNkX\nkNbX\nn;n0X\n\u001b;\u001b0f(\u000fkn\u001b)[1\u0000f(\u000fkn0\u001b0)]\n\u000fkn\u001b\u0000\u000fkn0\u001b0\u0002\nhkn\u001bjHSOjkn0\u001b0ihkn0\u001b0jHSOjkn\u001bi (5)\nwhere\u001b;\u001b0stand for the spin indices.\nThe second order formula given by Eq. 5 is applica-\nble only to a non-degenerate ground state. A degener-\nate ground state in an extended metallic system happens\nwhen there is a band crossing at a certain k-point pre-\ncisely at the Fermi level and this band pair is coupled\nbyHSO(i.e. the corresponding matrix element is not\nzero) This can happen eventually, and in this situation\nthe eigenstate pair should be treated separately by \frst-\norder degenerate state perturbation theory. However, in\na calculation with a large Nkthe contribution to \u0001 E(2)\nSO\nof these exactly-degenerate states would be negligible,\nsince only a handful of band crossings are expected at\nthe Fermi level, and they contribute with a factor of order\n\u0018=Nk29. A su\u000eciently \fne k-grid can map the spin-orbit\nband splitting e\u000bect nearby the crossing, so that Eq. 5\ncan be safely used.\nIn practice, it is convenient to express Eq. 5 in a basis\nwhose elements have well de\fned lmquantum numbers\n(spherical harmonics Ylm). A natural choice are atomic\norbitals (AOs), which already constitute the basis set in a\nnumber of DFT-based packages, leading to Bloch Kohn-\nSham eigenstates of the form:\njkn\u001bi=1pNkX\nReik\u0001RN\u000bX\n\u000bcn\u001b\n\u000b(k)j\u000b(R);\u001bi(6)\nwhere Rruns over the lattice vectors, j\u000b(R)idenotes an\nAO located in unit cell RandN\u000bis the total number of\nAOs in the basis set. Inserting Eq. 6 into 5 yields:\n\u0001E(2)\nSO=1\nNkNkX\nkX\n\u001b\u001b0N\u000bX\nn;n0f(\u000fkn\u001b)[1\u0000f(\u000fkn0\u001b0)]\n\u000fkn\u001b\u0000\u000fkn0\u001b0\u0002\nN\u000bX\n\u000b\f;\u000b0\f0h\u000bjH\u001b\u001b0\nSO(k)j\fih\u000b0jH\u001b0\u001b\nSO(k)j\f0innn0\u001b\u001b0\n\u000b\f (k)nn0n\u001b0\u001b\n\u000b0\f0(k)\n(7)\nwhere the k-space HSO(k) matrix elements are given by:\nh\u000bjH\u001b\u001b0\nSO(k)j\fi=1\nNkX\nReik\u0001Rh\u000b(0);\u001bjHSOj\f(R);\u001b0i\n(8)4\nand the generalized projected charges by:\nnnn0\u001b\u001b0\n\u000b\f (k) = (cn\u001b\n\u000b(k))\u0003cn0\u001b0\n\f(k) (9)\nIt is usual to further assume the so-called on-site ap-\nproximation, whereby the li\u0001sioperators only mix states\nwithin the same l-shell of a given atom contained in the\norigin unit cell. Eq. 8 then becomes:\nh\u000bjH\u001b\u001b0\nSO(k)j\fi\u0019h\u000b(0);\u001bjHSOj\f(0);\u001b0i\n\u0019\u0018\u000b;lh\u000blm\u001bjl\u0001sj\fl0m0\u001b0i\u000e\u000b\f\u000ell0(10)\nwhere indices \u000b;\fin the last term now refer to the prin-\ncipal quantum number in a given atom (in a multi- \u0010\nscheme, also the particular \u0010), andlmstand for the or-\nbital and magnetic quantum numbers of each AO. \u0018\u000b;l\nis the SOI strength for this l-shell resulting from the in-\ntegration of the radial part in the h\u000blm\u001bjHSOj\u000blm0\u001b0i\nmatrix elements (independent of mm0and\u001b\u001b0). The an-\ngular part of these matrix elements, h\u000blm\u001bjl\u0001sj\u000blm0\u001b0i,\ntake simple analytical expressions and tabulated formu-\nlas as a function of the spherical harmonics Ylminvolved\ncan be found, for example, in Refs.30,31. The on-site ap-\nproximation simpli\fes considerably the 2PT formula:\n\u0001E(2)\nSO=1\nNkNkX\nkX\n\u001b\u001b0N\u000bX\nn;n0f(\u000fkn\u001b)[1\u0000f(\u000fkn0\u001b0)]\n\u000fkn\u001b\u0000\u000fkn0\u001b0\u0002\nAnn0\u001b\u001b0(k)An0n\u001b0\u001b(k) (11)\nwhere we have de\fned:\nAnn0\u001b\u001b0(k) =X\n\u000bl\u0018\u000bl2l+1X\nmm0h\u000blm\u001bjl\u0001sj\u000blm0\u001b0innn0\u001b\u001b0\n\u000blm\u000blm0(k)\n(12)\nWe observe that the 2PT Eqs. 7 and 11 are perturbative\nin the SOI strength, which appears explicitly in the form\nof the parameters \u0018\u000blin this equation.\nNext we consider the case when the unperturbed spin-\npolarized calculation is realized employing a plane-wave\nbasis set, as many DFT codes do. Instead of using\na Bloch-function representation of the the Kohn-Sham\neigenstates, we use a set of Nwmaximally localized Wan-\nnier functions (MLWF) as formulated in Ref.32. MLWFs\nare constructed to yield the exact eigenvalues as the ab\ninitio calculation. Usually, a previous disentanglement\nprocedure is carried out, whereby a handful of relevant\nbands within an energy window are isolated from the\nrest33. For the systems under study, we focus on the\nbands that originate from the d-valence electrons of both\nmetal atoms (allowing also for some degree of s\u0000dhy-\nbridization) and belong to a window of about 10 eV be-\nlow and 5 eV above the Fermi energy. Afterwards, it is\nstraightforward to obtain new Bloch functions on a k-grid\nas dense as desired by interpolation34. This procedure al-\nlows us to estimate the 2PT MAE using as input solely\na scalar-relativistic \frst-principles calculation and does\nnot require a highly dense 1BZ k-sampling.Thej-th MLWF localized at the the unit cell Rthat re-\nsults from band disentanglement and wannierization for\nstates with spin \u001bis:\njw\u001b\nj(R)i=1\nNkNkX\nke\u0000ik\u0001RNbX\nnQn\u001b\nj(k)jkn\u001bi (13)\nwherejkn\u001bistands for the Kohn-Sham eigenstates al-\nready interpolated in the dense k-grid34andQn\u001b\nj(k) are\nthe coe\u000ecients that relate the Wannier and Bloch func-\ntions.\nTypically, atomic-like wavefunctions, formed by a ra-\ndial function and spherical harmonics Ylmto describe the\nangular component, are used to initialize the wannieriza-\ntion procedure. Here, we use d-orbitals centred at the\natomic sites and a few s-waves at interstitial positions.\nThe purpose of the latter functions is to facilitate the\nwannierization and will not take part in the 2PT MAE\ncalculation. We \fx l= 2 and drop this index in the fol-\nlowing. Thus, the jlabel accounts for the \u000b-th atom in\nthe cell Rand them= 0;\u00061;\u00062 quantum number. If\nthe deviation of the MLWFs from actual atomic wave-\nfunctions is small, we can approximate the matrix ele-\nmentshw\u001b\n\u000bm(R)jHSOjw\u001b0\n\u000bm0(R)iby the ones in the AO\nrepresentationh\u000bm\u001bjl\u0001sj\u000bm0\u001b0iand take advantage of\ntheir simple analytic expressions30,31. Note that, in gen-\neral, the resulting MLWFs do not keep a well-de\fned\norbital character because they have to account for both\nthe intra- and inter-atomic orbital hybridization present\nin the system35. Nevertheless, a suitable choice of axes\nin the systems under study allows to obtain MLWFs that\nkeep the atomic orbital character and justify this approx-\nimation. Substituting Eq. 13 in Eq. 5 and using this ap-\nproach, the second order energy correction associated to\nthe SOI is\n\u0001E(2)\nSO=X\n\u000b1\u000b2X\n\u001b\u001b0X\nm1m2X\nm0\n1m0\n2\u0018\u000b1\u0018\u000b2F\u001b\u001b0\u000b1\u000b2\nm1m2m0\n1m0\n2\u0002\nh\u000b1m0\n1\u001b0jl\u0001sj\u000b1m1\u001bih\u000b2m0\n2\u001b0jl\u0001sj\u000b2m2\u001bi(14)\nwhere themiindexes run over the 5 d-orbitals of each\natom\u000biin the unit cell. The Fcoe\u000ecients contain the\ndetails of the basis change from Kohn-Sham states to\nMLWF states and, implicitly, they allow us to use a dense\nk-grid by interpolation:\nF\u001b\u001b0\u000b1\u000b2\nm1m2m0\n1m0\n2=1\nNkNkX\nkNwX\nnn0f(\u000fkn\u001b)[1\u0000f(\u000fkn0\u001b0)]\n\u000fkn\u001b\u0000\u000fkn0\u001b0\u0002\nQn0\u001b0\n\u000b1m0\n1(k)(Qn\u001b\n\u000b1m1(k))?Qn\u001b\n\u000b2m2(k)(Qn0\u001b0\n\u000b2m0\n2(k))?\n(15)\nEq. 14 is similar to the 2PT MAE formula for extended\nsystems developed by van der Laan24, and here we gener-\nalize the result to the case of more than one atom per unit\ncell. It is also straigthforward to show that the on-site\napproximation Eq. 11 reduces to the above expression\nafter replacing the Bloch coe\u000ecients cn\u001b\n\u000b(k) in Eq. 9 by5\ntheQn\u001b\nj(k) ones and restricting the summation over the\nangular momentum numbers to the l=l0= 2 case.\nNote that Eqs. 14 (or Eq. 7) is a \\four-legged\" expres-\nsion in the sense that is contributed by two di\u000berent e\u0000h\npairs. There are other 2PT formulations based on the use\nof a localized basis set of orbitals22,23,25,27. However, it is\nworth to mention that our formulation of the MAE does\nnot neglect spin-\rip contributions, unlike the one pro-\nposed by Bruno22. Formulas that neglect wavefunction\nphase e\u000bects in the Kohn-Sham-to-local-basis projection\nhave also been proposed25,27. By doing so, the \\four-\nlegged\" Eq. 14 becomes only \\two-legged\" and Eq. 15\ncan be written in simpler terms, namely the projected\ndensities of states on the local orbitals.\nIII. COMPUTATIONAL DETAILS\nIn all the procedures described in this section, to pre-\nvent the MAE values from being biased by the structural\nparameters, we have kept the lattice constants a;c\fxed\nin all calculations (see Fig. 1 and Table I). The model for\nFeCo has the Fe bccunit cell volume and c=a= 1:2 to\nmaximise the anisotropy, as suggested by the literature\non strain e\u000bects12. For FeCu, we keep the Cu-Cu as in\nbulkfccCu andc=a= 1:3436. Finally, we use published\nlattice constants for FePd, FeAu37and FePt38.\nTwo DFT codes have been used in this work:\nSIESTA-Green39,40(SG) and VASP41,42. The former\nuses multi- \u0010non-orthogonal strictly localized numeri-\ncal AOs as basis set and replaces core electrons by\npseudo-potentials, while the latter uses plane-waves and\nprojector-augmented wave potentials to describe the ion\ncores43. Both codes feature SOI implementations40,44\nthat allow to obtain the MAE values directly from Eq. 2\nor the SOI-corrected eigenvalues of Eq. 3. By working\nwith both codes we ensure that the conclusions about\nthe MCA are not biased by the basis set type. The ex-\nchange and correlation functional used in all calculations\nis that of Perdew, Burke and Ernzerhof (PBE)45.\nIn the VASP calculations the psemi-core states are\nincluded as valence electrons. We have set the plane-\nwave energy cut-o\u000b to 400 eV in all the systems, except\nfor 420 eV in FeAu, and suitable k-point Monkhorst-Pack\ngrids46according to each lattice dimensions and calcula-\ntion type [(24\u000224\u000220) for FeCo and (24 \u000224\u000218) for\nothers]. The tetrahedron method with Bl ochl corrections\nis used to obtain the Fermi level47. For the SG calcula-\ntions we have adopted a double- \u0010polarized scheme to\ngenerate the AO basis set using a con\fnement energy\nof 100 meV although, as opposed to VASP, no psemi-\ncore states are considered. Pseudo-core corrections are\nincluded for all the atoms involved, while a very \fne real\nspace mesh is employed by setting the mesh cut-o\u000b to\n4000 Ry.\nIn the SG case, SOI matrices are calculated under\nthe fully-relativistic pseudo-potential (FR-PP) method\ndescribed elsewhere40. This approach goes beyond theon-site approximation48(Eq. 10) as it takes into account\nintra- and inter-atomic interactions between di\u000berent l-\nshells. Although the o\u000b-site terms tend to be small, test\ncalculations show that neglecting them can induce errors\nin the MAE of around 0.2 meV or even larger (around\n1 meV) in particular cases. Nevertheless, the on-site\napproximation allows to extract the SOI strengths \u0018\u000bl,\nwhich we provide in Table II for the dorbitals.\nEven when working with the SCF method for SOI,\nthe calculation parameters must be carefully tested. The\nFermi energy smearing is a decisive technical factor for\nthe MAE of some of the systems. This issue has been\naddressed with both codes for the SCF and NSCF meth-\nods. We \fnd satisfactory convergence when the tetrahe-\ndron method is used for the SCF MAE with VASP47. By\ndoing so, we avoid kT-dependence in the resulting MAE\nvalues (we have checked that the total energy extrapo-\nlation tokT= 0 gives, in general, good agreement with\nthe results of the tetrahedron method). With SG, since\nthe use of \fner k-point grids can be a\u000borded at a reason-\nable computational and memory cost, a high convergence\ncould be systematically achieved in the k-grid. Conver-\ngency values below 0.01 meV are obtained with kTvalues\nentering the Fermi-Dirac distribution function as low as\n1 meV. In the NSCF calculations we \fnd that the smear-\ning function, whether Fermi-Dirac or Methfessel-Paxton\nof a given order49, has a non-negligible in\ruence, but it\nbecomes less important for su\u000eciently \fne k-grids and\nsmallkT. A detailed convergence analysis for all phases\ncan be found in the Supplementary Information (see ta-\nbles in sections I,II and Figs. 1 and 2).\nFor the more elaborate 2PT methodology, we have im-\nplemented Eq. 7 for the SG calculations and the semi-\nanalytical form of Eq. 14 for VASP in combination with\nWannier9032,50. Thek-grids needed to obtain a faith-\nful representation of the electronic structure with ML-\nWFs can be less dense than the ones typically needed\nto obtain the MAEs. The latter, also used in the ex-\nplicit evaluation of Eqs. 14 and 15, can be chosen as\ndense as desired by MLWF interpolation34. Besides, due\nto the strongly hybridized d-bands in the L10Fe-alloys,\nprior to wannierization it is useful to perform a disen-\ntanglement of the bands33within an energy window that\ncontains the relevant states. A numerical drawback in\nthe whole procedure is that the quality of the wannier-\nization is system-dependent. Therefore, for each alloy\nwe have chosen suitable settings, shown in Table I, to\nmeet the usual sanity requirements of a wannierization,\nnamely, little overlap between MLWFs (at least between\nthe functions that emulate the d-orbitals), bandstructure\nreproducibility, and spatial localization.\nIt is worth to mention that the \fve atomic d-orbital\nwavefunction geometries, i.e. the representations of the\nangular functions Y2m(^r) in cartesian coordinates, de-\npend on the convention taken for the OX;OY;OZ axes\nin each code. Obviously, the electronic structure that\nresults from hybridization of the atomic wavefunctions\nmust be independent of those conventions. However, if6\nTABLE I. Unit cell lattice parameters (see Fig. 1) of the considered model systems and calculation parameters used in the\nwannierization. nsis the number of s-wave-like Wannier functions, introduced in addition to the d-orbital-like ones, placed\nat interstitial sites of the structure. [ w0;w1] are the frozen windows used for disentanglement with respect to the Fermi\nenergy. Two intervals are shown for FeAu that correspond to di\u000berent windows for the spin-majority and spin-minority bands,\nrespectively. k1is the grid used in the reference DFT calculation and k2the interpolated grid used to evaluate the MAE in the\n2PT approximation. pminis the minimum projection factor p\u000bm\u001b (Eq. 16) found for each system.\nFeCo FeCu FePd FePt FeAu\na(\u0017A) 2.680 2.553 2.751 2.722 2.885\nc=a 1.2 1.339 1.327 1.364 1.328\nns 3 4 3 2 3,4\n[w0;w1] (eV) [ \u00005:2;2:8] [ \u000010:0;3:5] [ \u00006:6;2:4] [ \u00005:8;2:2] [ \u00003:4;\u00001:1];[\u00001:4;1:6]\nk1 16\u000216\u000214 16 \u000216\u000212 16 \u000216\u000212 16 \u000216\u000212 16 \u000216\u000212\nk2 40\u000240\u000233 36 \u000236\u000228 36 \u000236\u000227 40 \u000240\u000230 36 \u000236\u000227\npmin 0.88 0.82 0.82 0.90 0.85\nTABLE II. SG values of the atomic SOI strength parameter \u0018Me(Me=Co,Cu,Pd,Pt,Au) and VASP values of the atomic spin\nand orbital magnetic moments, \u0016Sand\u0016L, respectively, in Bohr magnetons ( \u0016Svalues are obtained from calculations without\nSOI). The last column shows an approximated MAE obtained from the expression \u0000P\n\u000b\u0018\u000b\n4[\u0016x\nL(\u000b)\u0000\u0016z\nL(\u000b)], where the index\n\u000bruns over the Fe and Me atoms, with \u0018Fe= 59:65 meV.\n\u0018Me(meV) \u0016S(Fe)\u0016S(Me) \u0016x\nL(Fe)\u0016x\nL(Me) \u0016z\nL(Fe)\u0016z\nL(Me) MAE (meV)\nFeCo 74.12 2.74 1.67 0.053 0.077 0.064 0.088 0.37\nFeCu 110.44 2.55 0.16 0.045 0.009 0.055 0.011 0.20\nFePd 191.36 3.00 0.38 0.061 0.030 0.069 0.027 -0.02\nFePt 537.18 2.93 0.40 0.057 0.056 0.060 0.044 -1.57\nFeAu 615.05 2.98 0.03 0.045 0.037 0.065 0.029 -0.93\nwe align the crystallographic directions and the cartesian\naxes of VASP and Wannier90 as shown in Fig. 1, we can\nrepresent the bonding states as overlapping Y2m(dr\u0000R\u000b)\nfunctions localized at atomic positions R\u000b. Otherwise,\nthe overlaps would happen for linear combinations of\nY2m(^r) functions at each site. In such case, the MLWFs\nthat reproduce the electronic structure do not resemble\nthe initial spherical harmonics and Eq. 14 cannot be ap-\nplied directly: an intermediate step would be needed to\naccount for the linear combinations of Y2m(^r) functions.\nThe axes choice of Fig. 1 makes this step unnecessary.\nIndeed, we can trace the MLWFs back to individual Fe\nand Med-orbitals by visual inspection (see Supplemen-\ntary Information Fig. 3), deviations being the result of\nthe inter-atomic hybridization only, i.e. an e\u000bect of a\npurely physical origin, and not an spurious one caused\nby the axes convention. To quantify those deviations, we\nde\fne the projections\np\u000bm\u001b=jhw\u001b\n\u000bm(R)j\u000bm\u001bij2(16)\nwhich range between zero and one. As shown in Table I,\ndespite the dispersive character of the bandstructures, in\nthis work we \fnd projections above 0.80 after wannier-\nization.IV. RESULTS AND DISCUSSION\nA. Neutral systems\nTable III contains the collection of the MAE values cal-\nculated with the methodologies presented in Section III.\nAll the approaches provide the same behaviour in the\nMCA of each alloy, albeit there are some quantitative\ndi\u000berences in the corresponding MAE values, which will\nbe discussed below. The easy magnetization axis is OZin\nthe \fve studied systems. Overall, MAE values are smaller\nthan 0.5 meV except for FePt, a well-known prototype\nof large MCA, which shows a MAE of nearly 3 meV in\ngood agreement with other ab-initio values available in\nthe literature37,38,40,51.\nThe \frst important observation is that the MAE of Fe-\nbasedL10alloys is not directly correlated with the SO\nstrength of the alloying metal. Indeed, it is the electronic\nstructure what governs the MCA of these alloys, overrul-\ning the e\u000bect of the SO strength. We \fnd larger MAE\nvalues for FeCo and FeCu than for FePd and FeAu in\ndespite of\u0018Pd;Aubeing larger than \u0018Co;Cu(see Table II).\nIn the case of FeCo it is known that a large MAE can be\nachieved by a strain on the cell along OZ. Forc=a= 1:2,\nthe geometry chosen for this work, a maximum is ob-\ntained because degenerate states coupled by HSOlie on7\nTABLE III. MAE values in meV for the neutral systems. Labels SG and V indicate SIESTA-Green and VASP calculations,\nrespectively, with the cut-o\u000b energy and k-point samplings discussed in the text. In the 2PT SG (V) calculations a Fermi-Dirac\nsmearing with kT= 10 meV ( kT= 50 meV) has been used.\nSCF (SG) SCF (V) NSCF (SG) NSCF (V) 2PT (SG) 2PT (V)\nFeCo 0.42 0.39 0.45 0.55 0.44 0.35\nFeCu 0.38 0.45 0.42 0.45 0.42 0.38\nFePd 0.22 0.16 0.20 0.13 0.19 -0.11\nFePt 2.93 2.59 2.93 2.78 2.92 0.63\nFeAu 0.20 0.50 0.36 0.62 0.41 0.76\nthe Fermi level12. As a matter of fact, FeCo is not an\nisolated case27,52.\nThe SCF values obtained in the VASP and SG calcu-\nlations are in good agreement for FeCo, FeCu, and FePd,\nwhere discrepancies of 0.07 meV or smaller are found.\nFor FePt and FeAu larger deviations of around 0.3 meV\nexist. The reason for this discrepancy is di\u000ecult to iden-\ntify. Small quantitative di\u000berences in the band structures\nprovided by both codes would be unimportant for most\nphysical properties but, unfortunately, they become sig-\nni\fcant for the estimation of the MAEs. First, we note\nthat the VASP values could not be converged in k-grid\nat the same level of accuracy as the SG MAEs. Another\nsource of error could be associated to limitations of the\nSG calculations, such as the basis set size and AO exten-\nsions, the neglect of semi-core pstates or even the actual\nFR-PP approximation.\nThe MAE values predicted by the NSCF method (see\nTable III) are, in general, very close to the SCF values,\nwith typical deviations well below 0.1 meV. It is striking\nto \fnd such a good agreement even for the heaviest metal\nsystems, where the charge density change is expected\nto be larger. There are only a few exceptions, namely\nthe VASP calculation for FePt, FeCo and FeAu and the\nSG one for FeAu for which, nonetheless, di\u000berences re-\nmain smaller than 0.2 meV. For the formers, we assign\nthe discrepancy to calculation details rather than to the\nlimitation of using the non-self-consistent charge density.\nAmong the technical details, it is the smearing method of\nthe Fermi level the crucial one, since the NSCF approach\nis sensitive to the small Fermi energy shift when magneti-\nzation occurs in one or other direction. In the FeAu case\nwith SG, where full k-convergence is achieved, we may\nconsider the 0.12 meV deviation as an upper limit to the\naccuracy of the NSCF, probably due to the larger \u0018Au\nSOI strength. Notwithstanding, the HSOterm is fully\naccounted for by this approach.\nWe next address the performance of the 2PT approxi-\nmation to the MAE. If we \frst focus on the SG MAEs in\nTable III, they are in almost perfect agreement with the\nSCF and NSCF MAE values in all cases except, again,\nthe FeAu alloy which results 0.2 meV larger than the SCF\nvalue following a similar trend as under the NSCF. Thus,\nthe same behaviour between the NSCF and 2PT methods\nindicates that their accuracy is very similar despite theneglect of high-order HSOterms in the latter, both pro-\nviding excellent results at a much lower computational\ncost compared to the reference SCF calculations.\nThe 2PT-derived MAE is determined by eigenstates\naround the Fermi level within an energy range given by\nthe SOI strength \u0018and the (\u000fkn\u001b\u0000\u000fkn0\u001b0)\u00001factors. In-\ndeed, Eq. 7 gives less weight to the e\u0000hpairs coupled\nbyHSOmatrix elements that lie farther in energy (the\ncontributions of the energy prefactors are shown in Sup-\nplementary Information Fig. 4). Intuitively, deep states\nin the valence band might be regarded as negligible for\nthe MAE. However, a third factor needs to be considered:\nthe avaible number of states. To analyze the competi-\ntion of these three e\u000bects, we have calculated the 2PT\nMAE allowing only initial (\fnal) states within an energy\nwindow below (above) the Fermi energy when evaluat-\ning Eq. 7. The results are shown in Fig. 2 for the SG\ncase, while those with VASP are qualitatively the same\n(not shown). When imposing a window on the occupied\nstates, a plateau in the MAE value is not reached until\n5-7 eV, depending on the system. These energy ranges\ncomprise the d-band widths below the Fermi level (see\nthe densities of states (DOS) projected on the d-orbitals\nin the Supplementary Information Fig. 5). With heav-\nier atoms, deeper states contribute non-trivially to the\nMAE, even reversing its sign, as it is the case of FeAu.\nTherefore, the \fnal value of the MAE of each system\ndepends on its electronic structure details, since the dis-\npersion and binding energies of the individual band pairs\ncoupled by HSOlargely vary from one system to another.\nThe e\u000bect of constraining the accessible empty states in\nFig. 2 is less dramatic and reveals a common behaviour\nin all the systems. With a window above the Fermi level,\nthe MAE plateau is reached at \u00182:5 eV. As shown in\nthe DOS plots in the Supplementary Information (Fig. 5),\nthis energy range corresponds to the empty spin-minority\nstates of Fe in all the alloys, and it is also contributed to\na lesser extent by the other metal empty d-states if avail-\nable (Co and Pt).\nIt is a general trend in these systems that the high\nDOS counterbalances the decay of the ( \u000fkn\u001b\u0000\u000fkn0\u001b0)\u00001\nfactors. In brief, Fig. 2 shows that states distant from the\nFermi level by as much as several eV (that is, spanning\nthe wholed-band of the alloy) cannot be neglected in\na 2PT calculation, not even in the case of atoms with8\nFIG. 2. Dependence of the MAE calculated in the 2PT ap-\nproximation with Eq. 7 (SG calculation) on the energy win-\ndow of allowed occupied (solid) and unoccupied (dashed) one-\nelectron states.\nweak SOI. We conclude that the accessibility of empty Fe\nspin-minority states is a common feature that allows for\nsizeable MCAs in the Fe-based alloys, while the details\nof the occupied d-bands of each case determine the \fnal\nMAE value.\nThe 2PT MAE values obtained with the wannierized\nbands from the VASP calculation yield worse agreement\nthan the other methods becoming more pronounced the\nheavier the metal in the Fe alloy (see Table III). Specif-\nically, the easy axis for FePd is switched to OX while\nthe MAE of FePt is considerably underestimated and\nthat of FeAu overestimated. Although this method al-\nlows the use of very dense k-point grids by interpolation,\nthe quality of the wannierization procedure is crucial for\nnumerical accuracy. Nonetheless, the main factor that\nundermines the \fnal MAE values is the approximation\nin theHSOmatrix elements: the assumption that the\nMLWFs correspond to atomic orbitals with well-de\fned\nlmquantum numbers and, to a lesser extent, the on-site approximation (Eq. 10) and the neglect of sp-orbital\ncontributions. All in all Eq. 14 is a coarse approxima-\ntion. Despite the apparent resemblance of MLWFs with\natomic wavefunctions by visual inspection (see Supple-\nmentary Information Fig. 3), the deformations of these\n\\d-orbitals\" are signi\fcant, due to the fact that Fe-alloys\nhave strongly-hybridized bands. For FeCo and FeCu,\nnevertheless, the MAE behaviour in the energy windows\nanalyzed is similar to the SG ones (not shown).\nAnother appealing aspect of the 2PT formulation is\nthat it allows to split the MAE into contributions arising\nfrom pairs of atoms in the metallic alloy: Fe-Fe, Fe-Me\nand Me-Me. This is straighforward when using MLWFs\nand Eq. 14 (see Supplementary Information Fig. 6), while\nif Eq. 7 is used, the decomposition can be realized by re-\nstricting (\u000b;\u000b0) to a given pair of atoms and performing\nthe summation over all the other ( \f;\f0) AO contribu-\ntions. The result is shown in Fig. 3. As expected, Fe\nand Co have a balanced weight on the \fnal FeCo MAE,\nwhile Cu hardly contributes to that of FeCu. In the rest\nof alloys the decompositions show the contribution we\ncould expect from the knowledge of the electronic struc-\nture, at least qualitatively. The Pt-Pt and Fe-Pt terms\nare the main contributors in the FePt alloy, because \u0018Pt\nis an order of magnitude larger than \u0018Feand because the\ndelectrons of both species are strongly hybridized. We\n\fnd, in agreement to Ref.26, that the large contribution\nof the SOI of Pt atoms to the perpendicular anisotropy\n(OZeasy axis) is partially balanced by the e\u000bect of the\nFe-Pt hybrid bands, which favour in-plane anisotropy. In\nFeAu, the Au-Au term is also very large due to the mag-\nnitude of\u0018Au, but now we see that the contribution of\nFe-Au terms is weaker, since there is little hybridization\nwith Au-delectrons.\nFinally, we compare our perturbative results with the\nwidely used 2PT equation by Bruno for bands of d-orbital\ncharacter22, which assumes that all the accesible holes are\nminority spin states. Therefore, it neglects e\u0000hexcita-\ntions that involve spin-\rip, which leads to \u0001 E(2)/\u0018hLi.\nTable II shows the MAE values obtained in this approxi-\nmation with the DFT-calculated atomic orbital moment\nvalues\u0016Lfor each magnetization direction. Although\nthe density of majority-spin states above the Fermi level\nis marginal (shown in the Supplementary Information\nFig. 5), the results of the Bruno formula di\u000ber signi\f-\ncantly from the other methods, and in some cases it does\nnot even predict the correct easy magnetization axis. In\naddition to the breakdown of the approximation itself,\nwe can ascribe the discrepancies to the tendency of DFT\nto underestimate orbital moment values.\nB. Non-neutral systems\nFigure 4 shows the MAE as a function of the num-\nber of electrons Necalculated with the NSCF and the\n2PT approaches. Here, we have followed a rigid band ap-\nproach, in which the Fermi level is recalculated for each9\nFIG. 3. Contributions of the atom-pair terms of Eq. 7 (SG\ncalculation) to the 2PT MAE.\nNeemploying the eigenvalues and eigenstates of the un-\nperturbed neutral calculation. Hence, the plots represent\nthe MAE behaviour of each alloy under conditions of dop-\ning or application of a bias voltage, which are common\nworking conditions in magnetic devices. Note, however,\nthat due to the rigid band approach only small devia-\ntions from the neutral situation are physically meaning-\nful. As expected from the disccusion in the previous sec-\ntion, there are only small di\u000berences between the NSCF\ncurves calculated with SG and VASP.\nSwitching of the easy magnetization axis occurs several\ntimes as a function of band \flling in all cases. Interest-\ningly, a MAE reversal already takes place by removal or\naddition of one or two electrons per unit cell. Further-more, the MAE undergoes drastic changes in magnitude,\neven attaining values which are one order of magnitude\nlarger than the neutral ones, specially in the Ne\u001910\nregion where the d-bands show the highest density of\nstates.\nWith a methodological aim in mind, the study of the\nMAE in the non-neutral case allows us to study the va-\nlidity the 2PT perturbative approach with greater con\f-\ndence than in the neutral case, since now we can compare\na MAE curve with a rich structure instead of just a single\nvalue. The SG 2PT curves (dashed lines in the left-hand\npanel of Fig. 4) are in very good agreement with the\nNSCF curves for FeCo, FeCu, and FePd, while large dis-\ncrepancies appear when heavier atoms are present. This\nis particularly evident in FeAu for Ne= 5\u000010, which\ncorresponds to the spectrum region where the d-bands of\nAu lie. Considering the strong dependence of the MAE\non the electron band structure details discussed in the\nprevious section and the fact that the agreement with\nNSCF is not homogeneous as Nechanges, both facts sug-\ngest that 2PT loses its validity for elements with strong\nSOI. Thus, the coincidence in the neutral-case FePt and\nFeAu MAEs could be considered to be fortuitous, in the\nsense that the coincidence is a consequence of the speci\fc\nband structure of the alloy, as it is the case of FePt (also\nobserved in Ref.53) and FeAu.54\nThis restriction of the 2PT validity to light atoms is\nnot a serious disadvantage. This approach facilitates\nthe MAE evaluation with \fne k-point grids and narrow\nsmearing widths at a lower computational cost, since it\nrequires a single DFT calculation without SOI. We recall\nthat a weak MCA does not necessarily follow from a small\n\u0018, as we see in the systems under study. In extended sys-\ntems like the current ones, band dispersion governs the\nMCA.\nBoth NSCF and 2PT describe SOI with a perturba-\ntive treatment of either the charge density changes or\nthe\u0018parameter, respectively. At this point, it is impor-\ntant to note another fundamental di\u000berence between the\nNSCF and 2PT formalisms. In the former method, the\neigenvalues change with the magnetization axis and some\ndegeneracies may be broken. In the latter method, the\nunperturbed band structure is not explicitely modi\fed.\nInstead,e\u0000hpair excitations of the GS j\t(0)iare in-\nduced byHSO, with probabilities given by their matrix\nelements. In other words, 2PT is a many-body approach,\nwhile NSCF is a one-electron approach. Thus, if we take\nthe limit of single ions and uniaxial anisotropy, the 2PT\napproach described here tends to the widely-used for-\nmalism of the spin hamiltonian for non-degenerate states\nHion=DS2\nz+E(S2\nx\u0000S2\ny)55,56.\nVisual evidence of the inherent di\u000berence between\nNSCF and 2PT is presented in Fig. 5. This \fgure shows,\nfor the case of FeCo and the VASP-Wannier calculation,\nthe MAE densities as a function of Nein the reciprocal\nspace along a few high-symmetry directions inside the\n1BZ. To guide the eye, the bands without SOI have been\ntransformed from energies to the corresponding \flling Ne10\nFIG. 4. MAE as a function of the number of valence electrons Ne. The vertical line indicates the position of the Fermi\nlevel in the neutral systems. Solid (dashed) lines correspond to the NSCF (2PT). Left: SG calculations, i.e. the dashed line\nis obtained from Eq. 7, with a Fermi-Dirac smearing of kT= 10 meV is used here. Right: VASP calculations, i.e. the dashed\nline is obtained from Eq. 14, with a Fermi-Dirac smearing of kT= 50 meV.\nand superimposed on the MAE density graph. As ex-\npected, for the NSCF case (top panel) the regions of non-\nzero MAE are localized close to the unperturbed bands,\nand take positive or negative values depending on the\nrelativeHSOinduced shifts in the eigenvalues between\ntheOZandOXmagnetization directions. The map also\nreveals abrupt switching of the MAE densities close to\nseveral band-crossings. For example, this happens near\nthe \u0000 point and Ne'9, where a pair of majority spin\nbands is split by a few meV for magnetization along OZ.\nSince the splitting is nearly symmetric in energy about\nthe non-perturbed bands, the contribution to MAE is\nnegative as the bottom band is \flled and changes sign\nwhen the upper one starts to become occupied. When\nboth bands are completely \flled the MAE vanishes. In\nthe 2PT approach the MAE density (lower panel) takes a\nvery di\u000berent aspect since the bandstructure is not mod-\ni\fed and, as shown by Fig. 2, e\u0000hpairs with an en-\nergy di\u000berence as large as a few eV can contribute to\nMAE at a given Ne(see also Supplementary Information\nFig. 4). Thus, the map presents broad plateaus with\na non-negligible MAE density in areas devoid of bands,\nwhile sign changes are always localized precisely at the\nbands since they take place when the combined e\u0000hdis-tribution functions (term f(\u000fkn\u001b)[1\u0000f(\u000fkn0\u001b0)] in Eq. 5)\nalso changes sign as the band is crossed. Nevertheless,\nfor FeCo the two approaches yield a similar overall de-\nscription of the MAE( Ne) behaviour, as seen in the k-\nintegrated curve of Fig. 4 (right), in spite of treating\nbandstructures in a di\u000berent way by construction.\nWith the 2PT calculation that uses MLWFs poorer\nagreement is found in the pro\fle of the MAE( Ne) curves,\ndue to the limitations of this methodology, still the qual-\nitative behaviour is reproduced in all alloys except FeAu,\nwhere similar deviations as for the SG case are found\n(see right-hand panel of Fig. 4). Therefore, it could be\nused under suitable conditions to make predictions on\nthe MAE behaviour as a function of doping or bias volt-\nage at a low computational cost from a DFT calcula-\ntion which does not include SOI. Those conditions are\n(i) weak SOI strength and (ii) resemblance between the\nMLWFs and Y2mspherical harmonics. When the \frst\ncondition is met, the MAE obtained in the on-site ap-\nproximation (Eq. 10) performs as accurately as NSCF.\nThis has been checked with SG calculations (see Sup-\nplementary Information Fig. 7), where the drawback of\npoint (ii) is not present. Interestingly, the e\u000bect of inter-\natomicd-orbital hybridization on the MAE is well cap-11\nFIG. 5.k-resolved MAE as a function of the number of va-\nlence electrons Nealong high-symmetry directions inside the\n\frst Brillouin zone for the FeCo alloy. The top and bottom\npanels show the NSCF and 2PT approaches, respectively, ob-\ntained from Eq. 14 (VASP calculation) with a Fermi-Dirac\nsmearing of kT= 50 meV. The correspondence between the\nWannier-interpolated energy eigenvalues without SOI and the\nnumber of valence electrons is shown as black (majority spin)\nand green (minority spin) bandstructures. The horizontal line\nindicates charge neutrality. In the color scale, red and blue re-\ngions of the spectrum account for a contribution to anisotropy\neasy axis along OZ or OX, respectively.\ntured by this methodology via the Ffactors in Eq. 15,\nwhile the crude approximation done for the SOI matrix\nelements seems less important, as it can be deduced from\nthe good agreement with the NSCF curves in the FeCo\nand FeCu panels of Fig. 4 (right).V. CONCLUSIONS\nIn this work, we have included the spin-orbit interac-\ntion (SOI) in DFT calculations at di\u000berent levels of ap-\nproximation to obtain the magnetocrystalline anisotropy\nenergies (MAE). As reference, we use fully-relativistic\nfully-self-consistent (SCF) calculations. We have ex-\namined the accuracy of (i) the so-called force theorem\nmethod, where the SOI is applied once to the charge\ndensity of a scalar-relativistic calculation, without sub-\nsequent self-consistency cycles (NSCF), and (ii) a many-\nbody second-order perturbation (2PT) treatment of the\nSOI on the scalar-relativistic ground state. With a com-\npuational aim in mind, we have com\frmed that an accu-\nrate Fermi level determination is crucial to obtain con-\nverged MAE values.\nAs case studies, we have taken several FeMe tetrago-\nnal ferromagnetic alloys with L10structure. By choos-\ning Me=Co, Cu, Pd, Pt, and Au, we cover the scenar-\nios ofs\u0000dandd\u0000dhybrid band e\u000bects and a range\nof atomic SOI strength parameters \u0018. We \fnd NSCF\nto provide reliable MAEs in general, whereas the 2PT\napproximation describes accurately the MAEs of FeCo,\nFeCu, (\u0018\u00180:05 eV) and satisfactorily that of FePd\n(\u0018\u00180:1 eV), but fails for the alloys containing 5 dmetals\n(\u0018\u00180:5 eV). The di\u000berence in the performance of the\ntwo approximations has the following origin: NSCF is\nperturbative in the charge density changes by SOI, while\n2PT is perturbative in \u0018.\nWe \fnd that the MAE in this family of alloys is de-\ntermined not only by the empty spin-minority Fe states,\nwhich lie about 2 eV above the Fermi level, but also by\nthe whole valence band, which lies several eV below the\nFermi level. Thus, the magnetocrystalline anisotropy is\ndetermined by electronic states that lie from the Fermi\nlevel much further than the SOI strength parameter. The\ndetails of the bandstructure are, in essence, responsible\nfor the \fnal MAE value.\nFinally, the 2PT approximation is sound enough to\naccount for the anisotropy behaviour of the alloys under\ndeviations from charge neutrality. We show that mag-\nnetic switching can be induced by addition or removal\nof electrons. This e\u000bect could be tuned by, for example,\ndoping or strain, to \fnd an scenario under a minimal\nbias voltage. These properties have ample applications in\nmagnetoelectric and magnetostrictive devices. The 2PT\napproximation to the SOI, when valid, can be used to\nstudy large numbers of these cases e\u000eciently, as it uses\nas sole input a scalar-relativistic DFT calculation.\nACKNOWLEDGMENTS\nDiscussions with G. Teobaldi and M. dos Santos Dias\nare acknowledged. M.B.-R. and A.A. thank \fnancial\nsupport from MINECO (grant number FIS2016-75862-\nP), the University of the Basque Country (UPV/EHU)\nand the Basque Government (IT-756-13). 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Georgii,1, 4and C. Pfleiderer1\n1Physik Department, Technische Universit¨ at M¨ unchen, D-85748 Garching, Germany\n2Institute for Theoretical Physics, Universit¨ at zu K¨ oln, D-50937 K¨ oln, Germany\n3Institut f¨ ur Theoretische Physik, Technische Universit¨ at Dresden, 01062 Dresden, Germany\n4Heinz Maier-Leibnitz Zentrum (MLZ), Technische Universit¨ at M¨ unchen, D-85747 Garching, Germany\n(Dated: December 3, 2018)\nWe report high-precision small angle neutron scattering of the orientation of the skyrmion lattice\nin a spherical sample of MnSi under systematic changes of the magnetic field direction. For all field\ndirections the skyrmion lattice may be accurately described as a triple- /vectorQstate, where the modulus\n|/vectorQ|is constant and the wave vectors enclose rigid angles of 120◦. Along a great circle across /angbracketleft100/angbracketright,\n/angbracketleft110/angbracketright, and/angbracketleft111/angbracketrightthe normal to the skyrmion-lattice plane varies systematically by ±3◦with respect\nto the field direction, while the in-plane alignment displays a reorientation by 15◦for magnetic\nfield along/angbracketleft100/angbracketright. Our observations are qualitatively and quantitatively in excellent agreement with\nan effective potential, that is determined by the symmetries of the tetrahedral point group Tand\nincludes contributions up to sixth-order in spin-orbit coupling, providing a full account of the effect\nof cubic magnetocrystalline anisotropies on the skyrmion lattice in MnSi.\nMagnetocrystalline anisotropies (MCA) are of central\nimportance for the character and detailed properties of\nmagnetic order on all scales. Well-known examples in-\nclude the easy and hard axis of long-range magnetic or-\nder, spin-flop transitions in antiferromagnets, or the for-\nmation of magnetic domains. Yet, despite of their inher-\nent importance, the determination of MCAs on a purely\nexperimental level has been very challenging for many\ndecades, in particular for complex magnets with a spa-\ntially modulated magnetization.\nA particularly intriguing showcase of MCAs was pre-\ndicted nearly three decades ago in chiral magnets, when\nBogdanov and Yablonskii suggested that skyrmion lat-\ntices, as derived in terms of solitonic (particle-like) spin\nconfigurations, could be stabilized in an applied magnetic\nfield under moderate easy-plane anisotropies, i.e., terms\nthat are second order in spin-orbit coupling (SOC) [1, 2].\nIt was, in turn, a major surprise, when skyrmion lattices\nwere first identified experimentally in the B20 compounds\nMnSi [3], Fe 1−xCoxSi [4], FeGe [5] and related systems\n[6–9], featuring cubic MCAs that are fourth and sixth or-\nder in SOC and therefore much weaker. Moreover, it was\nshown [3, 10, 11] that, instead of the MCAs, stabiliza-\ntion in these systems arises from entropic effects due to\nthermal fluctuations, where the skyrmion lattice is well\nrepresented by a multi- /vectorQmagnetic texture [12].\nWhile these seminal studies suggested that cubic\nMCAs play a sub-leading role for the global understand-\ning of skyrmion lattices in cubic chiral magnets, several\nbreak-through discoveries have recently identified a num-\nber of new directions of research. Examples include the\nformation of topological defects, such as disclinations, as\na new avenue for spintronics applications [13, 14], and\nthe first observation of two independent skyrmion phases\nin the same material [15]. Moreover, cubic MCAs are\nalso held responsible for the anisotropy of the tempera-\nture and field range of skyrmion phases [3, 12, 16, 17],deviations from an ideal hexagonal lattice under uniaxial\nstrain [18–20], the formation of different skyrmion lat-\ntice morphologies [21, 22], as well as the formation, size,\nand shape of skyrmion lattice domains [3, 4, 8, 23–27].\nFurther, cubic MCAs are also essential for the motion of\nskyrmion lattices under spin currents [28–30]. Last but\nnot least, MCAs provide an important point of reference\nfor the understanding of superconducting vortex lattices,\nwhere different morphologies [31, 32] and the associated\ntopological character [33] attract great current interest,\nwhile higher-order contributions in the superconducting\norder parameter of the underlying Ginzburg-Landau the-\nories are still not known.\nFrom a technical point of view accurate determina-\ntion of MCAs in modulated structures is difficult as the\nspins point in various crystallographic directions, reflect-\ning a balance of varying contributions. Thus, the effect\nof MCAs on modulated structures cannot simply be de-\ntermined by means of torque magnetometry representing\nthe standard technique used in ferromagnets. Moreover,\nthe MCA may not only affect the propagation direction\nof the modulations but also specific details of the mod-\nulations causing changes of pitch and harmonicity. In\nturn, a given material might even appear to change its\neasy axis under applied magnetic fields [15, 34]. In com-\nparison, imaging methods such as magnetic force [35, 36]\nor scanning x-ray microscopy [25, 37], while capable of\ndetecting modulated structures and the associated tex-\ntures, are sensitive to surface effects. Taken together,\nthis underscores the need for controlled microscopic mea-\nsurements of MCAs in bulk samples which minimize the\ninfluence of the sample shape.\nIn this Letter we report the determination of the cu-\nbic MCAs in MnSi up to sixth order in SOC by means\nof high-precision small angle neutron scattering (SANS)\n[13, 38]. Using a spherical sample we minimized parasitic\neffects of the sample shape. Covering systematically aarXiv:1811.12439v1  [cond-mat.str-el]  29 Nov 20182\nlarge range of field directions, we determined the equilib-\nrium orientation of the skyrmion lattice in excellent qual-\nitative and quantitative agreement with an effective po-\ntential as inferred from the associated Ginzburg-Landau\ntheory and the symmetries of the tetrahedral point group\nT[39]. Remarkably, for all field directions the skyrmion\nlattice may be accurately described as a triple- /vectorQstate,\nwhere the modulus |/vectorQ|is constant and the wave vectors\nenclose rigid angles of 120◦, i.e., the MCAs have a mi-\nnor influence on the modulated state as such and mainly\naffect its orientation. This provides an important bench-\nmark of the response of skyrmion lattices under cubic\nMCAs. On a more general note, our study demonstrates\nthat systematic neutron diffraction experiments are ide-\nally suited to track the effects of MCAs in materials fea-\nturing modulated magnetic structures.\nFor our study a single crystal sample was prepared\nfrom an optically float-zoned ingot [40]. The single crys-\ntal was carefully ground into a sphere with a diame-\nter of 5.75 mm. The sample shape was chosen to mini-\nmize inhomogenities of the demagnetization fields, known\nfrom previous work to cause parasitic deviations of the\nskyrmion lattice orientation [3, 12, 41]. The sample was\noriented by means of x-ray Laue diffraction to better\nthan a few degrees and attached to an aluminum sample\nholder. The precise orientation of the sample during the\nSANS measurements was inferred from the position of\nthe intensity maxima in the helical state, which from nu-\nmerous independent studies are known to be accurately\naligned along the /angbracketleft111/angbracketrightdirections at zero field [13, 42–44].\nThe SANS measurements were carried out at the\ndiffractometer MIRA [45, 46] at the Heinz Maier-Leibnitz\nZentrum (MLZ) using a neutron wavelength λ= (10.4±\n0.5)˚A. The distance between sample and detector\nwas 2.1 m. Two apertures with a distance of 1.5 m\nand an opening of 2 ×2 mm2were placed between the\nmonochromator and the sample. The azimuthal and\nradial resolution of our set up were ∆ αaz= 3.8◦and\n∆q= 0.0024 ˚A−1, respectively. A closed-cycle cryostat\nequipped with a rotatable sample stick was used to cool\nthe sample, while permitting rotation of the sample with\nrespect to the vertical axis. The applied magnetic field\nwas generated with a pair of Helmholtz coils.\nShown in Fig. 1 is a schematic illustration of the an-\ngles required to describe our experimental set-up and\ndata. The sample could be rotated with respect to\nthe vertical axis described by ˆR. A small misalign-\nment of ˆRwith respect to the [ ¯110] axis of the sam-\nple is described by the angles θrandφr, such that\nˆR= (−cosθrcos(φr−π/4),−cosθrsin(φr−π/4),sinθr).\nForθr=φr= 0 the rotation axis corresponded to\nthe crystallographic [ ¯110] direction; finite angles θrand\nφrdescribe tilts of the rotation axis towards [001] and\n[¯1¯10], respectively. The direction of the magnetic field,\nˆH= ˆe1cosβ+ ˆe2sinβwas perpendicular to ˆR, where\nH||n\nyz\nεδβ\nxH||nsample\nNH:\nn:\nR:magnetic field\nneutron beam\nrotation axis\nê2≈[110]êR≈[110]\nzdetector\nnormal vector of\nskyrmion lattice\nin-plane alignment\nof skyrmion latticeN:\nω:ê1≈[001]\nyωQ1\nQ2\nQ3\nQ4Q5Q6\nHFIG. 1. Schematic illustration of the neutron scattering setup.\nThe magnetic field Hwas applied parallel to the neutron\nbeam n. The sample could be rotated with respect to the\naxis ˆR, where the rotation angle is denoted by β. Deviations\nof the normal vector Nof the skyrmion lattice from Hare\ndescribed by the angles εandδ, accounting for the tilt towards\nˆRand ˆH׈R, respectively. The orientation of the sixfold\nscattering pattern within the plane perpendicular to Nis\ndescribed by ω, denoting the angle between the rotation axis\nˆRand the closest wavevector Q1of the Bragg pattern. In our\nexperiments, the rotation axis ˆRpointed along [ ¯110] up to a\nsmall misalignment described in the text, thus ˆ e1≈[001] and\nˆe2≈[110].\nˆe1= (cos(φr−π/4) sinθr,sin(φr−π/4) sinθr,cosθr)\nand ˆe2=ˆR׈e1. Rotations of the sample with re-\nspect to ˆRare described by the angle β, such that\nˆH= (sinβ/√\n2,sinβ/√\n2,cosβ) forθr=φr= 0.\nThe orientation of the sixfold diffraction pattern of\nthe skyrmion lattice involves three degrees of freedom,\nnamely the angles δ,ε, andω. The first two describe\nthe orientation of the skyrmion lattice plane as defined\nby the unit normal vector ˆN=ˆHcosεcosδ+ˆRsinε+\n(ˆH׈R) cosεsinδ.ˆNis approximately parallel to the\nmagnetic field, ˆH. Small tilts towards ˆRand ˆH׈Rare\ndescribed by εandδ, respectively. Ignoring these small\ntilts, the orientation of the diffraction peaks are described\nby six unit vectors ˆQn=ˆRcos(ω+ (n−1)π/3) + ( ˆH×\nˆR) sin(ω+ (n−1)π/3) within the plane, where ωis the\nangle between ˆQ1and the direction of the rotation axis\nˆR.\nIn our study, we determined the detailed orientation of\nthe skyrmion lattice for a large number of applied mag-\nnetic field orientations, β, covering half of a great circle\nin steps of 5◦. For each angle βthe sample was cooled\nin an applied magnetic field of 183 mT from well above\nTcto the center of the skyrmion lattice phase pocket at3\n/s91/s48/s49/s48/s930\n0.51.0/s40\n/s50/s41/s44/s40/s51/s41/s61540/s32/s61/s32/s48/s176/s61538 /s32/s61/s32/s57/s48/s176/s105/s110/s116/s101/s103/s114/s97/s116/s101/s100/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s40\n/s53/s41/s44/s40/s54/s4100.51.0/s40\n/s54/s41/s40 /s50/s41/s44/s40/s51/s41/s40/s53/s41/s61538/s32/s61/s32/s52/s53/s176/s61540/s32/s61/s32/s49/s46/s49/s176-\n3-2-1012300.51.0/s40\n/s49/s41/s40/s52/s41/s61541/s32/s61/s32/s48/s46/s50/s176/s61538 /s32/s61/s32/s57/s48/s176/s32\n/s114\n/s111/s99/s107/s105/s110/s103/s32/s97/s110/s103/s108/s101/s32/s40/s176/s41/s40/s52/s41/s40/s49/s41/s40\n/s53/s41/s40/s54/s41/s40\n/s51/s41/s40/s50/s41/s91/s49/s49/s48/s93/s99/s116/s115/s46/s32/s47/s32/s115/s116/s100/s46/s32/s109/s111/s110/s46/s40\n/s108/s111/s103/s32/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s32\n/s32/s32/s32/s52/s32/s32/s49/s48/s32/s32/s50/s49/s32/s32/s52/s54/s49/s48/s48/s32\n/s32/s32/s32/s50/s664/s32H/s32/s124/s124/s32/s91/s48/s48/s49/s93/s40/s97/s41/s40 /s99/s41/s40\n/s98/s41/s48\n/s48/s46/s48/s53/s45/s48/s46/s48/s53/s48/s46/s48/s53/s45\n/s48/s46/s48/s53/s48qz/s32/s40/s197/s45/s49/s41\nq\ny/s32/s40/s197/s45/s49/s41/s48/s46/s48/s53/s45\n/s48/s46/s48/s53/s48qz/s32/s40/s197/s45/s49/s41/s664/s32H/s32/s124/s124/s32/s91/s49/s49/s48/s93/s61559\n/s32/s61/s32/s49/s53/s176/s61538/s32/s61/s32/s48/s176/s61538/s32/s61/s32/s57/s48/s176/s91\n/s49/s49/s48/s93/s91/s48/s48/s49/s93/s91\n/s49/s49/s48/s93\nFIG. 2. Typical small-angle neutron scattering data and anal-\nysis of these data. (a),(b) Sum over horizontal and vertical\nrocking scans for magnetic field along [110] ( β= 90◦) and\n[001] (β= 0◦) showing the sixfold scattering pattern of the\nskyrmion lattice. For technical reasons the intensity distribu-\ntion is uneven; see [39] for details. (c) Integrated intensity of\nthe red and blue boxes in the SANS data [cf. panel (a)] as a\nfunction of the vertical and horizontal rocking angle, respec-\ntively. Solid lines represent Gaussian fits. The angles δandε\ncorrespond to the arithmetic means of the peak centers.\n28.3 K. Subsequently, we performed a horizontal ( ±3.5◦)\nas well as a vertical ( ±2◦) rocking scan, both with a step\nsize of 0.25◦with respect to the direction of the incoming\nneutron beam (see [39] for technical details on the uneven\nintensity distribution). At the end of each measurement\nthe sample was heated up and βwas changed to the next\nvalue.\nFor all field directions studied the sum over both\nrocking scans showed the characteristic sixfold scatter-\ning pattern of the skyrmion lattice state, as depicted\nin Figs. 2(a) and 2(b) [39]. However, the orientation of\nthe pattern in reciprocal space displayed subtle changes.\nTo track these changes we labeled the intensity maxima\nofQ1throughQ6counter-clockwise, starting at the top\nwith the maximum closest to the rotation axis, ˆR. In\norder to determine the normal vector of the skyrmion\nlattice, ˆN, we first considered the integrated intensity of\nthe maxima Q2,Q3,Q5, andQ6(red boxes) as a func-\ntion of the vertical rocking angle, see top and middle\npanel of Fig. 2(c) for β= 45◦andβ= 90◦. Data were\nwell-described by Gaussian profiles where the arithmetic\nmean of the rocking centers provided the azimuthal de-\nviation angle δ. Analogously, the integrated intensity of\nthe maxima Q1andQ4(blue boxes) as a function of the\nhorizontal rocking angle provided the polar deviation ε\nas shown for β= 90◦in the bottom panel of Fig. 2(c).\nThe in-plane alignment of the sixfold scattering pattern\nwas finally described by the angle ωbetween the rotation\naxis and the closest intensity maximum. It was extracted\nfrom Gaussian fits to the integrated intensity of the sum\n090-6-3036/s40/s100/s41Δ /s97/s110/s103/s108/s101/s32/s47/s32/s49/s50/s48/s176/s32/s40/s37/s41/s111\n/s114/s105/s101/s110/s116/s97/s116/s105/s111/s110/s32/s61538/s32/s40/s176/s41-6-3036/s40/s99/s41Δ q/s32/s47/s32q/s104/s32/s40/s37/s41Q\n/s49/s32Q/s51/s32Q/s53-\n300306090120-5051015/s40/s98/s41/s105\n/s110/s45/s112/s108/s97/s110/s101/s32/s97/s108/s105/s103/s110/s109/s101/s110/s116/s115\n/s107/s121/s114/s109/s105/s111/s110/s32/s108/s97/s116/s116/s105/s99/s101/s91/s48/s49/s48/s93/s91\n/s49/s49/s48/s93/s97/s108/s105/s103/s110/s109/s101/s110/s116/s32/s61559/s32/s40/s176/s41/s99\n/s114/s121/s115/s116/s97/s108/s32/s111/s114/s105/s101/s110/s116/s97/s116/s105/s111/s110/s32/s61538/s32/s40/s176/s41-3.0-1.501.53.0/s91 /s49/s49/s49/s93/s91 /s49/s49/s48/s93/s110\n/s111/s114/s109/s97/s108/s32/s118/s101/s99/s116/s111/s114/s115\n/s107/s121/s114/s109/s105/s111/s110/s32/s108/s97/s116/s116/s105/s99/s101/s91/s49/s49/s49/s93/s91 /s48/s48/s49/s93/s61540\n/s61541/s97/s110/s103/s108/s101/s115/s32/s61540, /s61541/s32/s40/s176/s41/s40/s97/s41FIG. 3. Orientation of the skyrmion lattice as a function of\nthe field direction as parametrized by β. Specific directions\nofHare indicated at the top of panel (a) ignoring the tiny\nmisalignment of R. (a) Deviation of the normal vector ˆN\nof the skyrmion lattice from the magnetic field orientation as\nparametrized by δandε. (b) Alignment of the sixfold scatter-\ning pattern in the skyrmion lattice plane specified by ω. For\nˆR= [¯110] atω= 0◦andω= 15◦one of the Bragg wavevec-\ntors is aligned along [010] and [ ¯110], respectively. Solid lines\nare fits that account for the small misalignment of the rotation\naxis from [ ¯110], see text.(c),(d) Variation of the modulus of\nthe Bragg vectors and their enclosed angles. Within the error\nbars the Bragg pattern remains rigidly sixfold symmetric.\nover both rocking scans as a function of the azimuthal\nangle (not shown).\nOur experimental data on the orientation of the\nskyrmion lattice as a function of the field direction are\nsummarized in Fig. 3. The error bars shown in Fig. 3 were\ndetermined as part of the fitting procedure described\nabove and reflect the high accuracy at which changes\nof the orientation of the skyrmion lattice could be deter-\nmined. Panel (a) shows the orientation of the skyrmion\nlattice plane. The deviation δof its normal from the\nfield direction within the rotation plane vanishes for fields\nalong major crystallographic axes. These configurations\nwere covered in previous experiments [3, 12, 27], unless\nbadly aligned samples were studied. For fields away from\nthe major orientations, the anisotropies caused a system-\natic meandering with δreaching values in the range ±3◦.\nIn comparison, the deviation with respect to the rotation\naxis,ε, was less than 0 .4◦.\nThe in-plane orientation ωas a function of field direc-\ntion is shown in Fig. 3(b). For most directions we observe\nonly a small deviation ω, implying that Q1is approxi-\nmately aligned with [ ¯110]. This is consistent with pre-\nvious reports [3, 12, 28] where fields were aligned along\n/angbracketleft110/angbracketright. However, for a magnetic field close to [001], i.e.,4\nβ= 0◦, a pronounced reorientation takes place such that\nω≈15◦. Here one of the Qvectors of the Bragg pattern\nis aligned along [010], see also Fig. 2(b). As depicted in\nFigs. 3(c) and 3(d), neither the modulus of the wave vec-\ntorsQ1,Q3, andQ5nor the angle they enclosed changed\nas a function of the applied field direction within exper-\nimental accuracy. This implies that the skyrmion lattice\nfor any orientation of the magnetic field is characterized\nby a rigid sixfold Bragg pattern.\nThe observation of a rigid skyrmion lattice allows\nto reduce the theoretical interpretation from the full\nGinzburg-Landau theory to a minimal model of a rigid\nskyrmion lattice in a potential that takes into account the\nsymmetries of the skyrmion lattice and the tetrahedral\npoint group T[3, 4, 13]. For information on the relation-\nship of the Ginzburg-Landau parameters with the coef-\nficients of this potential we refer to the supplementary\ninformation [39].\nWe first discuss the orientation of the unit normal vec-\ntorˆNas described by the potential\nV(ˆN) =−k\n2(ˆNˆH)2+µ(ˆN4\nx+ˆN4\ny+ˆN4\nz). (1)\nThe first term in kfavors an alignment of the normal with\nthe magnetic field, i.e., it represents the tilting energy\nin the absence of MCAs, whereas the second term in\nµaccounts for changes of the tilting energy due to the\nMCAs. In the limit of small µ/k/lessmuch1, one expects only\na slight deviation of the normal from the magnetic field\norientation. The contribution of the crystalline potential\ninµyields a small force that slightly tilts ˆNaway from\nˆH. One obtains ˆN=ˆHwhen this force vanishes which is\nthe case for directions where the potential displays either\nan extremum or a saddle point, i.e., for field along /angbracketleft111/angbracketright,\n/angbracketleft001/angbracketrightand/angbracketleft110/angbracketright. Perturbatively minimizing the potential\nin the limit of small µ,θr, andφrone obtains for the\ndeviations in lowest order\nδ≈µ\n2k[1 + 3 cos(2 β)] sin(2β), (2)\nε≈µ\n2k/parenleftBig\n8φrsin3β+θr[3 cosβ+ 5 cos(3β)]/parenrightBig\n.(3)\nThe polar deviation εvanishes exactly in case of perfect\nalignment of the rotation axis ˆRwith [ ¯110], i.e.,θr=\nφr= 0.\nThis brings us to the orientation of the skyrmion lat-\ntice within the plane orthogonal to ˆN. Due to its sixfold\nsymmetry, the orientation of the skyrmion lattice is gov-\nerned by terms that are sixth order in the wavevectors\nQjand, therefore, sixth-order in spin-orbit coupling [4].\nThere are two independent sixth order invariants for a\nunit vector ˆQthat are consistent with the tetrahedral\npoint group T, notably: I1(ˆQ) = ˆQ6\nx+ˆQ6\ny+ˆQ6\nzand\nI2(ˆQ) = ˆQ2\nxˆQ4\ny+ˆQ2\nyˆQ4\nz+ˆQ2\nzˆQ4\nx. An effective potential\nfor the skyrmion lattice orientation ωis obtained by sum-\nming over the invariants of the sixfold symmetric Braggwavevectors, i.e., V(ω) =/summationtext6\nn=1/summationtext2\ni=1λiIi(ˆQn). Its ex-\nplicit form is given by\nV(ω) =AˆR(β) cos(6ω) +BˆR(β) sin(6ω), (4)\nwhere the coefficients AandBdepend on the rotation\naxis ˆRand on the rotation angle β.\nFor a perfect alignment of ˆRwith [ ¯110], i.e., for θr=\nφr= 0 where ˆH= (sinβ/√\n2,sinβ/√\n2,cosβ) the coeffi-\ncients simplify to\nA[¯110]=−9λ1\n1024[5 sinβ+ sin(3β)]2, (5)\nB[¯110]=−3λ2\n1024[66 cosβ−11 cos(3β) + 9 cos(5β)].(6)\nFor a magnetic field along [110], i.e., β=π/2, the coef-\nficientBvanishes and the potential is solely determined\nby the cos(6 ω) dependence. The Q1vector of the Bragg\npattern is then predicted to be aligned along [ ¯110] with\nω= 0 for positive λ1>0, butωis shifted by 30◦for\nnegativeλ1<0 so that one of the Qvectors is instead\naligned along [001]. A sign change of λ1as a function\nof temperature/field and pressure thus accounts for the\ntransition observed in Cu 2OSeO 3[24] and MnSi [18], re-\nspectively. For a magnetic field along [001], i.e., β= 0,\ntheAcoefficient vanishes and the potential is propor-\ntional to sin(6 ω). In this case, one of the Qvectors\naligns along [010] or [100] for positive λ2>0 or nega-\ntiveλ2<0, respectively. For MnSi at ambient pressure\nwe findλ1,λ2>0.\nAfter accounting for the various signs, the result for\nωinferred from a minimization of the potential depends\non the ratio λ2/λ1. For MnSi we obtain for the angle\nω(β) =1\n6arctan(B[¯110]/A[¯110]) +δωwhere the perturba-\ntive correction δωis due to the small misalignment of the\nrotation axis. To lowest order in θrandφrit is given by\nδω=A[¯110]δB−B[¯110]δA\n6(A2\n[¯110]+B2\n[¯110]), (7)\nδA=−9λ2\n2048/bracketleftBig\nθr(143 sin(2β)−44 sin(4β) + 3 sin(6β))\n+φr(154 + 77 cos(2 β) + 22 cos(4 β) + 3 cos(6β))/bracketrightBig\n,\n(8)\nδB=9λ1\n256(5 sinβ+ sin(3β))\n×/bracketleftBig\nθr(5−cos(2β)) + 4φrsin(2β)/bracketrightBig\n. (9)\nIn order to compare theory and experiment, we in-\ncludedφrandθras fitting parameters since the exper-\niment is particularly sensitive to a small misalignment\nof the rotation axis. In outstanding agreement with ex-\nperiment the theory accounts for the orientation of the\nskyrmion lattice observed as described by δ,ε, andω, as5\nshown by the solid lines in Figs. 3(a) and 3(b). From\na combined fit we obtain for the misalignment angles\nφr= 0.4◦andθr= 2.2◦as well as for the parameters\nof the effective potentials µ/k= 0.04 andλ2/λ1= 0.14.\nIn summary, our SANS measurements provide a de-\ntailed qualitative and quantitative account of the precise\nskyrmion lattice orientation in agreement with MCAs up\nto sixth order in spin-orbit coupling in accordance with\nthe tetrahedral symmetry group Tof MnSi. Interest-\ningly, the potential Eq. (4) for the in-plane orientation\nωvanishes for twelve directions of the magnetic field\nwhich, for the particular value λ2/λ1= 0.14, are specified\nbyˆH≈±(0.98,0.21,0) and symmetry-equivalent direc-\ntions. On the unit sphere spanned by ˆH, these points\nconstitute singularities with winding number w= 1/6.\nWe predict that encircling each of these points on\nthe unit sphere as a function of field orientation, will\ngenerate a rotation of the Bragg pattern by an angle\n∆ω= 2πw=π/3. This will provide a critical test of the\nso-called ”hairy ball theorem”, which has been discussed\nin the analogous context of the Abrikosov vortex lattice\nin type II superconductors [33]. A related open question\nto be addressed experimentally in the future concerns\nwhether these singular points account for the degenerate\nmultidomain configurations of the skyrmion lattice that\nhave been observed in some of the cubic chiral magnets\n[4, 25, 27, 36].\nWe wish to thank A. Rosch, P. B¨ oni, S. 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Hasegawa, and M. Kataoka,\n“The origin of the helical spin density wave in MnSi,”\nSolid State Commun. 35, 995 (1980).Supplementary information for:\nResponse of the skyrmion lattice in MnSi to cubic magnetocrystalline anisotropies\nT. Adams,1M. Garst,2, 3A. Bauer,1R. Georgii,1, 4and C. Pfleiderer1\n1Physik Department, Technische Universit¨ at M¨ unchen, D-85748 Garching, Germany\n2Institute for Theoretical Physics, Universit¨ at zu K¨ oln, D-50937 K¨ oln, Germany\n3Institut f¨ ur Theoretische Physik, Technische Universit¨ at Dresden, 01062 Dresden, Germany\n4Heinz Maier-Leibnitz Zentrum (MLZ), Technische Universit¨ at M¨ unchen, D-85747 Garching, Germany\n(Dated: December 3, 2018)\nWe report supplementary information on the relationship between the anisotropy potential used\nin the main text for the analysis of our experimental data and the Ginzburg-Landau theory of cubic\nchiral magnets. We also provide further information on the experimental methods, notably typical\nrocking scans and intensity patterns observed in small angle neutron scattering.\nPACS numbers: 75.40.-s, 74.40.-n, 75.10.Lp, 75.25.-j\nI. DERIVATION OF THE ANISOTROPY\nPOTENTIAL\nThe Ginzburg-Landau theory for cubic chiral magnets\nrepresents an effective low-energy theory in terms of the\nmagnetization M. Detailed introductions may, for in-\nstance, be found in Ref. 1–4. It is important to note that\nthe small spin-orbit coupling λSOCobserved in MnSi im-\nplies a hierarchy of interactions. Up to second order in\nλSOC, the Ginzburg-Landau theory includes the ferro-\nmagnetic exchange, the Dzyaloshinskii-Moriya interac-\ntion, as well as the dipolar interaction. On this level\nit is invariant with respect to a combined SO(3) rota-\ntion in real and spin space. In the presence of an ap-\nplied magnetic field, H, this symmetry is reduced to a\nO(2) rotation symmetry around the axis of the magnetic\nfieldH. Taking additionally into account terms that\nare fourth order and higher order in spin-orbit coupling,\nthe rotation symmetry is reduced by magnetocrystalline\nanisotropies (MACs). These are of central interest for\nthe study reported in the main text.\nAn analysis of the impact of MCAs on the skyrmion\nlattice in terms of the full Ginzburg-Landau theory for\nthe magnetization is a challenging task. In order to cap-\nture quantitatively the response of the modulated mag-\nnetization with respect to changes in the magnetic field,\nthe differential susceptibility tensor needs to be evaluated\nwhich is generally not accessible on the mean-field level.\nHowever, it turns out that the description simplifies con-\nsiderably in the limit of small spin-orbit coupling, which\nis applicable to MnSi.\nTo lowest order in spin-orbit coupling MCAs may be\nneglected, and the skyrmion lattice ground state pos-\nsesses a perfect six-fold symmetry characterized by a\nnormal unit vector ˆNdescribing the orientation of the\nskyrmion lattice plane and an angle ωthat accounts for\nthe orientation of the skyrmion lattice within this plane.\nIn equilibrium, the normal ˆNis aligned with the mag-\nnetic field direction. A tilt of ˆNaway from this direc-\ntion will cost energy and we demonstrate below that thisenergy cost is quantified by a single quantity, i.e., the\nstiffnesskthat may be expressed in terms of the homo-\ngeneous magnetization and the transverse susceptibility\nevaluated at fixed ˆN. To lowest-order in spin-orbit cou-\npling the angle ωremains undetermined due to the rota-\ntion symmetry around the applied magnetic field.\nThe effect of MCAs may then be captured by con-\ntributions to the effective potentials for ˆNandωthat\nare governed by the tetrahedral point group symmetry.\nThe coefficients µ,λ1andλ2of the resulting potentials\nmay be related to corresponding coefficients of MCAs\nin the Ginzburg-Landau theory by perturbatively evalu-\nating them within the approximate mean-field solution\nin the absence of MCAs, which possesses the six-fold\nsymmetry. This approach is borne out of our experi-\nmental observation of the modulus and angular relation-\nship of the Q-vectors, indicating that the distortion of\nthe skyrmion lattice for different field directions remains\nnegligible. The response of the skyrmion lattice is finally\naddressed by minimizing the effective potentials for ˆN\nandωas discussed in the mains text.\nAs a remark on the side, the same general strategy has\nalso been used to account for the reorientation transi-\ntion between the helix and conical phases in cubic chiral\nmagnets, which may be addressed in terms of an effec-\ntive potential for the orientation of the single- Qhelical\nmodulation [5].\nThe presentation is organized as follows. We begin\nin section I A with a discussion of the tilting stiffness of\nthe skyrmion lattice, i.e., changes of orientation of ˆN,\nin the absence of MCAs as expressed by the coefficient\nk. This is followed in section I B by a discussion of the\ntilting stiffness in the presence of MCAs, motivating the\nadditional parameter µ. Finally, the in-plane orientation\nwill be addressed in section I C, providing insights on the\nparameters λ1andλ2.arXiv:1811.12439v1  [cond-mat.str-el]  29 Nov 20182\nA. Skyrmion lattice tilting stiffness without MCAs\nWe begin with the free energy of a skyrmion lattice\nwith a fixed normal vector ˆNin the absence of mag-\nnetocrystalline anisotropies . For symmetry reasons the\nassociated energy density is given by\nF(ˆN) =−k\n2(ˆNˆH)2+F0, (1)\nwhere ˆH=H/Hdenotes the direction of the magnetic\nfield. The coefficient kintroduced here and in the context\nof the potential given in the main text are identical. It\nquantifies the tilting stiffness of the skyrmion lattice with\nrespect to changes of direction of ˆN. In turn, the coeffi-\ncientkas well as the contribution F0to the free energy\ndepend on the strength of the magnetic field H=|H|,\ni.e.,k=k(H) andF0=F0(H). In equilibrium and in the\nabsence of MCAs the free energy F(ˆN) of the skyrmion\nlattice is minimal for ˆN=ˆH.\nThe coefficient kmay be related to the differential mag-\nnetic susceptibility of the skyrmion lattice phase distin-\nguishing the longitudinal and transverse susceptibilities,\nχ/bardblandχ⊥, respectively. The longitudinal susceptibility\nis given by χ/bardbl=∂M/∂H , whereM=|M|is the mag-\nnitude of the magnetization. In contrast, for the trans-\nverse susceptibility, describing changes of the magnetiza-\ntion under a perpendicular perturbing field, two limits\nmust be considered. Namely, the susceptibility for un-\nconstrained changes of the direction of the normal vector\nˆN, denotedχ⊥,free, and the susceptibility when the direc-\ntion of the normal vector ˆNis fixed and cannot change,\ndenotedχ⊥,fixed. In comparison to the transverse sus-\nceptibility, the longitudinal susceptibility is the same for\nboth unconstrained and fixed normal vector ˆN.\nIf the normal vector is unconstrained and free to\nchange its orientation, it adopts its optimal value for\nˆN=ˆHinstantaneously under changes of field direction,\nand the magnetization Mis always parallel to the field,\nM=MˆH. For this unconstrained response the suscep-\ntibility tensor is given by the standard expression\n∂Mi\n∂Hj/vextendsingle/vextendsingle/vextendsingleˆNfree=∂M\n∂HˆHjˆHi+M\nH(δij−ˆHiˆHj),(2)\nand the associated unconstrained transverse susceptibil-\nity becomes\nχ⊥,free=M\nH(3)\nHowever, as shown in detail below, for constrained (fixed)\nnormal vector ˆNthe susceptibility tensor is given by\n∂Mi\n∂Hj/vextendsingle/vextendsingle/vextendsingleˆNfixed,ˆN→ˆH=∂M\n∂HˆHjˆHi+χ⊥(δij−ˆHiˆHj),(4)where the limit ˆN→ˆHwas taken after taking the deriva-\ntive and the transverse susceptibility is given by\nχ⊥,fixed=M\nH−k\nµ0H2. (5)\nThus, the tilting stiffness kin the absence of MCAs may\nbe inferred from the difference of the transverse suscepti-\nbilities for unconstrained and constrained (fixed) normal\nvector ˆN, notably\nk=µ0H2/parenleftbiggM\nH−χ⊥,fixed/parenrightbigg\n. (6)\nIt is now interesting to note that the transverse sus-\nceptibilityχ⊥,fixed may, in principle, be determined ex-\nperimentally by means of an ac susceptibility measure-\nment, where a small oscillating field component perpen-\ndicular to the applied static magnetic field stabilizing the\nskyrmion lattice generates an oscillating change of orien-\ntation H(t). In practice the direction of the oscillating\nfield component has to be chosen such that the effects\nof MCAs vanish. The response of the skyrmion lattice\nas described by the normal vector ˆN(t) then reflects to\nwhat extent the skyrmion lattice is capable of following\nany changes of the field direction as a function of time.\nIn fact, the relaxation of the normal vector ˆNinto its\nequilibrium position will require a relaxation time τthat\nmay be expected to be large as it involves the reorienta-\ntion of macroscopic skyrmion lattice domains. Thus, for\nrelatively large ac frequencies ωτ/greatermuch1, the normal vector\nˆNremains essentially unaffected. In contrast, ˆN(t) will\nbe able to follow the changes of field orientation for low\nfrequencies ωτ/lessmuch1. It is therefore expected that the\ntransverse ac susceptibility assumes the values χ⊥,fixed\nandχ⊥,free=M/H in the limits of very large and very\nlow frequencies, respectively. These consideration com-\npare with the frequency dependence of the ac suscepti-\nbility at the helix reorientation transition, which is also\ngoverned by very long relaxation times [5]. Thus, the rel-\nevant behavior for the skyrmion lattice may be accessible\nexperimentally.\nReturning now to the expression for χ⊥,fixed given in\nEq. (5), we present in the following a derivation starting\nwith the magnetization associated with the free energy\ndensityFin Eq. (1) at a fixed ˆN,\nµ0Mi(ˆN) =−∂F\n∂Hi\n=k/prime\n2(ˆNˆH)2ˆHi+k(ˆNˆH)ˆNj\nH(δji−ˆHjˆHi)−F/prime\n0ˆHi\n=/bracketleftBig\n(ˆNˆH)2/parenleftbiggk/prime\n2−k\nH/parenrightbigg\n−F/prime\n0/bracketrightBig\nˆHi+kˆNˆH\nHˆNi,(7)\nwherek/prime=∂HkandF/prime\n0=∂HF0. In equilibrium ˆN=ˆH,\nand the magnetization is parallel to the applied magnetic3\nfield\nµ0Mi(ˆN=ˆH) =/parenleftbiggk/prime\n2−F/prime\n0/parenrightbigg\nˆHi. (8)\nComputing the differential susceptibility at fixed ˆNwe\nfind\nµ0∂Mi(ˆN)\n∂Hj=µ0∂Mi(ˆN)\n∂HˆHj\n+µ0∂Mi(ˆN)\n∂ˆH/lscript1\nH(δ/lscriptj−ˆH/lscriptˆHj).(9)\nThis expression comprises two parts that originate in the\ndependences on Hand ˆH, i.e., the strength and the ori-\nentation of the magnetic field, respectively. Considering\nthese contributions separately, we obtain\nµ0∂Mi(ˆN)\n∂H=/parenleftbigg\n(ˆNˆH)2/parenleftbiggk/prime/prime\n2−k/prime\nH+k\nH2/parenrightbigg\n−F/prime/prime\n0/parenrightbigg\nˆHi\n+ (ˆNˆH)ˆNi/parenleftBigk/prime\nH−k\nH2/parenrightBig\n,\n(10)\nand\nµ0∂Mi(ˆN)\n∂ˆH/lscript=/parenleftbigg\n(ˆNˆH)2/parenleftbiggk/prime\n2−k\nH/parenrightbigg\n−F/prime\n0/parenrightbigg\nδi/lscript\n+ 2( ˆNˆH)/parenleftbiggk/prime\n2−k\nH/parenrightbigg\nˆHiˆN/lscript+k\nHˆN/lscriptˆNi.\n(11)\nIn the limit ˆN→ˆH, the susceptibility tensor simplifies\nto\nµ0∂Mi\n∂Hj/vextendsingle/vextendsingle/vextendsingleˆNfixed,ˆN→ˆH=/parenleftbiggk/prime/prime\n2−F/prime/prime\n0/parenrightbigg\nˆHjˆHi\n+1\nH/bracketleftbigg/parenleftbiggk/prime\n2−k\nH/parenrightbigg\n−F/prime\n0/bracketrightbigg\n(δij−ˆHiˆHj).(12)\nUsing the result for the magnetization given in Eq. (8),\nnotablyM=k/prime\n2−F/prime\n0, we obtain the expression for χ⊥,fixed\ngiven in Eq. (5).\nB. MCA potential of skyrmion lattice tilting\nHaving considered the tilting stiffness of the skyrmion\nlattice in the absence of MCAs, we turn next to the tilting\nstiffness in the presence of the MCAs. In the potential\nVthis is accounted for by the terms with the additional\nparameterµ. The magnetocrystalline anisotropies enter-\ning the Ginzburg-Landau theory of chiral magnets such\nas MnSi must satisfy the tetrahedral point group sym-\nmetries consisting of a threefold C3rotation symmetry\naround the crystallographic /angbracketleft111/angbracketrightaxes and a twofold C2\nrotation symmetry around crystallographic /angbracketleft100/angbracketrightaxes.Here one has to take into account contributions that\nbreak the continuous rotation symmetry both in real and\nspin space. For the tilting of the skyrmion lattice plane,\ndescribed by the normal vector ˆN(t), it is instructive to\nconsider the following three terms\nF4=K1(M4\nx+M4\ny+M4\nz)\n+K2M(∂4\nx+∂4\ny+∂4\nz)M\n+K3[(∂xMx)2+ (∂yMy)2+ (∂zMz)2].(13)\nWhereas the first term breaks the continuous rotational\nsymmetry in spin space, the second term breaks this sym-\nmetry in real space. The third term, finally, breaks this\nsymmetry simultaneously in spin and real space.\nIn order to classify these contributions according to\npowers of spin-orbit coupling λSOC, it is important to\nnote that each derivative eventually contributes a power\nofλSOCas the Dzyaloshinskii-Moriya interaction leads to\nspatial variations of the magnetization on a length scale\n∼1/λSOC. Whereas the coefficients themselves differ in\ntheir magnitude K1∼O (λ4\nSOC),K2∼O (λ0\nSOC), and\nK3∼ O (λ2\nSOC), all three terms essentially contribute\nonly at fourth order in spin-orbit coupling due to the\nderivatives. The situation is thus rather more involved\nthan in conventional ferromagnets without modulations\nwhere only the first term would be relevant. In the fol-\nlowing, we will refrain from providing an exhaustive list\nof all terms at a given order in spin-orbit coupling and\nrefer instead to the literature, see for example Ref. 6.\nTurning now to the relationship of the Ginzburg-\nLandau theory with the potential given in the main text,\nwe revisit the effective Landau potential for the normal\nvector ˆNof the skyrmion lattice at lowest order,\nVaniso(ˆN) =µ(ˆN4\nx+ˆN4\ny+ˆN4\nz), (14)\nwith the coefficient µ∼O(λ4\nSOC). As the normal vec-\ntorˆNcharacterizes an entire skyrmion lattice domain,\nit is spatially constant. The effective magnetocrystalline\npotential for ˆNis thus analogous to the that of a fer-\nromagnet with a spatially constant magnetization, i.e.,\nEq. (14) is the only term at this order of λSOC. For\nthe chiral magnets considered in our paper, the coeffi-\ncientµmay in principle be related to the corresponding\ncoefficients of the full Ginzburg-Landau theory for the\nmagnetization.\nTo lowest order in spin-orbit coupling λSOC, this\nmay be achieved by treating the magnetocrystalline\nanisotropies in lowest-order perturbation theory. This\nwill be illustrated in the following for the three terms\nlisted in Eq. (13). To zeroth order in the MCAs, the\nequilibrium magnetization of the skyrmion lattice may4\nbe well approximated by a triple- Qstructure [3],\nM(r) =M0ˆH\n+MQ3/summationdisplay\ni=1/parenleftBig\nˆe1icos(QˆQi+φi) + ˆe2isin(QˆQi+φi)/parenrightBig\n,\n(15)\nwhereM0andMQare the amplitudes of the mean and\nthe modulated magnetization. The three sets of unit vec-\ntors form an orthogonal frame each, ˆ e1i׈e2i=ˆQifor\ni= 1,2,3, and, in addition, ˆQ1+ˆQ2+ˆQ3= 0. The phases\nmay be chosen arbitrarily provided that φ1+φ2+φ3=π\n2.\nPlugging this Ansatz into Eq. (13) and averaging over a\nmagnetic unit cell, one obtains an effective anisotropy\npotential (14) with coefficient µ=µ1+µ2+µ3where\nµ1=K1M4\ns/parenleftBigg\n9\n64+39\n32m2−23\n64m4−√\n3\n6m(1−m2)3/2/parenrightBigg\n,\n(16)\nµ2=K2M2\nsQ43\n8(1−m2), (17)\nµ3=K3M2\nsQ23\n16(1−m2). (18)\nHere we identified Mswith the averaged magnitude of\nthe magnetization,/radicalbig\n/angbracketleftM2/angbracketright=/radicalBig\nM2\n0+ 3M2\nQ, andm=\nM0/Ms.\nAll three terms of Eq. (13) contribute to the coefficient\nµ. In fact, there are even more contributions account-\ning for magnetocrystalline anisotropies in the Ginzburg-\nLandau theory that are not listed in Eq. (13). This obser-\nvation is important in two respects. On the one hand, a\ndescription of the experimental behaviour of the normal\nvector ˆNon the level of the full Ginzburg-Landau the-\nory is not unique as there are different combinations of\nmagnetocrystalline terms that yield the same value of µ,\ni.e., the knowledge of ˆNis not sufficient to reconstruct all\nof these magnetocrystalline contributions. On the other\nhand, it is the beauty of emergent simplicity that the\nexperimental behavior of ˆNis captured by the effective\nLandau potential of Eq. (14) characterized by a single\ncoefficientµgiven the rather involved structure of the\nparent Ginzburg-Landau theory for the magnetization.\nTaken together, the tilting of the skyrmion lattice\nplane may be described by the combination of the param-\neterskandµ, wherekmeasures the tilting stiffness with-\nout MCAs, and µmeasures the strength of the anisotropy\npotential.\nC. MCA potential of skyrmion lattice rotation\nFor the orientation of the skyrmion lattice within the\nplane orthogonal to ˆN, magnetocrystalline anisotropiesof higher order than in Eq. (13) are required. Namely,\ndue to the sixfold symmetry of the skyrmion lattice, we\nneed to consider magnetocrystalline anisotropies at least\nof sixth order in spin-orbit coupling, O(λ6\nSOC). In order\nto keep the discussion simple, we consider only two such\ncontributions to the Ginzburg-Landau theory, namely\nF6=K4M(∂6\nx+∂6\ny+∂6\nz)M+\nK5M(∂2\nx∂4\ny+∂2\ny∂4\nz+∂2\nz∂4\nx)M.(19)\nThe coefficient of both terms are of the order K4,5∼\nO(λ0\nSOC). However, due to the derivatives they only con-\ntribute to orderO(λ6\nSOC) and may therefore be expected\nto be particularly weak. The first term, K4, possesses a\nhigher symmetry than the tetrahedral point group as it\nis invariant with respect to fourfold C4rotations around\nthe/angbracketleft100/angbracketrightaxes. The second term, K5, has a lower sym-\nmetry as it is invariant only with respect to C2rotations\naround/angbracketleft100/angbracketright.\nTreating these terms in lowest-order perturbation the-\nory with the Ansatz in Eq. (15), we obtain a potential\nthat may be expressed in terms of the three unit vectors\nˆQi, withi= 1,2,3, characterizing the orientation of the\nsix Bragg peaks of the skyrmion lattice phase,\nV=2λ13/summationdisplay\ni=1(ˆQ6\ni,x+ˆQ6\ni,y+ˆQ6\ni,z)\n+ 2λ23/summationdisplay\ni=1(ˆQ2\ni,xˆQ4\ni,y+ˆQ2\ni,yˆQ4\ni,z+ˆQ2\ni,zˆQ4\ni,x).(20)\nThe two terms in the brackets are the two sixth-order\ninvariants that are consistent with the symmetries of the\ntetrahedral point group. In the main text, we discussed\nthis potential in terms of six vectors ±ˆQ1,±ˆQ2and±ˆQ3\nand the factor of two in Eq. (20) ensures that the defini-\ntion of the coefficients λ1,2are consistent with the main\ntext. Perturbatively, we obtain for these coefficients from\nthe two terms in Eq. (19),\nλ1=K4M2\nsQ6/parenleftbigg\n−1\n6/parenrightbigg\n(1−m2), (21)\nλ2=K5M2\nsQ6/parenleftbigg\n−1\n6/parenrightbigg\n(1−m2). (22)\nNote that there are further contributions to λ1,2from\nMCAs not listed in Eq. (19) similar to the case of µin\nEq. (14).\nExperimentally it is found that λ2/λ1≈0.14, although\nboth terms are nominally of sixth order in λSOC. It is\ntempting to speculate that this small ratio reflects a rel-\natively weak breaking of the fourfold C4rotation symme-\ntry in MnSi that lowers the point group from octahedral\nto tetrahedral symmetry.5\nFIG. S1: Typical scattering patterns as recorded during a horizontal rocking scan in the A-phase of MnSi (the rotation axis is\nvertical). Data were recorded with a CASCADE detector measuring 200 ×200 mm2. The rocking scan shown here covered a\ntotal of 29 angular positions between -3.5◦and +3.5◦in steps of 0.25◦. Due to the very sharp rocking width of the individual\npeaks, intensity of individual peaks was only observed in a narrow angular range. Shown here are data for β= 90◦and the\nfollowing rocking angles: (a) -1.5◦, (b) -1.0◦, (c) -0.5◦, (d) 0◦, (e) +0.5◦, (f) +1.0◦, (g) +1.5◦, and (h) +2.0◦.\nII. EXPERIMENTAL METHODS\nFor the accurate determination of the orientation of the\nskyrmion lattice so-called rocking scans were performed\nwith respect to a vertical and a horizontal axis. In a\nrocking scan the sample and magnetic field are both ro-\ntated (rocked) with respect to the position of interest,\ni.e., the orientation for which the incoming neutron beam\nand the magnetic field were strictly parallel. This allows\nto determine the distribution of angles under which the\nscattering condition is satisfied and provides information\non the so-called mosaic spread.\nAs our study pursued the collection of a large system-\natic data set, rocking scans for different field orientations\nwere recorded automatically covering a large number of\nangular positions. Horizontal rocking scans were per-\nformed between -3.5◦and +3.5◦in steps of 0.25◦, where\ntypical data are shown in Fig. S1. Vertical rocking scans\nwere performed between -2◦and +2◦in steps of 0.25◦.\nThe sum over all horizontal and vertical rocking positions\nrepresents the so-called integrated scattering intensity.\nHowever, in our study all rocking scans were centered\nwith respect to the field direction. In turn, due to the\nsmall tilting of the skyrmion lattice plane, which was only\nidentified during the full analysis of the data after data\ncollection had been completed, not all rocking scans were\ncomplete in a strict sense covering all angular positionsunder which some scattering was still observable. While\nit was nonetheless possible to determine the direction of\nˆNat very high accuracy fitting all peaks simultaneously,\nthe intensity distribution in sums over rocking scans ap-\npears to be uneven. It is important to emphasize, that\nthis effect represents a purely technical consequence of\nthe incomplete range of the rocking scans.\nFor instance, the typical intensity patterns shown in\nFigs. 2 (a) and 2 (b) in the main text were selected to\nillustrate the definition and accuracy at which the tilting\nangles of the skyrmion lattice, δand/epsilon1, and the in-plane\nrotation angle, ω, could be determined. Unfortunately\nthe tilting angles for these positions was not aligned with\nthe main field direction, hence the intensity distribution\nas a function of azimuthal angle appears to be slightly\nuneven.\n[1] P. Bak and M. H. Jensen, J. Phys. C: Solid State 13, L881\n(1980), URL http://iopscience.iop.org/0022-3719/\n13/31/002 .\n[2] O. Nakanishi, A. Yanase, A. Hasegawa, and\nM. Kataoka, Solid State Commun. 35, 995 (1980), URL\nhttp://www.sciencedirect.com/science/article/pii/\n0038109880910042 .\n[3] S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch,6\nA. Neubauer, R. Georgii, and P. B¨ oni, Science 323, 915\n(2009), URL http://www.sciencemag.org/content/323/\n5916/915.full .\n[4] W. M¨ unzer, A. Neubauer, T. Adams, S. M¨ uhlbauer,\nC. Franz, F. Jonietz, R. Georgii, P. B¨ oni, B. Pedersen,\nM. Schmidt, et al., Phys. Rev. B 81, 041203 (R) (2010),\nURL http://link.aps.org/doi/10.1103/PhysRevB.81.041203 .\n[5] A. Bauer, A. Chacon, M. Wagner, M. Halder, R. Georgii,\nA. Rosch, C. Pfleiderer, and M. Garst, Phys. Rev. B\n95, 024429 (2017), URL https://link.aps.org/doi/10.\n1103/PhysRevB.95.024429 .\n[6] K. Everschor, Ph.D. thesis, Universit¨ at zu K¨ oln (2012)."    },    {        "title": "1812.03611v2.Noncollinearity_effects_on_magnetocrystalline_anisotropy_for__R_2_Fe___14__B_magnets.pdf",        "content": "Non-collinearity E\u000bects on Magnetocrystalline Anisotropy for\nR2Fe14B Magnets\nDaisuke Miura and Akimasa Sakuma\nDepartment of Applied Physics, Tohoku University, Sendai 980-8579, Japan\u0003\n(Dated: September 13, 2021)\nAbstract\nWe present a theoretical investigation of the magnetocrystalline anisotropy (MA) in R2Fe14B (R\nis a rare-earth element) magnets in consideration of the non-collinearity e\u000bect (NCE) between the R\nand Fe magnetization directions. In particular, the temperature dependence of the MA of Dy 2Fe14B\nmagnets is detailed in terms of the nth-order MA constant (MAC) Kn(T) at a temperature T. The\nfeatures of this constant are as follows: K1(T) has a broad plateau in the low-temperature range\nandK2(T) persistently survives in the high-temperature range. The present theory explains these\nfeatures in terms of the NCE on the MA by using numerical calculations for the entire temperature\nrange, and further, by using a high-temperature expansion. The high-temperature expansion for\nKn(T) is expressed in the form of Kn(T) =\u00141(T) [1 +\u000e(T)] [\u0000\u000e(T)]n\u00001, where\u00141(T) is the part\nwithout the NCE and \u000e(T) is a correction factor for the NCE introduced in this study. We also\nprovide a convenient expression to evaluate Kn(T), which can be determined only by a second-order\ncrystalline electric \feld coe\u000ecient and an e\u000bective exchange \feld.\n\u0003dmiura@solid.apph.tohoku.ac.jp\n1arXiv:1812.03611v2  [cond-mat.mtrl-sci]  2 Apr 2019I. INTRODUCTION\nR2Fe14B compounds ( Ris mainly a rare-earth element) have been research targets in\nthe \felds of not only engineering but also science; speci\fcally, the magnetism of these\ncompounds has been systematically investigated both experimentally and theoretically[1{\n6]. Present-day high-performance computers allow us to directly calculate these electronic\nand/or magnetic structures in complex and large calculation models for R2Fe14B-based sys-\ntems, which also becomes a motivation for developing new numerical methods such as high-\naccuracy \frst-principles calculation methods[6], constrained Monte Carlo methods[7], and\n\fnite-temperature Landau{Lifshitz{Gilbert analyses[8, 9]. Furthermore, the \felds of engi-\nneering strongly require theoretical guidelines to develop magnets, the performance of which\nexceeds that of Nd-Fe-B magnets. Most recently, the above-mentioned numerical methods\nhave been applied to a realistic model for rare-earth intermetallics, and quantitative results\ncomparable to the experimental results were obtained by \frst-principles calculations[10{15]\nand by Monte Carlo methods[16{20]. On the other hand, to date, simpler analyses have\nalso been conducted on the basis of phenomenological theory[21{24] or mean \feld theory\n(MFT)[3, 14, 25{27] to understand the mechanism of the coercive forces of rare-earth per-\nmanent magnets and to identify the factors dominating these mechanisms.\nThe magnetocrystalline anisotropy (MA) of a magnet refers to its free energy density as\na function of the magnetization direction. In simple magnets[28], the free energy density is\nwell expressed by the single term K1(T) sin2\u0002, whereK1(T) is the \frst-order MA constant\n(MAC) and \u0002 is the zenithal angle of the magnetization measured from the crystal axis.\nHowever, in rare-earth (RE) magnets, the angle dependence of the free energy density has\na more complex form, especially in the low-temperature range[29{32]; assuming tetragonal\nsymmetry such as that of R2Fe14B compounds, the free energy density can be expressed as\n1X\nm=0bm=2cX\nn=0K(0)n\nm(T) sin2m\u0002 cos 4n\b; (1)\nwhereK(0)n\nm(T) is themth-order MAC and \b is the azimuthal angle of the magnetization.\nThe expansion presents a convergence problem and it is di\u000ecult to uniquely determine\nK(0)n\nm(T) because of the non-orthogonality of the basis set of fsin2m\u0002 cos 4n\bgas reviewed\nby Kuz'min[33]. Thus, MACs have been evaluated by assuming convergence or by using\n\ftting methods that assume a \fnite-expansion form for practical purposes [29{32, 34{40].\n2In our previous studies on the MA of Nd 2Fe14B magnets[24{26], the total magnetization\nwas assumed to be collinear to the Fe magnetization. However, this assumption raises a\nserious error in evaluations for the MA of the R2Fe14B magnet, the Rmagnetization of which\nis highly non-collinear to its Fe magnetization. For example, Dy 2Fe14B magnets exhibit\nthe non-collinearity e\u000bect (NCE) on the temperature dependence of the MA. Recently,\nIto et al.[27] have calculated K1(T) of Dy 2Fe14B magnets without the NCE, and then they\ndemonstrated that the resultant K1(T) rapidly decays with increasing temperature; however,\nas is well known in experiments, K1(T) of Dy 2Fe14B magnets has a broad plateau in the\nlow-temperature range. In addition, they pointed out the importance of the NCE via an\nMFT analysis of the magnetization curves of Dy 2Fe14B magnets. In Sect. III, we clearly\nshow that the NCE on K1(T) is the origin of the disagreement between the MFT and the\nexperiments on Dy 2Fe14B magnets.\nIn this study, we theoretically investigated the NCE on the temperature-dependent MA\nofR2Fe14B magnets by using numerical calculations for the entire temperature range. Fur-\nthermore, in the high-temperature range, we provide explicit expressions to describe the\nNCE and clarify our understanding of how the NCE appears in the MA. We also provide a\npractical expression to estimate the temperature dependence of the MA in RE intermetallics.\nThe present article is constructed as follows: In Sect. II, we review previous theoretical work\non the temperature dependence of Kn(T) without NCEs in RE intermetallics. In Sect. III,\nwe show how the NCEs on the MA appear by taking the Nd 2Fe14B and Dy 2Fe14B magnets\nas examples, and we develop microscopic expressions for the MA with NCEs in R2Fe14B\nmagnets in the high-temperature range. In addition, we apply the results to R=Tb, Dy, Ho,\nEr, Tm, and Yb. In Sect. IV, we summarize this study.\nII. PRESENT UNDERSTANDING OF THE TEMPERATURE-DEPENDENT MA\nINR2Fe14BMAGNETS WITHOUT NCES\nFirst, let us brie\ry recall important theoretical studies on temperature-dependent MA\nin two-sublattice systems. Although it is di\u000ecult to directly express the temperature-\ndependent MACs under general conditions[33], several explicit expressions have been ob-\ntained for limited situations. Here, we consider the temperature-dependent MA of an\nR2Fe14B magnet as an example of the two-sublattice system. Assuming that the Fe mag-\n3netization is collinear to the total magnetization in the magnet, we can evaluate the MACs\nfrom the MA as a function of the Fe magnetization angle, and then the total mth-order\nMAC is separated into the R- and Fe-sublattice contributions as\nK(0)n\nm(T) =KR(0)n\nm(T) +KFe(0)n\nm(T): (2)\nHere, we assume that KFe(0)n\nm(T) is obtained from the experimental results for R2Fe14B\nmagnets with a nonmagnetic Relement such as R=Y, and therefore, we focus only on\nKR(0)n\nm(T). A qualitative (but simple) understanding of KR(0)n\nm(T) can be obtained from the\npower-law scenario derived by Zener[41]; most recently[24], we have explicitly expressed the\nextended form of the Akulov{Zener{Callen{Callen law[41{43] (or today simply known as\nthe Callen{Callen law) up to the third order, as\nKR\n1(T) =KR\n1(0)\u0016R(T)3\n+8\n7KR\n2(0)\u0002\n\u0016R(T)3\u0000\u0016R(T)10\u0003\n+8\n7KR\n3(0)\u0012\n\u0016R(T)3\u000018\n11\u0016R(T)10+7\n11\u0016R(T)21\u0013\n; (3a)\nKR\n2(T) =KR\n2(0)\u0016R(T)10\n+18\n11KR\n3(0)\u0002\n\u0016R(T)10\u0000\u0016R(T)21\u0003\n; (3b)\nKR0\n2(T) =KR0\n2(0)\u0016R(T)10\n+10\n11KR0\n3(0)\u0002\n\u0016R(T)10\u0000\u0016R(T)21\u0003\n; (3c)\nKR\n3(T) =KR\n3(0)\u0016R(T)21; (3d)\nKR0\n3(T) =KR0\n3(0)\u0016R(T)21; (3e)\nwhere\u0016R(T) is the normalized Rmagnetization with \u0016R(0) = 1, and we demonstrated that\nthe experimental results for Nd 2Fe14B magnets well obey the extended Callen{Callen law\n[Eq. (3)]. This view of the power law enables us to immediately establish that a narrow\nplateau appears in the low-temperature range and that higher-order MACs rapidly decay\nwith increasing temperature compared with lower-order ones. These features describe the\ngeneral behavior of on-site MA in homogeneous local moment systems[21, 24, 27, 44].\nOn the other hand, to re\rect the material individuality, a microscopic description for the\nMA is appropriate. Many authors have reported the microscopic theory for temperature-\ndependent MACs[1, 3]. At zero temperature, Yamada et al.[45] reported the explicit relation\n4between MACs and crystalline electric \felds (CEFs) under the conditions of a weak CEF,\nstrong e\u000bective exchange \feld (EXF), and strong spin-orbit interaction (SOI) (i.e., only the\ngroundJmultiplet is considered) on the Rsites:\nKR\n1(0) =\u00003f2B0\n2\u000040f4B0\n4\u0000168f6B0\n6; (4a)\nKR\n2(0) = 35f4B0\n4+ 378f6B0\n6; (4b)\nKR0\n2(0) =f4B4\n4+ 10f6B4\n6; (4c)\nKR\n3(0) =\u0000231f6B0\n6; (4d)\nKR0\n3(0) =\u000011f6B4\n6; (4e)\nwherefk:= 2\u0000k(2J)!=(2J\u0000k)! andBq\nkis the CEF coe\u000ecient. Under the same conditions\nexcept at zero temperature, in 1992, Kuz'min reported directly comparable results with the\npower law [Eq. (3)] as\nKR\n1(T) =KR\n1(0)~B2\nJ(x)\n+8\n7KR\n2(0)h\n~B2\nJ(x)\u0000~B4\nJ(x)i\n+8\n7KR\n3(0)\u0012\n~B2\nJ(x)\u000018\n11~B4\nJ(x) +7\n11~B6\nJ(x)\u0013\n; (5a)\nKR\n2(T) =KR\n2(0)~B4\nJ(x)\n+18\n11KR\n3(0)h\n~B4\nJ(x)\u0000~B6\nJ(x)i\n; (5b)\nKR0\n2(T) =KR0\n2(0)~B4\nJ(x)\n+10\n11KR0\n3(0)h\n~B4\nJ(x)\u0000~B6\nJ(x)i\n; (5c)\nKR\n3(T) =KR\n3(0)~B6\nJ(x); (5d)\nKR0\n3(T) =KR0\n3(0)~B6\nJ(x); (5e)\nwhere ~Bk\nJ(x) :=JkBk\nJ(x)=fkand Bk\nJ(x) is Kuz'min's generalized Brillouin function (GBF)[3,\n44, 46, 47] with x:= 2Jjg\u00001jHexf(T)=(kBT), wheregis the Land\u0012 e gfactor,Hexf(T) is\nthe magnitude of the EXF, and kBis Boltzmann's constant. Kuz'min derived the relation\nbetween the GBF and the Akulov{Zener power law for the low-temperature range:\n~Bk\nJ(x)'\u0016R(T)k(k+1)=2: (6)\nThis approximation was also referred to in terms of MFT by Ke\u000ber in 1955[43, 48]. Although\nthe Akulov{Zener power law is no longer quantitatively supported by microscopic theory\n5in the high-temperature range, it has been con\frmed that Eq. (6) is qualitatively satis\fed\n[27, 44]. The reason for this \fnding is simple: both monotonically decrease and take the same\nvalues atT= 0 (both are 1) and T=TC(both are 0), where TCis the Curie temperature.\nHere, it is necessary to note that Kazakov and Andreeva[49] derived results equivalent to\nEq. (5) in 1970 (see Ref. 11 in Ref. 33).\nIf the low-angle limit \u0002 !0 is considered, then the series in Eq. (1) converges, and there-\nfore, it becomes possible to obtain the exact expressions for temperature-dependent MACs.\nWe have recently derived these expressions and applied them to the case of Nd 2Fe14B[26],\nwhere the expressions reproduce Eq. (5) within the limits of the strong EXF and SOI.\nAs seen above, the expressions for the temperature-dependent MA are connected under\nappropriate conditions, although some expressions have been reported in di\u000berent forms.\nOne of our aims is to re\rect the NCEs in the previous results, and this work is presented in\nSect. III B.\nIII. NON-COLLINEARITY EFFECTS ON MA OF R2Fe14BMAGNETS\nIn this section, we reveal the NCEs on the MA of R2Fe14B magnets on the basis of a\nstandard ligand-\feld theory. First, we de\fne the theoretical model used in this study.\nThe crystal structure of R2Fe14B compounds is tetragonal with P42=mnm , and there\nare eightRions in the unit cell. The Rion sites are classi\fed into two types: i= f or g.\nThese two types are distinguished crystallographically, and therefore the CEF Hamiltonian\nfor the 4f electrons is written as Vi\nCEFdepending on i[1, 14, 15, 50]. Here, we consider the\n4f electrons described by\nHi(\u0012;\u001e;T) :=Vi\nCEF+\u0015~L\u0001~S\u00002~S\u0001HEXF(\u0012;\u001e;T); (7)\nwhere\u0015is the strength of the SOI and ~L(~S) is the operator of the total angular (spin) mo-\nmentum of the 4f electrons. The EXF is assumed to be proportional to the Fe magnetization\ngiven by\nMFe(\u0012;\u001e;T) :=MFe\u0016Fe(T)mFe(\u0012;\u001e); (8)\nmFe(\u0012;\u001e) := (sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012); (9)\nthat is,\nHEXF(\u0012;\u001e;T) :=HEXF\u0016Fe(T)mFe(\u0012;\u001e); (10)\n6whereMFeandHEXF, respectively, are the saturation magnetization and the strength of\nthe EXF at zero temperature, and \u0016Fe(T) describes the temperature dependence of the\nmagnitude of the Fe magnetization.\nIn this study, the parameters we use for the R2Fe14B magnets were determined systemat-\nically by Yamada et al.[45]. The temperature dependence of the saturated magnetization of\nthe Y 2Fe14B magnets was employed as MFe\u0016Fe(T), and its value at zero temperature is given\nbyMFe= 31:4\u0016B/f.u.[29, 31]. To obtain its continuous values at nonzero temperature, the\nKuz'min formula has been widely used[24, 51{55], which is given as\n\u0016Fe(T) =\"\n1\u0000s\u0012T\nTC\u00133=2\n\u0000(1\u0000s)\u0012T\nTC\u0013p#1=3\n: (11)\nHere, the Curie temperature TCis de\fned for the target R2Fe14B magnet. We con\frmed[24]\nthat selecting the values of s= 1=2 andp= 5=2 for the shape parameters provides a good \ft\nwith the experimental result of the Y 2Fe14B magnet[29, 31]. Then, the total magnetization\nof theR2Fe14B magnet is given by\nM(\u0012;\u001e;T) :=MFe(\u0012;\u001e;T) +MR(\u0012;\u001e;T); (12)\nwhere MR(\u0012;\u001e;T) is theRmagnetization in units of [ \u0016B/f.u.] induced by the presence of\nthe Fe magnetization and is de\fned by\nMx\nR(\u0012;\u001e;T) :=1\n2X\ni=f;g[mx\nRi(\u0012;\u001e;T) +my\nRi(\u0012;\u001e+\u0019=2;T)]; (13a)\nMy\nR(\u0012;\u001e;T) :=1\n2X\ni=f;g[my\nRi(\u0012;\u001e;T)\u0000mx\nRi(\u0012;\u001e+\u0019=2;T)]; (13b)\nMz\nR(\u0012;\u001e;T) :=1\n2X\ni=f;g[mz\nRi(\u0012;\u001e;T) +mz\nRi(\u0012;\u001e+\u0019=2;T)]: (13c)\nHere, we de\fned the magnetic moment of the Rion on a site ias\nmRi(\u0012;\u001e;T) :=\u0000Tr e\u0000[Hi(\u0012;\u001e;T)\u0000fi\nR(\u0012;\u001e;T)]=(kBT)(~L+ 2~S) (14)\nand the free energy of the Rion as\nfi\nR(\u0012;\u001e;T) :=\u0000kBTln Tr e\u0000Hi(\u0012;\u001e;T)=(kBT): (15)\nBy rotating MFe(\u0012;\u001e;T) from the z-axis by hand, the direction of MR(\u0012;\u001e;T) deviates from\nMFe(\u0012;\u001e;T) in the presence of the CEF; to describe this non-collinearity, we introduce the\nnew symbols of \u0002 and \b as the zenithal and azimuthal angles of the total magnetization,\nrespectively, as shown in Fig. 1.\n7xyz\nMFeM\nφθΘ\nΦFIG. 1. De\fnitions of the zenithal and azimuthal angles of the Fe magnetization MFe(\u0012;\u001e;T) and\nthe total magnetization M(\u0012;\u001e;T).\nA. Numerical analyses of Nd 2Fe14B and Dy 2Fe14B magnets across the entire tem-\nperature range\nWe explored the MA across the entire temperature range by computing the temperature-\ndependent magnetization and the temperature-dependent free energy density as functions\nof\u0012and\u001e. On the basis of our results, we show how the non-collinearity between the R\nand Fe magnetizations appears and how it e\u000bects the MA.\nFirst, we take Nd 2Fe14B magnets [56] as an example of magnets, the NCE of which is\nsmall. Figure 2 shows the angular di\u000berence de\fned by\n\u0001\u0002(\u0012;T) := \u0002(\u0012;\u001e= 0;T)\u0000\u0012; (16)\nas a function of \u0012for the compound Nd 2Fe14B at several temperatures. In the low-\ntemperature range below the spin-reorientation transition (SRT) temperature ( T\u0018130\nK[45]), these compounds exhibit complex behavior as shown by the lines for T= 0, 100,\nand 135 K. Above the SRT temperature, we can observe that \u0001\u0002<0 because MNd(\u0012;\u001e;T)\ntends to naively orient along the +z-axis, and that j\u0001\u0002jmonotonically decreases with\nincreasing temperature. In Nd 2Fe14B magnets,j\u0001\u0002jhas an extremely low value over the\nentire temperature range, and thus we conclude that the NCE is negligibly small as assumed\nin our previous studies[24{26].\nIn contrast, the Dy 2Fe14B magnets [57] exhibit high non-collinearity, which is shown\nin Fig. 3. Because Dy 2Fe14B magnets do not have the SRT, MDy(\u0012;\u001e;T) tends to be\noriented along the \u0000z-axis over the entire temperature range; that is, \u0001\u0002>0 for any\n8FIG. 2. Calculated angular di\u000berence, \u0001\u0002(\u0012;T), between the total and Fe magnetizations as a\nfunction of the zenithal angle \u0012of the Fe magnetization in Nd 2Fe14B compounds. The number on\neach line denotes the temperature T, and the dashed line is the result near the SRT temperature.\nFIG. 3. Calculated angular di\u000berence, \u0001\u0002(\u0012;T), between the total and Fe magnetizations as a\nfunction of the zenithal angle \u0012of the Fe magnetization in Dy 2Fe14B compounds. The number on\neach line denotes the temperature T, and the dotted line represents \u0019=2\u0000\u0012.\ntemperature. Here, let us focus on the intersection(s) of \u0001\u0002 and the dotted line \u0019=2\u0000\u0012\nin Fig. 3. At the intersection(s), \u0002 is equal to \u0019=2. In particular, in the temperature\nrange below approximately 100 K, the intersection exists at an angle \u0012=\u0012o< \u0019= 2. That\nis, \u0002 overshoots \u0019=2 at\u0012=\u0012o, after which \u0002 > \u0019= 2 for\u00122(\u0012o;\u0019=2). This fact becomes\nimportant when evaluating the MA of Dy 2Fe14B magnets in the low-temperature range.\nHere, it is shown that a large j\u0001\u0002jremains even in the high-temperature range compared\nwith the Nd 2Fe14B case.\n9free energy \ndensity(𝜃,ϕ)(Θ,Φ)\nFFΩFIG. 4. Schematic view of the method used to obtain the map Ffrom (\u0002;\b) to the free energy\ndensity, where the maps \nandFcan be calculated directly.\nIn accordance with the above results, we consider the NCE on the MA. We de\fne the\ntotal free energy density of R2Fe14B compounds as\nF(\u0012;\u001e;T) :=FR(\u0012;\u001e;T) +\u0014Fe(T) sin2\u0012; (17)\nwhereFR(\u0012;\u001e;T) is the contribution from the Rsublattice, which is given by\nFR(\u0012;\u001e;T) :=2\nVcellX\ni=f;g\u0002\nfi\nR(\u0012;\u001e;T) +fi\nR(\u0012;\u001e+\u0019=2;T)\u0003\n; (18)\nwhereVcellis the volume of the unit cell. The second term in Eq. (17) is the contribution\nfrom the Fe sublattice, where we use the \frst-order MAC of the Y 2Fe14B magnet as \u0014Fe(T),\nwhich is expressed by a \ftting form as[24]\n\u0014Fe(T) =\u0014Fe\n1\u0016Fe(T)3\n+8\n7\u0014Fe\n2\u0002\n\u0016Fe(T)3\u0000\u0016Fe(T)10\u0003\n+8\n7\u0014Fe\n3\u0014\n\u0016Fe(T)3\u000018\n11\u0016Fe(T)10+7\n11\u0016Fe(T)21\u0015\n; (19)\nwhere the \ftted parameters are given by \u0014Fe\n1= 0:77 MJ/m3,\u0014Fe\n2= 1:21 MJ/m3, and\u0014Fe\n3=\n0:11 MJ/m3. To determine the NCE on the MA in R2Fe14B magnets, we compute the total\nfree energy density as a function of \u0012and\u001e, and next obtain this energy density as a function\nof \u0002 and \b by the following process (see Fig. 4): (i) calculate F(\u0012;\u001e;T) as a function of ( \u0012;\u001e)\nin Eq. (17); (ii) calculate \u0002 and \b as a function of ( \u0012;\u001e); (iii) regard F(\u0012;\u001e;T) asF(\u0002;\b;T),\nby noting that if the values ( \u0012i;\u001ei) exist such that (\u0002 ;\b) = \n(\u00121;\u001e1) =\n(\u00122;\u001e2) =\u0001\u0001\u0001,\nthen we putF(\u0002;\b;T) = min[F(\u00121;\u001e1;T);F(\u00122;\u001e2;T);:::], where \nis the map from ( \u0012;\u001e)\nto (\u0002;\b).\n10FIG. 5. Calculated zenithal angle dependence of the free energy density of the Dy 2Fe14B compounds\nat each temperature. The solid and dashed lines represent F(\u0012;\u001e;T) andF(\u0002;\b;T), respectively,\nat\u001e= 0.\nThe calculated angle dependence of the total free energy density of the compound\nDy2Fe14B is shown in Fig. 5, where the solid and dashed lines are the \u0012and \u0002 dependen-\ncies, respectively. Across the entire temperature range, the stabilization angle is \u0012= 0; the\nDy magnetization tends to be oriented along the \u0000z axis; hence, \u0002\u0015\u0012in 0\u0014\u0012\u0014\u0019=2 as\nshown in Fig. 3. In the low-temperature range below approximately 100 K, Fig. 3 indicates\nthat \u0002>\u0019= 2 when\u0012varies from \u00120to\u0019=2, and therefore, it is clear from Fig. 5 (upper) that\nthe stabilization energy is lower than the \fctitious stabilization energy estimated from the\ndependence on \u0012. Moreover, the dashed lines in Fig. 5 (upper) almost completely overlap,\nwhich suggests that Dy 2Fe14B compounds have a magnetic anisotropy that is robust against\na rise in temperature. In contrast, as shown in Fig. 5 (lower), the stabilization energy es-\ntimated from the \u0002 dependence (dashed lines) is equal to the \fctitious stabilization energy\nfrom the\u0012dependence (solid lines) in the high-temperature range. However, the initial\nrise in the\u0012dependence is clearly larger than that of the \u0002 dependence, and therefore, the\nMACs would be overestimated if the estimation were to be based on the \u0002 dependence.\n11FIG. 6. MACs of the Dy 2Fe14B compounds as a function of the temperature T. The solid lines\nare the results calculated as part of this study, and the solid circles are the experimental results of\nthe \frst-order MAC[35].\nLastly, in this section, let us consider the MACs derived from \u0001F(\u0002;\b;T) :=F(\u0002;\b;T)\u0000\nF(0;0;T) in Dy 2Fe14B magnets. We introduce a third-order \ftting function for \u0001F(\u0002;\b;T)\nas\n\u0001F\ft(\u0002;\b;T) :=K\ft\n1(T) sin2\u0002\n+K\ft\n2(T) sin4\u0002 +K\ft\n3(T) sin6\u0002; (20)\nwhereK\ft\nn(T) is thenth-order \ftted MACs, and we have ignored the cases in which Dy 2Fe14B\ndepends on \b because this dependence is su\u000eciently small. Here, note that we have per-\nformed the \ftting calculations in a \u0002 range near \u0002 \u00180 (corresponding to \u00122[0;\u0019=20])\nbecause the \ftting form in Eq. (20) is clearly not appropriate for the singular shape of\nthe dashed lines near \u0002 = \u0019=2 in Fig. 5 (upper). Figure 6 shows the values obtained for\nK\ft\nn(T) by \ftting \u0001F\ft(\u0002;\b;T) to\u0001F(\u0002;\b;T). It is noteworthy that the broad plateau\nin the low-temperature range is reproduced, although the calculated K\ft\n1(T) is larger than\nthe experimental results by approximately 2 MJ/m3; for quantitative comparison, we notice\nthat the experimental \frst-order MAC has been estimated from experimental anisotropy\n\felds assuming the absence of higher-order MACs. This indicates that this broad plateau\noriginates from the robustness of the MA against a rise in temperatures as mentioned in\nthe previous paragraph. That is, the presence of the plateau re\rects the e\u000bects of the non-\ncollinearity; in fact, under the assumption of collinear magnetizations, such a broad plateau\nis not obtained for Dy 2Fe14B magnets as demonstrated by Ito et al.[27]. Furthermore, no\n12less important is that the damping of K\ft\n2(T) andK\ft\n3(T) is slow in the high-temperature\nrange. The slow damping of the higher-order MACs can also be understood in terms of the\nNCE, which is considered in the next section. To conclude this section, we emphasize that\nthe non-collinearity of Dy 2Fe14B magnets is not negligible, especially when evaluating the\nMA.\nB. Perturbative expressions for MACs with NCEs in the high-temperature range\nThe importance of considering the e\u000bect of the non-collinearity between the Dy and Fe\nmagnetizations is explained in the previous section. In this section, we \frst derive explicit\nmicroscopic expressions for the MA by taking into account the NCE in the high-temperature\nrange, and subsequently apply the result to Dy 2Fe14B and other R2Fe14B magnets. Here,\nwe consider only the ground Jmultiplet and ignore the J-mixing e\u000bects. Although some\nlightRions such as Pr, Nd, and Sm exhibit a large J-mixing e\u000bect on MA as pointed\nout by several authors[45, 58{62], we determined the order in which the non-collinearity\ncan be ignored to be 0 \u00142j\u0001\u0002Ndj=\u0019 < 0:03 as is evident from the previous section, and\n0\u00142j\u0001\u0002Prj=\u0019 < 0:04 and 0\u00142j\u0001\u0002Smj=\u0019 < 0:01 by further numerical calculations with\n\fnite SOI. Thus, we discuss the NCE only for heavy Relements by using the CEF and EXF\nparameters reported by Yamada et al.[45].\nThe 4f total Hamiltonian at a site iwithin the ground multiplet Jcan be expressed in\nterms of the total angular momentum operator of the 4f electrons, ~J, on the basis of the\nWigner-Eckart theorem, and ~J2is the constant J(J+ 1):\nVi\nCEF!\u0016Vi\nCEF:=X\n(`;m)Bm\n`(i)Om\n`(~J); (21a)\n\u0015~L\u0001~S!\u0015\n2h\nJ(J+ 1)\u0000~L2\u0000~S2i\n= const:; (21b)\n\u00002~S\u0001HEXF(\u0012;\u001e;T)!\u0000 2(g\u00001)~J\u0001HEXF(\u0012;\u001e;T); (21c)\nthat is,\nHi(\u0012;\u001e;T)!\u0016Vi\nCEF\u00002(g\u00001)~J\u0001HEXF(\u0012;\u001e;T) + const:; (22)\nwhereOm\n`(~J) is the Stevens operator[63], and the range of the ( `;m) summation is limited\nto (`;m) = (2;0), (2;\u00002), (4;0), (4;\u00002), (4;4), (6;0), (6;\u00002), (6;4), and (6;\u00006) by the\n13symmetry of the CEF. On the basis of the de\fnitions Eqs. (10) and (11), we can perform\nthe perturbative expansion for the free energy density of the Rions,FR(\u0012;\u001e;T), with respect\nto the dimensionless parameter \u0016Fe(T) in the high-temperature range. In this expansion,\nFR(\u0012;\u001e;T) only has even powers of \u0016Fe(T) owing to the time inversion symmetry, and thus,\nthe lowest contribution arises from the second-order of \u0016Fe(T) as\n\u0001FR(\u0012;\u001e;T) :=FR(\u0012;\u001e;T)\u0000FR(0;0;T) =\u0014R(T) sin2\u0012+O(\u0016Fe(T)4); (23)\nwhere\n\u0014R(T) :=8\nVcell[2(g\u00001)HEXF]2\n2\u0016Fe(T)2[\u001fz(T)\u0000\u001fx(T)]; (24)\nand the factor 8 =Vcellrepresents the concentration of the Rions. Furthermore, we introduced\n\u001f\u000b(T) :=1\n2X\ni=f;gZ(kBT)\u00001\n0d\u001cD\ne\u001c\u0016Vi\nCEFJ\u000be\u0000\u001c\u0016Vi\nCEFJ\u000bE\nT; (25)\nwhereh\u0001\u0001\u0001iTdenotes the statistical average in \u0016Vi\nCEFat a temperature T, and further, we\nused the symmetry \u001fx(T) =\u001fy(T) in the derivation of Eq. (24). Thus, the total free energy\ndensity is given by\n\u0001F(\u0012;\u001e;T) :=F(\u0012;\u001e;T)\u0000F(0;0;T) =\u00141(T) sin2\u0012+O(\u0016Fe(T)4); (26)\nwhere\n\u00141(T) :=\u0014Fe(T) +\u0014R(T): (27)\nIf one considers a magnet with a low non-collinearity between the Rand Fe moments such\nas a Nd 2Fe14B magnet, then it becomes possible to conclude that K1(T)'\u00141(T). However,\nas we have mentioned, this assumption is not always satis\fed.\nWe describe the total magnetization within the same framework with the aim of taking\nthe NCE into account. By perturbatively expanding Eq. (13) with respect to \u0016Fe(T) again,\nMR(\u0012;\u001e;T) only has odd powers of \u0016Fe(T), and we obtain\nM\u000b\nR(\u0012;\u001e;T) =\u00004g(g\u00001)HEXF\u0016Fe(T)m\u000b\nFe(\u0012;\u001e)\u001f\u000b(T) +O(\u0016Fe(T)3): (28)\nThen, we determine the relationship between the directions of M(\u0012;\u001e;T) andMFe(\u0012;\u001e;T)\nas\nsin2\u0012=[1 +\u000e(T)] sin2\u0002\n1 +\u000e(T) sin2\u0002+O(\u0016Fe(T)2); (29)\n14where we de\fned the non-collinearity factor as\n\u000e(T) :=\u00008g(g\u00001)HEXFfMFe\u00002g(g\u00001)HEXF[\u001fz(T) +\u001fx(T)]g\n[MFe\u00004g(g\u00001)HEXF\u001fx(T)]2[\u001fz(T)\u0000\u001fx(T)];(30)\nwhere we notice that \u000e(TC) does not vanish. Substituting Eq. (29) into Eq. (26) and\nassumingj\u000e(T)j<1, the MACs, including the NCE up to the second-order \u0016Fe(T), are\nexpressed as\nKn(T) =\u00141(T)[1 +\u000e(T)][\u0000\u000e(T)]n\u00001forn\u00151: (31)\nBecause the e\u000bect of the CEF on both \u0014R(T) and\u000e(T) is re\rected through \u001f\u000b(T), let us\ntry to expand \u001f\u000b(T) with respect to an expansion parameter y:=Bm\nl=(kBT) to determine\nthe relation between the MACs and the CEF:\n\u001f\u000b(T) =1X\nn=0x(n)\n\u000b\n(kBT)n+1=x(0)\n\u000b\nkBT+x(1)\n\u000b\n(kBT)2+x(2)\n\u000b\n(kBT)3+O(y3)\nkBT; (32)\nwhere the temperature-independent coe\u000ecients are given by\nx(0)\n\u000b=J(J+ 1)\n3; (33a)\nx(1)\n\u000b=\u00001\n2X\ni=f;gTr\u0016Vi\nCEF(J\u000b)2\n2J+ 1; (33b)\nx(2)\n\u000b=1\n2X\ni=f;gTrh\n2(\u0016Vi\nCEF)2(J\u000b)2+ (\u0016Vi\nCEFJ\u000b)2\u0000(\u0016Vi\nCEF)2~J2i\n6(2J+ 1); (33c)\n\u0001\u0001\u0001:\nSubstituting Eq. (32) into Eqs. (24) and (30), we obtain\n\u0014R(T) =\u0014(1)\nR(T)\u0012\n1 +\u0010\nT+O\u0000\ny2\u0001\u0013\n; (34)\n\u000e(T) =\u000e(1)(T) [1 +O(y)]; (35)\nwhere\n\u0014(1)\nR(T) :=4\nVcell[2(g\u00001)HEXF]2\u0016Fe(T)2x(1)\nz\u0000x(1)\nx\n(kBT)2(36)\n\u000e(1)(T) :=\u00008g(g\u00001)HEXF\nMFe\u00004g(g\u00001)HEXFJ(J+ 1)=(3kBT)x(1)\nz\u0000x(1)\nx\n(kBT)2; (37)\n15TABLE I. Calculated values of \u0010in units of [K].\nTb Dy Ho Er Tm Yb\n13:24\u00003:52\u00001:50 1:54 3:45\u000012:56\nand\n\u0010:=1\nkBx(2)\nz\u0000x(2)\nx\nx(1)\nz\u0000x(1)\nx: (38)\nIf the calculation of Kn(T) takes into consideration \u000e(1)(T), which describes the NCE in the\nleading order, then there is no reason to ignore a correction from \u0010=Tin Eq. (34) in general\ncases, because both are of the same order of y. However, if the targets are limited to R2Fe14B\nmagnets, we can conclude that the \u0010=Tterm is negligible in the high-temperature range as\nin Table I. As a result, ignoring \u0010in the assumption of \u0010=T\u001c1 allows us to estimate the\nMACs, up to the second-order \u0016Fe(T), by using\nKn(T)'h\n\u0014Fe(T) +\u0014(1)\nR(T)i\u0002\n1 +\u000e(1)(T)\u0003\u0002\n\u0000\u000e(1)(T)\u0003n\u00001forn\u00151; (39a)\nand explicit forms are given as\n\u0014(1)\nR(T) =\u00004(g\u00001)2J(J+ 1)(2J\u00001)(2J+ 3)\n5Vcell\u0012\u0016Fe(T)HEXF\nkBT\u00132X\ni=f;gB0\n2(i); (39b)\n\u000e(1)(T) =2g(g\u00001)J(J+ 1)(2J\u00001)(2J+ 3)HEXF\n5 [MFe\u00004g(g\u00001)J(J+ 1)HEXF=(3kBT)] (kBT)2X\ni=f;gB0\n2(i): (39c)\nThe main results of this study are expressed by Eq. (39). Near the Curie temperature,\nthe MA exhibits explicit temperature dependence: \u0016Fe(T)=T'21=3(1\u0000T=T C)1=3=TC,\n\u000e(1)(T)'\u000e(1)(TC), and\u0014Fe(T)=\u0014(1)\nR(T)\u001c1; thus, the temperature dependence of the MACs\nis proportional to (1 \u0000T=T C)2=3. The high-temperature expansion [Eq. (39b)] was \frst\nderived by Kuz'min[33]. If the NCEs are ignorable, i.e., j\u000e(T)j\u001c1, then, on the basis of\nEq. (39), it can be con\frmed that K1(T)'\u0014(1)\n1(T) and the higher-order MACs are negli-\ngible at high temperatures. In this sense, we have naturally extended Kuz'min's result in\nconsideration of the NCEs.\nLastly, in this section, let us compare the present results with the \ftted MACs. The\ncase for Dy 2Fe14B magnets is shown in Fig. 7, where the solid lines are K\ft\n1(T),K\ft\n2(T),\nandK\ft\n3(T), the same as in Fig. 6, and the dashed lines are K1(T),K2(T), andK3(T)\n16FIG. 7. Comparison of the MACs calculated by the \ftting method (solid lines) with those obtained\nby the high-temperature expansion (dashed lines) in the Dy 2Fe14B magnet; the solid lines represent\nthe same result as in Fig. 6. The number on each ring denotes the value of n.\nby using Eq. (39) with g= 4=3,J= 15=2,B0\n2(f)=kB=B0\n2(g)=kB=\u00001:392 K, and\nHEXF=kB= 145 K[45]. Then, we can observe that the high-temperature expansion provides\na good approximation for the solid lines. As mentioned in Sect. III A, the NCE in Nd 2Fe14B\nmagnets is low but high in Dy 2Fe14B magnets. If the \frst-order MAC of Dy 2Fe14B magnets\nis evaluated without the NCE, then it is overestimated by approximately 20% at 500 K as\nillustrated by \u00141(T) in Eq. (27) with Eq. (39b). Therefore, the decomposition of\u00141(T) into\nK1(T),K2(T),:::by the NCE, as expressed by Eq. (39a), is continued up to the Curie\ntemperature because \u000e(TC) does not vanish, and this is the reason that K2(T) survives\neven near the Curie temperature. As a consequence, we can understand that the non-\ncollinearity arises from the non-negligible K2(T) of Dy 2Fe14B magnets at high temperatures,\nas mentioned in Sect. III A, and also from B0\n2(i) andHEXF. In contrast, in a small non-\ncollinearity system, K2(T) is mainly induced by B0\n4(i) and/orB0\n6(i)[33]; thus, the mechanism\nessentially di\u000bers.\nC. Practical expressions for MACs in rare-earth intermetallics\nThe simple expression at zero temperature [Eq. (4a)],\nKR\n1(0) =\u00003f2B0\n2\u000040f4B0\n4\u0000168f6B0\n6; (40)\nmotivated the evaluation of B0\n2,B0\n4, andB0\n6, especially from \frst principles[6]. However, as\nis well known, the temperature dependence of the MA of RE magnets is complex, and thus,\n17FIG. 8. Temperature dependence of the non-collinearity factor \u000e(T) as a function of temperature\nTcalculated using Eq. (39c), for R2Fe14B magnets ( R=Tm, Er, Yb, Ho, Dy, and Tb).\nthis expression is inappropriate to evaluate the MA in the high-temperature range, which\nis important in practical situations used in electric vehicle motors. Here, for the reader's\nconvenience, we provide a useful form of the expressions obtained in the previous section\nto allow us to immediately estimate the temperature-dependent MA for two sublattice RE\nmagnets consisting of RE and transition elements.\nAlthough the three CEF coe\u000ecients are needed to evaluate the zero-temperature MA\nbecause they have the same order, we do not exert e\u000bort to evaluate the higher-order CEF\ncoe\u000ecients in the high-temperature range. This is because the high-temperature MA is\ndominated only by B0\n2as explained in the previous section. Now, when one has a CEF\ncoe\u000ecient, \u0016B0\n2[K], which is the average value of B0\n2over the total RE ions, and an EXF,\nHRE[K], by which the e\u000bective exchange energy is represented as \u00002(g\u00001)JzHREat zero\ntemperature, the nth-order MACs in the high-temperature range can be estimated as\nKTotal\nn(T) = [\u0014T(T) +\u0014RE(T)] [1 +\u000eRE\u0000T(T)] [\u0000\u000eRE\u0000T(T)]n\u00001; (41)\n18in units of [K/ Vcell], where\u0014T(T) is the experimental \frst-order MAC within the transition\nmetal sublattice, and\n\u0014RE(T) =\u0000AnRE\u0012HRE\u0016T(T)\nT\u00132\n\u0016B0\n2; (42)\n\u000eRE\u0000T(T) =BnREHRE\u0016B0\n2\nT(TM T\u0000CnREHRE); (43)\nwherenREis the number of RE ions in the unit cell, MT[\u0016B=Vcell] is the saturated magneti-\nzation of the transition metal sublattice at zero temperature, and A,B, andCare geometric\ncoe\u000ecients determined by gandJof each of the rare-earth elements listed in Table II. The\nde\fnitions are immediately obtained from Kuz'min's result [Eq. (39b)] and the present\nresult [Eq. (39c)]. \u0016T(T) can be expressed by the Kuz'min formula[51] as\n\u0016T(T) =\"\n1\u0000s\u0012T\nTC\u00133=2\n\u0000(1\u0000s)\u0012T\nTC\u0013p#1=3\n; (44)\nwheresandpare previously reported shape parameters[24, 51, 52]. For example, \u000e(T)\nfor Dy 2Fe14B magnets in Fig. 8 can be reproduced by setting nRE= 8,HRE= 145 K,\n\u0016B0\n2=\u00001:392 K,MT= 31:4\u00024\u0016B=Vcell,B= 2856,C= 170=9,s= 1=2,p= 5=2, and\nTC= 598 K. In addition to Dy, the calculated non-collinearity factors of other magnets\nare shown in Fig. 8. The values of the CEF and EXF parameters are those of Yamada et\nal.[45]. The sign of \u000e(T) is equal to ( g\u00001)P\ni=f;gB0\n2(i) as shown in Eq. (39c), and the sign\nofB0\n2(i) is equal to \u00122A0\n2(i), where\u00122is the Stevens factor and A0\n2(i) is the CEF parameter.\nBecauseA0\n2(i)>0 forR2Fe14B magnets, the Rdependence of the sign of \u000e(T) is determined\nby (g\u00001)\u00122, in whichg\u00001>0 for the heavy Rions, and\u00122>0 for Er, Tm, and Yb, and\n\u00122<0 for Tb, Dy, and Ho. We can observe that Dy and Tb especially exhibit a large NCE.\nNote that the present results do not include the e\u000bects of J-mixing. In general, elements\nfrom the light RE series have a larger J-mixing e\u000bect than the heavy ones[45, 58, 61, 62].\nWhether the J-mixing e\u000bect becomes serious for the NCE in general RE intermetallics is\nnot a trivial matter, although we were able to ignore the NCE for R= Pr, Nd, and Sm in\nthe case of R2Fe14B. For several light R, theJ-mixing e\u000bect on \u0014RE(T) cannot be ignored,\nespecially for Sm. This problem was detailed by Kuz'min[61] and Magnani et al.[62], and\nexplicit expressions for \u0014RE(T) were provided.\n19TABLE II. Calculated geometric coe\u000ecients for each RE ion\nRE ionA B C\nCe3+8=7\u000096=7\u00005=7\nPr3+308=25\u00002464=25\u000032=15\nNd3+1944=55\u000010368=55\u000036=11\nPm3+1232=25\u00003696=25\u000016=5\nSm3+200=7\u0000160=7\u000025=21\nGd3+189 756 21\nTb3+693=2 2079 21\nDy3+357 2856 170 =9\nHo3+513=2 2565 15\nEr3+3213=25 38556=25 51=5\nTm3+77=2 539 49 =9\nYb3+27=7 432=7 12=7\nIV. SUMMARY\nWe showed that Dy 2Fe14B magnets have a large NCE on the MA compared with Nd 2Fe14B\nmagnets, and that the NCE in Dy 2Fe14B magnets yields a plateau of K1(T) in the low-\ntemperature range and a non-negligible K2(T) in the high-temperature range. Further-\nmore, we derived microscopic expressions [Eq. (39)] for Kn(T) with NCEs by using the\nhigh-temperature expansion, and showed that these expressions were in a form extending\nKuz'min's collinear result [Eq. (39b)]. In homogeneous local moment systems, B0\n4andB0\n6\nare important for the rise of K2(T), andK2(T) rapidly decays with increasing temperature\nas represented by Eq. (5b). However, interestingly, in high non-collinear system, K2(T)\nsurvives even in the high-temperature range because of the presence of B0\n2as given in Eq.\n(39).\nIn terms of Eqs. (39b) and (39c), the main contribution to the MA comes from both\nB0\n2(s) andHEXFin the high-temperature range, whereas the higher-order CEF parameters\nare not e\u000bective. This is also an interesting result for the \feld of materials science, because\nB0\n2can be evaluated with relatively high accuracy compared with the other higher-order\n20CEF coe\u000ecients.\nACKNOWLEDGMENTS\nWe would like to thank Prof. H. Kato, Dr. Y. Toga, and Mr. D. Suzuki for useful\ndiscussions and information. This work was supported by JSPS KAKENHI Grant Nos.\n16K06702, 16H02390, 16H04322, and 17K14800.\n[1] J. F. Herbst, Rev. Mod. 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Magn. 53, 1300604 (2017).\n[56]TC= 586 K for the Nd 2Fe14B magnet[29, 31], and the parameters included in Hi(\u0012;\u001e;T) are\nlisted in Ref. 45.\n[57]TC= 598 K for the Dy 2Fe14B magnet[29, 31], and the parameters included in Hi(\u0012;\u001e;T) are\nlisted in Ref. 45.\n[58] M. D. Kuz'min and J. M. D. Coey, Phys. Rev. B 50, 12533 (1994).\n[59] N. Magnani, G. Amoretti, A. Paoluzi, and L. Pareti, Phys. Rev. B 62, 9453 (2000).\n[60] N. Magnani, G. Amoretti, A. Paoluzi, and L. Pareti, IEEE Trans. Magn. 37, 2540 (2001).\n23[61] M. D. Kuz'min, J. Appl. Phys. 92, 6693 (2002).\n[62] N. Magnani, S. Carretta, E. Liviotti, and G. Amoretti, Phys. Rev. B 67, 144411 (2003).\n[63] K. W. H. Stevens, Proc. Phys. Soc. A 65, 209 (1952).\n24"    },    {        "title": "1901.01207v1.How_to_accurately_determine_a_saturation_magnetization_of_the_sample_in_a_ferromagnetic_resonance_experiment_.pdf",        "content": "How to accurately determine a saturation magnetization of the sample\nin a ferromagnetic resonance experiment?\nP. Tomczak\u0003\nQuantum Physics Division\nFaculty of Physics, Adam Mickiewicz University ul. Umultowska 85,\n61-614 Pozna\u0013 n, Poland\nH. Puszkarski\nSurface Physics Division\nFaculty of Physics, Adam Mickiewicz University ul. Umultowska 85,\n61-614 Pozna\u0013 n, Poland\n(Dated: March 11, 2022)\nThe phenomenon of ferromagnetic resonance (FMR) is still being exploited for determining the\nmagnetocrystalline anisotropy constants of magnetic materials. We show that one can also deter-\nmine accurately the saturation magnetization of the sample using results of FMR experiments after\ntaking into account the relationship between resonance frequency and curvature of the spatial dis-\ntribution of free energy at resonance. Speci\fcally, three examples are given of calculating saturation\nmagnetization from FMR data: we use historical Bickford's measurements from 1950 for bulk mag-\nnetite, Liu's measurements from 2007 for a 500 mn thin \flm of a weak ferromagnet (Ga,Mn)As, and\nWang's measurements from 2014 for an ultrathin \flm of YIG. In all three cases, the magnetization\nvalues we have determined are consistent with the results of measurements.\nIntroduction | For a very long time, the ferromag-\nnetic resonance has been used successfully to \fnd mag-\nnetocrystalline anisotropy constants and spectroscopic\nsplitting factors of ferromagnets, see e.g., Ref. [1]. Re-\ncently, we have shown [2] that using this classic experi-\nmental technique completed with a cross-validation of the\nnumerical solutions of Smit-Beljers equation [3{6], not\nonly makes it possible to determine very precisely the\nspectroscopic splitting factor gand magnetocrystalline\nanisotropy constants, e.g., up to fourth order for the di-\nluted magnetic semiconductor (Ga,Mn)As, but also to\nstate which anisotropy constants are necessary to prop-\nerly describe the spatial distribution of the energy which\nis related to magnetocrystalline anisotropy. In what fol-\nlows we present that one can interpret the results of FMR\nexperiment in such a way that it is possible to determine\nthe saturation magnetization of the sample under inves-\ntigation, i.e, that it is possible to \fnd the spatial depen-\ndence of magnetocrystalline energy stored in a ferromag-\nnet using data collected in a single FMR experiment.\nWe will show how to determine saturation magneti-\nzation for three cases: for bulk magnetite, Fe 3O4, we\nwill use the historical Bickford's measurements [7] from\n1950, for a 500 mn \flm of a weak diluted ferromagnet,\n(Ga,Mn)As, we will use Liu's measurements [8] from 2007\nand for an ultrathin \flm of YIG we will use Wang's mea-\nsurements [9] from 2014.\nWhat does one measure while performing FMR\nexperiment? | The most simple answer is that the\nspatial distribution of resonance \feld Hris measured,\ni.e., its dependence on angles #Hand'H, see Fig. 1.\nHowever, let us consider two possible ways of performing\nthe FMR experiment. First, while \fxing a frequency !of\nan alternating microwave \feld one measures the depen-\nFIG. 1: The geometry of the FMR experiment. The\norientation of the applied magnetic \feld His usually\ndescribed in the coordinate system attached to the\ncrystallographical axes of the sample. The \feld\ndirection is represented by angles #Hand'Hand\nequilibrium direction of the sample magnetization Mis\nrepresented by angles #and'.\ndence of the static magnetic \feld Hron angles#Hand\n'Hat resonance, and the second | a static magnetic\n\feld is \fxed and one changes the frequency of microwave\n\feld to \fnd a frequency at which the precession of the\nmagnetic moment of the sample occurs. The resonance\ncondition, for both measurement methods, is given by\nthe following Smit-Beljers equation [3{6]\n\u0012~!\ng\u0016B\u00132\n=1\nsin2#(f##f\u001e\u001e\u0000f2\n#\u001e); (1)\nwherefis free energy of the sample divided by its satu-\nration magnetization M, i.e., free energy is expressed in\nterms of real (Zeeman term) and \fctitious (demagnetiz-\ning and anisotropy terms) \felds. f#\u001e=@f\n@#@f\n@',gis the\nspectroscopic splitting factor, \u0016B{ the Bohr magneton\nand~{ the Planck constant.\nLet us recall, that the Gaussian curvature of the sur-\nfaceS(#;') at one of its points, being the product of twoarXiv:1901.01207v1  [cond-mat.str-el]  4 Jan 20192\nprincipal curvatures \u00141and\u00142, is the determinant of the\nHessian matrix calculated at this point. One obtains in\nspherical coordinates\n\u00141\u00142= det\"\n@2S\n@#21\nsin#@2S\n@S#@'\u0000cot#\nsin#@S\n@'\n1\nsin#@2S\n@'@#\u0000cot#\nsin#@S\n@'1\nsin2#@2S\n@'2+ cot#@S\n@##\n:\n(2)\nLet us now assume that we are interested in the curva-\nture of the the free energy landscape of the sample at\nresonance and put S=fin Eq. (2). Remembering that\nMprecesses around its eqilibrium position, that is, the\nequations\n@f\n@#= 0;@f\n@'= 0 (3)\nare satis\fed, we arrive at Eq. (1) if we interpret\u0000~!\ng\u0016B\u00012\nas a Gaussian curvature of the spatial distribution of free\nenergy.\nReturning to the two ways of measuring the resonance\n\feld we see, that by \fxing a static \feld and subsequent\nmeasuring the frequency !of a microwave \feld for di\u000ber-\nent angles at resonance, we see directly a spatial distri-\nbution of the curvature of the free energy landscape. If,\nhowever, we set the microwave \feld, and then by chang-\ning the static \feld we hit the resonance ( Hr) for di\u000ber-\nent angles#Hand'Hthen we will maintain a constant\nGaussian curvature of the energy landscape during the\nmeasurement.\nThe answer to the question posed at the beginning\nof this section is, that while performing a typical FMR\nexperiment by \fxed !, we measure such a \feld Hrat\nwhich the Gaussian curvature \u00141\u00142of the free energy\nlandscape is constant, i.e., does not depend on #Hand\n'H. As we shall see, this observation will not only allow\n\fnding contributions from various types of anisotropy to\nthe free energy but also will enable the determining of\nsaturation magnetization of the sample.\nWhat determines the spatial distribution of the\nGaussian curvature of the free energy at res-\nonance? | Assume that the free energy density\nF(#H;'H) of a ferromagnet placed in a magnetic \feld,\nexpressed in terms of magnetic \felds, is given by\nF(#H;'H)\nM=f(#H;'H) =\u0000HX\n\u000bn\u000bnH\n\u000b\n+MX\n\u000b;\fN\u000b\fn\u000bn\f+1\n2HcX\n\u000b6=\fn2\n\u000bn2\n\f:(4)\n\u000b;\f =x;y;z . The \frst term on the right-hand side of\nEq. (4) is the Zeeman energy, the second term is the\ndemagnetizing energy, with N\u000b\fbeing the demagnetiza-\ntion tensor. The last term describes the energy related\nto cubic magnetocrystalline anisotropy. The components\nof vectorsnH\n\u000b=H=Handn\u000b=M=Mare de\fned as\nusuall, see e.g., Eqs. (3a) and (3b) in Ref. [2] and Fig. 1.Note that with ful\flled relations (3), the free energy of a\nsample with a speci\fc geometry and saturation magneti-\nzation in a magnetic \feld does not depend on angles #;'\nexplicitly.\n(a)\n (b)\n (c)\n (d)\nFIG. 2: Terms in Eq. (4) representing \fctitious\nmagnetic \felds and their Gaussian curvatures\n(a)MP\n\u000b;\fN\u000b\fn\u000bn\f(ellipsoid) (b) its curvature\n(c)1\n2HcP\n\u000b6=\fn2\n\u000bn2\n\f(d) its curvature.\nEach component of Eq. (4) describes a spatially vary-\ning magnetic \feld with a certain curvature. For exam-\nple the spatial dependencies of the \frst two terms and\ntheir curvatures are shown in Fig. 2. Surface (a) - el-\nlipsoid - represents the \fctitious demagnetizing \feld for\nthe sample oriented along axes [100], [010] and [001]. Its\ncurvature is shown in Fig. 2b. Note that it is non-zero\nin the (001) plane, although this is not visible due to\nthe used scale. The surface (c) represents the \frst order\ncubic magnetocrystalline \fctitious \feld and in Fig. 2d\nis shown its curvature. The curvature of the Zeeman\nterm in Eq. (4) depends on the static magnetic \feld and\ncan be changed during the measurement while the cur-\nvatures associated with the \fctitious demagnetizing and\ncubic anisotropy \felds remain constant during the mea-\nsurement.\nHow to \fnd saturation magnetization and\nanisotropy \felds numerically? | During the FMR\nexperiment the total Gaussian curvature of the free en-\nergy remains constant: contributions of curvatures from\nvarious \fctitious magnetic \felds (anisotropy, demagne-\ntizing), related to the sample itself, are compensated by\nthe curvature of the Zeeman term. To \fnd numerically\nvalues of those \felds, g-factor and saturation magnetiza-\ntion we apply a procedure to some extent reverse to the\nmeasurement: for a given set of measured Hr(#H;'H)\ndata, collected in a single experiment, the constant g,M\nand anisotropy and demagnetizing \felds should be cho-\nsen so that Eq. (1) should be met for all measured static\nresonance \felds Hr(#H;'H). It means, that minimizing\nan appropriate objective function, which measures the\ndeviation from the constant curvature with respect to un-\nknown anisotropy \felds, gandMlets us \fnd them. This\nprocedure is described in detail in Ref. [2]. The key point\nfor \fnding saturation magnetization is that its change\nleads to the change of the curvature of the demagnetizing\n\feld. But to ensure that the determined magnetization\nvalue is unambiguous, the constraint Nx+Ny+Nz= 1\nshould be met during the minimization procedure.\nTo carry out the minimization procedure, it is neces-3\nsary to know the functional form of free energy. Usu-\nally one assumes its speci\fc form taking into account the\nsymmetry of the system | free energy should be invari-\nant under relevant symmetry transformations. Then it is\npossible to expand it into basis functions (orthogonal or\nnon-orthogonal) with the same symmetry. Typically this\nexpansion is limited to some low-order terms of system-\natically decreasing basis functions. It should be remem-\nbered, however, that in the case of examination of cur-\nvature, the higher-order basis functions may give greater\ncontributions to total curvature of free energy than the\nlower-order ones.\nNumerical example I: Saturation magnetization\nof bulk magnetite | Bickford [7] measured the reso-\nnance \felds for disk-shaped samples of magnetite cut in\n(100) and (110) planes. The results of his measurements\nare shown as blue squares in Figs 3a and 3b, respectively.\nEach sample was oriented di\u000berently with respect to the\ncrystallographic planes and thus it has di\u000berent demag-\nnetization tensor. We will assume, however, that both\ntypes of samples have the same g-factor and the same\nsaturation magnetization. The demagnetizing tensor is\nassumed to be diagonal for samples cut in (100) plane\nN(100) =2\n4Nx0 0\n0Ny0\n0 0Nz3\n5: (5)\nThe curvature of free energy distribution, given by\nEq. (1), depends now on six parameters which we denote\ncollectively by the vector hB= (g;M;Nx;Ny;Nz;Hc):\nUsing the procedure described in Ref. [2], we get the\nfollowing coordinates of the vector hB\nhB= (2:220(88);5:906(150);0:2216(267);0:2212(268);\n0:5572(535);\u00000:2392(95)):\n(6)\nBoth,MandHcare given in kOe. The errors (in brack-\nets) refer to the last signi\fcant digits and were calculated\nassuming that the measurement errors are subject to nor-\nmal distribution. Having determined the vector hB, we\nsolve Eq. (4) numerically with respect to Hrand get the\nresonance \feld dependence on the #Hangle (black line in\nFig. 3a). To consider measurements of samples cut in the\nplane (110) let us rotate the coordinate system in such a\nway, that the zaxis is perpendicular to the (110) plane.\nThe demagnetizing tensor is given by\nN(110)=2\n641\n4(Nxy+ 2Nz)1\n4(Nxy\u00002Nz)p\n2\n4(Ny\u0000Nx)\n1\n4(Nxy\u00002Nz)1\n4(Nxy+ 2Nz)p\n2\n4(Ny\u0000Nx)p\n2\n4(Ny\u0000Nx)p\n2\n4(Ny\u0000Nx)1\n2Nxy3\n75;\n(7)\nNxystands forNx+Ny. Solving again Eq. (4) for such a\ntensorN(110), we reach the dependence of Hr(#H) shown\nin Fig. 3b. In Table I the relevant anisotropy constants\nfound by Bickford are presented compared with those\nobtained in the way as described in the text.\nFIG. 3: Resonance \feld versus out-of-plane angle #H\nfor bulk magnetite for samples cut in (100) { (a) and\n(110) planes { (b). Squares - Bickford's measurements\n[7], black line - solution of Smit-Beljers equation for the\nvector hBgiven by Eq. (6). Figs (a) and (b) correspond\nto Figs 3 and 4 in Ref. [7], respectively.\nTABLE I: Comparison of g-factor, saturation\nmagnetization Mand cubic anisotropy constant Kc\nvalues found by Bickford [7]with those obtained by the\nmethod described in the text. The errors (in brackets)\nrefer to the last signi\fcant digits.\ng M [kA/m] Kc[kJ/m3]\n2.220(88) 470(12) -14.12(92)\nRef. [7] 2.07 472a-11\naMeasurement result, [7].\nTo summarize this example, let us stress two things.\nFirst, we observe a fairly large statistical errors that are\nthe result of the scattering of original measurements re-\nported in Ref. [7]. Second, in fact, the Bickford's samples\nhad the shapes of a circular \rattened disks, since in (100)\ngeometryNx\u0019Ny6=Nz.\nNumerical example II: Saturation magnetization\nof the (Ga,Mn)As thin \flm | In Ref. [2] we showed,\nanalyzing Liu's measurements [8] of resonance \felds for\nthe weak ferromagnet (Ga,Mn)As, that in order to prop-\nerly describe the free energy spatial distribution at reso-\nnance, one should expand it up to the fourth order with\nrespect to cubic anisotropy \felds. Here we add to this ex-\npansion demagnetizing \felds, and, since the sample was\noriented along crystallographical axes of (Ga,Mn)As we\nuse a diagonal form od the demagnetizing tensor. The\nvector hLdepends now on eleven parameters\nhL\u0011(g;M;Nx;Ny;Nz;Hc1;:::;Hc4;H[001];H[110]):(8)\nAnisotropy \felds are denoted like in Ref. [2]. The mini-\nmization procedure leads to (\felds are given in Oe)\nhL= (1:984(3);383:5(2:2);0:08104(90);0:06796(95);\n0:8510(20);76:86(0:42);\u0000539:5(10:0);42:86(0:92);\n1412(70);4213(24);68:08(0:64)):\n(9)\nThe numerical solutions of Eq. (1) for such an hLvector\nis shown in Fig. 4. They are equivalent to those obtained4\nin Ref. [2] but now the use of the demagnetizing \felds\nmade it possible to determine saturation magnetization\nM= 30:52(0:18) [emu/cm3].\nFIG. 4: Resonance \feld versus out-of-plane angle #H\n(a) and versus in-plane angle 'H(b) for (Ga,Mn)As\n\flm. Squares - Liu measurements [8], black line -\nsolution of Smit-Beljers equation for the vector hL\n(Eq. 9). Figs (a) ad (b) correspond to Figs 5 and 6 in\nRef. [8], respectively.\nNumerical example III: Saturation magnetization\nof the YIG ultrathin thin \flm | The measurements\nof the resonance \feld of the 9.8 nm thin \flm are taken\nfrom Ref. [9] and shown in Figs 5a and 5b, respectively, as\nblue squares. To determine the anisotropy \felds let us as-\nsume that the free energy is given by Eq. (1) from Ref. [9]\nand that the demagnetizing \felds, as in the (Ga,Mn)As\ncase, are calculated using a diagonal demagnetizing ten-\nsor. Then the vector hHdepends on nine parameters\nhH= (g;M;Nx;Ny;Nz;H2?;H4?;H2k;H4k):(10)\nFactorg= 2 for YIG [10], while saturation magnetization\nM= 1851 [Oe] was determined for this sample using\na vibrating sample magnetometer [9].\nThe components of the vector hHcorresponding to the\nconstant curvature of free energy, i.e., for the sample at\nresonance, are collected in Table II for four cases. In the\n\frst case, we take gandMvalues as measured experi-\nmentally and \fnd remaining ones. In the second case we\n\fx only experimental value of M, in the third case - only\nvalue ofg, and \fnally, we treat both gandMas unknown\n(last row of in Table II). In the last column of Table II the\nerror function E1\nRMS(hH) is given, as de\fned in Ref. [2].\nIts value informs us about the quality of the prediction\nof new experimental results of resonance \felds calculated\nfrom a given free energy formula with speci\fc values (row\nof Table II) of anisotropy \felds. We see that analysis\nof FMR measurements of the YIG ultrathin \flm givesworse results when we treat measured gandMvalues\nas known. This points out that the form of free energy\nused for this analysis is not chosen optimally, because it\nwas not possible to reproduce the experimental values g\nandMwith satisfactory accuracy. Thus the presented\nmethod may also serve as a test for the correctness of\nthe assumed free energy form. Note also, that although\nthe investigated \flm was ultrathin, we obtained non-zero\nvalues of the demagnetization \felds in the (001) plane.\nThe dependencies of resonance \felds on angles #Hand\n'Hare shown, for hHfrom the \frst row of Table II, in\nFig. 5.\nFIG. 5: Resonance \feld for 9.8 nm ultrathin YIG \flm\nversus out-of-plane angle #H(a) and in-plane angle \u001eH\n(b). Squares - Wang's measurements [9], black line -\nsolution of Smit-Beljers equation for the vector hH\ngiven by Eq. (10). Figs (a) and (b) correspond to\nFigs 2(b) and 2(c) in Ref. [9], respectively .\nConclusion | We presented in this article one more ap-\nproach to the interpretation of the classical FMR experi-\nmental data. It is based on the analysis of the curvature\nof the spatial dependence of free energy of the sample at\nresonance and makes it possible, after assuming the func-\ntional form of free energy, to determine (with an accu-\nracy of 0.5-4% in the presented numerical examples) the\nsample saturation magnetization and, consequently, the\nspatial dependence of magnetocrystalline energy stored\nin the sample in a single FMR experiment. The method\npresented here can also be a test for the correctness of the\nassumed form of free energy of the sample at resonance.\nThis may be important for people working in the \feld of\nmagnetic resonance.\nAcknowledgment | This study is a part of the project\n\fnanced by Narodowe Centrum Nauki (National Science\nCentre of Poland) No. DEC-2013/08/M/ST3/00967. Nu-\nmerical calculations were performed at Pozna\u0013 n Super-\ncomputing and Networking Center under Grant No. 284.\n\u0003e-mail: ptomczak@amu.edu.pl\n1M. Farle, Reports on Progress in Physics 61, 755 (1998).\n2P. Tomczak and H. Puszkarski, Phys. Rev. B 98, 144415(2018).\n3J. Smit and H. G. Beljers, Philips Res. Rep. 10, 113 (1955).\n4J. O. Artman, Phys. Rev. 105, 74 (1957).5\nTABLE II: Comparison of g-factor, saturation magnetization Mand and anisotropy \feld values found in Ref. [9]\nwith those obtained by the method described in the text. The errors (in brackets) refer to the last signi\fcant digits.\ng M [Oe] Nx Ny Nz H2?[Oe] H4?[Oe] H2k[Oe] H4k[Oe] E1\nRMS(hH)\n2.000a1851b0.08429(10) 0.08721(11) 0.8285(10) -497.7(2.0) 372.3(1.1) 62.23(0.27) 49.02(0.22) 0.815\n2.014(1) 1851b0.08795(17) 0.08985(10) 0.8222(15) -466.5(1.5) 276.9(0.9) 61.83(0.23) 47.27(0.18) 0.691\n2.000a1800(11) 0.08102(15) 0.08408(11) 0.8349(16) -538.9(1.7) 372.6(1.0) 62.16(0.11) 49.05(0.16) 0.815\n2.013(1) 1772(12) 0.08263(8) 0.08467(13) 0.8327(13) -526.2(1.6) 277.5(0.9) 61.82(0.09) 47.42(0.18) 0.691\nRef. [9] 1851(37) -1253(25) 60.4(1.2) 42.0(1.2)\naMeasurement result, see Apendix A of Ref. [10].\nbMeasurement result, Ref. [9].\n5L. Baselgia, M. Warden, F. Waldner, S. L. Hutton, J. E.\nDrumheller, Y. Q. He, P. E. Wigen, and M. Mary\u0014 sko,\nPhys. Rev. B 38, 2237 (1988).\n6A. Morrish, The Physical Principles of Magnetism (Wiley-\nIEEE Press, 2001).\n7L. R. Bickford, Phys. Rev. 78, 449 (1950).8X. Liu, Y. Y. Zhou, and J. K. Furdyna, Phys. Rev. B 75,\n195220 (2007).\n9H. Wang, C. Du, P. C. Hammel, and F. Yang, Phys. Rev.\nB89, 134404 (2014).\n10D. Stancil, A. Prabhakar, Spin Waves. Theory and Appli-\ncations (Springer, 2009)."    },    {        "title": "1901.01750v1.Field_induced_phases_in_a_heavy_fermion_U_Ru___0_92__Rh___0_08______2__Si___2___single_crystal.pdf",        "content": "Field-induced phases in a heavy-fermion U(Ru 0:92Rh 0:08)2Si2single crystal\nK. Proke\u0014 s,1,\u0003T. F orster,2Y.-K. Huang,3and J.A. Mydosh4\n1Helmholtz-Zentrum Berlin f ur Materialien und Energie, Hahn-Meitner Platz 1, 14109 Berlin, Germany\n2Hochfeld-Magnetlabor Dresden (HLD), Helmholtz-Zentrum Dresden-Rossendorf and TU Dresden, D-01314 Dresden, Germany\n3Van der Waals-Zeeman Institute, University of Amsterdam, 1018XE Amsterdam, The Netherlands\n4Kamerlingh Onnes Laboratory and Institute Lorentz,\nLeiden University, 2300 RA Leiden, The Netherlands\n(Dated: November 16, 2021)\nWe report the high-\feld induced magnetic phases and phase diagram of a high quality\nU(Ru 0:92Rh0:08)2Si2single crystal prepared using a modi\fed Czochralski method. Our study, that\ncombines high-\feld magnetization and electrical resistivity measurements, shows for \felds applied\nalong thec-axis direction three \feld-induced magnetic phase transitions at \u00160Hc1= 21.60 T, \u00160Hc2\n= 37.90 T and \u00160Hc3= 38.25 T, respectively. In agreement with a microscopic up-up-down arrange-\nment of the U magnetic moments the phase above Hc1has a magnetization of about one third of\nthe saturated value. In contrast the phase between Hc2andHc3has a magnetization that is a factor\nof two lower than above the Hc3, where a polarized Fermi-liquid state with a saturated moment Ms\n\u00192.1\u0016B/U is realized. Most of the respective transitions are re\rected in the electrical resistivity\nas sudden drastic changes. Most notably, the phase between Hc1andHc2exhibits substantially\nlarger values. As the temperature increases, transitions smear out and disappear above \u001915 K.\nHowever, a substantial magnetoresistance is observed even at temperatures as high as 80 K. Due to\na strong uniaxial magnetocrystalline anisotropy a very small \feld e\u000bect is observed for \felds apllied\nperpendicular to the c-axis direction.\nPACS numbers: 75.25.-j, 75.30.-m\nI. INTRODUCTION\nDespite numerous studies, the exact nature of an or-\nder phase appearing in the heavy fermion compound\nURu 2Si2below T HO= 17.5 K is still unclear. This phase,\ncalled hidden order (HO), is one of the most debated top-\nics in heavy-fermion physics research1,2. Studies are often\ndevoted to states emerging from the HO state. At low\ntemperatures, starting from the HO state, di\u000berent mag-\nnetic phases can be induced by pressure, magnetic \feld\nor a moderate substitution. The determination of such\nphases is a subject of many current studies3{7.\nThis HO state coexists below T sc= 1.5 K with super-\nconductivity (SC) and is linked to a parasitic antiferro-\nmagnetic (AF)8{10order characterized by a propagation\nvector Q0= (1 0 0) with very small dipolar magnetic\nmoments (0.01 - 0.03 \u0016B10). These moments that seem\nto be related to lattice imperfections or strain are not\ncompatible with a large entropy and a lambda-type spe-\nci\fc heat anomaly at T HOassociated with this transition.\nConsequently, the phase is called the small moment an-\ntiferromagnetic (SMAF) phase. Single crystals of higher\nquality with a higher residual electrical resistivity show\nas a rule smaller dipolar U moments. These moments,\nhowever, can develop either under pressure or with light\ndoping of di\u000berent dopants (Re, Rh, Fe, Os, Tc)3,7,11{15.\nA common e\u000bect is that the HO and SC states are sup-\npressed fairly quickly and new types of a magnetic order\nappear. In particular, Rh for Ru at a level of \u00192 %\nstabilizes the SMAF and the HO re-appears at elevated\n\felds but \u00194 % Rh destroys the HO completely. Short-\nrange AF correlations start to develop around 5 % of Rhdoping and lead to AF order at higher concentrations11.\nAbove \u001910 %, a long-range AF order with Q3= (1\n21\n21\n2)\nexists. By studying these emergent phases one hopes to\nbe able to deduce valuable information on how the HO\nrelates to the formation of U magnetic moments.\nWhile a moderate pressure stabilizes a long-range AF\norder of the Q0type16(called large moment AF, LMAF),\nthe applied magnetic \feld induces yet new phases. Polar-\nized Fermi-liquid state with large saturated moments can\nbe reached above a critical \feld \u00160Hc\u001938 T that seems\nto be independent on the Rh-doping level17{20. This\nphase is reached via a series of transitions that, in con-\ntrast, depends signi\fcantly on the Rh content. Concomi-\nnant with a destruction of the HO state, the \frst critical\n\feld shifts to lower values11,21{23. Challenging high-\feld\nneutron di\u000braction experiments showed that in the case\nof the pristine system the \frst \feld induced phase adopts\npropagation vector Q'2= (0.60 0 0), close to the Fermi\nlevel nesting vector20. This structure is reported to be\nsine-wave modulated and perhaps of a multi- Qnature.\nIn contrast, for Rh-doped systems a commensurate up-\nup-down magnetic structure with propagation vector Q2\n= (2\n30 0) has been detected22,23.\nRecently we have shown that\nU(Ru 0:92Rh0:08)2Si2does not exhibit any sign of\nHO, SC, SMAF or LMAF states down to 0.2 K6. A\nheavy-fermion behavior, however, remains intact resem-\nbling above T HOvery strongly properties of the pristine\nURu 2Si26,24. Nevertheless, in contrast to URu 2Si2, it\nexhibits a short-range order (SRO) at Q325. The SRO\nsignal appears in zero \feld at temperatures comparable\nto T HOand it is initially strengthened by the applied\n\feld lower than \u001922 T where the \frst metamagneticarXiv:1901.01750v1  [cond-mat.str-el]  7 Jan 20192\ntransition (MT) occurs. Above this \feld the SRO signal\ndisappears, being replaced by new Bragg re\rections\nindexable by Q222,23.\nUp to date high-\feld experiments on\nU(Ru 0:92Rh0:08)2Si2above ordinary laboratory \felds\nwere limited to a single magnetization curve up to 55\nT recorded at 1.4 K and to neutron di\u000braction up to\n24 T. In this work we report the high-\feld magnetic\nphase diagram up to 58 T deduced from combined\nmeasurements of the magnetization and electrical re-\nsistivities. We show that the response of the material\nto the external magnetic \feld is very anisotropic with\nthec-axis being the easy magnetization direction along\nwhich three MT's are detected. Sharp transitions\nindicate \feld-induced phases of a di\u000berent kind, the\n\frst being an uncompensated AF with magnetization \u0019\nMs/3, the second exhibiting magnetization of about \u0019\nMs/2 before entering the fully polarized state. However,\nthe intermediate Ms/2 phase is not clearly resolved in\nthe electrical transport properties.\nCombining the available data the magnetic phase dia-\ngram is constructed documenting existence ranges of sev-\neral magnetic phases and regimes of U(Ru 0:92Rh0:08)2Si2.\nResults are discussed in a context of pristine and lightly\ndoped URu 2Si2systems.\nII. EXPERIMENTAL\nThe details of our U(Ru 0:92Rh0:08)2Si2single crystal\npreparation, quality and other physical properties are\npresented in our recent work6. Magnetization M(T) mea-\nsurements in \felds up to 58 T generated by discharg-\ning a capacitor bank producing a 25 ms long pulse were\nperformed at the Hochfeld-Magnetlabor Dresden (HLD),\nHelmholtz-Zentrum Dresden-Rossendorf. A small 12.5\nmg single crystal used in the magnetization experi-\nment has been oriented using LAUE X-ray backscatter\nmethod. The edges of the sample were cut along di-\nrections parallel to principal crystallographic axes using\nspark erosion. Measurements were carried out between\n1.4 K and 80 K with the \feld oriented along and perpen-\ndicular to the c-axis. The magnetic signal was detected\nusing compensated pick-up coils and and scaled to match\nprevious magnetization measurements obtained in static\n\felds up to 14 T6.\nElectrical resistivity measurements were performed be-\ntween 1.4 and 150 K on two bar-shaped single crystals\n(one of them used also for the magnetization measure-\nments) with dimensions of about 0.7 mm x 0.7 mm x 3\nmm cut along the principal a-andc-axes. We have used a\nstandard four-wire AC method with frequencies between\n16 and 25 kHz and excitation currents between 10 and\n20 mA \rowing along the longest dimension. The \feld\nhas been applied along and perpendicular to the c-axis\nand both, longitudinal and transverse resistivities were\nrecorded. Fields up to 62 T produced by a magnet with\nthe pulse length of 150 ms were applied.\nFIG. 1. (Color online) Field dependence of the\nU(Ru 0:92Rh0:08)2Si2single crystal magnetization measured at\nseveral di\u000berent temperatures with pulsed \felds applied along\nthec-axis (a) and perpendicular to the tetragonal axis (b). In-\nsets in (a) show the \feld dependence of the \feld derivation of\nthe magnetization at 1.4 K around metamagnetic transitions\ntaken with increasing and decreasing \feld applied along the\nc-axis, respectively.\nIII. RESULTS\nA. Magnetization\nThe magnetic response of U(Ru 0:92Rh0:08)2Si2to the\nmagnetic \feld is extremely anisotropic. In Fig. 1 (b) we\nshow magnetization curves obtained at 1.4 K and at 29.3\nK with \feld applied along the a-axis. The magnetization\nis small and increases linearly as a function of the applied\n\feld. No anomaly is present up to 58 T. Magnetic curves\nare very weakly temperature dependent, in agreement\nwith low-\feld PPMS data.6\nIn contrast to the direction perpendicular to the tetrag-\nonal axis, magnetization measured with \feld along the c-\naxis exhibits strong \feld and temperature dependences.\nIn Fig. 1 (a) we show several representative magnetiza-\ntion curves measured with \feld applied along the c-axis\nbetween 1.4 and 31 K with increasing \feld. For the lowest\ntemperature we show also the descending \feld magneti-\nzation curve. At low temperatures, steep changes around\n\u001922 T and around \u001938 T, respectively, are observed\nmarking metamagnetic transitions (MTs). The \frst MT,\nfrom the low-\feld heavy fermion liquid (HFL) to the \frst\n\feld-induced phase appears with increasing \feld at 22.0\nT. As it is shown in insets of Fig. 1 (a), all the MTs\nappear with decreasing \feld at slightly lower \felds. The\n\frst MT appears with decreasing \feld at 21.2 T leading\nto a hysteresis of \u00190.8 T and the average of \feld-up and\ndown transitions at \u00160Hc1= 21.6 T. The signi\fcant hys-\nteretic behavior is in agreement with \frst-order type of\nthe phase transition found in the pristine system and 43\n% Rh-doped system transition18,26,27.\nA closer look at the transition around 38 T reveals that\nit consists from two anomalies suggesting a presence of\ntwo individual MT's. As it is illustrated in the upper in-\nset of Fig. 1 (a), the former MT appears with increasing\n\feld at 38.05 T and the latter at 38.40 T, respectively. On\nthe decreasing direction, they are found at 37.75 T and\n38.10 T, respectively suggesting a somewhat smaller hys-\nteresis of 0.3-0.4 T. The average of up and down sweeps\namount to\u00160Hc2= 37.90 T and \u00160Hc3= 38.25 T.\nAbove H c3, the magnetization exhibits a gradual ten-\ndency towards a saturation at a level of M s= 2.1\u0016B/U,\nwhich appears to be larger than in the pure and 4 % Rh-\ndoped systems.22Here a Fermi-liquid polarized state is\nestablished. The magnetization step across the \frst MT\namounts to 0.46 \u0016B/U and across the second MT at 38\nT to 0.94\u0016B/U leading to a total magnetization change\nacross MTs of 1.40 \u0016B/U. The increase at the former MT\namounts thereby to one third of the total magnetization\nincrease across all the transitions. This value is in accord\nwith our recent high-\feld single crystal neutron di\u000brac-\ntion on this system showing that the \frst \feld-induced\nphase is a commensurate uncompensated AF phase of\n1.45 (9)\u0016BU moments directed along the c-axis. The U\nmoments are arranged in an up-up-down sequence propa-\ngating along the a-axis23. We denote therefore this phase\nasMs/3 phase. On the contrary, compared to both pris-\ntine system and lightly Rh-doped systems, this phase ex-\nists over much larger range of \felds between \u00160Hc1=\n21.6 T and \u00160Hc2= 37.90 T28.\nThe magnetization between Hc2andHc3amounts to\nabout one half of the magnetization increase due to all\nMTs. We denote this phase as Ms/2 phase. The range of\n\felds where this phase exists is very small, about 0.35 T.\nThis is to be compared with larger ranges of existence of\ndi\u000berent \feld-induced phases in the pure and lightly Rh-\ndoped systems that were studied in high-\feld magnetic\n\felds18{20,22,23,28.\nThe high-\feld magnetic susceptibility de\fned as\n\u001f(H) =@M(H)/@H, whereHdenotes the \feld strength\nare at 1.4 K distinctly di\u000berent for di\u000berent \feld-induced\nphases. While below \u00160Hc1= 21.6 T the \u001f(H) increases\nprogressively in the vicinity of the transition, it is con-\nstant between Hc1andHc2on a lower level than below\nHc1. AboveHc3, the\u001f(H) at 1.4 K steadily decreases\nwith increasing \feld towards a gradual saturation.\nWith increasing temperature all MTs along the c-axis\ndirection smear and the hysteresis between increasing\nand decreasing \feld branches becomes reduced. The\n\frst MT, de\fned by the maximum on the @M(H)/@H\nas function of H, moves with increasing temperature\nsteadily to higher \felds and disappears above \u001915-16\nK. In contrast, the upper two MT's shift with increasing\ntemperature to lower \felds. Although all the transitions\nare relatively well de\fned we could follow the splitting of\nthe transitions in the \feld derivatives of M(H) around 38\nT only up to \u00199 K. Above this temperature we observe\naround 38 T a single MT.\nFIG. 2. (Color online) Magnetic phase diagram of\nU(Ru 0:92Rh0:08)2Si2single crystal deduced from magnetiza-\ntion measurements with \feld applied along the tetragonal\naxis. Para, SRO, M s, Ms/2 and M s/3 denote various phases\nexisting in U(Ru 0:92Rh0:08)2Si2(see the main text).\nBetween \u00199 K and \u001916 K, well de\fned maxima in\n@M(H)/@Hdenote the \feld range where the Ms/3 phase\nexists. Between these maxima the M(H) exhibits an S-\nshape type increase (see Fig. 1). Above \u001916 K up to\n\u001950 K no clear anomalies are visible from the M(H)\ndependences. However, one still observes an S-shaped\nmagnetization curve. We interpret the in\rection point as\na \feld above which a \feld-induced polarized state exists\nin U(Ru 0:92Rh0:08)2Si2. At yet higher temperatures the\nM(H) increases linearly with \feld.\nIn Fig. 2 the magnetic phase diagram for \feld applied\nalong the tetragonal axis is constructed from high-\feld\nmagnetization data. In the inset we show the detail of the\nrange where the Ms/2 phase exists and can be traced up\nto up to \u00199 K. The magnetic phase diagram also clearly\nestablishes that the Ms/3 phase occurs as an inclusion\nbetween the low-\feld state where only the short-range\norder (SRO) de\fned by Q3exists and the Ms/2 phase,\nor, at temperatures between \u00199 K and \u001915 K, between\nthe SRO and the polarized state.\nB. Electrical Resistivity\nElectrical resistivity measurements with \feld applied\nalong thea-axis show no anomalies (not shown). This\n\fnding is in agreement with magnetization data which\nshow no appreciable \feld e\u000bect for this orientation. The\nobserved change of the resistivity with current both along\nthe tetragonal axis and perpendicular to it is qualita-\ntively similar to data published for the pure system29,30.\nIn the pure system, however, larger changes at low tem-\nperatures are found. In our sample, for the current along4\nthecaxis, the resistivity increases by less than \u00194 %\nof its zero-\feld value, indicating that the \feld applied\nperpendicular to c-axis direction does not alternate sig-\nni\fcantly the scattering of conduction electrons.\nIn Fig. 3 (a) the \feld dependence of the electrical resis-\ntivity with current along the a-axis,\u001aa(H==c ), and \feld\napplied along the tetragonal axis for selected tempera-\ntures between 1.7 K and 80 K is shown. These data are\ntaken with decreasing \feld. At 1.5 K, the \frst MT at\nHc1manifests itself as a sudden increase of the resistiv-\nity. The transition to the polarized state causes, in con-\ntrary, a signi\fcant decrease. We are not able to resolve\nthe two transitions at Hc2andHc3that were indicated\nin the magnetization measurements. As the temperature\nincreases, both MTs smear out, the one at lower \feld\nfaster than the upper one.\nIt should be noted that at the lowest temperature, be-\nlow the \frst MT and between MTs the resistivity cannot\nbe approximated by the expression \u001aH=\u001a0T+aH2,\nwhere\u001a0Tdenotes the electrical resistivity at zero \feld.\nTo get a reasonable description of data, an inclusion of a\nterm linear in His necessary documenting beyond Fermi\nliquid type behavior. This type of \feld dependence that\nis in contrast to the pure system where the H2behavior\nis observed8,31. Such behavior has been previously iden-\nti\fed for the low-\feld range at various temperatures for\nthis sample. The agreement of the \ftted parameters with\nliterature is good23. Above the Hc3, the electrical resis-\ntivity depends quadratically on the magnetic \feld. This\ntype of \ft yields \u001a0T= 25.8 (5)\u0016\ncm and suggests that\nmuch of the electron scattering is caused by processes\nthat are quenched in the high \feld limit.\nThe \feld dependence of the resistivity changes with\ntemperature signi\fcantly. For instance, the behavior be-\nlow the \frst MT is at high temperatures reverted to that\nat low temperatures and the resistivity decreases with\nincreasing \feld. Up to \u001915 K the measured curves have\na dome-like dependence with higher resistivities between\nHc2andHc3. At higher temperatures one can discern a\nreduction of the electrical resistivity values up to 20-30 T\nfollowed by a subsequent increase at higher \felds. At the\nhighest temperatures the increase in the high-\feld limit\nis nearly linear.\nIn Fig. 3 (b) we show the \feld dependence of the elec-\ntrical resistivity \u001ac(H==c ) with \feld and current along\nthec-axis. Also these data are taken with descending\n\felds. At 1.7 K the MTs are clearly visible and manifest\nthemselves as sudden changes in the electrical resistivity\nvalues. Starting from the relatively high resistivity at\nzero \feld, the resistivity increases moderately in agree-\nment with PPMS data6, \frst by few percents until \u001920\nT where it starts to increase signi\fcantly up to \u001924 T.\nHere it attains twice as high resistivity with respect to the\nzero \feld. The interval across which it increases is signif-\nicantly broader than the transition indicated in magnetic\nbulk measurements. However, it still coincides with the\nthe \frst MT centered at \u00160Hc1= 21.6 T. In the M s/3\nphase the resistivity increases nearly linearly by very few\nFIG. 3. (Color online) Field dependence of the\nU(Ru 0:92Rh0:08)2Si2single crystal electrical resistivity mea-\nsured at several di\u000berent temperatures with pulsed \felds\n(sweeps down) applied along the c-axis with electrical cur-\nrent perpendicular (a) and along (b) the tetragonal axis.\n% only to drop above \u001936 T. The reduction over a \feld\ninterval of \u00192 T across this transition is signi\fcant and\nthe resistivity reaches to about 1/3 of its zero \feld value\nwith a further weak decrease at even higher \felds. At\n1.7 K, the electrical resistivity reduces at 60 T by 74 %\nwith respect to its zero \feld value. Extrapolation of data\ntaken at the lowest temperature assuming a linear depen-\ndence towards zero \feld suggests \u001a0T= 32 (1)\u0016\ncm, i.e.\na value comparable with the extrapolation of the data at\n1.7 K for electrical current along the a-axis.\nWhile a hysteresis of a comparable width as in the mag-\nnetization measurements has been detected across Hc1\nin electrical resistivity, no signi\fcant hysteresis between\n\feld up and \feld down sweeps has been detected in the\nHc2-Hc3region. However, the critical \feld of MTs de-\n\fned from electrical resistivity for both orientations are\nfound at di\u000berent \felds. Also the width of the transi-\ntion is di\u000berent - the electrical transport yield broader\ntransitions.\nAs it is shown in Fig. 4, with increasing \feld the elec-\ntrical resistivity \u001ac(H) reaches its reduced value at \felds\nwhere the magnetization only starts to increase. Sim-\nilarly, upon decreasing the \feld \frst the magnetization\nreaches a reduced value below Hc2before the resistiv-\nity starts to increase. At the same time, the transition\nrange seen on the electrical resistivity is about three times5\n020406080100120140160ρc (µΩcm) ρc field up \nρc field down3\n63 73 83 90.81.21.62.0µ\n0H // c axis M\n (µB/U)µ\n0H (T) M field up \nM field downT = 1.7 K\nFIG. 4. (Color online) Enlarged portion of the \feld depen-\ndence around the Hc2-Hc3transitions of the \u001ac(H) for in-\ncreasing (dashed line) and decreasing (full line) \feld sweeps\nas compared to the magnetization curve, measured with \feld\napplied along the c-axis. Note di\u000berent onsets of relevant\nchanges and width of transitions.\nbroader (10 % - 90 % rule) than the transition across\nbothHc2-Hc3transitions seen on the magnetization.\nThis is valid for both, ascending and descending \felds,\nruling out a possibility that the di\u000berence is due to a\ndi\u000berent preferred way of data recording (the electrical\nresistivity presented above have been measured as a rule\nwith descending \feld while the magnetization with in-\ncreasing \feld). Similar observation, albeit with smaller\ndi\u000berences between the magnetization and resistivity, is\nfound for the \frst MT. The electrical resistivity starts to\nchange before the relevant change starts on the magneti-\nzation (not shown).\nWith increasing temperature the zero \feld resistiv-\nity increases in agreement with previous data6and the\nchanges at respective critical \felds are smeared out. One\nobserves a dome-like structure for temperatures up to \u0019\n15 K. Similar to the a-axis direction, no clear indication\nof the M s/2 phase s found.\nComparing the measurements with current along the\na-andc-axis one realizes several signi\fcant di\u000berences.\nFirst, the electrical resistivity along the a-axis is larger\nthan along the tetragonal axis at all temperatures except\nfor a small \feld interval at lowest temperatures. Second,\nin the M s/3 phase\u001ac(H) increases at 1.7 K nearly lin-\nearly with \feld, changing at intermediate temperatures\nbetween \u00195 K and \u001913 K to a dome-like dependence\nsimilar to the a-axis direction. Finally, at high enough\ntemperatures (above \u001920 K) the\u001ac(H) decreases with\n\feld, in contrast to \u001aa(H) that \frst decreases and then\nincreases in the high \feld limit. This di\u000berence results\nfrom a di\u000berent geometry of the current with respect to\nthe applied \feld and is due to cyclotron motion of elec-\ntrons that leads to higher scattering rate in the tranverse\ngeometry.\nA more comprehensive picture of the electrical resis-\n204 06 08 0010203040µ0Η (Τ)i//aB\n//cρ\n (µΩcm)T\n (K)80.002\n30.03\n80.05\n30.0FIG. 5. (Color online) Color-coded transverse electrical re-\nsistivity\u001aa(H) of the U(Ru 0:92Rh0:08)2Si2single crystal with\ncurrent along the a-axis as a function of the temperature and\nmagnetic \feld applied along the c-axis.\n204 06 08 0010203040T\n (K)µ0Η (Τ)0.0001\n00.02\n00.02\n80.0ρ (µΩcm)i//cB\n//c\nFIG. 6. (Color online) Color-coded longitudinal electrical re-\nsistivity\u001ac(H) of the U(Ru 0:92Rh0:08)2Si2single crystal mea-\nsured along the c-axis as a function of the temperature and\nmagnetic \feld applied along the c-axis.\ntivity behavior can be obtained form color-coded maps\nin Fig. 5 and Fig. 6 that show a portion of the transverse\n\u001aa(H) and longitudinal \u001ac(H) electrical resistivities with\ncurrent along the a-axis and the c-axis, respectively as a\nfunction of temperature and magnetic \feld applied along\nthec-axis. Clearly, the Ms/3 manifests itself as an island\nof enhanced resistivity for both current directions. Com-\nmon for both current directions is a rather signi\fcant\nreduction of resistivity values at low temperatures that\nis clearly present for both orientations also at high \felds.\nThis observation is similar to measurements on the pure\nsystem. An exception is the c-axis orientation at high\n\felds above the Ms/3 phase, where the low-resistivity re-\ngion extends as compared to the a-axis direction also to6\nhigher temperatures. Also common for both orientations\nis a decrease of the resistivity with increasing \feld up to\n\u001925 T at elevated temperatures, where the reduction is\nprogressively larger in the low- temperature limit. Above\nthis \feld only the c-axis resistivity continues to decrease.\nFor thea-axis direction the resistivity increases in the\nhigh-\feld limit leading to a minimum around 25 T. The\nlatter observation indicates di\u000berent contributions to the\nscattering of conduction electrons.\nIV. DISCUSSION\nPresented data show unambiguously that the response\nof U(Ru 0:92Rh0:08)2Si2to magnetic \feld is extremely\nanisotropic. This is documented by the \feld induced\nmetamagnetic transitions for the \feld applied along the\nc-axis observed at \u00160Hc1= 21.60 T, \u00160Hc2= 37.90 T\nand\u00160Hc3= 38.25 T, respectively and the absence of\nany anomalies on the magnetization curve for the direc-\ntion perpendicular to the tetragonal axis. This behavior\nresembles very much properties of the pure system. The\nmagnetocrystalline anisotropy that has Ising character is\ncaused by the non-Kramers doublet in \u0000 5that couples\nonly to the c-axis component of the magnetic \feld. The\ntwo 5f2states, having both electric quadrupolar and spin\ndegrees of freedom (spin-orbital liquid32) are proposed to\nbe responsible for the hidden order in the pristine system\nand the \frst-order phase transition at Hc127,33. In our,\n8 % Rh-doped system, no HO exists. However, the bare\nphysical properties remains the same - absence of a long-\nrange magnetic order and heavy-fermion behavior at low\ntemperatures, very strong uniaxial anisotropy and \feld-\ninduced phases for \feld along the c-axis before arriving to\na polarized state. It is therefore to be expected that in the\nlow-T limit at low \felds also in U(Ru 0:92Rh0:08)2Si2are\n5felectrons in the 5 f2con\fguration. These become itin-\nerant upon application of strong magnetic \feld. A Zee-\nman splitting causes one of the subbands to be shifted\nbelow the Fermi energy causing the reconstruction of\nits topology, reducing the hybridization between the 5 f\nstates and conduction electrons and thereby creating siz-\nable and stable magnetic moments. These appear at low\ntemperatures above Hc1. As the application of \feld per-\npendicular to the c-axis does not reconstruct the Fermi\ntopology, the 5 felectrons are still in the vicinity of the\nFermi surface, hybridized with conduction electron.\nThis is also the reason why the anisotropy is re\rected\nvery strongly in the electrical transport properties. While\nfor the \feld applied along the a-axis no signi\fcant modi\f-\ncations are observed in agreement with the magnetization\nbehavior, the application of the \feld along the tetragonal\naxis leads to drastic changes of the resistivity along both\nthea-andc-axes. At low temperatures, the state between\nHc1andHc2leads generally to a larger resistivity than\nbelow the MT at Hc1where merely a short-range order\nexists and in the high \feld limit where the polarized state\nis established. The simplest way to interpret this obser-vation is that it is due to the appearance of a long-range\nmagnetic order (phase M s/3) with Q2= (2\n30 0) lead-\ning to a modi\fcation of the Fermi surface topology and\nthe appearance an additional superzone boundary. As\nthe details of the entire Fermi surface topology in\ruence\nconduction electron scattering (also due to strong hy-\nbridization with other electron states), it is not surpris-\ning that one observes signatures of the M s/3 phase (and\ngenerally the in\ruence of \feld applied along the c-axis)\nfor both current orientations. The signi\fcant reduction\nof the electrical resistivity upon entering the polarized\nphase, called giant magnetoresistance, is very common in\nuniaxial U-based systems34,35and is present also in the\npure system29,30. It is usually interpreted as being due to\nanisotropic reconstruction of the Fermi surface topology\nand a reduction of anisotropic magnetic \ructuations34,36.\nFurther in the high \feld limit, especially at elevated\ntemperatures, the electrical resistivities along and per-\npendicular to the tetragonal axis behave di\u000berently. For\nthec-axis direction the resistivity values above Hc3are\nat all temperatures signi\fcantly lower than in zero \felds.\nIn contrast, the resistivity for the a-axis direction, i.e.\nwith the current perpendicular to the applied \feld, in-\ncreases at high \felds. This di\u000berent behavior could be\nexplained considering the combined e\u000bects of the mag-\nnetocrystalline anisotropy and the Lorentz force acting\non the conduction electrons causing classical magnetore-\nsistance e\u000bect. Higher cyclotron frequencies lead to re-\nduction of the mean free path of conduction electrons\nand higher scattering rates for the transverse geometry.\nComparison with available data for the pure system30\nwe recognize that the e\u000bect of the magnetoresistance is\nstronger in our sample.\nAt this point we should mention a possible non-\nnegligible in\ruence of the magnetocaloric e\u000bect (MCE)\nand/or eddy currents. While the former e\u000bect can lead\nto either increase or decrease of the sample's temper-\nature, the latter one always increases the sample tem-\nperature. MCE has been clearly identi\fed in previous\nhigh \feld experiments in the pure and 4 % Rh doped\nsystems18,26,27. This e\u000bect is rather important as it shifts\nthe phase boundaries and obscures the exact determina-\ntion of the existence regions of various phases. Also, it\nmay hamper the comparison between results obtained\nusing di\u000berent techniques. In fact, our preliminary mea-\nsurements using a much larger single crystal indicate that\nalso U(Ru 0:92Rh0:08)2Si2exhibits MCE37. It is therefore\nalmost sure that MCE plays a role also in our measure-\nments and would lead to modi\fcations of the phase tran-\nsition values. However, we argue that even if the phase\nboundaries are determined with lesser precision, the main\nmessage of the current work remains intact. Question can\nbe raised whether the double structure of the MT at Hc2\n-Hc3cannot be either due to variations in the Rh concen-\ntration or due to temperature inhomogeneities caused by\nfast \feld sweeps. Indeed, concentration inhomogeneities\nwould lead to a broadening or even splitting of MTs.\nHowever, no such structure is observed around Hc1that7\nis more sensitive to the Rh concentration. In addition,\nthe MTs are sharp and neutron di\u000braction observation\nwith resolution limited Bragg re\rections did not reveal\nany structural inhomogeneities6. Therefore we conclude\nthat the double MT at Hc2-Hc3is real. For analogical\nreasons we rule out also a possible in\ruence of temper-\nature inhomogeneities due to fast \feld sweeps. On the\nother hand, both mechanisms probably contribute to the\nobserved hysteresis of all MTs.\nIn general, the electrical resistivity follows closely the\n\feld-induced changes of the U(Ru 0:92Rh0:08)2Si2mag-\nnetic state. Metamagnetic transitions at Hc1andHc2\nare clearly visible both in magnetization and electrical\nresistivity. An exception here is the Hc3transition that\nremains invisible in the electrical resistivity measure-\nment. In the transport properties only Hc2MT is visible,\nbeing completed before the magnetization changes (see\nFig. 4). As the resistivity experiments were performed\nwith longer \feld pulses, eddy currents and MCE play less\nimportant role with the temperature of the sample be-\ning more constant. As eddy currents always increase the\ntemperature of the sample, it is to expected that the sam-\nple during magnetization experiment (with higher \feld\nsweep rate) is higher than in the resistivity measurement\ncase. Considering the magnetic phase diagram shown\nin Fig. 2, this would lead to a lower critical \felds of\nthe upper transition(s) than determined from magneti-\nzation. However, it is the electrical resistivity that leads\nlower critical \feld around 38 T. This observation sug-\ngests that both, the polarized and the Ms/2 phases are\ncreated due to reconstruction of the Fermi surface lead-\ning to polarization of individual Fermi surface pockets38\nand consequently to reduction of the 5 f-conduction elec-\ntron hybridization that short-cuts the sample just below\nHc2.\nFrom the available data we have constructed the mag-\nnetic phase diagram for the \feld applied along the c-axis\nthat is shown in Fig. 7. Besides the phase transition\nboundaries determined from the \feld and temperature\nanomalies of the magnetization and electrical resistivity\nwe identify also regions of di\u000berent electrical resistivity\nbehavior as a function of temperature or the applied \feld.\nFirst of all, by the red dashed line we denote position\nof the in\rection point in the temperature dependence of\nthe electrical resistivity that agrees with the occurrence\nof the SRO25. It is interesting to note that the SRO\nappears approximately at the same temperature as the\nHO in the pristine system. This may suggest that both\nground states are on the same energy scale. Both ground\nstates are itinerant heavy-fermion liquids. The di\u000ber-\nence is, however, a lesser degree of coherence of the SRO\nstate and thereby magnetic disorder that is re\rected also\nby much higher residual resistivity in Rh doped systems.\nThe loss of coherence is most probably also the reason for\ndestruction of the HO. The light blue dashed line limits\nthe quadratic \feld dependence of the electrical resistiv-\nity found for current along the a-axis due to classical\nmagnetoresistance. This type of dependence changes at\n01 02 03 04 05 06 0020406080P\nFLdecr. ρa decr. ρc m\ninimum ρadecr. ρc i\nncr. ρa decr. ρc ρ\na ∼ Β2MSMS/2M\nS/3S ROParaBc1 upBc1 downB\nc2 upBc2 downB\nc3 upBc3 downC\nritical Field (T)T (K)ρ\n ∼ ΒnFIG. 7. (Color online) Magnetic phase diagram of\nU(Ru 0:92Rh0:08)2Si2for magnetic \feld applied along the c-\naxis documenting the ranges of existence of various phases\nand di\u000berent regions with typical \feld and temperature be-\nhaviors of the electrical resistivity. SRO denotes region with\na short-range magnetic order described by Q3, PFL denotes\nPolarized Fermi liquid state, Para a paramagnetric state and\nMs/3 andMs/2 \feld induced phases with one third and one\nhalf saturated magnetization Ms, respectively. Dashed lines\nrepresent approximate boarders of di\u000berent resistivity behav-\nior regions (see the main text).\nhigher temperatures to a nearly linear one. The green\nline shows approximately the high-temperature low-\feld\nregion where the electrical resistivity along both a-axis\nandc-axis directions decreases suggesting a reduction of\nconduction electron scattering due to magnetic \ructua-\ntions. Finally, the dark blue line de\fnes a region where\nthea-axis resistivity alone increases leading to a mini-\nmum between the two regions. Comparing our magnetic\nphase diagram with those published for the pristine and\nlightly doped Rh systems1,2,17{19,28one realizes immedi-\nately that they are very similar, especially in the high\n\feld region. Biggest di\u000berences concern the low-\feld re-\ngion at low temperatures. In the pristine system, where\nthe HO exists the longitudinal resistivity strongly in-\ncreases to fall down upon entering the \frst induced phase.\nIn our system there is no HO and also the electrical re-\nsistivity does not show any anomalies below Hc1that\nis in addition shifted signi\fcantly to lower \feld values.\nThis points to a signi\fcant interplay between conduction\nelectron states and the HO order which is removed when\nreplaced by SRO in our system. Yet, the SRO and the\nHO sets in at similar temperatures. Interesting is also\nthe fact that the uppermost MT appears in the pristine\nand Rh-doped systems nearly at the same critical \felds\nsuggesting yet another common energy scale in these sys-\ntems.\nIn conclusion, we show that the behavior of\nU(Ru 0:92Rh0:08)2Si2is very anisotropic and the electri-\ncal transport properties are dictated by the Fermi sur-8\nface topology changes caused by the magnetic \feld. The\neasy magnetization direction is found along the tetrago-\nnal axis, similar to other systems stemming from the pure\nURu 2Si2. We identify several distinct \feld induced mag-\nnetic phases and ranges with di\u000berent \feld/temperature\nbehavior of the electrical resistivity pointing to vari-\nous di\u000berent scattering mechanisms of conduction elec-\ntrons. Complementary information should become avail-able from the Hall e\u000bect experiments.\nACKNOWLEDGMENTS\nWe acknowledge the support of the HLD at HZDR,\nmember of the European Magnetic Field Laboratory\n(EMFL).\n\u0003prokes@helmholtz-berlin.de\n1J. A. Mydosh and P. M. Oppeneer, Rev. Mod. Phys. 83,\n1301 (2011).\n2J. A. Mydosh and P. M. Oppeneer, Phil. Mag. 94, 3642\n(2014).\n3T. J. Williams, Z. Yamani, N. P. Butch, G. M. Luke, M. B.\nMaple, and W. J. L. Buyers, Phys. Rev. B 86, 235104\n(2012).\n4T. J. Williams, A. A. Aczel, M. B. Stone, M. N. Wilson,\nand G. M. Luke, Phys. Rev. B 95, 104440 (2017).\n5K. R. Shirer, M. 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Petrovic\nCondensed Matter Physics and Materials Science Department ,\nBrookhaven National Laboratory, Upton, New York 11973, USA\n(Dated: January 10, 2019)\nIntrinsic, two-dimensional (2D) ferromagnetic semicondu ctors are an important class of materials\nfor spintronics applications. Cr 2X2Te6(X= Siand Ge) semiconductors show2D Ising-like ferromag-\nnetism, which is preserved in few-layer devices. The maximu m magnetic entropy change associated\nwith the critical properties around the ferromagnetic tran sition for Cr 2Si2Te6−∆Smax\nM∼5.05 J\nkg−1K−1is much larger than −∆Smax\nM∼2.64 J kg−1K−1for Cr 2Ge2Te6with an out-of-plane\nfield change of 5 T. The rescaled −∆SM(T,H) curves collapse onto a universal curve indepen-\ndent of temperature and field for both materials. This indica tes similar critical behavior and 2D\nIsing magnetism, confirming the magnetocrystalline anisot ropy that could preserve the long-range\nferromagnetism in few-layers of Cr 2X2Te6.\nI. INTRODUCTION\nLayered ferromagnets such as Cr 2Ge2Te6, CrI3, and\nFe3GeTe2have recently attracted considerable atten-\ntion since long-range ferromagnetism (FM) persists in\natomically thin devices.1–6Intrinsic magnetic order is\nnot allowed at finite temperature in the two-dimensional\n(2D) isotropic Heisenberg model by the Mermin-Wagner\ntheorem7, however large magnetocrystalline anisotropy\nin van der Waals (vdW) magnets would lift this restric-\ntion.\nBulk Cr 2X2Te6(X = Si and Ge) exhibit FM below\nthe Curie temperature ( Tc) of 32 K for Cr 2Si2Te6and 61\nK for Cr 2Ge2Te6, respectively, and show large magne-\ntocrystalline anisotropy as a result of strong spin-orbit\ncoupling (SOC).8–12Neutron scattering measurements\nshowed that bulk Cr 2Si2Te6is a strongly anisotropic 2D\nIsing-like ferromagnet with a critical exponent β= 0.17\nand a spin gap of ∼6 meV.13On the other hand, re-\ncently observed β= 0.151 and a much smaller spin\ngap of∼0.075 meV argue that the spins in Cr 2Si2Te6\nare Heisenberg-like.14Cr2Ge2Te6is proposed to be a\n2D Heisenberg ferromagnet based on spin wave theory,4\nbut was also found to follow the tricritical mean-field\nmodel,15calling for further studies. The magnetocaloric\neffect (MCE) in the FM vdW materials is also of inter-\nest since it can give insight into the magnetic properties.\nFe3−xGeTe2withTc= 225 K shows the maximum value\nof magnetic entropy change −∆Smax\nMabout 1.1 J kg−1\nK−1at 5 T.16CrI3exhibits anisotropic −∆Smax\nMwith\nvalues of 4.24 and 2.68 J kg−1K−1at 5 T for H//cand\nH//ab, respectively.17\nIn this work we studied the anisotropic magnetocaloric\neffect associated with the critical behavior of Cr 2X2Te6\n(X = Si and Ge) single crystals. The magnetocrystalline\nanisotropy constant Kuis temperature-dependent, and\nis evidently larger for Cr 2Si2Te6when compared to\nCr2Ge2Te6. The maximum magnetic entropy change in\nout-of-plane field up to 5 T −∆Smax\nM∼5.05 J kg−1\nK−1for Cr 2Si2Te6is nearly double of −∆Smax\nM∼2.64 J\nkg−1K−1for Cr 2Ge2Te6. Critical exponents β,γ, andδ\nand critical isotherm analysis suggest 2D Ising-like spins.This is further confirmed by the scaling analysis of mag-\nnetic entropychange −∆SM(T,H), in which the rescaled\n−∆SM(T,H) collapse on a universal curve. Our work\nprovides evidence for magnetocrystalline anisotropy that\ndrives the 2D Ising ferromagnetic state in few layers of\nCr2X2Te6(X = Si and Ge).\nII. EXPERIMENTAL DETAILS\nSinglecrystalsofCr 2X2Te6(X=SiandGe)werefabri-\ncated by the self-flux technique starting from an intimate\nmixture of pure elements Cr (3N, Alfa Aesar) powder, Si\nor Ge (5N, Alfa Aesar) pieces and Te (5N, Alfa Aesar)\npieces with a molar ratio of 1 : 2 : 6. The starting mate-\nrialswerevacuum-sealedin a quartztube, heated to 1100\n◦C over 20 h, held at 1100◦C for 3 h, and then cooled to\n680◦C at a rate of 1◦C/h. The x-ray diffraction (XRD)\ndata were taken with Cu Kα(λ= 0.15418 nm) radia-\ntion of a Rigaku Miniflex powder diffractometer. The dc\nmagnetization was collected in Quantum Design MPMS-\nXL5system. Themagneticentropychange −∆SM(T,H)\nfrom the dc magnetization data was estimated using the\nMaxwell relation.\nIII. RESULTS AND DISCUSSION\nA. Structural and basic magnetization data\nBulk Cr 2X2Te6(X = Si and Ge) were first synthesized\nby Carteaux et al..9,10They crystalize in a layered struc-\nture [Fig. 1(a)]. The Cr ions are located at the centers of\nslightly distorted octahedra of Te atoms. The short X-X\nbonds result in X-X dimers forming an ethane-like X 2Te6\ngroups, similar to P-P dimers in CdPS 3.18Figure 1(c)\npresents the single crystal x-ray diffraction (XRD) data.\nThe observed (00l) peaks distinctly shift to higher angles\nin Cr2Ge2Te6when compared to Cr 2Si2Te6indicating\na smaller vdW gap in Cr 2Ge2Te6. The powder XRD\ndata can be indexed in the R¯3hspace group [Fig. 1(d)].\nThe determined lattice parameters are a= 6.772(2)˚A2\nFIG. 1. (Color online) Crystal structure of Cr 2X2Te6(X =\nSi and Ge) from (a) side and (b) top views. (c) Single crys-\ntal x-ray diffraction (XRD) and (d) powder XRD patterns of\nCr2X2Te6(X = Si and Ge). The vertical tick marks represent\nBragg reflections of the R¯3hspace group.\nTABLE I. The parameters obtained from fits of the 1 /Mvs\nTdata for Cr 2X2Te6(X = Si and Ge) single crystals.\nFitTrange C θ pµeff\n(K) (emu K/mol) (K) ( µB/Cr)\nX = Si\nH//c130≤T≤300 1.70(1) 63(1) 3.68(2)\nH//ab130≤T≤300 1.60(1) 63(1) 3.57(1)\nX = Ge\nH//c150≤T≤300 1.22(1) 114(2) 3.12(1)\nH//ab150≤T≤300 1.54(2) 101(1) 3.51(2)\nandc= 20.671(2)˚A for Cr 2Si2Te6[a= 6.826(2)˚A and\nc= 20.531(2)˚A for Cr 2Ge2Te6], in agreement with the\nreported values.9,10\nFigure 2(a) presents the temperature dependence of\nthe zero field cooling (ZFC) magnetization M(T) mea-\nsured in H= 10 kOe applied in the abplane and parallel\nto thecaxis, respectively. The FM transition stems from\nthe near-90◦Cr-Te-Cr superexchange interaction and is\nobserved in both materials. An apparent bifurcation at\nlow temperature is observed in Cr 2Si2Te6. The absence\nof bifurction in Cr 2Ge2Te6indicates smaller magnetic\nanisotropy. The smaller vdW gap and largerin-plane Cr-\nCr distance in Cr 2Ge2Te6contribute to the enhancement\nof theTcfrom 32 K for Cr 2Si2Te6to 63 K for Cr 2Si2Te6.\nThe 1/MvsTcurves at high temperature follow the\nCurie-Weiss law, χ(T) =M/H=C/(T−θp), where χ\nis magnetic susceptibility, Mis magnetization, Cis the\nCurie constant, and θpis the Weiss temperature. The\nobtained parameters Candθpare listed in Table I. The\npositive Weiss temperatures θp, nearly twice the values\nofTc, for both directions suggest strong short-range FM/s48 /s55/s48 /s49/s52/s48 /s50/s49/s48 /s50/s56/s48 /s51/s53/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s45/s53/s48 /s45/s50/s53 /s48 /s50/s53 /s53/s48/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51\n/s45/s53/s48 /s48 /s53/s48/s45/s50/s48/s50/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s49/s48/s50/s48/s51/s48\n/s67/s114\n/s50/s71/s101\n/s50/s84/s101\n/s54/s77/s32/s40/s101/s109/s117/s32/s103/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s32/s72/s32/s47/s47/s32/s97/s98\n/s32/s72/s32/s47/s47/s32/s99/s40/s97/s41\n/s72/s32/s61/s32/s49/s48/s32/s107/s79/s101/s67/s114\n/s50/s83/s105\n/s50/s84/s101\n/s54\n/s67/s114\n/s50/s71/s101\n/s50/s84/s101\n/s54/s67/s114\n/s50/s83/s105\n/s50/s84/s101\n/s54/s77/s32/s40\n/s66/s47/s67/s114/s41\n/s72/s32/s40/s107/s79/s101/s41/s32/s72/s32/s47/s47/s32/s97/s98\n/s32/s72/s32/s47/s47/s32/s99/s84/s32/s61/s32/s50/s32/s75/s40/s98/s41/s48/s50/s52/s54/s56/s49/s48\n/s49/s47/s77/s32/s40/s103/s32/s101/s109/s117/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s48/s50/s52/s54/s56/s49/s48\nFIG. 2. (Color online) (a) Temperature dependence of zero\nfield cooling (ZFC) magnetic susceptibility χ(left axis) and\ncorresponding 1 /χ(right axis) for Cr 2X2Te6(X = Si and Ge)\nmeasured in in-plane and out-of-plane field of H= 10 kOe.\n(b) Field dependence of magnetization measured at T= 2 K.\ncorrelation in Cr 2X2Te6(X = Si and Ge) above Tc. The\neffective magnetic moment µeff≈√\n8Cis also listed in\nTable I. The values are close to the theoretical value ex-\npectedforCr3+of3.87µB. Theisothermalmagnetization\natT= 2 K is shown in Fig. 2(b). We estimate the satu-\nration magnetization Msfrom the intercept of a linear fit\nofM(H) at high field and the saturation field Hsas the\npoint of deviation from the linear behavior. The derived\nMs≈2.86(1)µB/Crwithout-of-planefieldforCr 2Si2Te6\nis larger than that of 2.40(2) µB/Cr for Cr 2Ge2Te6. The\nsaturation field Hs≈3 kOe with out-of-plane field is\nsmaller than Hs≈5 kOe with in-plane field and is much\nsmaller than that of 25 kOe for Cr 2Si2Te6. These results\nconfirm the easy c-axis and smaller magnetic anisotropy\nin Cr2Ge2Te6, in agreement with previous reports.9–123\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s49/s48/s50/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s49/s48/s50/s48/s67/s114\n/s50/s83/s105\n/s50/s84/s101\n/s54\n/s72/s32/s47/s47/s32/s99\n/s56/s48/s32/s75/s53/s32/s75/s40/s98/s41/s77/s32/s40/s101/s109/s117/s32/s103/s45/s49\n/s41\n/s72/s32/s40/s107/s79/s101/s41/s77/s32/s40/s101/s109/s117/s32/s103/s45/s49\n/s41/s40/s97/s41/s53/s32/s75\n/s56/s48/s32/s75/s72/s32/s47/s47/s32/s97/s98/s67/s114\n/s50/s83/s105\n/s50/s84/s101\n/s54/s67/s114\n/s50/s71/s101\n/s50/s84/s101\n/s54/s40/s99/s41\n/s52/s52/s32/s75\n/s56/s54/s32/s75\n/s72/s32/s47/s47/s32/s97/s98\n/s67/s114\n/s50/s71/s101\n/s50/s84/s101\n/s54\n/s72/s32/s47/s47/s32/s99/s56/s54/s32/s75/s52/s52/s32/s75/s40/s100/s41\n/s72/s32/s40/s107/s79/s101/s41\nFIG. 3. (Color online) Typical initial isothermal magnetiz a-\ntion curves measured in (a,c) H//aband (b,d) H//caround\nTcfor Cr 2X2Te6(X = Si and Ge).\nB. Magnetocrystalline anisotropy\nFigure 3 shows the magnetization isotherms with field\nup to 50 kOe applied for both H//abandH//caround\nTcfor Cr 2X2Te6(X = Si and Ge). When H//ab, the\nsaturationfield Hsis associatedwith the uniaxialmagne-\ntocrystallineanisotropyparameter Kuandthesaturation\nmagnetization Ms, i.e., 2Ku/Ms=µ0Hs, whereµ0is the\nvacuum permeability.19The temperature dependence of\nKuas well as MsandHsfor Cr 2X2Te6(X = Si and Ge)\nare depicted in Fig. 4. The calculated Kufor Cr 2Si2Te6\nis about 61 kJ/m3at 5 K. It gradually decreases to 38\nkJ/m3atTc= 32 K, comparable with the Kuvalues in\nCrBr3.20The anisotropyparameter Kuis much lower for\nCr2Ge2Te6: about 12kJ/m3at 44K and 5.6kJ/m3atTc\n= 63 K. For clarity, only the Kuvalues from Tcto 44 K\narepresented forCr 2Ge2Te6. TheKuof20kJ/m3at 2 K\nfor Cr 2Ge2Te6estimated from Fig. 2(b) is much smaller\nthan that of 65 kJ/m3for Cr 2Si2Te6, in line with the\nmagnetization data. The observed decrease of Kuwith\nincreasing temperature arises solely from a large num-\nber of local spin clusters fluctuating randomly around\nthe macroscopic magnetization vector and activated by\na nonzero thermal energy, whereas the anisotropy con-\nstants are temperature-independent.21,22This provides/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48/s75\n/s117/s32/s40/s107/s74/s32/s109/s45/s51\n/s41\n/s84/s32/s40/s75/s41/s32/s75\n/s117/s45/s67/s114\n/s50/s83/s105\n/s50/s84/s101\n/s54\n/s32/s75\n/s117/s45/s67/s114\n/s50/s71/s101\n/s50/s84/s101\n/s54/s72\n/s115/s32/s40/s84/s41\n/s84/s32/s40/s75/s41/s32/s72\n/s115/s45/s67/s114\n/s50/s83/s105\n/s50/s84/s101\n/s54\n/s32/s72\n/s115/s45/s67/s114\n/s50/s71/s101\n/s50/s84/s101\n/s54\n/s32/s77\n/s115/s45/s67/s114\n/s50/s83/s105\n/s50/s84/s101\n/s54\n/s32/s77\n/s115/s45/s67/s114\n/s50/s71/s101\n/s50/s84/s101\n/s54/s77\n/s115/s32/s40/s101/s109/s117/s32/s103/s45/s49\n/s41\n/s84/s32/s40/s75/s41\nFIG. 4. (Color online) Temperature dependence of the calcu-\nlated anisotropy constant Ku, the estimated saturation field\nHsand the saturation magnetization Ms(insets) below Tcfor\nCr2X2Te6(X = Si and Ge).\ninsight into the understanding of FM in few-layers of\nCr2X2Te6(X = Si and Ge). While in pure 2D system no\nlong-range magnetic order is expected,7mechanical cor-\nrugationsand magnetic anisotropyare possible pathways\nto establish magnetism in few-layers samples.\nC. Magnetic entropy change\nWe estimate the magnetic entropy change\n∆SM(T,H) =/integraldisplayH\n0/parenleftbigg∂S\n∂H/parenrightbigg\nTdH=/integraldisplayH\n0/parenleftbigg∂M\n∂T/parenrightbigg\nHdH,\n(1)\nwhere/parenleftbig∂S\n∂H/parenrightbig\nT=/parenleftbig∂M\n∂T/parenrightbig\nHis based on Maxwell’s relation.\nIn the case of magnetization measured at small discrete\nfield and temperature intervals [Fig. 3], ∆ SMcan be\napproximated:\n∆SM(Ti,H) =/integraltextH\n0M(Ti,H)dH−/integraltextH\n0M(Ti+1,H)dH\nTi−Ti+1.\n(2)\nFigures 5(a) and 5(b) present the calculated\n−∆SM(T,H) as a function of temperature with in-\nplane and out-of-plane fields. All the −∆SM(T,H)\ncurves show a pronounced peak at Tc, and the peak\nbroads asymmetrically on both sides with increasing\nfield. The maximum value of −∆SMis 4.9 J kg−1K−1\nfor Cr 2Si2Te6and 2.6 J kg−1K−1for Cr 2Ge2Te6with\nin-plane field change of 5 T. These slightly increase to\n5.05 and 2.64 J kg−1K−1, respectively, with out-of-plane\nfield change of 5 T. The obtained −∆SMvalues are com-\nparable and larger than that of Fe 3GeTe2and CrI 3.16,17\nThe rotational magnetic entropy change ∆ SR\nMis cal-\nculated as ∆ SR\nM(T,H) = ∆SM(T,Hc)−∆SM(T,Hab).4\no\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s49/s50/s51/s52/s53\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s49/s50/s51/s52/s53/s54/s48 /s56/s48\n/s54/s48 /s56/s48\n/s54/s48 /s56/s48/s32/s67/s114\n/s50/s83/s105\n/s50/s84/s101\n/s54/s45 /s83\n/s77/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41/s32/s49/s32/s84\n/s32/s50/s32/s84\n/s32/s51/s32/s84\n/s32/s52/s32/s84\n/s32/s53/s32/s84/s40/s97/s41\n/s72/s32/s47/s47/s32/s97/s98\n/s32/s67/s114\n/s50/s83/s105\n/s50/s84/s101\n/s54/s32/s49/s32/s84\n/s32/s50/s32/s84\n/s32/s51/s32/s84\n/s32/s52/s32/s84\n/s32/s53/s32/s84/s45 /s83/s82 /s77\n/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s40/s99/s41/s32/s67/s114\n/s50/s83/s105\n/s50/s84/s101\n/s54/s32/s49/s32/s84\n/s32/s50/s32/s84\n/s32/s51/s32/s84\n/s32/s52/s32/s84\n/s32/s53/s32/s84/s40/s98/s41/s45 /s83\n/s77/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41\n/s72/s32/s47/s47/s32/s99/s32/s67/s114\n/s50/s71/s101\n/s50/s84/s101\n/s54\n/s32/s67/s114\n/s50/s71/s101\n/s50/s84/s101\n/s54\n/s32/s67/s114\n/s50/s71/s101\n/s50/s84/s101\n/s54\nFIG. 5. (Color online) Temperature dependence of isother-\nmal magnetic entropy change −∆SMobtained from magne-\ntization at various magnetic fields change (a) in the abplane\nand (b) along the caxis, respectively, for Cr 2X2Te6(X = Si\nand Ge). (c) Temperature dependence of −∆SR\nMobtained by\nrotating from the abplane to the caxis in various fields.\nAs shown in Fig. 5(c), the value of ∆ SR\nMfor Cr 2Si2Te6\nis larger than that for Cr 2Ge2Te6, in line with the\ncalculated Ku[Fig. 4]. The anisotropy is gradually\nsuppressed in higher field, and interestingly, it splits into\ntwo peaks on both sides of Tcwith field above 3 T for\nCr2Si2Te6.\nD. Critical behavior\nAccording to the scaling hypothesis, the second-order\nphase transition around Tccan be characterized by a set\nof interrelated critical exponents and magnetic equation\nof state.24The exponents βandγcan be obtained from\nspontaneous magnetization Mspand inverse initial sus-\nceptibility χ−1\n0, below and above Tc, respectively, while δ\nis a critical isotherm exponent at Tc. The mathematical/s51/s46/s50 /s52/s46/s48 /s52/s46/s56/s49/s46/s50/s49/s46/s52/s50/s48 /s50/s52 /s50/s56 /s51/s50 /s51/s54 /s52/s48\n/s84\n/s99/s32/s61/s32/s51/s50/s46/s50/s40/s51/s41/s32\n/s32/s61/s32/s49/s46/s51/s51/s40/s56/s41/s84\n/s99/s32/s61/s32/s51/s49/s46/s56/s40/s49/s41/s32\n/s32/s61/s32/s48/s46/s49/s54/s57/s40/s52/s41\n/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s77\n/s115/s32/s40 /s47/s67/s114/s41\n/s50/s48 /s50/s52 /s50/s56 /s51/s50 /s51/s54 /s52/s48/s45/s54/s48/s45/s52/s48/s45/s50/s48/s48\n/s84\n/s99/s32/s61/s32/s54/s50/s46/s56/s40/s49/s41/s32\n/s32/s61/s32/s49/s46/s50/s56/s40/s51/s41\n/s84/s32/s40/s75/s41/s77\n/s115/s112/s40/s100/s77\n/s115/s112/s47/s100/s84/s41/s45/s49\n/s32/s40/s75/s41\n/s84\n/s99/s32/s61/s32/s51/s50/s46/s50/s40/s52/s41/s32\n/s32/s61/s32/s48/s46/s49/s55/s56/s40/s57/s41/s48/s51/s54/s57/s49/s50\n/s84\n/s99/s32/s61/s32/s54/s50/s46/s55/s40/s49/s41/s32\n/s32/s61/s32/s49/s46/s51/s50/s40/s53/s41/s84\n/s99/s32/s61/s32/s54/s50/s46/s54/s40/s49/s41/s32\n/s32/s61/s32/s48/s46/s49/s57/s54/s40/s51/s41/s40/s97/s41\n/s40/s107/s79/s101/s47 /s41\n/s53/s54 /s53/s56 /s54/s48 /s54/s50 /s54/s52 /s54/s54 /s54/s56\n/s84\n/s99/s32/s61/s32/s51/s50/s46/s49/s40/s50/s41/s32\n/s32/s61/s32/s49/s46/s51/s50/s40/s52/s41/s84\n/s99/s32/s61/s32/s54/s50/s46/s55/s40/s49/s41/s32\n/s32/s61/s32/s48/s46/s50/s48/s48/s40/s51/s41\n/s48/s50/s52/s54/s56\n/s40/s100 /s47/s100/s84/s41/s45/s49\n/s32/s40/s75/s41/s40/s98/s41/s32/s61/s32/s55/s46/s55/s51/s40/s50/s41/s108/s111/s103/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s108/s111/s103/s72/s32/s40/s79/s101/s41/s32/s51/s50/s32/s75\n/s32/s54/s51/s32/s75\n/s32/s61/s32/s57/s46/s50/s56/s40/s51/s41\nFIG. 6. (Color online) (a) Temperature dependence of the\nspontaneous magnetization Msp(left axis) and the inverse\ninitial susceptibility χ−1\n0(right axis) in out-of-plane field with\nsolid fitting curves for Cr 2X2Te6(X = Si and Ge). Inset\nshows log Mvs logHcollected at Tcwith linear fitting curves.\n(b) Kouvel-Fisher plots of Msp(dMsp/dT)−1(left axis) and\nχ−1\n0(dχ−1\n0/dT)−1(right axis) with solid fitting curves.\ndefinitionsoftheexponentsfrommagnetizationmeasure-\nment are given below:\nMsp(T) =M0(−ε)β,ε <0,T < T c,(3)\nχ−1\n0(T) = (h0/m0)εγ,ε >0,T > T c,(4)\nM=DH1/δ,T=Tc, (5)\nwhereε= (T−Tc)/Tcis the reduced temperature, and\nM0,h0/m0andDare the critical amplitudes.25\nThecriticalexponents β,γ, andδ, aswellastheprecise\nTccan be obtained by the modified Arrott plot of M1/β\nvs (H/M)1/γin the vicinity of Tcwith a self-consistent\nmethod.26,27This gives χ−1\n0(T) andMsp(T) as the inter-\ncepts on the H/Maxis and the positive M2axis, respec-\ntively. Figure 6(a) presents the final Msp(T) andχ−1\n0(T)\nas a function of temperature. According to Eqs. (3) and\n(4), the criticalexponents β= 0.169(4)with Tc= 31.8(1)\nK [β= 0.196(3) with Tc= 62.6(1) K], and γ= 1.33(8)5\nTABLE II. Critical exponents of Cr 2X2Te6(X = Si and Ge).\nThe MAP, KFP and CI represent the modified Arrott plot,\nthe Kouvel-Fisher plot and the critical isotherm, respecti vely.\nβ γ δ n m\nX = Si\n−∆Smax\nM 0.52(2)\nRCP 1.09(1)\nMAP 0.169(4) 1.33(8) 8.9(3) 0.45(3) 1.112(4)\nKFP 0.178(9) 1.32(4) 8.4(2) 0.45(3) 1.119(3)\nCI 9.28(3) 1.108(1)\nX = Ge\n−∆Smax\nM 0.51(1)\nRCP 1.13(1)\nMAP 0.196(3) 1.32(5) 7.7(2) 0.47(2) 1.130(3)\nKFP 0.200(3) 1.28(3) 7.4(1) 0.46(1) 1.135(2)\nCI 7.73(2) 1.129(1)\nwithTc= 32.2(3) K [γ= 1.32(5) with Tc= 62.7(1) K],\nare obtained for Cr 2Si2Te6[Cr2Ge2Te6].\nBased on the Kouvel-Fisher (KF) relation:28\nMsp(T)[dMsp(T)/dT]−1= (T−Tc)/β, (6)\nχ−1\n0(T)[dχ−1\n0(T)/dT]−1= (T−Tc)/γ. (7)\nLinearfittingtotheplotsof Msp(T)[dMsp(T)/dT]−1vsT\nandχ−1\n0(T)[dχ−1\n0(T)/dT]−1vsT, as shown in Fig. 6(b),\nyieldβ= 0.178(9) with Tc= 32.2(4) K [β= 0.200(3)\nwithTc= 62.7(1) K], and γ= 1.32(4) with Tc= 32.1(2)\nK [γ= 1.28(3) with Tc= 62.8(1) K]. The third expo-\nnentδcan be calculated from the Widom scaling relation\nδ= 1 +γ/β. From βandγobtained with the modi-\nfied Arrott plot and the Kouvel-Fisher plot, δ= 8.9(3)\nand 8.4(2) [7.7(2) and 7.4(1)] for Cr 2Si2Te6[Cr2Ge2Te6],\nwhich are close to the direct fits of δtaking into account\nthatM=DH1/δatTc[δ= 9.28(3) at 32 K and 7.73(2)\nat 63 K, inset in Fig. 6(a)]. The critical exponents of\nCr2X2Te6(X = Si and Ge) are summarized in Table II.\nThey are close to but not identical to values expected\nfor the 2D-Ising model ( β= 0.125, γ= 1.75 and δ=\n15). This deviation is most likely associated with non-\nnegligible interlayer coupling and spin-lattice coupling in\nthis system.11,13\nE. Scaling analysis of the ∆SMdata\nFor a material displaying a second-order transition,29\nthe field dependence of the maximum magnetic entropy\nchange shows a power law −∆Smax\nM=aHn,30wherea\nis a constant and the exponent natTcis related to the\ncritical exponents as n(Tc) = 1 + ( β−1)/(β+γ). An-\nother important parameter is the relative cooling power\n(RCP):RCP=−∆Smax\nM×δTFWHMwhere−∆Smax\nM/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48/s32/s67/s114\n/s50/s83/s105\n/s50/s84/s101\n/s54/s32/s32 /s32/s67/s114\n/s50/s71/s101\n/s50/s84/s101\n/s54\n/s72/s32/s47/s47/s32/s99\n/s32/s49/s32/s84\n/s32/s50/s32/s84\n/s32/s51/s32/s84\n/s32/s52/s32/s84\n/s32/s53/s32/s84/s110\n/s84/s32/s40/s75/s41/s40/s98/s41/s48/s52/s48/s56/s48/s49/s50/s48/s49/s54/s48\n/s109/s32/s61/s32/s49/s46/s49/s51/s40/s49/s41/s109/s32/s61/s32/s49/s46/s48/s57/s40/s49/s41\n/s110/s32/s61/s32/s48/s46/s53/s50/s40/s50/s41\n/s32/s82/s67/s80\n/s32/s82/s67/s80 /s61/s32/s98/s72/s109\n/s82/s67/s80/s32/s40/s74/s32/s107/s103/s45/s49\n/s41\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s49/s50/s51/s52/s53\n/s72/s32/s47/s47/s32/s99/s40/s97/s41\n/s32/s45 /s83/s109/s97/s120\n/s77\n/s32/s45 /s83/s109/s97/s120\n/s77/s32/s61/s32/s97/s72/s110\n/s72/s32/s40/s84/s41/s45 /s83/s109/s97/s120 /s77\n/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41\n/s110/s32/s61/s32/s48/s46/s53/s49/s40/s49/s41\nFIG. 7. (Color online) (a) Field dependence of the maxi-\nmummagnetic entropychange −∆Smax\nMand therelative cool-\ning power RCPwith power law fitting in red solid lines for\nCr2X2Te6(X = Si and Ge). (b) Temperature dependence of\nnin various fields.\nis the maximum entropy change near TcandδTFWHM\nis the full-width at half maximum.31The RCP also de-\npendsonthemagneticfieldwith RCP=bHm, wherebis\na constant and mis associated with the critical exponent\nδ,m= 1+1/δ.\nFigure 7(a) presents the summary of the out-of-plane\nfield dependence of −∆Smax\nMand RCP. The calculated\nvalues of RCP areabout 114and 87 J kg−1for Cr2Si2Te6\nand Cr 2Ge2Te6, respectively, with out-of-plane field\nchange of 5 T. Fitting of the −∆Smax\nMgives that n=\n0.52(2) and 0.51(1) for Cr 2Si2Te6and Cr 2Ge2Te6, re-\nspectively [Fig. 7(a)], which is quite close to that of\nn= 0.53 for the 2D-Ising model ( β= 0.125,γ= 1.75).\nFitting of the RCP generates m= 1.09(1) and 1.13(1)\n[Fig. 7(a)], which is also close to the expected value of\n1.07 for 2D-Ising model ( δ= 15). Figure 7(b) displays\nthe temperaturedependence of n(T) in variousfields. All\nn(T) curves follow an universal behavior.32At low tem-\nperatures, well below Tc,nhas a value around 1. On the\nother side, well above Tc,nis close to 2 as a consequence6\nFIG. 8. (Color online) The normalized ∆ SMas a function of\nthe reduced temperature θwith (a) out-of-plane and (b) in-\nplane field for Cr 2X2Te6(X = Si and Ge). Scaling plot for (c)\nCr2Si2Te6and (d) Cr 2Ge2Te6based on the critical exponents\nβandγobtained in out-of-plane field.\nof the Curie-Weiss law. At T=Tc,n(T) has a minimum.\nScaling analysis of −∆SMcan be built by normalizing\nall the−∆SMcurves against the respective maximum\n−∆Smax\nM, namely, ∆ SM/∆Smax\nMbyrescalingthereduced\ntemperature θ±as defined in the following equations,33\nθ−= (Tpeak−T)/(Tr1−Tpeak),T < T peak,(8)\nθ+= (T−Tpeak)/(Tr2−Tpeak),T > T peak,(9)\nwhereTr1andTr2are the temperatures of two reference\npoints that corresponds to ∆ SM(Tr1,Tr2) =1\n2∆Smax\nM.\nFollowing this method, all the −∆SM(T,H) curves invarious fields collapse into a single curve in the vicinity\nofTcfor Cr 2X2Te6(X = Si and Ge), as shown in Figs.\n8(a)and8(b). Thevaluesof Tr1andTr2depend on H1/∆\nwith ∆ = β+γ[inset in Fig. 8(a)].\nIn the phase transition region, the scaling analysis of\n−∆SMcan also be expressed as\n−∆SM\naM=Hnf(ε\nH1/∆), (10)\nwhereaM=T−1\ncAδ+1Bwith A and B representing the\ncritical amplitudes as in Msp(T) =A(−ε)βandH=\nBMδ, respectively, and f(x) is the scaling function.34\nIf the critical exponents are appropriately chosen, the\n−∆SMvsTcurvesshouldbe rescaledinto a singlecurve,\nconsistent with normalizing the −∆SMcurves with two\nreference temperatures ( Tr1andTr2). As shown in Figs.\n8(c) and 8(d), the rescaled −∆SMfor Cr 2X2Te6(X =\nSi and Ge) with out-of-plane field collapse onto a single\ncurve, confirming the reliable critical exponents and 2D\nIsing behavior for Cr 2X2Te6(X = Si and Ge).\nIV. CONCLUSIONS\nIn summary, we have studied the critical behavior and\nmagnetocaloric effect around the FM-PM transition in\nCr2X2Te6(X = Si and Ge) single crystals. 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Stauntona\naDepartment of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom\nbInstitute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan\nAbstract\nWe present a computational study of the compound Y(Co 1\u0000x\u0000yFexCuy)5for\n0\u0014x;y\u00140:2. This compound was chosen as a prototype for investigating\nthe cell boundary phase believed to play a key role in establishing the high\ncoercivity of commercial Sm-Co 2:17 magnets. Using density-functional theory,\nwe have calculated the magnetization and magnetocrystalline anisotropy at zero\ntemperature for a range of compositions, modeling the doped compounds within\nthe coherent potential approximation. We have also performed \fnite temper-\nature calculations for YCo 5, Y(Co 0:838Cu0:162)5and Y(Co 0:838Fe0:081Cu0:081)5\nwithin the disordered local moment picture. Our calculations \fnd that substi-\ntuting Co with small amounts of either Fe or Cu boosts the magnetocrystalline\nanisotropy K, but the change in Kdepends strongly on the location of the\ndopants. Furthermore, the calculations do not show a particularly large di\u000ber-\nence between the magnetic properties of Cu-rich Y(Co 0:838Cu0:162)5and equal\nFe-Cu Y(Co 0:838Fe0:081Cu0:081)5, despite these two compositions showing di\u000ber-\nent coercivity behavior when found in the cell boundary phase of 2:17 magnets.\nOur study lays the groundwork for studying the rare earth contribution to the\nanisotropy of Sm(Co 1\u0000x\u0000yFexCuy)5, and also shows how a small amount of\ntransition metal substitution can boost the anisotropy \feld of YCo 5.\nKeywords: Permanent magnets, Doping, Coercivity, Coherent potential\napproximation\n1. Introduction\nOf the wide variety of magnetic materials that can be formed by alloy-\ning rare-earth elements with transition metals (RE/TM) [1], the permanent\nmagnet market is dominated by those based either on Nd-Fe-B [2, 3] or Sm-\nCo [4, 5]. The performance of a permanent magnet is usually quanti\fed by\n\u0003Corresponding author\nEmail address: c.patrick.1@warwick.ac.uk (Christopher E. Patrick)\nPreprint submitted to Elsevier November 5, 2021arXiv:1901.04185v1  [cond-mat.mtrl-sci]  14 Jan 2019its maximum energy product ( BH)max, which measures the energy stored in\nthe air gap of the associated magnetic circuit [6]. At temperatures up to ap-\nproximately 120\u000eC, the Nd-Fe-B materials have the highest ( BH)maxof all\navailable permanent magnetic materials, but above this temperature their ex-\ncellent performance sharply diminishes [7]. By contrast, Sm-Co magnets do not\nhave the same dramatic sensitivity to temperature as Nd-Fe-B, showing superior\nperformance above 120\u000eC [7] and even operating at temperatures in excess of\n400\u000eC [8]. Sm-Co magnets are therefore the materials of choice for applications\nwhere high-temperature performance is critical, e.g. sensing in manufacturing\nprocesses [9].\nThe Sm-Co magnets can be further partitioned into the 1:5 and 2:17 classes\nbased on their nominal crystal structures, with the highest-performing magnets\nfalling into the 2:17 class [10]. As well as Sm and Co, commercial 2:17 magnets\nalso contain Fe, Cu and Zr at an approximate stoichiometry Sm(Co 1\u0000x\u0000y\u0000uFexCuyZru)z,\nwherez\u00187:5,x;y\u00180:1 andu\u00180:01 [11]. As illustrated by the value of z, the\n2:17 magnets also do not simply consist of a single Sm 2TM 17phase but rather\nadopt a multi-phase structure [12]. This structure consists of a cellular phase\ncomposed of 2:17 cells surrounded by thin ( \u001810 nm) cell boundaries with an\napproximate 1:5 stoichiometry, and a lamellar, Zr-rich \\Z\" phase [13].\nIt has long been accepted that this complex multi-phase structure is es-\nsential to maintaining the excellent high-temperature performance of the 2:17\nmagnets [10]. Recent work has highlighted the importance of the Z phase in\naiding the formation of the cellular phase [14]. The critical role played by the\ncellular phase is then revealed by electron microscopy experiments, which show\nthe pinning of magnetic domain walls at 2:17/1:5 boundaries [15]. This domain\nwall pinning inhibits magnetization reversal and thus provides a coercive force.\nOver the years a number of theories have been proposed to explain the\npinning of the domain walls [15{25]. According to micromagnetic theory [26],\nthe energy of a domain wall depends both on the strength of the exchange\ninteraction Aand the magnetic anisotropy Kas/p\nAK. Assuming that the\n2:17 cells and 1:5 cell boundary phases have di\u000berent AandK, there will be an\nenergy barrier associated with a domain wall moving between these regions [16].\nInterestingly, this argument does not rely on the domain wall energy being larger\nor smaller in the cell boundary phase compared to the cell [17]. In models based\non \\repulsive\" pinning, the domain wall energy is higher in the cell boundary\nphase, so the domain wall gets stuck in the cell [16, 18, 19], while in \\attractive\"\npinning models the domain walls have higher energies in the cell, so they get\npinned in the cell boundary phase instead [15, 20{22]. More recently, models\nhave been proposed where it is the variation of Kwithin the cell boundary\nregion that determines the coercivity [23{25].\nThe existence of di\u000berent models re\rect the complicated nature both of\n2:17 magnets and of coercivity in general. Indeed, the small size of the cell\nboundary phase and of the domain walls themselves already presents a challenge\nto continuum-based micromagnetics [20]. However, assuming micromagnetics\ncan be used to gain insight into the magnetization reversal process, a basic\nquestion is what values of A,Kand magnetic polarization Jshould be used as\n2input in the simulations. Focusing on the magnetic anisotropy K, based upon\nthe values measured for SmCo 5and Sm 2Co17one would expect a much larger\nanisotropy in the 1:5 cell boundary phase compared to the 2:17 cells. Indeed,\nthere are reports of measurements on a commercial 2:17 sample which support\nthis view [16]. However, other experimental studies concluded that the 1:5 cell\nboundary was actually softer (smaller anisotropy) than the cell [21]. Another\nstudy found similar anisotropy energies for the two phases, but explained the\npinning of domain walls in terms of a large di\u000berence in exchange energy A\nbetween the phases [20].\nOf course, a crucial property of the commercial magnets is the presence of the\nadditional elements Cu, Fe and Zr. A recent 3D atom probe study measured the\nchemical compositions of the cell boundary phase for 2:17 magnets showing both\nhigh and low coercivities, depending on heat treatment [25]. This study reported\nthat the high-coercivity sample coincided with an enhanced Cu and diminished\nFe content in the 1:5 cell boundary, as well as a sharp interface between the cell\nand cell boundary phases. Conversely, having a similar Fe and Cu content in\nthe 1:5 cell boundary, as well as having a di\u000busive interface between the cell and\ncell boundary phases, was correlated with low coercivity [25]. The Zr content\nwas found to be very small in both cases.\nIt would be desirable to be able to establish a link between chemical compo-\nsition and magnetic properties. In Ref. [25] it is pointed out that, although there\nis some data on binary Sm(Co,Cu) 5(e.g. Ref. [27]), there is a gap in the litera-\nture considering the ternary compound Sm(Co 1\u0000x\u0000yFexCuy)5. Indeed we note\nthat even the experimental data of Ref. [27] on Sm(Co 1\u0000yCuy)5only reports\nanisotropy \felds measured for y\u00150:24 (\u001520% by atom) which is already larger\nthan the\u001815% measured in the cell boundary phase in Ref. [25]. The modi\fca-\ntion of TM content by the substitution of Co with Cu and Fe can be expected\nto change the magnetic anisotropy of SmTM 5in a number of ways, namely: (a)\na\u000becting the single ion anisotropy of Sm by modifying the crystal \feld [28] (b)\na\u000becting the temperature dependence of this single ion anisotropy by modify-\ning the exchange \feld felt by the RE due to the RE/TM interaction [29] and\n(c) modifying the contribution of the TM-3 delectrons to the anisotropy[30{34].\nThe introduction of Fe and Cu can also be expected to a\u000bect the other key\nmicromagnetic parameters Aand polarization J, leading for example to mod-\ni\fed Curie temperatures [35]. Furthermore, changing the local environment of\nthe Sm atoms may even modify their valency and orbital hybridization, further\na\u000becting the anisotropy [36].\nAs a \frst step towards obtaining an improved understanding of how the mag-\nnetic properties of the cell boundary phase are a\u000bected by chemical composi-\ntion, we have performed \frst-principles (density-functional theory) calculations\non the ternary compound Y(Co 1\u0000x\u0000yFexCuy)5(Fig. 1). The reasons for \frst\ninvestigating Y rather than Sm are twofold: \frst, although Y has the same s2d\nvalence structure as Sm, the absence of 4 felectrons means that we can isolate\nthe Tm-3dcontribution to the anisotropy [point (c) above] from the single ion\ncontribution [(a) and (b)]. Second, YCo 5remains an interesting magnet in its\nown right, being free of lanthanide elements [37], having an anisotropy \feld of\n3Figure 1: Schematic representation of RETM 5crystal structure (space group 191, P6=mmm ),\nshowing the RE site (purple) and two inequivalent TM sites: the 2 cposition (in plane with\nRE, dark grey), and the 3 gposition (out of plane, light grey).\norder 20 T [38] and potentially having a coercivity comparable to traditional\nSmCo 5magnets [39]. Indeed, Fe-doped YCo 5magnets are the subject of active\nresearch as potential intermediate-performance permanent magnets [40].\nOur study consists of two parts. In the \frst part, we calculate the zero-\ntemperature properties (magnetization and magnetocrystalline anisotropy) of\nY(Co 1\u0000x\u0000yFexCuy)5for 0\u0014x;y\u00140:2. The studied compositions fall into the\nranges previously investigated in experiments on binary compounds [41, 42].\nThe chemical disorder is modelled within the coherent potential approxima-\ntion (CPA) [43]. In the second part, we concentrate on the compounds YCo 5,\nY(Co 0:838Cu0:162)5and Y(Co 0:838Fe0:081Cu0:081)5and calculate their \fnite tem-\nperature properties within the disordered local moment (DLM) picture [44].\nThese particular concentrations were chosen based on the experimentally-measured\ncompositions of the cell boundary phases of (Sm-Co) 2:17 magnets which showed\nhigh and low coercivity respectively in Ref. [25].\nThe current manuscript aims to address the gap in the literature concerning\nthe intrinsic properties of the ternary Y(Co 1\u0000x\u0000yFexCuy)5compound. From a\ntechnical aspect, due to the current lack of studies which have used the CPA\nto model the doping, the manuscript includes some technical discussion, such\nas a comparison of the CPA with the simpler \\rigid band\" approach. However,\nwe also aim to make a practical connection to Ref. [25] by reporting values of\nthe anisotropy \feld and micromagnetic parameters A,KandJfor the repre-\nsentative high and low coercivity cell boundary compositions. We hope that\nsuch parameters might be useful for future micromagnetics calculations like\nthose originally performed in Ref. [25]. In this way we follow recent works on\nRE/TM magnets which have demonstrated how microscopic quantities can be\nincorporated into large scale simulations [45, 46].\nInterestingly, the zero temperature calculations \fnd that a low level of sub-\nstitution enhances the magnetocrystalline anisotropy, regardless of whether Co\nis substituted with Fe or Cu. In particular, substituting \u001815% of Co yields\n4the largest anisotropy energies. The calculations also demonstrate how the\nanisotropy is very sensitive to the location of the dopant atoms.\nWe \fnd that the calculated di\u000berence between Cu-rich and equal Cu-Fe\nsubstitution is not particularly large. The current calculations therefore do not\nindicate that the TM-3 dcontribution to the anisotropy of the cell boundary\nphase is an important factor in determining the coercivity of the 2:17 magnets.\nNonetheless, our study lays the groundwork for studying the RE contribution\nto the anisotropy of Sm(Co 1\u0000x\u0000yFexCuy)5, and highlights the route of boosting\nthe anisotropy \feld of YCo 5through TM substitution.\nOur manuscript is organized as follows. In Section 2 we outline the methods\nused to calculate magnetic properties at zero and \fnite temperature. The cal-\nculated results are presented in Section 3 and analyzed in Section 4. We present\nour conclusions and discuss future directions for study in Section 5.\n2. Calculation details\nAll calculations were performed within the multiple-scattering, Korringa-\nKohn-Rostocker (KKR) formulation of density-functional theory (DFT) [47],\ntreating exchange and correlation e\u000bects within the local spin-density approxi-\nmation (LSDA) [48]. Scalar-relativistic calculations were performed within the\natomic sphere approximation (ASA) for the charge density and potential using\ntheHutsepot KKR code [49], solving the scattering problem up to a maximum\nangular momentum quantum number lmax= 3 and sampling the Brillouin zone\non a 20\u000220\u000220 grid. The Y-4 pelectrons were treated explicitly as valence\nstates. The self-consistent potentials were obtained for the T=0 K, ferromag-\nnetic arrangement of magnetic moments.\nSubstitutional doping of Co with Fe or Cu was modeled within the CPA [43,\n47]. For all compositions the lattice parameters were kept \fxed to the values\na,c, = 4.950, 3.986 \u0017A, as measured experimentally for YCo 5at 300 K [50].\nTo calculate magnetic properties, the \\frozen\" scalar-relativistic potentials\nwere inserted into the Kohn-Sham-Dirac equation in order to solve the fully-\nrelativistic scattering problem [51]. Spin and orbital magnetic moments were\ncalculated by tracing the appropriate operators with the Green's function [52].\nThe magnetocrystalline anisotropy energy was obtained via the torque, i.e. the\nchange in free energy on rotation of the magnetization vector [53, 54]. An\nadaptive algorithm for the Brillouin zone integration was used to ensure high\nnumerical precision [55].\nFinite temperature properties were calculated within the disordered local\nmoment picture, which treats the temperature-induced \ructuations of the local\nmoments at the level of the CPA [44]. The temperature-dependent Weiss \felds\nwere determined using an iterative procedure [34, 35]. Both an overview of\nDFT-DLM and the detailed procedure of evaluating the Weiss \felds and torque\ncan be found elsewhere, e.g. Ref. [54].\nPrevious computational studies on YCo 5found the LSDA to yield values of\nboth the orbital magnetic moments and the magnetocrystalline anisotropy which\n5are smaller than measured experimentally, but including an orbital polarization\ncorrection (OPC) [56] on the TM- dorbitals corrects this discrepancy [30{32].\nAlthough the relativistic DFT-DLM calculations do allow an OPC to be in-\ncluded [29, 57, 58], in the current work we do not do so due to the large number\nof (Fe,Cu) compositions considered. Therefore our calculated anisotropy ener-\ngies are underestimates compared to experiment. Test calculations for selected\ncompositions found the same qualitative trends to be obeyed by OPC and non-\nOPC calculations, but more work is necessary to perform a full comparison of\nthe two approaches.\n3. Results\n3.1. Anisotropy at zero temperature: rigid band model\nFor a crystal with hexagonal symmetry, the expected variation of the free\nenergy with magnetization angle \u0012isK1sin2\u0012+K2sin4\u0012+O(sin6\u0012), where\n\u0012is given with respect to the caxis. Evaluating the torque @F=@\u0012 at\u0012=\n45\u000eyieldsK1+K2, which we label K. Previous experimental and theoretical\nstudies [38, 58] have determined K2to be an order of magnitude smaller than K1\nin pristine YCo 5, soK\u0019K1. A positive value of Kcorresponds to out-of-plane\nanisotropy.\nThe most straightforward method of simulating the substitutional doping of\nCo with Fe or Cu is to use the rigid band approximation, i.e. simply shift the\nFermi level in the DFT calculation of pristine YCo 5so that the total number of\nelectrons in the unit cell matches that expected for the doped system. YCo 4Fe\ncorresponds to a change in electron number of \u0001 Ne=\u00001, while YCo 4Cu cor-\nresponds to \u0001 Ne= +2.\nThe anisotropy Kcalculated in this way is shown as the dotted line in\nFig. 2(a). As noted in previous works [30{32, 34] there is a pronounced de-\npendence of Kon the band \flling. As discussed at length in Ref. [31], the\nanisotropy energy originates from the splitting of otherwise degenerate states by\nthe spin-orbit interaction, with the strongest contributions coming from states\nwith energies close to the Fermi level. Shifting the Fermi level changes the\nweights of the contribution of each state, which may overall lead to an increase\nor decrease in K.\nFrom the shape of the curve in Fig. 2(a) we see that the rigid band model\npredicts that adding Fe would increase K, up to a maximum close to YCo 4Fe.\nBy contrast, adding instead a small amount of Cu to form YCo 4:75Cu0:25would\nreduceKto zero and yield a perfectly soft magnet. Increasing the Cu content\n(e.g. YCo 4Cu) would again result in an enhanced K1compared to the pristine\ncase.\n3.2. Anisotropy at zero temperature: CPA, non-preferential substitution\nWe now consider modeling the doping within the coherent potential approx-\nimation (CPA), a more sophisticated approach than the rigid band model [43].\nIn these calculations it is necessary to specify the location of the dopants, i.e.\n6(a)\n(b)\n0.0 0.1 0.2 0.3 0.40.00.51.0\nx+yK(meV/FU)\n0.00\n0.05\n0.10\n0.15\n0.20x y-1.0 0.0 1.0 2.00.00.51.01.5\n∆NeK(meV/FU)\n0.00\n0.05\n0.10\n0.15\n0.20x yFigure 2: (a) Anisotropy energy Kper formula unit (FU) as a function of change in electron\nnumberNe, calculated for Y(Co 1\u0000x\u0000yFexCuy)5either in the rigid band approximation (black\ndotted line) or with the CPA (crosses). Each cross lies on the intersection of a solid ( x) and\ndashed (y) line which allows the composition to be deduced. (b) The same CPA calculations\nas (a) replotted as a function of dopant content x+y.\n7either the 2 cor 3gcrystallographic sites (Fig. 1). Here we choose that Fe and\nCu occupy 2 cand 3gsites with equal probability, i.e. non-preferential substitu-\ntion, and explore the case that di\u000berent sites are preferred by di\u000berent dopants\nin Section 3.5.\nWe have calculated Kfor Y(Co 1\u0000x\u0000yFexCuy)5within the CPA for x;y=\n[0:00;0:05;0:10;0:15;0:20]. The data are shown in Fig. 2 as crosses. The com-\nposition of each data point can be deduced by noting each cross lies at the\nintersection of a solid and dashed line. For instance, the composition with the\nhighestK, Y(Co 0:85Fe0:15)5, lies at the intersection of the x= 0:15 (blue solid)\nandy= 0:00 (red dashed) lines. The change in electron number \u0001 Neis -0.75.\nComparing the CPA and rigid band calculations shows two key di\u000berences.\nFirst, the predicted variation in Kis smaller for the CPA case, with the CPA\nvalues occupying a range of 0.8 meV/FU compared to 1.7 meV/FU for the rigid\nband model. Second, according to the CPA the addition of dopants almost al-\nways increases K, with only Y(Co 0:65Fe0:20Cu0:15)5and Y(Co 0:60Fe0:20Cu0:20)5\nhaving a (slightly) reduced anisotropy energy compared to the pristine case.\nTherefore, the rigid band and CPA calculations strongly disagree regarding the\ne\u000bect of, for instance, adding a small amount of Cu. Another example of the dis-\nagreement between the two models is seen in con\fgurations with the same num-\nber of electrons, like YCo 5, Y(Co 0:85Fe0:10Cu0:05)5and Y(Co 0:70Fe0:20Cu0:10)5.\nAccording to the rigid band model, Kcalculated for each of these compounds\nshould be the same, but in the CPA, Kvaries over a range of 0.5 meV/FU. In\nSection 4.2 we return to the comparison of the rigid band model and CPA.\nIt is interesting to replot the CPA data as a function of total dopant content,\nx+y, which is done in Fig. 2(b). In this case, the data for di\u000berent ratios of\nFe and Cu doping follow a more general trend, which is an increase in Kup to\na maximum for x+y= 0.15. Plotting the data in this way indicates that the\nsize of the Co de\fcit is an important contributor to the variation in anisotropy.\nFurthermore, replacing Co with Fe rather than Cu yields the largest Kvalues,\ni.e. the compositions with Cu content y= 0.00.\n3.3. Magnetization and anisotropy \feld at zero temperature: CPA, non-preferential\nsubstitution\nAccording to micromagnetic theory, the theoretical maximum for the co-\nercive \feld of a ferromagnet is the anisotropy \feld, \u00160Ha= 2K=M [59, 60].\nTherefore, it is also important to investigate the dependence of the magnetiza-\ntionMon doping, which is shown in Fig. 3(a).\nIn this case, there is quite close agreement between the rigid band model\n(black dotted line) and the CPA calculations. The moment calculated for pris-\ntine YCo 5is 8.3\u0016B/FU, where \u0016Bis the Bohr magneton. Adding electrons\nto YCo 5(Cu-doping) increases the population of the minority spin band, and\ntherefore decreases the magnetization. The reverse applies when electrons are\nremoved (Fe-doping). We note that, although the agreement is generally good,\nthe magnetization decreases faster with Cu doping than predicted by the rigid\nband model. For instance, for YCo 4Cu the magnetization calculated with the\nCPA is 6.1 \u0016B/FU, compared to the rigid band prediction of 6.6 \u0016B/FU.\n8(a)\n(b)-1.0 0.0 1.0 2.06.07.08.09.010.0\n∆NeM(µ B/FU)\n0.0 0.1 0.2 0.3 0.41.02.03.04.0\nx+yµ0Ha(T)\n0.00\n0.05\n0.10\n0.15\n0.20x y\n0.00\n0.05\n0.10\n0.15\n0.20x yFigure 3: (a) Zero-temperature magnetization Mcalculated as a function of electron number\nwithin the rigid band model (black dotted line) or using the CPA. (b) Anisotropy \feld \u00160Ha\n(= 2K=M ) obtained in the CPA, plotted as a function of dopant content x+y.\n90.0 0.05 0.10.51.0\nyK(meV/FU)\nCu-doped 2c\nNo pref.\n3g0.0 0.05 0.10.51.0\nxK(meV/FU)\nFe-doped\n2cNo pref.3gFigure 4: Anisotropy energies of Y(Co 1\u0000xFex)5and Y(Co 1\u0000yCuy)5calculated with no pref-\nerential (pref.) substitution at particular sites, compared to exclusive substitution at the 2 c\nor 3gcrystal sites.\nFigure 3(b) shows the anisotropy \feld calculated from the anisotropy en-\nergies and magnetizations in Figs. 2(b) and 3(a), respectively. Comparing to\nFig. 2(b), we \fnd a smaller scatter in Hacompared to Kfor di\u000berent composi-\ntions. The reason for this smaller scatter is that the increased Kfrom doping\nwith larger amounts of Fe is partly o\u000bset by the increased M; similarly, in-\ncreasing the amount of Cu weakens Mand therefore helps to boost Ha. The\nclearest example is YCo 4Cu, whose value of Kis smaller than YCo 4Fe by a fac-\ntor of 1.4 (0.61 vs. 0.86 meV/FU), but whose anisotropy \feld is actually larger\n(3.4 vs. 3.3 T). Nonetheless, the purely Fe-doped Y(Co 0:85Fe0:15)5has both the\nhighest anisotropy energy (0.93 meV/FU) and anisotropy \feld (3.6 T).\n3.4. Dependence of anisotropy on site occupation\nAs stated already, the CPA calculations presented above were performed as-\nsuming both Cu and Fe substitute onto the 2 cand 3gsites with equal probability.\nOn the other hand, it is possible that the dopants may prefer to substitute at\nparticular sites, and that the value of Kmight be a\u000bected. Therefore, in Fig. 4\nwe compare the previous calculations of the anisotropy energy of Y(Co 1\u0000xFex)5\nand Y(Co 1\u0000yCuy)5with the case where the dopants are placed exclusively at\nthe 2cor 3gcrystal sites (Fig. 1).\n10Interestingly, there is indeed a strong dependence of Kon the location of\nthe dopants, with the site that gives the largest enhancement to the anisotropy\ndepending on the dopant. Substituting Co with Cu at the 3 gsite has essentially\nno e\u000bect on the anisotropy energy, while substituting with Fe at the 2 csite also\ndoes not result in a large change. By contrast, placing Cu at the 2 csite or Fe\nat the 3gsite has a large e\u000bect on K. Substituting the dopants equally at both\nsites e\u000bectively interpolates between the two limiting cases.\nWe are unaware of experimental data directly characterizing the location\nof dopant atoms in Y(Co 1\u0000xFex)5or Y(Co 1\u0000yCuy)5. Neutron di\u000braction mea-\nsurements on the related compounds Th(Co 1\u0000xFex)5and Y(Co 1\u0000yNiy)5found\npreferential Fe/Ni substitution at the 3 g/2csites, respectively [61, 62]. However\nbased on the changes in lattice parameters in Y(Co 1\u0000xFex)5with Fe doping it\nwas argued in Ref. [63] that Fe preferred to occupy 2 csites. Our own CPA\ncalculations performed at zero temperature (ferromagnetic state) \fnd 2 csubsti-\ntution to be preferred for Fe, Cu, and Ni [35] but calculations on stoichiometric\nsystems including optimization of the lattice parameters found Fe to prefer 3 g\ndoping and Cu to prefer 2 c[64]. Furthermore our test calculations and previ-\nous work [65] have shown the energetics to be sensitive to the modeling of the\nmagnetic state (ferromagnetic versus nonmagnetic). For de\fniteness, in what\nfollows we place Fe at 3 gsites and Cu at 2 csites in line with the neutron\ndata [61, 62], the calculations including geometry optimization [64] and with\nprevious theoretical works [66, 67].\n3.5. Anisotropy at zero temperature: CPA, preferential 3 g/2csubstitution\nIn Fig. 5(a) we plot Kcalculated for Y(Co 1\u0000x\u0000yFexCuy)5with the CPA,\nplacing Fe at the 3 gsites and Cu at the 2 csites. For comparison we show again\nthe band-\flling behavior predicted by the rigid band approximation (Sec. 3.1).\nReferring to the data shown in Fig. 2(a), we see that the anisotropy en-\nergy is enhanced compared to non-preferential site substitution. The largest\nvalue ofK, calculated for Y(Co 0:85Fe0:09Cu0:06)5, is 1.18 meV/FU, whilst the\nsmallest value (apart from pristine YCo 5) is calculated to be 0.70 meV/FU for\nY(Co 0:79Cu0:21)5.\nAs with the calculations with non-preferential site substitution, the agree-\nment between the rigid band and CPA calculations is poor. For instance, it is\ninteresting to compare the sensitivity of Kto the level of Cu doping, at low\nand high Fe content. For Y(Co 1\u0000yCuy)5(red solid line), Kvaries between\n0.31 meV/FU at y= 0:00 to a peak value of 0.92 meV/FU at y= 0:09. How-\never, including Fe at the level Y(Co 0:79\u0000yFe0:21Cuy)5(purple solid line) reduces\nthe variation in Kto just 0.03 meV/FU over the entire range of y.\nReplotting the CPA calculations against x+y[Fig. 5(b)] does not show\nas clear a trend as for the non-preferential doping case [Fig. 2(b)]. However,\nagain it is found that the largest values of the anisotropy energy are calculated\nfor concentrations with x+y\u00180:15. Around this optimal level, the largest\nvalues ofKhave similar Fe and Cu concentrations. By contrast, the Cu-rich\nY(Co 0:85Cu0:15)5has a relatively low K. However, at the lowest dopant con-\n11(a)\n(b)-1.0 0.0 1.0 2.00.00.51.01.5\n∆NeK(meV/FU)\n0.0 0.1 0.2 0.3 0.40.00.51.0\nx+yK(meV/FU)\n0.00\n0.06\n0.09\n0.15\n0.21x y\n0.00\n0.06\n0.09\n0.15\n0.21x yFigure 5: (a) Anisotropy energy Kof Y(Co 1\u0000x\u0000yFexCuy)5calculated either in the rigid\nband approximation (black dotted line) or with the CPA (crosses) (cf. Fig. 2). The CPA\ncalculations were performed placing Fe (Cu) at 3 g(2c) sites. (b) The same CPA calculations\nas (a) replotted as a function of dopant content x+y.\n120.0 0.1 0.2 0.3 0.41.02.03.04.05.0\nx+yµ0Ha(T)\ny = x\ny = 5x0.00\n0.06\n0.09\n0.15\n0.21x yFigure 6: Anisotropy \felds calculated within the CPA placing Fe (Cu) at 3 g(2c) sites.\nAdditionally, calculations with equal Fe/Cu doping ( x=y) and Cu-rich ( y= 5x) doping are\nshown with thick blue and red lines, respectively.\ncentrations ( x;y) = 0:06 the e\u000bect of replacing Co with either Cu or Fe is very\nsimilar.\n3.6. Anisotropy \felds at zero temperature: Cu-rich vs. equal Cu/Fe content\nUnlike the anisotropy energy, the magnetization Mis not particularly sen-\nsitive on the location of the dopants, following the same behavior as shown in\nFig. 3(a). In Fig. 6 we plot the anisotropy \feld for preferential substitution\nof Fe/Cu at the 3 g/2csites. As for the non-preferential case [Fig. 3(b)] the\nanisotropy \feld is enhanced (weakened) for large Cu (Fe) content, due to e\u000bect\nof the dopants on M. This can be seen most clearly for the composition dis-\ncussed above, Y(Co 0:79\u0000yFe0:21Cuy)5, which has Ke\u000bectively independent of y\nbutHawhich increases with Cu content due to the corresponding reduction in\nM.\nMotivated by the observations made at the end of the previous section,\nFig. 6(b) shows some additional data points, calculated for equal Fe/Cu dop-\ning (x=y; thick red line) and Cu-rich doping, which we de\fne as y= 5x\n(thick blue line). Here, it can be seen that despite the boost to Haby the\nsmaller magnetization of Cu, the compositions with equal amounts of Fe and\nCu have larger anisotropy \felds than the Cu-rich compositions. Of all of the\ncompositions considered, Y(Co 0:82Fe0:09Cu0:09)5is found to have the highest\nanisotropy \feld of \u00160Ha= 5.3 T. The Cu-rich composition with the same x+y,\nY(Co 0:82Fe0:03Cu0:15)5, has\u00160Ha= 4.9 T.\n3.7. Anisotropy at \fnite temperature: YCo 5, Y(Co 0:838Cu0:162)5and Y(Co 0:838Fe0:081Cu0:081)5\nIn order to make a tentative connection to the 2:17 Sm-Co magnets, we now\nfocus on speci\fc compositions similar to those reported for the cell-boundary\n130 100 200 3000.00.51.0\nT(K)K(meV)\nY(Co Fe Cu )5 0.838 0.081 0.081\nY(Co Cu )5 0.838 0.162 YCo5Figure 7: Anisotropy energy Kcalculated for YCo 5(black circles), Y(Co 0:838Cu0:162)5(red\nsquares) and Y(Co 0:838Fe0:081Cu0:081)5(blue circles) as a function of temperature.\n(x;y)K(meV/FU) M(\u0016B/FU)\u00160Ha(T)TC(K)\n(0.000,0.000) 0.31 ; 0.12 8.14 ; 7.16 1.3 ; 0.6 841\n(0.000,0.162) 0.83 ; 0.57 6.42 ; 5.52 4.4 ; 3.6 711\n(0.081,0.081) 1.10 ; 0.80 7.63 ; 6.66 5.2 ; 4.1 788\nTable 1: Anisotropy energy K, magnetization M, anisotropy \feld Haand Curie temperature\nTCcalculated for Y(Co 1\u0000x\u0000yFexCuy)5. The two values given for K,MandHacorrespond\nto calculations at T= 0 and 300 K.\nphase in high and low coercivity samples in Ref. [25] and calculate their proper-\nties between 0{300 K. In Ref. [25], high coercivity was correlated with a Cu-rich\ncell-boundary phase, while low coercivity was correlated with equal Cu and Fe\ncontent. We model the two cases with compositions ( x,y) = (0.00,0.162) and\n(0.081,0.081). We also compare to pristine YCo 5. As in Secs. 3.5 and 3.6 we\nplace the Fe and Cu dopants at the 3 gand 2csites, respectively.\nFigure 7 shows the anisotropy energy Kcalculated as a function of tem-\nperatureT, for the three cases. As expected from Fig. 5, at zero temperature\nYCo 5has the lowest Kvalue of 0.31 meV/FU, followed by Y(Co 0:838Cu0:162)5\n(0.83 meV/FU) and then Y(Co 0:838Fe0:081Cu0:081)5(1.10 meV/FU). Up to at\nleastT= 300K, this ordering is unchanged, with each composition showing a\nmonotonic decrease in Kwith temperature with a similar slope. At 300 K, the\nKvalues for the three con\fgurations are 0.12, 0.57 and 0.80 meV/FU respec-\ntively.\nThe magnetization and anisotropy \feld (not shown) display the same mono-\ntonic decrease with temperature. We also calculated the Curie temperatures TC,\n\fnding the largest value (841 K) for pristine YCo 5. TheTCof Y(Co 0:838Cu0:162)5\nis found to be lower by over 100 K (711 K), while Y(Co 0:838Fe0:081Cu0:081)5lies\nin between (788 K). The temperature-dependent results are summarized in Ta-\nble 1.\n14(x;y)Km(MJ/m3)Jm(T)Am(pJ/m)\n(0.000,0.000) 0.59 ; 0.23 1.12 ; 0.99 8.3 ; 6.6\n(0.000,0.162) 1.57 ; 1.08 0.88 ; 0.76 7.0 ; 5.4\n(0.081,0.081) 2.08 ; 1.52 1.05 ; 0.92 7.7 ; 6.1\nTable 2: Anisotropy energy Km, magnetic polarization Jmand exchange sti\u000bness constant\nAmcalculated for Y(Co 1\u0000x\u0000yFexCuy)5as discussed in the text. The two values given corre-\nspond to calculations at T= 0 and 300 K.\n3.8. Deriving parameters for micromagnetics simulations\nThe quantities K,MandTClisted in Table 1 may be obtained directly\nfrom the DFT-DLM calculations. However micromagnetics simulations in fact\nrequire the magnetic polarization Jmand sti\u000bness constant Amin addition\nto the anisotropy Km[26]. Denoting the volume per formula unit as \n(=p\n3a2c=2), the anisotropy and polarization can be straightforwardly related to\nthe quantities given in Table 1:\nKm=K=\n (1)\nJm=\u00160M=\n (2)\nWe again note that the derived Kmvalues are likely to be underestimates since\nno orbital polarization correction terms were included (Sec. 2).\nWe have not yet established a formal framework to extract the exchange\nsti\u000bness constant Amfrom the DFT-DLM calculations. A similar challenge is\nencountered when performing calculations based on atomistic spin models [45].\nIn the current work we use a basic approximation [26, 27]:\nAm= (kBTC=\n)\u00102[mCo(T)]2(3)\nwhere\u0010is the nearest neighbor distance between transition metal atoms and\nmCo(T) is the calculated order parameter of the Co moments at temperature\nT. The results of using equations 1{3 to express the quantities in Table 1 as\nmicromagnetics parameters are shown in Table 2.\n3.9. Temperature in DFT-DLM calculations\nWe conclude this section by noting that the classical statistical mechanics\nused in the DLM picture leads to a faster decay of the the magnetic order param-\netermCo(T) with temperature than observed experimentally [68]. As a result,\nthe DFT-DLM \\temperature\" may in fact correspond to a higher temperature\nin experiment. To illustrate this aspect, we note from our DFT-DLM calcu-\nlations on YCo 5that at a calculated temperature of 300 K, mCohas a value\nof 0.90. However, the experimental data shown in Ref. [69] shows mCo= 0:90\nin fact corresponds to a temperature of 465 K. Conversely, at an experimental\ntemperature of 300 K the order parameter is 0.95 [69], which in the DFT-DLM\ncalculations corresponds to a temperature of 200 K. As illustrated here, it is\nrelatively straightforward to correct for this e\u000bect if required by mapping the\nDFT-DLM order parameters onto the experimental magnetization versus tem-\nperature curves [69].\n154. Discussion\nHere we discuss the speci\fc results presented in Sec. 3 in more general terms.\n4.1. Modelling doping\nWe \frst highlight some limitations of our chosen method of simulating the\ndoping. First, we have not attempted to include structural e\u000bects in our cal-\nculations, instead using the same lattice parameters for all concentrations. The\nanisotropy has previously been shown to be sensitive to the c/aratio [70], so\ndi\u000berent-sized dopants may a\u000bect Kindirectly in this way. In the context of un-\nderstanding the cell-boundary phase in the 2:17 magnets modeling this e\u000bect is\nespecially di\u000ecult, since there may be local strains which invalidate predictions\nbased on Vegard's law or total energy minimization [64].\nAlso, the CPA assumes the limit of a dilute alloy, with the dopants dis-\npersed homogeneously throughout the host structure. As a result, e\u000bects from\nshort-range ordering are not included [71], which may be especially important\nin understanding the low coercivity 2:17 Sm-Co sample which had a di\u000busive\ninterface between the cell and cell boundary phase [25].\nFinally, we note that we have assumed a perfect YCo 5structure and not ex-\nplored the role of point defects, e.g. the \\dumbbell\" substitution which replaces\nY with a pair of Co atoms [10]. It is possible that Fe and Cu may interact with\nthese defects in di\u000berent ways.\n4.2. Rigid band or CPA?\nWithin the limitations of the above model, we explored two methods of\nmodeling doping, namely the rigid band model or the CPA. While the two\nmodels produce similar values for the magnetization M, the predicted values of\nKdi\u000ber substantially. The rigid band model predicts much larger variations in\nthe anisotropy energy Kthan the CPA. Indeed the rigid band model predicts\nthat the addition of a small amount of Cu should reduce K, potentially yielding\na perfectly soft magnet for YCo 4:75Cu0:25. By contrast the CPA almost always\npredictsKto increase regardless of the dopant species, especially if there is\npreferential substitution at particular crystal sites.\nFrom the theoretical point of view, the CPA is the more rigorous approach [43].\nAs suggested by its name, the rigid band model cannot account for changes in\nthe bandstructure induced by the addition of dopants. Furthermore if we take\nthe view that the anisotropy depends not only on the dopant species but also on\nwhich site it occupies (as argued experimentally a number of decades ago [72]\nand observed in the CPA calculations, e.g. Fig. 4), we see that the rigid band\nmodel cannot provide a full account of the behavior of K.\n4.3. Comparison to experiment\nAs noted in Sec. 2, since the current calculations do not include a correction\nfor orbital polarization, we expect the calculated values of Kto be smaller than\nobserved experimentally. Therefore we restrict our comparison to trends in K\n16with doping. We have not found experimental data on the ternary compound\nY(Co 1\u0000x\u0000yFexCuy)5, but studies on the binaries Y(Co 1\u0000xFex)5[41, 63, 73] and\nY(Co 1\u0000yCuy)5[42] are available.\nConsidering Fe-doping \frst, Ref. [63] reports anisotropy constants at room\ntemperature for x= 0.0{0.3. Here Kincreases from a value of 4.5 MJm\u00003forx\n= 0.0 to a maximum of 5.5 MJm\u00003atx= 0.10, before reducing again. Interest-\ningly a follow-up work [41] states that there were calibration errors the original\ndata, even though later papers continued to cite the original reference [72]. A\nmore recent work [73] measured Kforx= 0.0{0.15 and found an increase from\n4.2 to 5.0 MJm\u00003, (again at room temperature), reasonably consistent with\nRef. [63].\nSo, for Fe doping there is at least qualitative agreement between di\u000berent\nexperiments and the current calculations in predicting an increased Kon the\naddition of a small amount of Fe. Quantitatively, the change of 0.8 MJm\u00003\nbetweenx= 0:0 andx= 0:15 found in Ref. [73] corresponds to an absolute\nincrease of approximately 0.4 meV/FU, or a relative increase of 19%. The (zero\ntemperature) calculations predict an absolute increase of 0.7 meV/FU over the\nsame composition range (Fig. 5), but a much larger relative increase of over\n300%.\nNow considering Cu doping, Ref. [42] reports anisotropy \felds for Y(Co 1\u0000yCuy)5\nfory= 0.0, 0.2 and 0.4, at temperatures T >200 K (lower temperature data are\nreported for y= 0:4). At the studied temperature range, YCo 5is reported to\nhave the highest anisotropy \feld. However the di\u000berence between the pristine\ncase andy= 0:2 (YCo 4Cu) gets smaller with decreasing temperature, and a\nstraightforward linear extrapolation of the data indicates the anisotropy \feld of\nYCo 4Cu would exceed YCo 5at temperatures below 50 K. We were unable to\n\fnd experimental data measured for lower Cu concentrations which could be\ncompared more directly to our calculations. Such data, particularly at very low\nCu concentration ( y\u00180:05) would be useful e.g. in comparing the rigid band\nmodel with the CPA, since the former predicts a perfectly soft magnet at this\nconcentration (Fig. 2).\nFinally, we note that although our calculations should ideally be compared\nto anisotropy constants measured for single crystals, there are a number of\nexperiments which report coercivity enhancement in RE/TM magnets upon\naddition of Fe or Cu [74{76].\n4.4. 2:17 Sm-Co magnets\nWe return to the original motivation of our work, to quantify the e\u000bect on\nthe TM contribution to the anisotropy with Fe and Cu doping. As shown\nby the blue and red lines in Figs. 6 and 7, and the numbers reported in\nTable 1 the di\u000berences between doping purely with Cu or with equal quan-\ntities of Cu and Fe are not particularly large. The anisotropy energy and\n\feld is calculated to be slightly smaller for Y(Co 0:838Cu0:162)5compared to\nY(Co 0:838Fe0:081Cu0:081)5. This fact, combined with a lower exchange sti\u000bness\nconstant for Y(Co 0:838Cu0:162)5(Table 2) gives a lower domain wall energy in\nthis phase compared to Y(Co 0:838Fe0:081Cu0:081)5[26].\n17Applying this observation to a repulsive pinning model (where the 2:17\ncells have smaller domain wall energies than the boundaries), one would expect\nstronger pinning for boundaries with the Y(Co 0:838Fe0:081Cu0:081)5composition.\nUnfortunately however, in Ref. [25] this composition (equal Fe and Cu content)\ncorresponded to the low, not high, coercivity sample. Therefore, according to\nour calculations the TM contribution to the anisotropy does not account for the\ncorrelation of coercivity with the composition of the cell boundaries reported in\nRef. [25]. Of course, as described in the Introduction the larger Sm contribution\nto the anisotropy may provide this missing link. In the \fnal section we outline\nfuture work aimed at exploring the 2:17 magnets further.\n5. Conclusions and outlook\nThe aim of this work has been to study how the magnetic anisotropy and\nmagnetization of YCo 5is a\u000bected when Co is substituted with Fe and/or Cu.\nWe have demonstrated how two di\u000berent approaches to modeling dopants | the\nrigid band model and the coherent potential approximation (CPA) | give rather\ndi\u000berent predictions. We also showed how the results of the CPA calculations\ndepend very strongly on which crystal site the dopant atoms occupy.\nOur CPA calculations found that the anisotropy of YCo 5could be enhanced\nby adding reasonably small amounts of Fe and/or Cu, with the largest anisotropy\n\feld at zero temperature observed for the composition Y(Co 0:81Fe0:09Cu0:09)5.\nOf the compositions studied, assuming preferential substitution of Fe and Cu at\n3gand 2csites respectively, Fe-rich samples (i.e. Y(Co 0:79\u0000yFe0:21Cuy)5) had\nthe lowest anisotropy \felds, but these still exceeded the anisotropy \feld of YCo 5\nat zero temperature.\nOn the theoretical side, the obvious next step is to study the RE contribution\nto the anisotropy in Sm(Co 1\u0000x\u0000yFexCuy)5. In this case it will be essential to\nproperly account for the crystal \feld e\u000bects in the calculations [77] and analyse\nthe e\u000bects of hybridization of the 4 fstates with their environment [29, 36].\nIt will be interesting to see whether addition of a small quantity of Fe or Cu\nboosts the anisotropy like in YCo 5, or whether Kdecreases for all compositions.\nIn addition to the 1:5 calculations, a DFT-DLM characterization of the pristine\nbulk Y 2Co17and Sm 2Co17will be necessary to build a full picture of the cellular\nphase.\nOn the experimental side, \frst considering YCo 5, in our view the question\nof the site preference of the Fe-dopants has still not been conclusively answered.\nHaving knowledge of this aspect would be a useful test both of using the CPA to\ncalculate magnetic anisotropy, and also of total energy calculations in general to\npredict the preferred location of dopants. Furthermore, additional data explor-\ning the behavior of Kfor low Fe and Cu content, particularly for the ternary\nsystem, would also be useful. Complementary measurements on SmCo 5are also\nrequired. In particular, as pointed out in Sec. 1, the experimental data explor-\ning Cu-doped SmCo 5does not extend to the critical y<0:2 region [27]. A full\ncharacterization of (Y,Sm)(Co 1\u0000x\u0000yFexCuy)5for (x;y)<0:2 would therefore\nbe a valuable contribution to the permanent magnet literature.\n18Acknowledgments\nThe present work forms part of the PRETAMAG project, funded by the UK\nEngineering and Physical Sciences Research Council, Grant No. EP/M028941/1.\nMMs work was partly supported by JSPS KAKENHI Grant Number 15K13525.\nWe thank H. Sepehri-Amin for useful discussions.\nReferences\n[1] K. H. J. Buschow, Rep. Prog. Phys. 40 (1977) 1179. doi:10.1088/\n0034-4885/40/10/002 .\n[2] M. Sagawa, S. Fujimura, N. Togawa, H. Yamamoto, Y. Matsuura, J. 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B 53 (1996) 3272{3289. doi:10.1103/\nPhysRevB.53.3272 .\n23"    },    {        "title": "1902.01561v2.Intersublattice_magnetocrystalline_anisotropy_using_a_realistic_tight_binding_method_based_on_maximally_localized_Wannier_functions.pdf",        "content": "Intersublattice magnetocrystalline anisotropy using a realistic tight-binding method\nbased on maximally localized Wannier functions\nLiqin Ke1,\u0003\n1Ames Laboratory, U.S. Department of Energy, Ames, Iowa 50011\n(Dated: February 22, 2019)\nUsing a realistic tight-binding Hamiltonian based on maximally localized Wannier functions, we\ninvestigate the two-ion magnetocrystalline anisotropy energy (MAE) in L10transition metal com-\npounds. MAE contributions from throughout the Brillouin zone are obtained using magnetic force\ntheorem calculations with and without perturbation theory. The results from both methods agree\nwith each other, and both re\rect features of the Fermi surface. The intrasublattice and intersub-\nlattice contributions to MAE are evaluated using a Greens function method. We \fnd that the sign\nof the intersublattice contribution varies among compounds, and that its amplitude may be signi\f-\ncant, suggesting MAE can not be resolved accurately in a single-ion manner. The results are further\nvalidated by scaling spin-orbit-coupling strength in density functional theory. Overall, this realistic\ntight-binding method provides an e\u000bective approach to evaluate and analyze MAE while retaining\nthe accuracy of corresponding \frst-principles methods.\nI. INTRODUCTION\nMagnetocrystalline anisotropy (MA) arises from the\ninterplay between spin-orbit coupling (SOC) and crystal-\n\feld e\u000bects and is one of the most fundamental intrin-\nsic magnetic properties [1]. Materials with high MA\nhave been used in many applications including perma-\nnent magnets [2] and magnetic recording media. One\nof the latest examples is the recently realized magnetic\ntwo-dimensional (2D) van der Waals (vdW) class of ma-\nterials, in which MA is required to stabilize the long-\nrange magnetic ordering down to atomically thin dimen-\nsions. These materials can be exploited as platforms for\ntrue 2D magnetism and for innovative applications such\nas energy-e\u000ecient, ultra-compact, spin-based electronics.\nIn general, it is of great interest to evaluate and resolve\nMA, and to unravel the underlying mechanisms in a given\nsystem. Ultimately, such understanding will guide the\ncontrol and tuning of MA, accelerating the development\nof new materials and their applications [3].\nDensity functional theory (DFT) has proven to be\na valuable tool to investigate and predict MA energy\n(MAE) in various systems. The relativistic e\u000bects of va-\nlence electrons are often treated using various approxima-\ntions to reduce computational complexity and cost. In-\nstead of directly solving the four-component Dirac equa-\ntion self-consistently, one usually treats SOC as pertur-\nbation and starts \frst with the two-component scalar-\nrelativistic (SR) Hamiltonian [4], omitting SOC but in-\ncluding all other relativistic e\u000bects such as mass-velocity\nand Darwin terms. SOC can be added directly into\nthe SR Hamiltonian or included in a subsequent step\nusing the basis (often a subset of it) of SR wave func-\ntions (second variation) [5, 6]. Because the charge- and\nspin-density variations caused by SOC vanish to \frst or-\nder in the SOC strength [7], the magnetic force theo-\n\u0003liqinke@ameslab.govrem (MFT) [8{10] is often applied on top of the second-\nvariation method to calculate MAE as the di\u000berence of\none-electron band energies. Finally, perturbation theory\n(PT) [11{18] is also often used with the MFT to compute\nand analyze the change of band energies due to SOC.\nOverall, depending on the system size and approxima-\ntions used, MAE computation can be quite demanding,\nbecause of the enlarged dimension of Hamiltonian and\nthe reduced symmetry due to SOC, and the denser k\nmesh needed for high accuracy.\nEmpirical or semi-realistic tight-binding (TB) meth-\nods [14] were widely used to study MAE long before MAE\nbecame accessible to more sophisticated DFT meth-\nods [10]. Pioneering work [14, 15] using TB provided a\nfundamental understanding of MAE in various systems.\nHowever, empirical TB Hamiltonians are generally hard\nto parametrize and often have insu\u000ecient accuracy to de-\nscribe band structure, usually limiting TB to obtaining\nqualitative results in systems with large MAE. The re-\ncently developed maximally localized Wannier functions\n(MLWFs) method [19{21] has been widely used to ef-\nfectively construct TB Hamiltonians to compute many\nproperties such as Fermi surfaces, Berry curvature, and\ntransport. With a smaller basis, it can describe an iso-\nlated set of bands and/or entangled bands in a given\nenergy window. Indeed, this method is also very suit-\nable for MAE calculations, considering that MAE is, af-\nter all, a ground-state quantity, determined by the oc-\ncupied states. However, due to the minuteness of MAE,\nit is not clear how accurately the realistic TB can de-\nscribe MAE in systems such as transition metal bulk\ncompounds. Here, we demonstrate that the realistic TB\nframework based on the MLWFs method can produce\naccurate MAE in comparison to DFT, thereby providing\nan e\u000ecient framework to compute and analyze MAE.\nVarious decomposition schemes have been used to re-\nsolve MAE into kspace, atomic sites, orbital, and spin\nchannels, providing insight and guidance on tuning MAE.\nThe MFT enables resolution of MAE into individualarXiv:1902.01561v2  [cond-mat.mtrl-sci]  21 Feb 20192\nbands on each kpoint in reciprocal space, allowing\na band-structure-origin analysis of MAE. Site-resolved\nMAE is often calculated using methods such as evalu-\nation of the on-site SOC energy [22], second-order PT,\nscaling the SOC strength, or others [23, 24]. For exam-\nple,R2Co17(withR= Y or Ce) compounds have very\nsmall MAE and ab initio analysis found that one par-\nticular Co sublattice, the so-called dumbbell sites, has\na large negative contribution to uniaxial anisotropy [25].\nThus, proper substituents that preferentially occupy the\ndumbbell sites and eliminate the negative contributions\nwill signi\fcantly improve the uniaxial anisotropy, as ob-\nserved in experiments. Another interesting example is\nthe oscillating MAE behavior in diluted nitridometalates\nLi2[(Li1\u0000xTx)N], withT= Mn, Fe, Co, or Ni. The\nMAE can be solely attributed to individual Tatoms and\nthe band-\flling e\u000bect on MAE can be quantitatively de-\nscribed in a single-ion MAE model [17, 18].\nSite-resolved MAE values, together with exchange\nparameters, can also be used as inputs for subse-\nquent large-scale atomic spin simulations to calculate\ntemperature-dependent magnetic properties. When in-\nterfacing ab initio results with atomic spin simulation,\nMAE is usually treated in the single-ion model in the\natomic spin Hamiltonian. However, it has been found\nthat a two-ion anisotropy model is needed to properly de-\nscribe properties such as temperature-dependent MAE in\nsome systems [26]. Thus, it is of great interest to resolve\nMAE into sites and, in particular, beyond the single-ion\nmodel.\nIn this work, we investigate two-ion MAE in various\nL10compounds, one of the most widely studied systems,\nwith MAE values ranging from several tens of \u0016eV to a\ncouple of meV per formula unit [27{29]. Our approach\nis based on second-order PT using a Green's function\nmethod implemented within the realistic TB framework.\nWe demonstrate that this approach achieves accuracy\nsimilar to DFT and provides a highly e\u000ecient means to\ncompute and analyze the two-ion anisotropy in transition\nmetal systems.\nII. COMPUTATIONAL DETAILS\nWe \frst construct the real-space scalar-relativistic TB\nHamiltonian using the MLWFs method. The correspond-\ning Green's function is also constructed for use in the\nPT approach. MAE is calculated using the MFT in TB,\nwith and without PT, referred to hereafter as PT and\nMFT, respectively, for simplicity. DFT methods, includ-\ning both vasp [30, 31] and an all-electron full-potential\nLMTO (FP-LMTO) method [32], are used to calculate\nMAE and compare with TB. To compare with the two-\nion MAE values obtained using PT in TB, we also calcu-\nlate intersublattice MAE contribution by scaling the SOC\nstrength in vasp . All DFT calculations are carried out\nwithin the generalized gradient approximation (GGA)\nusing the functional of Perdew, Burke, and Ernzerhof(PBE) [33] unless local density approximation (LDA) [34]\nis speci\fed.\nA. TB Hamiltonian and SOC\nThe MLWFs are constructed through a postprocessing\nprocedure [19{21] using the output of a self-consistent\nscalar-relativistic vasp calculation. For each L10com-\npound, 18 MLWFs corresponding to s-,p-, andd-type\norbitals for each of the two atoms in the unit cell were\ngenerated using wannier90 [35]. The spread functional\nfor entangled energy bands is minimized by a two-step\nprocedure [21]. An outside energy window with a larger\nnumber of bands was selected to ensure good descrip-\ntion of the band structure of the \\frozen\" inner energy\nwindow, spanning from the bottom of the valence band\nto a few eV above the Fermi level. A real-space Hamil-\ntonianH(R) with dimensions 18 \u000218 is constructed to\naccurately represent the band structures in this speci-\n\fed \\frozen\" energy window. Then, H(k) is obtained\nby Fourier transformation. The energy bands are recal-\nculated within TB to ensure that DFT bands can be\naccurately reproduced before further MAE calculations.\nThe TB Hamiltonian is represented in a basis of or-\nthonormalized atomic functions ji;l;m;\u001bi, whereilabels\natomic sites, l;mangular and magnetic quantum num-\nbers (in cubic harmonics), and \u001bthe spin. The SOC part\nof the Hamiltonian, which can be directly added into H\nor included using PT, can be written as\nVso=\u0018L\u0001S=~2\n2M2c21\nrdV\ndrL\u0001S; (1)\nwhere L\u0001Sdepends explicitly on the direction of spin\nquantization axis (details can be found in Appendix A).\nThe radial part of Vso, the SOC constants \u0018\u001b\u001b0\ni;l, are calcu-\nlated using FP-LMTO. For simplicity, we ignore the en-\nergy and spin dependence of \u0018. Furthermore, the occupa-\ntion numbers of the Pt- porbitals, which have a large SOC\nconstant, are overestimated in TB in comparison to DFT.\nWe renormalize the Pt- poccupation numbers based on\nthe DFT value when adding SOC into the Hamiltonian.\nB. MAE\nTurning on SOC lowers the system energy. Here, we re-\nfer to this energy change as the SOC energy Eso, which\ndepends on the spin direction. For uniaxial geometry,\nMAE can be de\fned as K=Eso\n110\u0000Eso\n001, withEso\n110\nandEso\n001indicating the SOC energies along the spin di-\nrections [110] and [001], respectively. We use [110] as\nthe reference direction for the basal plane. A positive K\nvalue indicates the system has uniaxial anisotropy with\nthe easiest spin direction being out of plane.3\n1. Magnetic force theorem\nAfter SOC is added into the TB Hamiltonian, tetra-\nhedron integration with Bl ochl correction [36] is used to\ndetermine the Fermi level \u000fF, band weights, and band\nsums. MAE is calculated as\nK=X\nk;b\u0000\n\u000f110\nk;bf110\nk;b\u0000\u000f001\nk;bf001\nk;b\u0001\n; (2)\nwhere,bis the band index, krefers to the wave vector in\nthe \frst Brillouin zone (BZ), and fk;bis the corresponding\nband occupancy. In order to resolve the MAE into k\nspace, Eq. (2) needs to be modi\fed. A grand-canonical\nensemble version [37] is used:\nK=X\nk;b\u0000\n(\u000f110\nk;b\u0000\u000f0\nF)f110\nk;b\u0000(\u000f001\nk;b\u0000\u000f0\nF)f001\nk;b\u0001\n;(3)\nwhere\u000f0\nFis the Fermi level calculated without SOC. The\ntotal MAE value does not depend on the reference energy\nbecause the total number of valence electronsPfk;bis\nconserved when spin is along di\u000berent directions; how-\never, thek-resolved MAE does depend on the choice of\nreference energy [24, 38]. We use Eq. (3) to properly\ndecompose MAE because kresolution of MAE based on\nEq. (2) will only re\rect the change of band occupancy\n\u0001fk;i. The Eq. (2) result will be dominated by the relax-\nation e\u000bect of the Fermi surface due to turning on SOC\nwith magnetization along various directions, and MAE\ncontributions will only be signi\fcant near the Fermi sur-\nface.\n2. Perturbation theory\nUsing second-order PT, we can express orbital mo-\nment, SOC energy, and their anisotropies in terms of the\nsusceptibility [17] calculated using the unperturbed band\nstructure. The SOC energy Esodue to the spin-orbit in-\nteractionVsocan be written as\nEso=\u00001\n2\u0019ImZ\u000fF\n\u00001d\u000fTrh\neG(\u000f)Vsoi\n; (4)\nwhereeG(\u000f), the full Green's function, includes SOC and\ncan be constructed from the non-perturbed Green's func-\ntionG(\u000f). Using second-order PT (here we consider only\nsystems with a uniaxial geometry), the SOC energy can\nbe written as\nEso=\u00001\n2\u0019ImZ\u000fF\n\u00001d\u000fTr [G(\u000f)VG(\u000f)V]; (5)\nwhere the Green's function is constructed using\nG(\u000f) = (\u000f\u0000H)\u00001: (6)A complex contour integration on an elliptical path [39]\nis used for the integration. By exploiting the fact that G\nis spin diagonal and Vis (i;l) diagonal, the SOC energy\ncan be written as\nEso(^ n) =X\nil;jl0;\u001b\u001b0E\u001b;\u001b0\nil;jl0(^ n); (7)\nwhereE\u001b;\u001b0\nil;jl0(^ n) is the contribution from the sublattice-\norbital-spin pair ( il\u001b;jl0\u001b0). We have\nE\u001b\u001b0\nil;jl0(^ n) =\u00001\n2\u0019ImZ\u000fF\n\u00001d\u000fZ\ndkTrE\u001b\u001b0\nil;jl0(k;\u000f;^ n);(8)\nand\nE\u001b\u001b0\nil;jl0(k;\u000f;^ n) =G\u001b\nil;jl0(k;\u000f)V\u001b\u001b0\njl0(^ n)G\u001b0\njl0;il(k;\u000f)V\u001b0\u001b\nil(^ n):\n(9)\nV=\u0018\u001b\u001b0\ni;l(\u000f)L\u0001S(^ n) couples states within the same l\nchannel at the same site. Here, for simplicity, we treat\nSOC strength as a constant \u0018ilfor eachlchannel at site\ni, ignoring its energy and spin dependence. L\u0001Scan be\nwritten as a function of magnetization direction ^ n. The\nMAE can be written as\nK=X\nij;\u001b\u001b0K\u001b\u001b0\nij=X\nij;\u001b\u001b0E\u001b\u001b0\nij(^ n110)\u0000E\u001b\u001b0\nij(^ n001):(10)\nWe de\fne the isotropic and anisotropic parts of V(^ n)\nasUandA, respectively:\n2U=V(^ n110) +V(^ n001);\n2A=V(^ n110)\u0000V(^ n001):(11)\nThen, Eq. (10) can also be written as\nK=X\nij;\u001b\u001b0eK\u001b\u001b0\nij (12)\nwith\neK\u001b\u001b0\nij=\u00002\n\u0019ImZ\u000fF\n\u00001d\u000fZ\ndk\nTrh\nG\u001b\nij(k;\u000f)U\u001b\u001b0\njG\u001b0\nji(k;\u000f)A\u001b0\u001b\nii\n:(13)\nHere, we have eK\u001b\u001b0\nij=K\u001b\u001b0\nijwhen\u001b=\u001b0. Unlike\nK\"#\nij=K#\"\nij, we haveeK\"#\nij6=eK#\"\nij, however,\neK\"#\nij+eK#\"\nij=K\"#\nij+K#\"\nij: (14)\nAccording to Eq. (13), the strength of the intersublat-\ntice MAE contribution Kijdepends on the SOC strength\nof both sublattices \u0018iand\u0018j(contained in VsoorUandA)\nand on the intersublattice Green's function Gij. The ele-\nment types of the sublattices determine the SOC strength4\n\u0018whileGijis relevant to the hopping or hybridization be-\ntween two sublattices and depends on the detail of elec-\ntronic structure. If there is no hybridization between\nthe two sublattices or they are coupled only through s\norbitals (i.e., the angular parts of UandAvanish) ele-\nments, the corresponding intersublattice contribution be-\ncomes negligible.\n3. Scaling SOC strength\nInstead of using Eq. (13), the intrasublattice and in-\ntersublattice MAE contributions can also be obtained by\nscaling the SOC strength \u0018ion each site iby a factor \u0015i,\nand by \ftting the MAE as a function of scaling vector\n\u0015= [\u00151;\u00152;:::;\u0015n]|:\nVso(\u0015) =X\ni\u0015i\u0018iL\u0001S; (15)\nK(\u0015) =X\nij\u000bij\u0015i\u0015j+O(\u00154):(16)\nComparing Eq. (16) to Eqs. (10), (12), and (13), the\ncoe\u000ecients \u000bijare nothing but the corresponding terms\ncontaining\u0018i\u0018jinK. Thus, we have\n\u000bij=Kij=X\n\u001b\u001b0K\u001b\u001b0\nij=X\n\u001b\u001b0eK\u001b\u001b0\nij: (17)\nThe SOC-scaling procedure is often used in DFT, prob-\nably due to its rather straightforward implementation.\nObviously, the scaling procedure can be generalized from\nsites to orbitals to obtain contributions from individual\norbitals.\nC. Crystal structure of L10compounds\nFIG. 1. Schematic representation of the CuAu-type L10\nstructure.We chose to focus on L10magnetic compounds because\nthey are one of the most widely studied systems [27{\n29]. Their simple CuAu-type crystal structure is shown\nin Fig. 1. The primitive cell is body-centered tetragonal\n(bct) and contains one formula unit (f.u.) while the con-\nventional cell is face-centered tetragonal ( fct) and con-\ntains two f.u. For all L10magnetic compounds that we\nstudy in this work, experimental lattice parameters have\nbeen used. The c=aratio values (with respect to the bct\nprimitive cell) are in the range of 1.28{1.414. Consider-\ning that hypothetical bct-FeCo structures with di\u000berent\nc=aratios have received signi\fcant attention in the past\nfew years, we also investigated a hypothetical bct-FeCo\nstructure with a c=aratio of 1.1.\nIII. RESULTS\nA. Electronic structure and magnetic moment\nFor all compounds we investigated in this work, the\nscalar-relativistic band structures recalculated in TB are\nessentially in perfect agreement with those obtained from\nDFT, within the speci\fed energy window. In both TB\nand DFT, the tetrahedron integration with Bl ochl cor-\nrection method [36] is used to determine the Fermi level.\nHere, we use FePt as an example to illustrate our imple-\nmentation of SOC in the realistic TB framework. The\nnew Fermi level obtained in TB with charge neutrality\ndeviates from the original DFT Fermi level by less than\n0:01 eV. As shown in Fig. 2, without SOC, the band\nstructures are nearly identical between TB bands and\nthe all-electron full potential bands calculated using FP-\nLMTO. The vasp bands (not shown) are essentially the\nsame also. The SOC constants \u0018ilcalculated using FP-\nLMTO are used as input parameters to construct rela-\ntivistic TB bands.5\n−10−8−6−4−2024\nXΓ MXRAΓZ108642024\nX MXRA Z108642024\nX MXRA ZE (eV)(a) (b) (c)\nFIG. 2. (a) Scalar-relativistic band structure of FePt using a tight-binding Hamiltonian and FP-LMTO (black dotted). The\ntight-binding bands are in color, with blue identifying the sstates, red the (Fe- p, Pt-p) states, and green everything else. The\nband structure with spin-orbit coupling is calculated with spin along the [001] (b) and [110] (c) directions.\nDirectly using the SOC constants calculated in FP-\nLMTO, the resulting band structure's SOC splitting is\noverestimated just above and below the Fermi level at R\nandMpoints, respectively. By investigating the eigen-\nvectors of the non-SOC band in FP-LMTO and TB, we\nfound that TB overestimates the eigenvector component\nof the Pt-pstates, which have a very large SOC constant\nof about 2:6 eV. (The site-and-orbital-resolved charges\nand moments are listed in Appendix B.) This is because\nporbitals are much more extended than dorbitals and\nwe do not include the interstitial sites into the projection,\nand as a result, the interstitial components also fold into\nthe atomic sites. Thus, we simply renormalize the occu-\npation numbers (or, equivalently, renormalize the SOC\nconstants) of the Pt- pchannel using the ratio of atomic\npcharge between vasp and TB. This adjustment im-\nproves agreement between DFT bands and TB bands.\nCorrespondingly, the MAE also improves as we discuss\nlater. In comparison to a previously reported empirical\nTB method [40] that used Slater-Koster parametrization,\nour TB method shows signi\fcantly better agreement with\nDFT.\nSOC parameters \u0018\"#\nilcalculated in FP-LMTO in vari-\nous compounds are summarized in Table I. Generally, \u0018\nintegration is only signi\fcant near a nucleus where elec-\ntrons move fast. As a result, for a given element in var-\nious compounds, \u0018lbarely changes, especially for the d\nchannel, enabling transferability. The dorbitals of 3 dele-\nments are more spin polarized than those of 5 delements,\nand the ratio of \u0018\"\"\ni=\u0018\"#\ni\u0019\u0018\"#\ni=\u0018##\niof 3delements varies\nbetween 1.07 and 1.13 in various L10compounds. For\nsimplicity, we use the value of \u0018\"#\nilfor all spin channels.\nThe SOC energy Esois proportional to \u00182within\nsecond-order PT. Among the L10compounds we stud-\nied, CoPt has the largest \u0018values as well as the largest\nEso. TheEsovalues are much larger than MAE, indi-\ncating that the isotropic part of Esois much larger than\nthe anisotropic part. In comparison to other compounds,TABLE I. Spin-orbit coupling constants \u0018\"#\ni(meV) in various\ncompounds calculated in FP-LMTO. On the 3 dsites (Mn, Fe,\nand Co),\u0018\"\"\ni=\u0018\"#\ni\u0019\u0018\"#\ni=\u0018##\nivaries between 1.07 and 1.13 for\nthedchannel. SOC energies (isotropic) Eso(meV) calculated\nin TB are also listed.\nCompound1st element 2nd elementEso\n\u0018p\u0018d\u0018p\u0018d\nFePt 197.2 55.0 2626.4 574.9 193.8\nCoPt 190.5 72.1 2793.1 580.8 210.6\nFePd 168.9 54.7 898.0 200.8 22.4\nFeNi 232.9 55.9 218.2 91.6 11.7\nMnGa 193.1 41.0 209.6 84.1 5.3\nMnAl 203.0 41.3 31.3 0.4 2.7\nMnGa and MnAl have a rather high anisotropic/isotropic\nratio, suggesting that they have a more \\e\u000ecient\" band\nstructure, in the sense that the Fermi level is close to the\nband\flling position that gives the largest MAE value,\nas we show in subsection III B. On the other hand, it is\nchallenging to analyze the relationship between MAE and\nband structure for FePt and CoPt as each MAE value is\nonly a small fraction of Eso.\nB. MAE and band-\flling e\u000bect\nAs shown in Table II, the MAE values calculated using\nTB agree well with DFT, especially with those obtained\nby the all-electron FP-LMTO method. They are also\ncomparable to previous DFT calculations using various\nmethods [22, 29]. We found that MAE values calculated\nusing PT generally agree very well with the MFT results.\nThe largest di\u000berence in MAE is in CoPt, in which PT\ngives a MAE 20% larger than the MFT result. Inter-\nestingly, the MAE of CoPt also strongly depends on the\nexchange-correlation functionals used. LDA increases the6\nTABLE II. MAE ( \u0016eV/f.u.) in L10systems calculated in our\ntight-binding framework with (denoted as TB-PT) and with-\nout (denoted as TB) perturbation approach. MAE values\ncalculated using FP-LMTO with both PBE (denoted as FP)\nand BH (denoted as FP-LDA) exchange-correlation function-\nals, and using vasp with PBE functional, are also listed for\ncomparison. The SOC constants used in TB were obtained\nfrom FP using PBE functionals.\nCompound TB FP TB-PT VASP FP-LDA\nFePt 2495 2556 2692 2656 2746\nCoPt 836 788 1098 858 1156\nFePd 175 164 186 194 173\nFeNi 66 68 65 80 68\nMnGa 399 381 415 431 421\nMAE values by 30%. Likely, this is because CoPt has a\nlarge SOC and its MAE depends on the detailed, subtle\nband features near the Fermi level. We will study CoPt\nin more detail in a later section.\nFigure 3 shows the band-\flling dependence of MAE\nin FePt, CoPt, FePd, and FeNi. The oscillation [41]\nof MAE can be associated with the local susceptibil-\nity [17, 42]. For the entire band\flling range, MAE values\ncalculated using DFT and TB agree well. On the other\nhand, although PT gives a good description of MAE at\nthe actual electron \flling for each compound, it disagrees\nwith the MFT result in certain band\flling ranges in FePt\nand CoPt. Speci\fcally, the disagreement is pronounced\nfrom four to eight electrons in FePt and from four to ten\nin CoPt. A similar \fnding was reported by a previous\nLMTO-ASA study [29]. In comparing results for 3 d, 4d,\nand 5dcompounds, second-order PT is best suited for the\nlower SOC strength found in the lighter compounds, and\nthat's where we obtained the best agreement between\nMFT and PT. In general, we found that we could not\nimprove agreement by using a denser kmesh.\nAlthough MnGa does not contain heavier 4 dand 5d\nelements, its MAE is larger than FePd. As shown in\nFig. 4, the Fermi level is located close to the \flling with\nmaximum MAE. PT generally agrees well with MFT.\nWe also consider the renormalization of the p(for both\nGa and Mn) charge when including SOC, and it slightly\ndecreases the MAE. All TB results are within \u000610% of\nthe FP-LMTO results. While MnGa MAE is about 10 %\nof the SOC energy, the ratio is much higher than in FePt\nand CoPt.\nC. Reciprocal-space resolved MAE and its\ncorrelation with Fermi surface\nAs shown in Table II, MAE values calculated using\nsecond-order TB with either MFT or PT generally agree\nwell with each other and with the corresponding DFT cal-\nculations. To further validate the applicability of PT, we\ncompare the k-resolved MAE using both methods for a\nmore stringent test. Remarkably, for all L10compounds\n0 2 4 6 8 10 12 14 16 18 20 22\n−8−404812\n  \nFePt (a) FP-LMTO\nTB-MFT\nTB-PT\n−8−404812 CoPt (b)\n−2024FePd (c)Anisotropy (meV/f.u.)\n0 2 4 6 8 10 12 14 16 18 20 22−2−1012\nNumber of electronsFeNi (d)FIG. 3. MAE in FePt, CoPt, FePd, and FeNi as a function of\nband \flling. The solid and dashed lines are the results calcu-\nlated using the magnetic force theorem and the second-order\nperturbation theory, respectively. The blue dots are from\nFP-LMTO. The vertical dashed-and-dotted lines indicate the\nactual number of valence electrons in each compound.\nwe study, the two methods produce very similar results,\nfurther suggesting that the overall e\u000bect of Fermi surface\nrelaxation is non-signi\fcant and that second-order PT\nis valid. CoPt has the largest di\u000berence of MAE values\nbetween MFT and PT among the compounds we study.\nFigure 5 shows the kzdependence of MAE contributions\ncalculated in FePd and CoPt. Although the kz-resolved\nMAE in CoPt shows a larger di\u000berence, the two meth-\nods still generally agree with each other. The larger dif-\nference is likely due to its rather large SOC for the Pt7\n8.8 9.2 9.6 10 10.4 10.8−0.4−0.200.20.4\nNumber of electronsAnisotropy (meV/f.u.)\n  \nFP-LMTO\nTB-MFT\nTB-PT\nTB2-MFT\nTB2-PT\nFIG. 4. MAE in MnGa as a function of band \flling calcu-\nlated in TB and FP-LMTO. The vertical dashed-and-dotted\nline atNe= 10 indicates the actual number of valence elec-\ntrons in MnGa. For TB calculations, we also consider the\nrenormalization of Ga- pand Mn-pelectron occupancy when\nincluding SOC (denoted as TB2).\n0 0.2 0.4 0.6 0.8 1−20246\nkzNormalized MAE\n  \nFePd (TB-MFT)\nFePd (TB-PT)\nCoPt (TB-MFT)\nCoPt (TB-PT)\nFIG. 5.kz-resolved normalized MAE of FePd and CoPt cal-\nculated in TB. Calculations are performed with and without\nusing perturbation theory. For the sake of comparison, MAE\nis normalized so that the average contribution from each kz\nplane is equal to 1.\natom and more complex Fermi surface, which results in\na larger Fermi surface relaxation e\u000bect. Similar to FePd,\nthekz-resolved MAE calculated using MFT and PT in\notherL10compounds (not shown) are also nearly iden-\ntical. And, the Z-R-A(kz= 0:5) plane has the largest\npositive contribution to MAE.\nThe (kx;ky)-resolved MAE at kz= 0 in CoPt calcu-\nlated using both MFT and PT are shown in Fig. 6, which\nalso includes the corresponding non-SOC Fermi contour.\nFigure 6(a) shows the resolved MAE calculated using\nMFT. Obviously, the four-fold symmetry is broken due\nto SOC. The twofold or mirror symmetry along the [110]\ndirection is simply an artifact of our choice of [110] as\nthe reference in-plane spin direction. The Fermi contour\nplot, as shown in Fig. 6(b), is calculated by integrating\n(a) (b)\n(c) (d)Г XMFIG. 6.k-resolved MAE and Fermi surface contour in CoPt\nforkz= 0 calculated in TB. Red (blue) color indicates pos-\nitive (negative) contributions to MAE. (a) k-resolved MAE\ncalculated via the magnetic force theorem. (b) Fermi surface\ncontour plot. (c) Symmetrized k-resolved MAE. (d) Sym-\nmetrizedk-resolved MAE calculated via perturbation theory.\nThe number of kpoints used in the full Brillouin zone is \u0018108.\nthe electron density at the Fermi level with a width of\n0:02 eV. To better compare with the Fermi surface and\nk-resolved MAE, we also symmetrized k-space contribu-\ntions with symmetry operations that are compatible with\nSOC when the spin quantization axis is along the zdi-\nrection. The symmetrized k-resolved MAE calculated in\nMFT and PT are shown in Figs. 6(c) and 6(d), respec-\ntively.\nAs shown in Figs. 6(b) and 6(d), the correlation be-\ntween the ( kx;ky)-resolved MAE and the Fermi contour\nis apparent and two features stand out. First, large\nchanges of MAE contributions occur at the Fermi con-\ntour. This is because MAE contributions are obtained\nby integrating Eq. (13) up to \u000fF. When the kpath\ncrosses the Fermi contour, some bands become occupied\nor unoccupied, and their MAE contributions appear or\ndisappear. Second, from the point of view of PT, the\nstrongest contributions to MAE are from those virtual\ntransitions between the unoccupied and occupied states\nnear the Fermi level that are coupled by SOC [17, 43].\nAs a result, the largest contributions are located at and\nnear the degenerate states across the Fermi contour, as\nexpected from PT, and as shown in Fig. 6(d).\nAs shown in Figs. 6(c) and 6(d), the overall ( kx;ky)-\nresolved MAE calculated using the two methods share\ngreat similarity. The largest di\u000berences exist around the\nFermi contour, where the relaxation e\u000bect is large. The\nlarge contributions (near the degenerate states) observed8\nin PT disappear in MFT. Indeed, the MFT results show\nmore complex features than the non-SOC Fermi contour,\nwhich corresponds to the Fermi surfaces when SOC is\non and the magnetization direction being along in- and\nout-of-plane directions. The SOC-induced lifting of band\ndegeneracy, especially near the \u000fF, is often discussed to\nexplain MAE in various systems [43{46].\nThus, we demonstrate that MFT and PT give very\nsimilar results for not only total MAE but also for k-\nresolved MAE. To achieve this, it is important to use the\nreference energy \u000f0\nFas in Eq. (3). Otherwise, if one just\nresolves MAE using Eq. (2), a very di\u000berent dependence\ncan be obtained, and k-resolved MAE will only manifest\nthe Fermi surface, near which the change of band occu-\npancy is signi\fcant. MAE has often been resolved into\nkspace along certain line paths between high-symmetry\npoints. Not surprisingly, using Eq. (2) to resolve MAE\nwill result in spikes at points where bands cross the Fermi\nlevel. Further resolution of MAE into atomic sites by\nprojecting eigenvectors of each kpoint may produce un-\nphysical results [24, 38, 47].\nD. Two-ion MA: Intersublattice contribution\nSite-resolved MAE values, together with exchange pa-\nrameters, can be used to construct an atomic spin Hamil-\ntonian for subsequent Monte Carlo or spin-dynamics\nsimulations to calculate the temperature dependence of\nmagnetic properties. Methods such as evaluating the\nanisotropy of on-site SOC energy [22], which is a local\nquantity, have been used to resolve the MAE contribu-\ntion from each individual sublattice. Here, we use PT\nin TB to resolve MAE into sublattices and validate the\ndecomposition using the SOC-strength-scaling approach\ninvasp .\nTABLE III. Sublattice-resolved MAE (normalized to 1) in\nL10systems calculated using perturbation theory in TB. For\neach compound AB, MAE is resolved into intrasublattice con-\ntributions, KA-AandKB-B, and intersublattice contribution\nKA-B. We also de\fne the contribution from individual sub-\nlatticeAasKA= (KA-A+KA-B=2). The hypothetical FeCo\nstructure with c=a= 1:1 is also included.\nAB K A-AKA-BKB-BKAKB\nFePt 0.11 -0.55 1.44 -0.16 1.16\nCoPt 0.17 -0.77 1.59 -0.21 1.21\nFePd 1.80 -2.58 1.78 0.51 0.49\nFeNi 4.51 -5.88 2.37 1.57 -0.57\nMnGa 0.66 0.24 0.10 0.78 0.22\nMnAl 0.98 0.02 0.00 0.99 0.01\nFeCo 0.10 0.63 0.28 0.41 0.59\nAs discussed above, PT can well describe the MAE\nin these systems. To quantify the single-ion and two-\nion contributions of MAE, we \frst use PT within TB\nto resolve MAE into intrasublattices and intersublatticecontribution. Results are summarized in Table III. Inter-\nestingly, all intrasublattice contributions are positive in\nall elements except for the s-like Al site in MnAl, where\nit vanishes. The sign of the intersublattice contribution\nvaries and its amplitude is generally comparable to or\neven larger than that of either individual intrasublattice.\nFor FePt and CoPt, the major contributions are from the\nPt sites. The intersublattice contributions are negative\nfor FePt, CoPt, FePd, and FeNi. Especially in FeNi, the\namplitude of the negative intersublattice contribution is\nlarger than each individual intrasublattice contribution.\nIn contrast, the intersublattice contribution is positive in\nMnGa. An even larger positive intersublattice contribu-\ntion is found in hypothetical FeCo with c=a= 1:1.\nTo validate TB results, we also investigate the intra-\nsublattice and intersublattice MAE contributions by scal-\ning the SOC strength in vasp . Figure 7 shows the nor-\nmalized MAE as a function of the SOC-scaling factors\n(between 0 and 1) for L10materials using the second-\nvariation method in vasp . For all compounds, the sign\nand relative amplitude of intrasublattice and intersublat-\ntice contributions agrees well with TB-PT results. Fur-\nthermore, we \ft MAE as a function of SOC-scaling fac-\ntors (with 0 :9< \u0015i<1:1) using Eq. (16) and further\ncon\frm that the second-order terms agree very well with\nTB results listed in Table III. The fourth-order terms are\ngenerally small especially for 4 dand 3dcompounds. Ow-\ning to stronger SOC, FePt and CoPt have larger and neg-\native fourth-order contributions: \u00188% of total MAE. A\nprevious study also found a small and negative high-order\ncontribution to MAE in FePt [48]. The good agreement\nbetween TB and vasp further validates the accuracy of\nthe PT approach for those systems. Unlike the scaling\nprocedure, the PT approach resolves all contributions in\na single calculation. Thus, for the same analysis, once\nthe TB Hamiltonian is constructed, the TB-PT approach\nis orders of magnitude faster than the SOC-scaling ap-\nproach in DFT.\nWe further investigate FeNi, in which the intersublat-\ntice MAE dominates. Along each direction, the intrasub-\nlattice SOC energy is large and dominated by the \u00152term\nwhile the intersublattice term is rather small. However,\nthe majority of intrasublattice terms cancel out between\nthe two directions, while the intersublattice term does\nnot. Hence, the intersublattice term becomes dominant\nin the anisotropy of SOC. In other words, the intrasub-\nlattice terms in SOC are large but more isotropic, while\nthe intersublattice terms are smaller but more anisotropic\nwith respect to the magnetization quantization direction.\nA large intersublattice MAE contribution may suggest\nthe need to go beyond the single-ion MAE model when in-\nterfacing ab initio methods with atomic spin simulation.\nTo simulate temperature-dependent MAE or other mag-\nnetic properties, exchange coupling is often included over\nthe \frst one or two nearest-neighbor shells while MAE\nis often included using the single-ion MAE term ki(sz\ni)2.\nFor systems with strong intersublattice contribution, one\nmay also need to include two-ion terms such as kijsz\nisz\nj9\n00.51\n00.5100.511.5\nPt Fe\n00.51\n00.5100.511.52\nPt Co\n00.51\n00.5100.51\nMn Al\n00.51\n00.5100.51\nMn Ga\n00.51\n00.5101234\nFe Ni\n00.51\n00.5100.51\nFe Co\nFIG. 7. Normalized MAE K(\u0015i;\u0015j)=K(\u0015i= 1;\u0015j= 1) in\nL10compounds as a function of SOC-scaling factors \u0015iand\n\u0015jcalculated using second-variation method in vasp . The\nSOC strengths are scaled between 0 and 1.\ninto the atomic spin Hamiltonian.\nIV. CONCLUSIONS\nUsingL10systems as a test case, we demonstrate that\nthe ab initio TB framework, constructed using the max-\nimally localized Wannier functions method, can be used\nto e\u000eciently and accurately compute and resolve MAE in\ntransition metal systems. With the magnetic force theo-\nrem, TB quantitatively reproduces DFT results over the\nfull band-\flling range from the bottom of valence band\nto a few eV above the Fermi level. When calculating\nk-resolved MAE in TB, the magnetic force theorem and\nperturbation theory results agree with one another, and\nboth yield MAE contour maps that are consistent with\nthe Fermi surface. We also resolve MAE into intrasublat-\ntice and intersublattice contributions using perturbation\ntheory in TB and a scaled spin-orbit strength procedure\nin DFT. The results using these two methods are in ex-\ncellent agreement. We found that the sign of the inter-\nsublattice contribution di\u000bers among compounds, and its\namplitude may be comparable to or even larger than the\nintrasublattice contributions, suggesting the need to go\nbeyond the single-ion MAE model. Depending on the\nsystem size, once the TB Hamiltonian is constructed, it\ncan speed up the calculation by orders of magnitude, pro-viding an e\u000ecient, accurate, and high-resolution method\nto calculate MAE. We expect that it can be applied to\nmore complex compounds and structures to compute and\nanalyze MAE. Finally, this realistic TB method can also\nbe interfaced with ab initio methods beyond DFT, such\nas the much more expensive self-consistent GW meth-\nods [49, 50], to greatly accelerate the calculations and\nanalysis of MAE or other SOC-related properties using\nthose methods.\nACKNOWLEDGMENTS\nI thank B. Harmon for helpful discussions. This work\nwas supported by the U.S. Department of Energy, Of-\n\fce of Science, O\u000ece of Basic Energy Sciences, Materials\nSciences and Engineering Division, and Early Career Re-\nsearch Program. The initial development of the software\nwas supported by the Laboratory Directed Research and\nDevelopment Program of Ames Laboratory. Ames Lab-\noratory is operated for the U.S. Department of Energy\nby Iowa State University under Contract No. DE-AC02-\n07CH11358.\nAppendix A: Spin-orbit coupling operators in real\nspherical harmonics representation\nL\u0001S=1\n2(J2\u0000L2\u0000S2) =~2\n2\u0012\nLzL\u0000\nL+\u0000Lz\u0013\n: (A1)\nIn the presentation of complex spherical harmonics\nYm\n`, the non-vanished matrix elements of Lz,L+, and\nL\u0000are\nhl;mjLzjl;mi=~m;\nhl;m\u00001jL\u0000jl;mi=~p\nl(l+ 1)\u0000m(m\u00001);(A2)\nhl;m+ 1jL+jl;mi=~p\nl(l+ 1)\u0000m(m+ 1):\nwannier90 uses the real spherical harmonics Y`m, also\nknown as tesseral spherical harmonics, which can be writ-\nten in terms of the complex spherical Ym\n`as\nY`m=8\n>>><\n>>>:ip\n2\u0010\nY\u0000jmj\n`\u0000(\u00001)mYjmj\n`\u0011\nifm< 0;\nY0\n` ifm= 0;\n1p\n2\u0010\nY\u0000jmj\n`+ (\u00001)mYjmj\n`\u0011\nifm> 0:\n(A3)\nThe angular momentum matrices in the real-spherical-\nharmonics representation, O(R), can be obtained by di-\nrectly evaluating the angular momentum operator on real\nspherical functions in Eq. (A3), or transforming O(C), the\ncorresponding operator matrix from complex representa-\ntion:\nO(R)=UR CO(C)Uy\nR C=UR CO(C)UC R:(A4)10\nTABLE IV. Site-and-orbital-resolved charge qand magnetic moment m(\u0016B/atom) inL10compounds calculated in TB and\nvasp . The moments of s-orbitals are negligible and not shown. Spin-orbit is not included in calculation.\nCompound Method1stelement 2 ndelement Total\nqsqpmpqdmdq1m1qsqpmpqdmdq2m2q m\nFePtVASP 0.41 0.45 -0.01 6.12 2.92 6.98 2.92 0.61 0.50 -0.05 7.73 0.43 8.84 0.36 15.82 3.28\nTB 0.83 0.81 -0.04 6.38 2.95 8.03 2.95 0.98 0.81 -0.09 8.18 0.45 9.97 0.34 18.00 3.29\nCoPtVASP 0.41 0.44 -0.01 7.20 1.91 8.05 1.89 0.61 0.50 -0.04 7.72 0.46 8.82 0.41 16.88 2.30\nTB 0.83 0.80 -0.04 7.48 1.92 9.11 1.87 0.96 0.78 -0.06 8.14 0.47 9.89 0.40 19.00 2.26\nFePdVASP 0.42 0.41 -0.01 6.09 2.97 6.92 2.96 0.37 0.31 -0.04 7.96 0.42 8.63 0.36 15.55 3.33\nTB 0.81 0.67 -0.04 6.35 3.05 7.83 3.00 0.85 0.70 -0.09 8.63 0.42 10.17 0.29 18.00 3.29\nFeNiVASP 0.47 0.49 -0.02 6.17 2.69 7.12 2.67 0.51 0.51 -0.07 8.31 0.72 9.33 0.62 16.45 3.29\nTB 0.80 0.70 -0.05 6.38 2.74 7.89 2.67 0.85 0.73 -0.09 8.53 0.71 10.11 0.57 18.00 3.24\nMnGaVASP 0.29 0.31 0.02 5.13 2.49 5.73 2.52 1.08 1.31 -0.11 0.16 0.02 2.56 -0.14 8.29 2.39\nTB 0.55 0.47 0.02 5.45 2.60 6.46 2.64 1.36 1.89 -0.15 0.28 0.03 3.54 -0.15 10.00 2.49\nMnAlVASP 0.33 0.34 0.02 5.12 2.37 5.78 2.41 0.40 0.42 -0.04 0.00 0.00 0.82 -0.06 6.60 2.35\nTB 0.56 0.47 0.03 5.49 2.49 6.52 2.55 1.21 1.87 -0.14 0.41 0.03 3.48 -0.17 10.00 2.39\nFrom Eq. (A3), the transfer matrix UC Rcan be writ-\nten as\nUC R=1p\n20\nBBBBBBB@i0 0 0 0 0 1\n0i0 0 0 1 0\n0 0i0 1 0 0\n0 0 0p\n2 0 0 0\n0 0i0\u00001 0 0\n0\u0000i0 0 0 1 0\ni0 0 0 0 0\u000011\nCCCCCCCA: (A5)\nForpanddorbitals, only the corresponding subblocks of\nEq. (A5) are needed.\nSimilarly, for quantization along other directions, the\nmatrix can be rotated by using Wigner matrix\nHso(^n) =\u0018\n2U(\u0012;')(L\u0001S)Uy(\u0012;'); (A6)\nand\nU(\u0012;') = \nei\u001e\n2cos\u0000\u0012\n2\u0001\ne\u0000i\u001e\n2sin\u0000\u0012\n2\u0001\n\u0000ei\u001e\n2sin\u0000\u0012\n2\u0001\ne\u0000i\u001e\n2cos\u0000\u0012\n2\u0001!\n; (A7)where\u0012and'are the angles of the direction of\nmagnetization when the unit vector is de\fned by ^n=\n(sin\u0012cos';sin\u0012sin';cos\u0012). When the spin quantization\naxis is along [110] direction, the SOC Hamiltonian can be\nwritten as\nHso(^ n110)\n=\u0018\n2 p\n2\n2(Lx+Ly)\u0000Lz+ip\n2\n2(Lx\u0000Ly)\n\u0000Lz\u0000ip\n2\n2(Lx\u0000Ly)\u0000p\n2\n2(Lx+Ly)!\n:\n(A8)\nAppendix B: Charge and moment calculated in TB\nand VASP\nTable IV lists the site-resolved charge and magnetic\nmoments in L10compounds calculated in TB and vasp .\n[1] J. H. van Vleck, Phys. Rev. 52, 1178 (1937).\n[2] R. McCallum, L. Lewis, R. Skomski, M. Kramer, and\nI. Anderson, Annual Review of Materials Research 44,\n451 (2014).\n[3] I. G. Rau, S. Baumann, S. Rusponi, F. Donati,\nS. Stepanow, L. Gragnaniello, J. Dreiser, C. Piamonteze,\nF. Nolting, S. Gangopadhyay, O. R. Albertini, R. M.\nMacfarlane, C. P. Lutz, B. A. Jones, P. Gambardella,A. J. Heinrich, and H. Brune, Science 344, 988 (2014).\n[4] D. D. Koelling and B. N. Harmon, Journal of Physics C:\nSolid State Physics 10, 3107 (1977).\n[5] C. Li, A. J. Freeman, H. J. F. Jansen, and C. L. 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B 76, 165106 (2007)."    },    {        "title": "1902.03536v1.Spin_dynamics_of_anisotropic_azimuthal_modes_in_heterogeneous_magnetic_nanodisks.pdf",        "content": "Spin dynamics of anisotropic azimuthal modes in heterogeneous \nmagnetic nanodisks \nHuirong Zhao1,2,3, Zhengyun Lei1,2,3 and Ruifang Wang1,2,3,4 \n1 Department of Physics, Xiamen University, Xiamen 361005, China \n2 Institute of Theoretical Physi cs and Astrophysics, Xiamen Unive rsity, Xiamen 361005, China \n3 Collaborative Innovation Centre f or Optoelectronic Semiconducto rs and Efficient Devices, Xiamen \nUniversity, Xiamen 361005, China \n4. Key laboratory of low dimensional condensed matter physics, Department of Education of Fujian Province, \nXiamen University, Xiamen 361005, China \nEmail: wangrf@xmu.edu.cn \nAbstract \nIt is well known that azimuthal spin wave modes of magnetic vor tex state in permalloy nanodisks \nhave circular symmetry. Intuitively, magnetic materials having magnetocrystalline anisotropy is not \ncompatible with the circular symmetry of the azimuthal modes. I n this article, however, we report cubic \nazimuthal modes in heterogeneous nanodisks consisting of a perm alloy core and a Fe shell. The fourfold \nsymmetry of azimuthal modes is due to the exchange, and magneto -static, interactions between the \npermalloy core and the Fe shell. In comparison to results of ci rcular azimuthal mode, the vortex \nswitching occurs considerably faster under the excitation of cu bic azimuthal mode. The gyration path \nof vortex core turns into square under the influence of induced  cubic anisotropy in the Py region. We \nfind out periodic oscillation of the vortex core size and the g yration speed as well. Our findings may \noffer a new route for spintronic applications using heterogeneo us magnetic nanostructures.  \n    \n \n \n 1. Introduction \nMagnetic vortex (MV) is a common ground state for soft ferromag netic thin-film \nstructures, ranging from a few microns down to 100 nanometers o r so.[1,2] In a MV, the \nmagnetization circulates in-plane around a core where the magne tization orients \nperpendicular to the sample plane. The vortex core (VC) magneti zation can be either \nupward or downward, denoting the vortex polarity p = +1 or p = −1, respectively. Spin \ndynamics of VC reversal under the excitation of ac magnetic fie ld[3-8] and spin current[9-\n11] receives intensive research interest recently, because of i ts importance in fundamental \nphysics and technological applications.[1,12] Spin wave excitat ion and its interaction with \nthe VC play a key role in the dynamic reversal of VC polarity. A MV state features three \nkinds of spin wave modes, namely the gyrotropic[13-15], azimuth al[16-18] and \nradial[16,17,19,20]. The sub-GHz gyrotropic mode is characteriz ed by the VC’s spiral \nmotion under the excitation of an in-plane ac field. The azimut hal and radial modes, on the \nother hand, have much higher eigenfrequencies (~10 GHz). Thus, the azimuthal and radial \nmodes can drive VC switching ten times faster than the gyrotrop ic one.[5,18]  \nThe azimuthal spin wave modes are characterized by their radial  (n = 1, 2, 3, …) and \nazimuthal mode numbers ( m = 1, 2, 3, ...). The clockwise (CW) and counterclockwise \n(CCW) azimuthal modes correspond to m < 0 and m > 0, respectively. According to \nconventional knowledge, only materials with zero magnetocrystal line anisotropy are \nsuitable for the azimuthal spin wave, since the existence of an isotropic energy will suppress \nthe transmission of this mode along azimuthal direction. Up to now, studies on azimuthal \nspin wave have been focused on permalloy (Py: Fe 0.2Ni0.8) nanodisks, which have no \nmagnetocrystalline anisotropy.[5,18,21,22] Here, we report nove l spin wave dynamics of \nazimuthal modes in a heterogeneous nanodisk consisting of a Py core and a Fe shell. In \ncontrast to the results in uniform Py disks, our micromagnetic simulations manifest clear \nfourfold symmetry for the azimuthal spin wave in the Py region,  due to the exchange and \nmagneto-static interactions between the Fe shell and Py core. F urthermore, the anisotropic \nazimuthal spin wave drives considerably faster VC switching, in  comparison to the results \nof homogeneous disks. Meanwhile, the size of VC and the gyratio n speed oscillate \nperiodically as the VC gyrates along a square path.    \n2. Micromagnetic simulation model \n      The core-shell nanodisk in study is illustrated in Fig. 1 (a). The Py core is 100 nm in \nradius and the surrounding Fe shell has width of 50 nm. The dis k thickness is set as 20 nm. \nWe use typical Py/Fe material parameters for micromagnetic simu lations[23] in this paper: \nsaturation magnetization Ms = 800/1714 KA/m, exchange stiffness constant Aex = 13/21 \npJ/m, magnetocrystalline anisotropy Kc =  0 / 4 7  K J / m3 and Gilbert damping constant = \n0.01/0.01. The exchange stiffness constant across the interface  of Py/Fe is set as int\nexA = 16 \npJ/m, which is the harmonic mean of Aex of Fe and Py. Experimentally, the core-shell \nnanodisks can be prepared using advanced nanofabrication techni ques.[24] The mesh cell \nsize for the simulations is 1.5 ⅹ1.5 ⅹ20 nm3.  Additional simulations using cell size of \n1.5 ⅹ1.5 ⅹ10 nm3 verified the reliability of re sults obtained in this paper. \nFig. 1 (a) Schematic view of the core-shell nanodisk with indic ated dimensions. The Py core (in magenta) has a radius \nof 100 nm and the Fe shell (in blue) is 50 nm in width. In its initial state, the magnetic vortex has counterclockwise \nmagnetization rotation and VC polarity p = +1. (b) FFT power spectrum of the damped oscillation of < mx> after the \nexcitation of an in-plane sinc-function field. (c) FFT amplitud e (top row) and phase (bottom row) images of the six \nazimuthal modes. (d) - (e) Distribution of the in-plane compone nts of the magnetocrystalline an isotropy field (d) and the \nanisotropy energy density (e). In (d), arrows and color shades denotes the amplitude and directions of the \nmagnetocrystalline anisotropy field, respectively. In (e), the color shade indicates the magnitude of the anisotropy energy \ndensity. The solid black lines correspond to locations with the  lowest magnetocrystalline anisotropy field in (d), and the \nhighest anisotropy energy density in (e). The cubic symmetry of  both the anisotropy field and anisotropy energy density \nof the Fe shell imprints into the Py core, as illustrated by th e dotted black lines that connect  with the solid black lines.  \n \n3. Azimuthal modes with fourfold symmetry \nThe sample is initially in a relaxed ground state with CCW magn etization curling \nsense and VC polarity p = +1, as shown in Fig. 1(a). The diameter of the VC is 11.9 nm , \nwhich is defined in such a way that the z component of magnetization ( mz = Mz / Ms) at the \nedge of core is 30% lower than that at the peak. To stimulate t he spin wave modes of the \nv o r t e x  s t a t e ,  w e  a p p l y  a n  i n - p l a n e  s i n c - f u n c t i o n  f i e l d  \n00 0 22 () [ ( ) ] ) ] e[ (  ρ\nx BtB s i n ftt ftt  with B0 = 1 mT, t0 = 1 ns and f = 50 GHz. We \nthen conduct fast Fourier transformation (FFT) on the subsequen t temporal oscillation of \n<mx> = < Mx> / Ms, namely the x-component of magnetization averaged over the entire disk. \nThe resultant FFT power spectrum shown in Fig. 1(b) contains se ven resonance peaks.  The \npeak denoted as G0 at 0.595 GHz corresponds to the gyrotropic mode. The other pea ks are \nfound at 9.0, 11.5, 15.8, 17.7, 21.0, and 22.7 GHz,  respectively, which correspond to the \neigenfrequencies of (n, m) = (1, 1), (1, +1), (2, 1), (2, +1), (3, 1), and (3, +1) azimuthal \nmodes,  respectively . The small peaks in Fig. 1(b) originate from irregular standin g waves, \nwhich will disappear if we set kc = 0 for the Fe shell.   \nFigure 1(c) displays the FFT amplitude and phase images of the azimuthal modes. To \nunderstand the fourfold symmetry shown in Fig. 1(c), we plot in  Figs. 1(d) and 1(e) the \nspatial distribution of the magnetocrystalline anisotropy field  and the anisotropy energy \ndensity[25] respectively, for the Fe shell in the relaxed groun d state.  The cubic anisotropy \nfield in the Fe shell is expressed as (2 )cos cos2  ani c sBk M , in which 𝜃 is the angle \nbetween magnetization and the nearest easy axis, which is defin ed as the x, y and z axes in \nthe sample. The maximum anisotropy field, i.e., 25 4 . 8  cskM m T , is found along the \neasy axes. Whereas the minimum, 0aniB, occurs at =/ 4  o r  3 / 4 . The density of \nanisotropy energy is defined as 22 22 2 2\n12 23 31()   ani cEk , where123, ,  are \nthe direction cosines of magnetization with respect to the easy  a x e s .  This directional \ndependence of the crystalline anisotropy energy is disadvantage ous for  transmission of \nazimuthal spin waves in the Fe shell, because of the energy bar riers at \n=/ 4  a n d  3 / 4 .  A s  a  r e s u l t ,  t h e  P y  c o r e  h a s  m u c h  l a r g e r  F F T  a m p l i t u d e ,  i . e .  \nstronger spin wave oscillation, than the Fe shell, as shown in Fig. 1(c). On the other hand, \nthe azimuthal modes in the Py core shows clear fourfold symmetr y in Fig. 1(c). This result indicates that the Fe shell induces cubic anisotropy in the Py core, due to the exchange and \nm a g n e t o - s t a t i c  c o u p l i n g  b e t w e e n  t h e  c o r e  a n d  t h e  s h e l l .  T h i s  i n duced cubic symmetry \nresults in nontrivial spin dynam ics that we will discuss in det ail below.  \n \n4. Dynamics of the anisotropic azimuthal mode \nAfter identifying the azimuthal modes in this core-shell nanodi sk, we apply an in-plane \nCW rotating field with amplitude of 8 mT and frequency of 9 GHz  to excite the ( n = 1, m \n= 1) mode. The snapshots of the mz spatial distribution shown in Fig. 2 illustrates the spin \ndynamics until the VC switching occurs. At t = 0 ps, the vortex is in the relaxed ground \nstate. After excitation by the in-plane rotating field, the VC moves away from the disk \ncenter due to the interaction of azimuthal spin wave with the g yrotropic mode.[5,22] For \nthe m = −1 mode in study, the gyrofield[26] and spin wave oscillatio n acts constructively, \nthus forms relatively broad negative and positive mz regions, to the left and right side of \nthe core, respectively, in relation to the direction of motion. [5] Note that the VC, along \nwith the negative and positive mz regions, deforms periodically whenever they travel across \nthe anisotropic energy barriers. Specifically, they are compres sed (or stretched) when they \nare approaching (or leaving) the barriers at = /4 and 3 / 4  , in order for the system \nto keep its total energy low. A vortex-antivortex pair[8,26] is  created at t = 689 ps, inside \nthe negative 𝑚௭ region  as the spin wave oscillation intensifies.  The antivortex then \nannihilates with the original vortex core at t = 751 ps, leaving the surviving vortex with \nreverse polarity in the dot. This completes the VC reversal pro cess. In comparison, VC \nreversal in a homogeneous Py disk of same dimensions takes 1056  ps, driven by a 9 mT \nresonant field that excites the ( n = 1, m = 1) mode.[18]  \nFig. 2 Snapshot images showing spin dynamics of the magnetic vo rtex after application of a resonant in-plane rotating \nfield to excite the ( n = 1, m = 1) mode. Below each snapshot is top view of a 150  150 nm2 area in the center of the \ndisk. The apex of VC is marked a s a white dot. The dotted diago nal lines indicate induced ani sotropic energy barriers in \nthe Py core.  \n \n      The gyration trajectory of VC, illustrated in Fig. 3(a), manifests an unusual feature of \nthe anisotropic azimuthal mode. We plot the trajectory by recor ding positions of the apex \nof the VC. It shows that the VC spirals around the disk center in a clockwise sense at a \nf r e q u e n c y  o f  8 . 8  G H z ,  s l i g h t l y  l e s s  t h a n  t h e  e i g e n f r e q u e n c y  o f  the 1m  mode. This \nbehavior is consistent with previous studies on homogeneous vor tices.[18] However, in \ncontrast to the circular gyration in homogeneous disks[5,18], t he VC trajectory in Fig. 3(a) \ntransforms from circular into square after t = 294 ps, when the VC is about 6.4 nm away \nfrom the disk center. Such transformation indicates that the VC  interacts with growing \ninduced cubic anisotropy as it moves toward the Py/Fe interface .  \n\nFig. 3 (a) The VC trajectory under the excitation of the ( n = 1, m = 1) mode. (b) Variations of gyration speed and the \nsize of VC with time. The circled numbers in (b) correspond to the snapshot images with same numbering in figure 2. \n \nFurthermore, we show temporal evolution of the gyration speed ( vcore) and diameter \n(dcore) of VC in Fig. 3(b). Before t = 294 ps, vcore increases in general with time, while \ns h o w i n g  s o m e  i r r e g u l a r  o s c i l l a t i o n s .  A t  t h e  s a m e  t i m e ,  dcore decreases almost \nmonotonically. These behaviors are similar to the results obtai ned in homogeneous \nnanodisks[5,18,21], since the induced cubic anisotropy is weak in close proximity to the \ndisk center. At t = 294 ps, the core velocity has increased to 351 m/s and the c ore diameter \nis reduced to 10.3 nm. After t = 294 ps, however, both vcore a n d  dcore oscillates with \nfrequency of 34.4 GHz, which is exactly four times as large the  gyration frequency of VC. \nThe reason is that induced cubic anisotropy in Py causes VC dec eleration and compression, \nwhen the VC is approaching energy barriers at = /4 and 3 / 4  ; the opposite is true \nwhen the VC is leaving the barriers.  \n\nFig. 4 Phase diagrams of VC reve rsal under the excitation of cl ockwise (left) and countercl ockwise (right) rotating \nfields, respectively. The logarithmic color scale denotes excit ation time until VC switching. \n \nFinally, Fig. 4 shows how long for the VC to reverse after a ro tating field is applied. \nThe field amplitude varies from 10 mT to 30 mT, and the field f requency keeps in the range \nof 6 to 24 GHz. As expected, the VC switching occurs faster for  higher field amplitude. \nThe local minima coincide with the eigenfrequencies of azimutha l modes, a clear indication \nof resonant behavior. Note that the gyrofield acts constructive ly with the spin wave for the \nm = 1 mode, but destructively for m = +1.[5] Therefore, VC switching is faster and occurs \nin wider frequency range under excitation of the m =  1 mode. However, for field \namplitude above 25 mT, the ( n = 1, m = +1) mode dominates and the unidirectional reversal \nof VC is lost.[5] \n      In summary, azimuthal modes in heterogeneous magnetic nan odisks show clear \nfourfold symmetry, due to interactions between the Fe shell and  the Py core. The large \ndifference in crystalline anisotropy across the Py/Fe interface  causes confinement of \nazimuthal spin wave inside the Py region. We observe markedly f aster VC switching, in \ncomparison to the results of homogeneous disks. The VC gyration  trajectory turns into \nsquare under the influence of induced cubic anisotropy in Py, w hich results in periodic \noscillations of both the gyration speed and size of the VC as w ell.  \nWe are grateful for the financial support from the National Nat ural Science \nFoundation of China under Grant Nos. 10974163 and 11174238. \n \n[1] R.Hertel,NatureNanotech. 8,318(2013).\n[2] A.Wachowiak,J.Wiebe,M.Bode,O.Pietzsch,M.Morgenster n,andR.Wiesendanger,\nScience298,577(2002).\n[3] B.VanWaeyenberge etal.,Nature444,461(2006).\n[4] B. Pigeau, G. de Loubens, O. Klein, A. Riegler, F. Lochner,  G. Schmidt, and L. W.\nMolenkamp,NaturePhys. 7,26(2010).\n[5] M.Kammerer etal.,Nat.Commun. 2,279(2011).\n[6] R.WangandX.Dong,Appl.Phys.Lett. 100,082402(2012).\n[7] M.E.Stebliy etal.,SciRep7,1127(2017).\n[8] R.Hertel,S.Gliga,M.Fähnle,andC.Schneider,Phys.Rev .Lett.98,117201(2007).\n[9] Y.Liu,S.Gliga,R.Hertel,andC.M.Schneider,Appl.Phy s.Lett.91,112501(2007).[10] Y.‐S.Choi,M.‐W.Yoo,K.‐S.Lee,Y.‐S.Yu,H.Jung,andS .‐K.Kim,Appl.Phys.Lett. 96,\n072507(2010).[11] S.‐K.Kim,Y.‐S.Choi,K.‐S.Lee,K.Y.Guslienko,andD.‐ E.Jeong,Appl.Phys.Lett. 91,082506\n(2007).\n[12] S.‐K.Kim,J.Phys.D:Appl.Phys. 43,264004(2010).\n[13] K.Y.Guslienko,B.Ivanov,V.Novosad,Y.Otani,H.Shima ,andK.Fukamichi,J.Appl.Phys.\n91,8037(2002).\n[14] Y.LiuandA.Du,J.Appl.Phys. 107,013906(2010).\n[15] F.A.Apolonio,W.A.Moura‐Melo,F.P.Crisafuli,A.R.P ereira,andR.L.Silva,J.Appl.Phys.\n106,084320(2009).\n[16] K.Y.Guslienko,W.Scholz,R.Chantrell,andV.Novosad, Phys.Rev.B 71,144407(2005).\n[17] M.Buess,R.Höllinger,T.Haug,K.Perzlmaier,U.Krey,D .Pescia,M.Scheinfein,D.Weiss,\nandC.Back,Phys.Rev.Lett. 93,077207(2004).\n[18] M.‐W.YooandS.‐K.Kim,J.Appl.Phys. 117,023904(2015).\n[19] Z.Wang,M.Li,andR.Wang,NJPh 19,033012(2017).\n[20] H.ZhaoandR.Wang,J.Magn.Magn.Mater. 465,495(2018).\n[21] M.Sproll etal.,Appl.Phys.Lett. 104,012409(2014).\n[22] K.Y.Guslienko,A.N.Slavin,V.Tiberkevich,andS.‐K.K im,Phys.Rev.Lett. 101,247203\n(2008).\n[23] WeusedLLGMicromagneticS imulatorver.3.14ddevelopedb yM.R.Scheinfeintocarry\noutmicromagneticsimulations.[24] G.Duerr,M.Madami,S.Neusser,S.Tacchi,G.Gubbiotti, G.Carlotti,andD.Grundler,Appl.\nPhys.Lett. 99,202502(2011).\n[25] J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge University Press,\nCambridge,2010).[26] K.Y.Guslienko,K.‐S.Lee,andS.‐K.Kim,Phys.Rev.Lett .100,027203(2008).\n\n "    },    {        "title": "1902.05120v1.Design_of_Face_Centered_Cubic_Co81_8Si9_1B9_1_with_High_Magnetocrystalline_Anisotropy.pdf",        "content": "1 Design of Face Centered Cubic Co 81.8Si9.1B9.1 with High Ma gnetocrystalline Anisotropy  \nDr. Jiliang Zhang, Dr. Guangcun Shan*, Dr. Yuefei Zha ng, Dr. Fazh u Din g and Prof.  Dr. \nChan Hung Shek* \nDr. Jiliang Zhang, Dr. Guangcun Shan*, Prof. Dr. Chan Hung Shek \naDepartment of Materials Science, City University of Ho ng Kong, Hong Kong Email:gcshan-2@my.cityu.edu .hk, apchshek@cityu.edu.hk \nDr. Guangcun Shan,  \nbSchool of Instrument Science and Opto-electronic Engineering, Beihang Univ ersity, Beijing 100191, China; \nDr. Yuefei Zhang \ncInstitute of Microstructure and Property of Advanced Materials, Beijing University of Technology, Beijing 100124, China \nDr. Fazhu Ding \ndInstitute of Electrical Engineering, Chinese Academy of Sciences, Beijing 1001 90, China \neUniversity  of Chinese Academy of Sciences, Beijing 100049, China \nKeywords: cobalt, mag netization, magnetocrystalline anisotro py, atomic cluster  \nDespite the composition close to glassy forming alloys, face centered cubic (FCC) \nCo81.8Si9.1B9.1, designed based on Co9B atomic cluster (po lyhedral) , are synthesized as single-\nphase ribbons successfully. These ribbons, with grain sizes of ca. 92 nm, show supreme \nductility and strong orientation along (111), which couples with shape aniso tropy leading to \nhigh magnetocrystalline aniso tropy comp arable to Co rich Co-Pt nanoscale thin films, with a \ncoercivity of 430 Oe and squareness of 0.82 at room temperatur e. The stability and magnetic \nbehaviors of the phase are discussed based on experimental electr onic structure. This work \nnot only develops low cost Co-based materials for hard magnetic applications, but also \nextends the atomic cluster model developed for amor phous alloys into the design of new \ncrystalline materials.   2 Meta\nllic cobalt (Co) has been widely studied as an important ma gnetic material and its \nmagnetic properties can vary significantly depending on its str uctures:[1] the face centered \ncubic (FCC, stable above 450 °C) phase shows good soft magnetic  behaviours, and \namorphous CoSiB materials based on the structure (with a genera l composition at ca. \nCo80(Si,B) 20) are one of best soft magnetic materials, used in many high-performance electric \ndevices;[2] the hexagonal close-packed (HCP, stable below 450 °C) Co shows hard magnetic \nbehavior because of the structural anisotropy, and its compound s with rare earth (RE ) \nelements (e.g. SmCo 5) or noble metals (e.g. CoPt) do show significantly enhanced hard \nmagnetic behaviours due to the hi gh anisotropy of RE elements or strong 3 d-5d \nhybridizations.[3,4] The high cost of these materials inhibits their applications a s permanent \nmagnets in many cases. On another hand, many of these compounds  show unsatisfactory \nmechanical performance. The fabrication of nanomaterials especi ally nanowires does improve \nthe hard magnetic behaviours significantly, [5, 6] but the complex process and oxidization are \nmain drawbacks. \nIn metallic glasses, it is generally agreed that there are one or two kinds of atomic clusters \n(short-range orders) dominating the structure and thus controll ing the physical properties. [7] It \nis reported that based on different dominant clusters, CoSiB me tallic glasses with close \ncompositions but significantly different structure and magnetic  behaviors (from \nferromagnetism to anitferromagnetism) can be achieved.[8, 9] In infinite solid state solutions, it \nis also found that their atomic structures are not completely d isordered. Instead, dominant \natomic clusters determining the magnetic properties also exist in these compounds, and can \nvary depending on the compositions. [5,6,10] Thus, it is possible to design this kind of magnetic \ncompounds based on the dominant atomic clusters by using the at omic cluster model \npreviously developed for amorphous alloys.[8] 3 CoB/CoSiB \nmetallic glasses are widely studied in both structure  and magnetic properties \nsince 1970s. The atomic clusters favouring glassy formation hav e to be avoided, because the \nabsence of anisotropy in glassy states will deteriorate the har d magnetism. According to Co-B \nphase diagram,[1] there is a Co rich FCC phase, Co 23B6, stable at high temperature. Thus the \ncluster favouring the FCC structure can be extracted from this compound,[11] a s  C o 9B \npolyhedral (see the inset structure in Figure 1a ). In order to reduce the cost and increase the \nmagnetic anisotropy, it is better to include Si and B as much as possible while remaining the \nFCC or HCP Co-type structure. Th e maximum solubility of Si is 1 0 at. % (Co:Si=9:1) in HCP \nCo and 18 at. % in FCC Co.[1] Si concentration more than the maximum solubility may favour \nthe amorphous state instead of Co structure. Thus the compositi on is designed as Co 9BSi, \nnormalized as Co 81.8Si9.1B9.1. The synthesized structure is expected to be FCC Co type, \nbecause the dominant cluster is from FCC Co 23B6. As a result, in our work, we obtained the \nFCC Co-type CoSiB ribbons successfully, with high magnetocrystalline anisotropy. The structural stability and magnetic behaviours of the compound ca n also be understood based on \nexperimental elect ronic structures. \nThe X-ray diffraction (XRD) pattern in Figure 1a  shows the dominant (111) reflection of the \nFCC phase with no visible trace of amorphous hump, affirming th e single-phase nature of the \nas-made ribbons. The absence of other reflections suggests the strong orientation along (111). \nThe morphologies of these ribbons indicate the good ductility o f these samples, as shown in \nthe inset of Figure 1(a). The lattice parameter estimated from the (111) reflection is a=3.514 \nÅ, smaller than the value 3.537-3.558 Å of FCC Co at room tempe rature.\n[12] It is in agreement \nwith the small atomic size of the substituting elements B and S i. \nThe (111) reflection peak is broadened obviously, suggestive of  the nanocrystalline nature. \nThe apparent crystallite size was estimated as ca. 92 nm using the Scherrer’s formula.[13] The 4 nanocrystals \nof Co 81.8Si9.1B9.1 were confirmed by transmission  electron microscopy (TEM), as \nshown in Figure 1b . All observed crystals are less than 100 nm in critical size, in agreement \nwith XRD results. No diffuse ring observed in selected area ele ctron diffraction (see the inset \nin Figure 1b ) confirms the absence of any amorphous phases. Most crystals are not spherical \nbut elongated along the spinning direction. The isotropic shape  surely benefits the hard \nmagnetic behaviours. \nMagnetic hysteresis loops were measured under applied fields along longitude direction and \nperpendicular to ribbons at room temperature. As shown in Figure 2 , Co 81.8Si9.1B9.1 exhibits \nhigh magnetocrystalline anisotr opy with (111) easy direction. S uch strong anisotropy was \nmainly observed in nanoscale thi n film or nanomaterials of anis otropic shapes. [6, 14] \n      The magnetization is almost saturated at 2000 Oe along th e easy direction close to the \nfield to saturate soft CoB/CoSiB amorphous alloys, and at 10000 Oe along the perpendicular \ndirection close to the field for bulk and nanoscale Co in both FCC and HCP structures.  \nHowever, the saturation magnetization ( Ms) is only 31.7 emu/g, corresponding to 28 emu per \nCo gram, much smaller than the reported Ms in bulk Co and most nanoscale Co.[15, 16] \nObviously the reduced Ms is not due to the dilution effect  of the substituted elements. It was \nreported that Co nanoparticles of 2 nm in size show a smaller Ms, very likely owing to the \nsurface oxidation. [17] Apparently, it is not the case here. Though antiferromagnetic CoSiB \namorphous alloys are also reported,  the small filed for saturat ion along (111) rules out the \npossibility. Thus the reduced Ms should result from the part ial delocalization of Co 3 d \nelectrons due to the strong interaction with B or Si. \nThe coercivity ( Hc) is ca. 430 Oe, in both easy and hard directions, as evident i n the inset of \nFigure 3 . The coercivity is much larger than bulk Co and most of reported nanoscale HCP \nand FCC Cobalt (10-300 Oe),[15, 17] and close to some Co-Pt thin films.[14] Despite the reduced 5 Co \nmagnetic moment, the saturation remanence ( Mr) is still up to 26 emu/g with a high \nsquareness ( Mr/Ms) of 0.82. With proper tuning, the materials will be promising hard \nmagnetic materials. \nTo reveal the origin of the improvement in hard magnetization, magnetic domains in the samples were observed by magnetic force microscaopy (MFM). The samples show (see \nFigure 3 ) distinctive feature of dark a nd bright stripe magnetic domian  structure. The stripe \ndomains are often observed in thin ferromagnetic films and plat elets with weak perpendicular \nor oblique anisotropy.\n[18] In ferromagnetic films, the period of strip domains is comparable to \nthe film thickness. [18] However, the period (width) of the strip domain in the Co 81.8Si9.1B9.1 \nribbon is ∼ 90 nm, which is much less tha n the thickness of the ribbon ( ∼ 20 µm). Instead, the \nperiod is very close to the longitudal size of the nanocrystal grains. In the view, the stripe \nmagnetic domains in the Co 81.8Si9.1B9.1 ribbons here should mainly result from the anisotropic \nshape of CoSiB nanocrystals. Thus the hard magnetic behaviour o f the materials can be \nfurther improved by tuning grain sizes. \nDespite the fact that the composition of the FCC Co 81.8Si9.1B9.1 is very close to compositions \nof many CoSiB amorphous alloys, neither amorphous nor crystalli ne impurities were \nobserved in XRD patterns and TEM observations. It is already re ported that two CoSiB \namorphous alloys having very close compositions show distinctiv e differences in their \nmagnetisms, due to different short-range orders or clusters. [8, 9] To understand the origin of \nthe special structure and magnetism of the present Co 81.8Si9.1B9.1 phase, electronic structures \naround fermi level ware measured on the present Co 81.8Si9.1B9.1 phase, HCP Co and a soft \nmagnetic CoSiB alloy Co 63.1B27Si9.9. Because the major phase after crystallization of soft \nmagnetic CoSiB amorphous alloy is generally the FCC Co structur e, the electronic structure \nof soft magnetic CoSiB amorphous alloy should be similar to tha t of FCC Co.[8, 19]  6 Com\npared with the electronic structures of HCP Co (see Figure 4 ), many soft magnetic \nCoSiB alloys show similar feature around the fermi level: peak electronic density of state \n(DOS, N(E)) at fermi level followed by a shoulder at 1.5 eV (bl ack arrows in Figure 4 ). [8, 20] \nBoth HCP Co and CoSiB amorphous alloy have large DOS close to t he peak value at their \nfermi level, while the FCC Co 81.8Si9.1B9.1 exhibits a small DOS at the fermi level. It is evident \nthat the FCC Co 81.8Si9.1B9.1 is more stable than both hcp Co and amorphous CoSiB alloys. As  \nthe electronic structure around the Fermi level determines the magnetic behaviours, the Co \natoms in both HCP Co and amorphous CoSiB show similar magnetic moment, in agreement \nwith literatures. The shoulder at 1.5 eV corresponds to the ele ctrons in the spin-down states, \nwhile the peak at Fermi level is from spin-up electrons.[20] F o r  t h e  F C C  C o 81.8Si9.1B9.1, the \nshift of peak towards high bindi ng energy and the absence of the shoulder indicate that the \nstrong interactions between Co and B/Si leads to the delocalisa tion of Co 3 d electrons and \nthus low Co moments.  It is also supported by the fact that the  hallmark peak of localized B \n2p-Co 3d hybridisation at ∼ 8 eV in soft magnetic CoSiB is not observed in the present FCC  \nCo81.8Si9.1B9.1 phase.[21] Such delocalization yielding significantly reduced magnetic mom ents \nwas also observed in FeBY glassy alloys.[22] Obviously, the compositions of FCC CoSiB \nphase are tuneable to achieve a balance between better magnetic  performance and structural \nstability.  \nBased on the Co9B cluster extracted from FCC Co 23B6, the low-cost FCC Co 81.8Si9.1B9.1 \nribbon was successfully produced. Despite the high content of m etalloid Si and B, the as –\nmade ribbons are ductile and show significantly enhanced hard m agnetic behaviours similar \nto Co-Pt nanoscale thin film, with a room-temperature coercivit y of 430 Oe, much larger than \nthe value of most reported bulk and nanoscale Co. The wall thic kness of magnetic domains is \nclose to the critical size of Co 81.8Si9.1B9.1 nanocrystals, indicating the important role of these \nnanocrystals on the magnetism.  The structural stability and sp ecial magnetism of the present 7 m\naterials are well explained based on the measured electronic s t r u c t u r e s .  T h e  r e s u l t s  a l s o  \nshow that the hard magnetic behaviours can be further improved without destruction of the \nstructure, although a balance between structural stability and magnetic performance has to be \ntaken into account. \nExperimental Section  \nSynthesis of FCC Co 81.8Si9.1B9.1: the mixtures (10 g) of pure constituent elements Co chunks \n(99.99 wt. %), Si chunks (99.99 wt. %), and B particles (99.9 wt. %) under a Ti-gettered \nargon atmosphere. The alloy ingots were remelted four times to improve compositional \nhomogeneity. Using these master ingots, ribbon samples with thi ckness of ∼30 µm and width \nof ∼2 mm were produced by a single r oller melting-spinning apparatu s  a t  a  w h e e l  s u r f a c e  \nvelocity of 40 m/s. Characterization and measurements : Phase identification was conducted on a Philips X’pert \nx-rays diffractometer (Cu Kα, 0.15406 nm). Microstructure obser vations were done with a\nhigh-resolution transm ission electron microscope (HRTEM; model No. TECNAI 20 G2, FEI \nCompany, Hillsboro, OR) operated at 200 kV.  Ribbons for TEM ob servations were thinned \nby twin-jet electropolishing using a HClO\n4-C2H6O solution (volume ratio 1:8) at ∼ −30 °C. \nThese specimens were finally cleaned by low-angle ion milling f or no more than 5 min for \nTEM observation. Magnetic measurements were carried out using a  Cryogenic vibrating \nsample magnetometer with a maximum field of up to 20000 Oe. Surface morphologies and magnetic domain structures of ribbons were observed using a Vec co atomic force microscope \nand a magnetic force microscope in the tapping/lift scanning mo de with a CoCr-coated tip \nmagnetized downward. Measurements of electronic structure were performed using x-ray \nphotoelectron spectroscopy with a  monochromatized Al Kα radiati on (1486.6 eV) in an \nULVAC-PHI 5802 system (Kanagawa, Japan). 8 Acknowledgements \nThe work was financially supported by National key research and developm ent program \n(no. 2016YFC1402504) by \nMOST of China and funded by a General Research Fund from \nResearch Grants Council, Hong  Kong SAR Government (Project number: CityU 11211114).      \n \nReferences \n[1]Binary Alloys Phase Diagrams , (Eds: T.B. Massalski), ASM I nternational, Materials Park,\nOH, 1990. \n[2] a) F. Alves, R. Lebourgeois, T. Waeckerle, International Transactions on Electrical\nEnergy Systems , 2005, 15, 467; b) P. Quintana, E. Amano, R. Valenzuela, J.T.S. Irvine, J. \nAppl. Phys. 1994, 75, 6940. [3] Y.L. Hou, Z.C. Xu, S. Peng, C.B. Rong, J.P. Liu, S.H. Sun, Adv. Mater. , 2007, 19 , 3349.\n[4] D. Alloyeau, C. Ricolleau, C . Mottet, T. Oikawa, C. Langois , Y. Le Bouar, N. Braidy, A.\nLoiseau, Nat. Mater.  2009 , 8, 940. \n[5] M. Vazquez, L.G. Vivas, Phys. Status Solidi B  2011, 248, 2368.\n[6] C. Bran, Yu.P. Ivanov, D.G. Trabada, J. Tomkowicz, R.P. del  Real, O. Chubykalo-\nFesenko, M. Vazquez, IEEE Trans. On Magnet.  2013, 49, 491. \n[7] a) D.B. Miracle, Nat. Mater.  2004 , 3, 697; b) C. Dong, Q. Wang, J.B . Qiang, Y.M. Wang,\nN. Jiang, G. Han, Y.H. Li, J. Wu, J.H. Xia, J. Phys. D Appl. Phys.  2007, 40, R273.\n[8] J. Zhang, Y. Wang, C.H. Shek, J. Appl. Phys.  2011, 110, 083919.\n[9] G. Shan, J. Zhang, J. Li, S. Zhang, Z. Jiang, Y. Huang, C.H . Shek, J. Mag. Mag. Mater.\n2013, 352, 49. [10] A. Froideval, R. Iglesias , M. Samaras, S. Schuppler, P. Na gel, D. Grolimund, M.\nVictoria, W. Hoffelner, Phys. Rev. Lett.  2007, 99 , 237201. \n[11] H. Hillebrecht, D. Kotzott, M. Ade, J. Solid State Chem.  2009, 182, 538.10 \nFigure 1.  PXRD pattern (a) and TEM\n observation (b) of as-cast Co 81.8Si9.1B9.1 ribbons. The \ninset in (a) shows pictures of r ibbons and the Co9B cluster ext racted from Co 23B6 (yellow \nballs for B and blue balls for Co ). The inset in (b) gives the selected area electron diffraction \ncorresponding to the observed zone. 11 \nFigure 2.  Hysteresis loops of Co 81.8Si9.1B9.1 ribbons measured with a field of 20000 Oe \napplied along (black) the longitu dinal direction and perpendicu lar (red) to ribbons. \nFigure 3 \nFigure 3.  Magnetic domain structure ( left, square: 5x5 µm\n) of the Co 81.8Si9.1B9.1 ribbon and \nits magnified views (r ight, square: 2x2 µm). \nFigure 4.  Electronic structures of the Co 81.8Si9.1B9.1 phase(black), bulk HCP Co (red) and a \ntypical soft magnetic CoSiB glassy alloy (Co 63.1B27Si9.9). The black arrows mark a shoulder \nnext to the peak electronic DOS, and the blue arrow indicates t he shoulder induced by B-Co \ninteractions. 12 Title \nDesign of Face Centered Cubic Co 81.8Si9.1B9.1 with High Magnetocrystalline Anisotropy  \nToC figure  \n"    },    {        "title": "1902.05241v1.Database_of_novel_magnetic_materials_for_high_performance_permanent_magnet_development.pdf",        "content": "Database of novel magnetic materials for\nhigh-performance permanent magnet development\nP. Nievesa, S. Arapana,b, J. Maudesc, R. Marticorenac, N. L. Del Br\u0013 \u0010oa, A.\nKovacsd, C. Echevarria-Bonete, D. Salazare, J. Weischenbergf, H. Zhangf, O.\nYu. Vekilovag, R. Serrano-L\u0013 opeza, J.M. Barandiarane, K. Skokovf, O.\nGut\reischf, O. Erikssong,h, H.C. Herperg, T. Schre\rd, S. Cuesta-L\u0013 opeza,i\naICCRAM, International Research Center in Critical Raw Materials and Advanced\nIndustrial Technologies, Universidad de Burgos, 09001 Burgos, Spain\nbVSB Tech Univ Ostrava, IT4Innovations, 17 Listopadu 15, CZ-70833 Ostrava, Czech\nRepublic\ncDepartment of Civil Engineering, Universidad de Burgos, 09006 Burgos, Spain\ndCenter for Integrated Sensor Systems, Danube University Krems, Lower Austria, 3500\nKrems, Austria\neBCMaterials, Basque Centre for Materials, Applications and Nanostructures, UPV/EHU\nScience Park, 48940 Leioa, Spain\nfDepartment of Materials and Geosciences, TU Darmstadt, Darmstadt, 64287, Germany\ngDepartment of Physics and Astronomy, Uppsala University, Box 516, 75121 Uppsala,\nSweden\nhSchool of Science and Technology, Orebro University, SE-70182 Orebro, Sweden\niICAMCyL Foundation International Center for Advanced Materials and Raw Materials of\nCastilla y Le\u0013 on, 24492 Cubillos del Sil, Le\u0013 on, Spain\nAbstract\nThis paper describes the open Novamag database that has been developed for\nthe design of novel Rare-Earth free/lean permanent magnets. The database soft-\nware technologies, its friendly graphical user interface, advanced search tools and\navailable data are explained in detail. Following the philosophy and standards\nof Materials Genome Initiative, it contains signi\fcant results of novel magnetic\nphases with high magnetocrystalline anisotropy obtained by three computa-\ntional high-throughput screening approaches based on a crystal structure pre-\ndiction method using an Adaptive Genetic Algorithm, tetragonally distortion\nof cubic phases and tuning known phases by doping. Additionally, it also in-\ncludes theoretical and experimental data about fundamental magnetic material\nproperties such as magnetic moments, magnetocrystalline anisotropy energy,\n\u0003Corresponding author\nEmail address: pnieves@ubu.es (P. Nieves)\nPreprint submitted to Elsevier February 15, 2019arXiv:1902.05241v1  [cond-mat.mtrl-sci]  14 Feb 2019exchange parameters, Curie temperature, domain wall width, exchange sti\u000b-\nness, coercivity and maximum energy product, that can be used in the study\nand design of new promising high-performance Rare-Earth free/lean permanent\nmagnets. The results therein contained might provide some insights into the\nongoing debate about the theoretical performance limits beyond Rare-Earth\nbased magnets. Finally, some general strategies are discussed to design possible\nexperimental routes for exploring most promising theoretical novel materials\nfound in the database.\nKeywords: Database, magnetic materials, permanent magnets, Materials\nGenome Initiative, High-throughput, VASP, Computer simulation,\nNOVAMAG\n1. Introduction\nPermanent magnets (PMs) are materials with an internal structure capable\nof creating an external magnetic \feld by themselves. Nowadays, these materials\nplay an important role in critical sectors of our advanced society as transport, en-\nergy, information and communications technology [1]. The magnetic \feld source\ngiven by PMs is widely used in many technological applications, e.g. for inducing\nmechanical motion in electric motors, making sound in loudspeakers, generating\nelectrical energy in wind turbines, data storage in computer engineering, mag-\nnetic resonance imaging in healthcare industry, holding, oil dewaxing, etc [2, 3].\nIn many of these technologies, most e\u000ecient designs are frequently achieved us-\ning PMs. For instance, the omission of gearboxes in wind turbines based on PM\ngenerators considerably reduces maintenance, service costs and overall material\nutilization [4]. Similarly, PM-based motors have many advantages compared to\ninternal combustion engines including high e\u000eciency, compact size, light weight,\nand high torque [1].\nAt macroscopic scale, a magnet is described by extrinsic properties that\nare deduced from the hysteresis loop and determine its performance. These\nproperties are the maximum energy product (BH) max (related to the maxi-\n2mum magnetostatic energy stored and obtained from the second quadrant of\nthe B(H) hysteresis loop), saturation magnetization M s(total magnetization\nthat is achieved in a strong external magnetic \feld H that aligns all magnetic\ndomains), remanence M r(net magnetization after H is removed) and coerciv-\nity H c(magnetic \feld that reverses a half of the magnetization reducing the\noverall magnetization to zero, it determines the \"resistance\" against demagne-\ntization by external magnetic \felds). The larger (BH) max, the smaller amount\nof material is needed to generate a required magnetic \feld strength, so high\nvalues of (BH) maxare desired. Energy product and coercivity depend on the\nmagnet's shape due to its own demagnetizing \feld. The theoretical upper limit\nof energy product (BH) max=\u00160M2\nr=4, where\u00160is vacuum permeability, can\nbe ideally reached for a magnet shape with demagnetizing factor D= 0:5 and\nHc> M r=2 [5]. Extrinsic properties also depend on temperature, in fact they\ntypically decrease as temperature increases, reducing the magnet's performance,\nespecially close to the Curie temperature T C(i.e. ferromagnetic-paramagenetic\ntransition).\nThe macroscopic behavior is tightly linked to the microscopic properties\ncalled intrinsic. Main magnetic intrinsic properties are atomic magnetic mo-\nment\u0016at(the magnetic moment per volume gives the maximum theoretical\nMs), exchange interactions J ij(which determine the magnetic order and T C)\nand magnetocrystalline anisotropy K1(that can enhance H cand it is indis-\npensable in modern magnets to get Hc> M r=2)[6]. PMs should have high\natomic mangetic moments per volume ( >0:1\u0016B=\u0017A3), strong ferromangetic ex-\nchange interactions (able to give TC>600 K) and high easy axis magnetocrys-\ntalline anisotropy ( K1>1 MJ/m3) in order to exhibit good extrinsic properties\nsuitable for PM applcations. In particular, magnetic materials with hardness\nparameter\u0014=p\nK1=(\u00160M2s)>1 (called \"hard\" magnets) are very valuable\nsince can be used to make e\u000ecient magnets of any shape [6]. At mesoscopic\nscale, intergranular structure between the grains and crystallographic defects\ncan strongly a\u000bect the performance of a magnet [7]. Therefore, the optimization\nof the material's microstructure is also very important in the design and devel-\n3opment of PMs, especially to approach the theoretical upper limit of (BH) max.\nPresently, a magnetic material needs to have quite demanding extrinsic prop-\nerties in order to be considered as a high-performance PM. Desirable values\nmight be\u00160Ms>1 T,\u00160Hc>1 T and (BH) max>200 kJ/m3, with cost lower\nthan 100 $/kg [8], but it really depends on the speci\fc application [1]. Develop-\ning new magnets that ful\fll these criteria is quite di\u000ecult. For example, it took\naround 1 century to go from modest Co-steel magnets with (BH) max=6 kJ/m3\n(old fashioned horseshoe shape) to current Rare-Earth (RE) based magnets like\nsintered Nd 2Fe14B with (BH) max=470 kJ/m3, which is close to its theoretical\nupper limit 515 kJ/m3. Note also that (BH) maxhas not improved much in the\npast 20 years. The great performance of RE-based PMs makes them essential\nin many technological applications, leading to a strong dependency on expen-\nsive RE elements like Nd, Sm or Dy that are monopolized by China's market.\nThis situation has forced PM industry to search for other viable alternatives\nbased on RE-free PMs as MnAl(C) \u001c-phase, FeNi L1 0, non-cubic FeCo alloys,\nMnBi,\u000b00-Fe16N2, exchange spring PM, etc, and also on RE-lean PMs as NdFe 12\n[9, 10, 11].\nAiming to \fnd new clues, the experimental exploration of new PMs begins\nto be assisted and guided by computational approaches thanks to their advances\nin calculation speed, accuracy and reliability [12, 13]. In 2011, USA launched\nthe Materials Genome Initiative (MGI) alongside the Advanced Manufacturing\nPartnership to help businesses discover, develop, and deploy new materials twice\nas fast [14, 15]. As a result, large open material databases, like AFLOW [16, 17]\nand Materials Project [18, 19], have been created, providing a powerful tool for\ndiscovering and designing novel materials through machine learning and data\nmining techniques [20, 21]. In Europe, NOMAD repository [22] was established\nto host, organize, and share materials data. At present, NOMAD contains\nab-initio electronic structure data from Density Functional Theory (DFT) and\nmethods beyond. Some examples of material properties that have been calcu-\nlated by high-throughput computational approach are elasticity [23], piezoelec-\ntricity [24] and electrolytics [25]. A review of some material databases can be\n4found in Ref. [26]. Concerning to magnetic materials, we highlight the MAG-\nNDATA database [27], that provides more than 500 published commensurate\nand incommensurate magnetic structures, and MagneticMaterials.org, an online\nrepository of auto-generated magnetic materials databases that makes use of the\nChemDataExtractor toolkit to extract magnetic properties automatically from\nscienti\fc journal articles and theses, e.g. recently an auto-generated materials\ndatabase of T Chas been created using this approach [28].\nCurrent databases based on MGI like AFLOW or Materials Project provide\ninformation of atomic magnetic moment, but other relevant magnetic properties\nare missing, like magnetocrystalline anisotropy energy (MAE), exchange inte-\ngrals, T C, coercivity, energy product, etc which are needed for screening and\ndiscovering new high-performance Rare-Earth free/lean PMs. In this work, we\ndescribe the Novamag database which aims to complement and extend these\ndatabases for the case of PMs by making publicly available many theoretical\nand experimental results of magnetic materials studied in the H2020 European\nproject Novamag [29, 30]. This paper is organized as follows: Section 2 explains\nthe software technologies used in the development of the database, while Section\n3 gives a detailed tutorial on how to use it. Next, Section 4 shows a general\noverview of the available data and the methods used to obtain them. In Section\n5, we discuss about possible ways to take advantage of Novamag database for\nexploring the novel promising phases uploaded into it. Finally, this paper ends\nwith a summary of the main conclusions.\n2. Methodology\nIn this section the technologies involved in the development of Novamag\ndatabase are described. One mandatory requirement present in each decision in\nthe design and implementation of this database is that every component has to\nbe free and widely supported by the community. The application consists of two\nmain modules: i) a web application that uses Phyton ecosystem technologies\nto show and search the materials and their related information and ii) a data\n5loader application that uses Java ecosystem technologies to insert new items in\nthe database (i.e., NOVAMAG Java Loader Application). These new items are\nin JSON format. Fig. 1 shows the general work\row of the Novamag database.\nBoth developments are publicly available in two GitHub projects. The Web\nApplication is in Ref. [31], while the Java data loader project is in Ref. [32].\nThese applications share the same PostgreSQL database [33]. PostgreSQL is\nthe chosen database management system because the following reasons:\n\u000fThe database schema and data items relationships are simple enough to\nbe represented by a relational system. In fact, the schema consists of\na main table (i.e., items) containing almost all information about the\nmaterials, and a very small set of tables containing information about\nchemical composition, attached \fles, and authors.\n\u000fRelational databases are the most extended database systems, and Post-\ngreSQL is one of the most popular. PostgreSQL is an open source database.\nIt has a graphic easy-to-use administration interface and advanced fea-\ntures that are used in the project (i.e., it supports JSON data, triggers,\nstored procedures and functions). PostgreSQL presents a high level of\nconformance to SQL standard ANSI/ISO speci\fcations.\nThe open Novamag database has a Free and Open Source Software (FOSS)\nlicensing and the link to access it is given by Ref. [30].\n2.1. The Novamag Java Loader Application\nThis application is a utility that can read JSON \fles containing information\nabout one or several materials and load it into the database. The JSON \fle\ncan reference some attached \fles, which are supposed to be located in the same\npath as the JSON \fle. The application also can cope with zip \fles containing a\nsubdirectories tree. This subdirectory tree can contain several JSON \fles and\ntheir attached \fles. Again, the attached \fles of a JSON \fle must be located in\nthe same path as the JSON \fle.\n6Figure 1: Work\row of the Novamag database.\nThe loader application is invoked by the database administrator from the\noperative system command line specifying the path of the JSON or ZIP \fle as\na command line argument. Because it is a Java application, it can be invoked\nboth within Unix or Windows operative systems. If remote execution for the\ndatabase administrator is needed, FTP with SSH for Unix, or a Windows re-\nmote terminal for Windows, can be used. JSON format was adopted because\nthe data is presented in form of aggregation. An item can contain several \fle\nreferences, some arrays like the magnetocrystalline anisotropy energy, or some\nmore complex structures like the atomic positions. XML and JSON are the\nmost extended solutions for this level of complexity in the data representation.\nXML is a much heavier format than JSON, therefore JSON was considered as\nthe best choice. Presently, e\u000borts are being made to develop a common format\nfor Computational Materials Science Data based on JSON \fle [22, 34].\nNowadays, Java is one of the most widespread programming languages.\nGiven that the Loader Application source code is free and available for the\n7community, a very popular language was needed. Application portability for\nLinux and Windows without recompilation is another issue that was taken into\naccount when Java was chosen. Java is an Object Oriented language. It is very\nmature and has a lot of libraries and resources. Once Java was chosen for the\ndevelopment of the loader, many of the rest of technologies were determined by\nsuch election (i.e., JDBC, JNDI and JUnit).\nA logger was used to record the messages the application throws into a log\n\fle. Some messages can be eventual error messages, and some others can be\ncon\frmations that everything was performed properly. Apache Log4j is one of\nthe most extended loggers for Java, Log4j2 is its newest version. It is a software\nfrom Apache Foundation, so its support over the next years is guaranteed. Even\nbeing Log4j a very popular logger, SLF4J was used as facade increasing the\ncompatibility with other loggers. That is, the Java loader program only interacts\nwith SLF4J. Then SLF4J receives the Java loader program invocations and\ntranslate them into Log4 invocations. In this way, the application does not\ndepend on Log4j and can work in the future with any other logger supported\nby SLF4J.\nFinally, Eclipse was adopted as IDE (Integrated Development Environment)\nbecause it is probably the Java most extended IDE. The Mars version was the\nnewest version at the beginning of the development of this application.\n2.2. The Novamag Web Application\nThe web application allows to query the information available from the pre-\nvious importation process. It has been developed using a web framework named\nFlask.\nFlask is a microframework for Python based on Werkzeug and Jinja 2, and it\nis BSD licensed. Python is one of the language with greatest growing in the last\nyears, as can be consulted in the Tiobe Ranking (see Ref. [35]). Their features\nmake it one of the most powerful programming languages including structured,\nobject-oriented and functional paradigm. On the other hand, the Flask and\nPython architecture, allows adding new modules in the development in a very\n8simple way.\nThe dynamic web pages are developed using the Model-View-Presenter de-\nsign pattern using the following Python/Flask libraries:\n\u000fWerkzeug: collection of various utilities for Web Server Gateway Interface\n(WSGI) applications and has become one of the most advanced WSGI\nutility modules (Presenter). WSGI is a simple calling convention for web\nservers to forward requests to web applications or frameworks written in\nthe Python programming language.\n\u000fJinja 2: a modern and designer-friendly templating language for Python,\nmodelled after Djangos templates. It is fast, widely used and secure with\nthe optional sandboxed template execution environment (View).\n\u000fSQLAlchemy: is the Python SQL toolkit and Object Relational Mapper\nthat gives application developers the full power and \rexibility of SQL. It\ngives data access model with generated SQL from the persistent model in\nthe database (Model).\nThe web requests are processed by a Werkzeug class, that loads the data and\nredirect the request to the appropriate Jinja2 template, that renders the HTML\ndynamically using the previous data. The data are loaded using SQLAlchemy\ncode. All the SQL code is automatically generated from a persistent model\nextracted from the PostgreSQL schema to Python code, using a tool named\nsqlacodegen (see Ref. [36]). All the developed code follows this well-known\npattern, allowing an easy maintenance and evolution of the code in the future.\nFrom the point of view of the web design, it has been loaded the Flask-\nBootstrap package. Bootstrap is an open source toolkit for developing with\nHTML, CSS, and JS providing a standard and responsive web design. This\nguarantees a correct and uniform visual design in all the web pages.\nAll this software has been developed using the PyCharm IDE (Integrated\nDevelopment Environment), version Professional 2017.3, built on November 28,\n2017. The used technologies are all multi-platform, since the product has been\n9Technology Role More information\nPosgreSQL v9.2.3 Relational database to store data [33]\nPython 3.0 Programming language used in Flask [37]\nFlask 0.12.2 Microframewok to build dynamical webpages [38]\nFlask-Bootstrap 3.3.7.1 Flask update for the design of the webpage [39]\nFlask-SQLAlchemy 2.3 Flask update (Object-Relational Mapping) [40]\nPgAdmin III Tool to control PostgresSQL [33]\nPsql Console of PostgreSQL [33]\nPyCharm 2017 2.3 Integrated development environment for Python [41]\nJSON Input data format to import data into the database [42]\nJava 8Programming language used in the application to\nimport JSON data into PostgreSQL database[43]\nJDBC 3API to program SQL insertions from Java into the\ndatabase in the data importation application[44]\nJNDIJava Naming and Directory Interface to store database\nconnection settings in the data importation application[45]\nApache Log4j2 Logger in the data importation Java application [46]\nSLF4JSimple Logging Faade for the data importation Java\napplication[47]\nJUnit 4Testing framework for Java used in the data\nimportation application[48]\nGitHubVersion control repository used both for the Java\ndata importation application and for the web application[49]\nEclipse Java MarsIntegrated development environment used in the\nproject for Java programming[50]\nTable 1: Summary of the main technologies used in Novamag database.\ndeveloped on Windows, but it is currently deployed and running on Linux. Table\n1 provides a summary of the main technologies used in the Novamag database\nand their role.\n2.3. Uploading procedure\nIn the \frst stage of the database development, a set of theoretical and exper-\nimental properties to be included in the database was agreed by the Novamag\nconsortium. Fig. 2 shows the full list of properties. These properties are clas-\nsi\fed into 5 categories following MGI standards and the Review of Material\n10Modelling (RoMM) vocabulary: (i) chemistry, (ii) crystal, (iii) thermodynam-\nics, (iv) magnetics and (v) additional information.\nFigure 2: List of theoretical and experimental properties included in Novamag database.\nIn order to deal and upload the large amount of data generated by compu-\ntational high-throughput approaches (see Sections 4.1 and 4.2), we created an\nautomated transfer procedure using metadata JSON \fles written by bash script-\ning. Once the data is in JSON format it can be uploaded into the database by\nthe NOVAMAG Java Loader Application. A JSON \fle can contain information\nof a solely material or of several materials. The attached \fles of these materials\nmust be allocated in the same path than the JSON \fle for the loading. The\nNOVAMAG Java Loader needs several jar libraries that provide methods to\nmanage JSON format, JDBC database connectivity, JNDI services and logging.\nIf a large data upload is needed, then Java Loader Application can cope with\nZIP compressed \fles. The ZIP \fles can contain a sub-directory tree. In each\nsub-directory new sub-directories or JSON \fles can be allocated.\n113. Graphical User Interface\nA friendly Graphical User Interface (GUI) has been developed to use and ex-\nplore the Novamag database easily. Fig. 3 shows the home webpage of Novamag\ndatabase, which contains two search modes: i) standard and ii) advanced.\nFigure 3: Screenshot of the home webpage of Novamag database GUI.\nThe standard mode allows to perform quick and general search through the\ndatabase by specifying the chemical name abbreviation[30]. For example, if one\nwants to search for compounds with iron and nickel, then one should type \"Fe\n& Ni\" inside the box called \"Search\" and then click on the button \"Submit\",\nsee Fig. 3. In this case, the system will show all available compounds that\ncontain the speci\fed chemical elements. Additionally, one can search for speci\fc\nstructures by typing the chemical formula, e.g. \"FeNi2\".\nAlternatively, users can also perform advanced search by clicking on the\nbutton Advanced Search [51], that is placed on the top of home webpage, see Fig.\n3. Next, after doing that, it will appear the advanced search site, see Fig. 4. At\npresent, it allows to \flter and screen compounds according to: Crystallographic\nspace group symmetry, saturation magnetization (M s), \frst magnetocrystalline\nanisotropy constant (K 1), unit cell formation enthalpy, atomic species, species\ncount and stoichiometry.\nThese \flters are very useful to identify promising materials for permanent\nmagnet development in a \frst screening stage. To this end, we recommend to\n12Figure 4: Screenshot of the advanced search tool of Novamag database GUI.\nselect uniaxial space groups (from 75 to 194) because they might exhibit a well-\nde\fned magnetocrystalline anisotropy easy axis, saturation magnetization above\n1 Tesla, negative unit cell formation enthalpy (in order to reject very unstable\nphases), atomic species like magnetic elements (Fe, Ni, Co) and non-Critical\nRaw Materials (that could be light elements to induce tetragonal distortions\nand phase stabilization like B, C, N, P, etc or/and transition metals with high\nspin-orbit coupling as Ta, Hf, etc), species count (binaries, ternaries, etc) and\nstoichiometry corresponding to Fe-rich compounds (with Fe content above 50 %\nat.) to enhance saturation magnetization and Curie temperature. Additional\n\flters as Curie temperature, coercivity or maximum energy product can also\nbe useful in a second screening stage, and might be implemented in the future.\nSome general search strategies through the database are discussed in Section 5.\nOnce the search settings are ready and submitted, then the list of items that\nful\fll these criteria are presented, see Fig. 5. By default, items are sorted by\nenthalpy of formation per atom, so that most stable phases for a given chemical\nformula can be identi\fed easily. In the list of items, it is shown some proper-\nties or \felds of each compound in di\u000berent columns like: material identi\fcation\nnumber (Mat.Id), chemical formula, summary (data available of each item),\napproach (theory or experiment), atomic formation enthalpy, space group, sat-\n13Figure 5: Screenshot of Novamag database GUI showing the list of materials found in an\nadvanced search. Items are sorted by enthalpy of formation per atom, so that most stable\nphases can be identi\fed easily.\nuration magnetization and \frst magnetocrystalline anisotropy constant.\nIn order to see more details of a compound that is in the list one should click\non the corresponding material identi\fcation number (called Mat.Id, at the \frst\ncolumn). Then, all available data of this compound will be shown, see Fig. 6.\nOn the left hand side of the name of each \feld or property, there is a small black\nicon with label \"i\", by clicking on it one can see more information about the\nde\fnition and physical units of that property. Finally, material metadata \fles\nlike crystallographic information \fle (.CIF), JSON \fle used in the database to\nupload it, CONTCAR (output \fle of VASP code that describes the relaxed unit\ncell), \fgure of the unit in portable network graphics format (.png), etc can be\nfound and downloaded in section \"Additional information\" (at the end of the\nwebpage of each material).\n4. Material Data Available\nThis section gives a brief overview of the data available in Novamag database.\nAdditional theoretical and experimental contributions from the Novamag con-\nsortium, not shown or mentioned here, are expected to be uploaded into the\ndatabase in the near future.\n14Figure 6: Screenshot of Novamag database GUI showing material properties of a selected\nitem.\n4.1. Theoretical structures calculated with an Adaptive Genetic Algorithm\nOne of the most important and di\u000ecult step in the computational high-\nthroughput technique for materials design is the prediction of new stable crys-\ntal phases. Here, we make use of an Adaptive Genetic Algorithm (AGA) [52]\nto explore the crystal phase space for many Rare-Earth-free/lean magnetic bi-\nnary and ternary compounds. This task was computationally performed using\nthe software USPEX [53] combined with Vienna Ab Initio Simulation Package\n(VASP) [54, 55, 56]. USPEX uses as an input the number and type of ions to\nbe considered within the unit cell only. At the \frst step, a set of structures is\ngenerated at random, by randomly choosing a crystal space group, correspond-\ning lattice vectors, and ion positions. These initial structures are far from their\nequilibrium, thus performing a structure relaxation is required to estimate ac-\ncurately the energy of each one, which serves as a \ftness criterion. A subset\nof \ftted structures is selected to generate a next generation of structures by\nmeans of genetic operations (crossover and mutations). The process of random\ngenerations of structures is performed also at each step to provide a diversity of\nstructures for each generation. The search for an optimal structure is performed\nuntil no new best structures are generated for a certain number of generations\nor the maximum number of generations is reached. We modi\fed and opti-\n15mized the interface between USPEX and VASP codes in order to improve the\nperformance of structural optimization as well as to perform calculations in a\nhigh-throughput manner. [57, 58, 59]\nFigure 7: Calculated formation enthalpy at zero-temperature for some low-energy Fe-Ge\nmetastable phases obtained in Novamag using AGA (red circle), and found in Material\ndatabases: Materials Project (blue square) and AFLOW (green triangle). Blue line repre-\nsents the enthalpy hull line.\nIn Novamag project, over 15,000 structures have been calculated with AGA\nmethodology, where we mainly explored Fe-rich binary and ternary compounds\nwithout RE-elements. Typically, more than 100 of structures are calculated for\na given chemical formula. At the moment, only around 10-20 most stable phases\n(i.e. with the lowest energy) per chemical formula are being uploaded to the\nNovamag database. The formation enthalpy of these structures is calculated as\n(binary phase)\n\u0001HF(XaYb) =E(XaYb)\u0000a\u0001E(X)\u0000b\u0001E(Y); (1)\nwhereE(:) is the energy, aandbare the number of atoms of XandYin the\nformula unit XaYb, respectively. Fig. 7 presents calculated formation enthalpy\n16at zero-temperature for some Fe-Ge metastable phases obtained in Novamag\nusing AGA, and found in MGI databases like Materials Project[18, 19] and\nAFLOW[16, 17]. We see that AGA is able to \fnd many structures close the\nenthalpy hull line (blue line), reproducing and extending the data available in\nother databases based on MGI.\nThe amount of generated data of each structure depends on the number of\natoms in unit cell and on the number of AGA generations. We may consider as\nrepresentative a compound with 12 atoms/unit cell, which may generate up to\n100GB of data for 650 structures, or about 150MB/structure. We can estimate\nthe overall amount of generated data to be about 2TB, that are stored and\nbacked-up in o\u000ece computers, external hard drives and Relational Database\n(PosgreSQL). A back-up of all data is done every month.\n4.2. High-throughput search for structures with high magnetocrystalline anisotropy\nOne of the key intrinsic properties of modern PMs is MAE because it can\ngreatly increase the coercivity. In this section, we discuss about three computa-\ntional high-throughput strategies performed in the Novamag project for \fnding\nmagnetic structures with high easy-axis MAE.\n4.2.1. Screening of metastable uniaxial phases from AGA and databases\nOnly a small amount of phases from the large set of theoretical uniaxial mag-\nnetic structures close the enthalpy hull line might exhibit a high easy-axis MAE.\nHence, a computational high-throughput approach is needed in order to analyze\nthem and be able to identify most promising ones. To this end, we devised a\nscript that controls the calculation of MAE in an automated way providing as\nthe input only the atomic position \fle. For each structure, calculations are per-\nformed in several steps: full structural optimization (volume, cell and atomic\npositions) with VASP standard precision, followed by more accurate calcula-\ntions with optimal code parameters obtained for a set of representative known\npermanent magnets. For the optimized structure, accurate charge density and\nwave function are calculated for ferromagnetic arrangement of spins via collinear\n17spin-polarized DFT calculations, which will serve as input for non-collinear spin\ncalculations (NCL) with spin-orbit coupling. NCL-calculations are performed\nin a non-self-consistent manner for a set of polar and azimuthal angles. This\nprocedure is fed by the crystallographic data of Rare-Earth free/lean theoretical\nstructures predicted by AGA (section 4.1) and found in open material databases\nthat are previously screened according to space group (only uniaxial phases are\nconsidered), formation enthalpy ( <0) and saturation magnetization ( &1T), see\nFig. 9.\nFigure 8: Work\row of the high-throughput calculation of magnetocrystalline anisotropy en-\nergy developed in Novamag project for the discovery of high-performance Rare-Earth-free/lean\npermanent magnets.\nOnce the energy of an uniaxial unit cell is calculated, constraining the atomic\nmagnetic moments at di\u000berent polar angles \u0012, then \frst and second anisotropy\nconstants K 1and K 2are obtained by \ftting the energy data to 2nd order func-\ntion\nE(\u0012) =K1sin2\u0012+K2sin4\u0012: (2)\nA \fnal procedure, written by bash scripting, was also prepared to upload the\nlarge amount of generated data from this approach to the database automati-\ncally. As an example, in Fig. 9 we show the high-throughput calculation of \frst\nanisotropy constant K 1at T=0K as a function of polarization \u00160Msfor some\ntheoretical uniaxial Fe-based binaries (Fe-X where X= Al, B, P and Hf) with\n\u0001HF<0 and easy axis, predicted by AGA in Novamag project and found in\n18AFLOW database. We observe this approach is capable to reveal hidden RE-\nfree theoretical phases with hardness parameter \u0014=p\nK1=(\u00160M2s)>1, such\nas hexagonal P-6m2 FeAl ( \u0014= 2:6), and tetragonal I4/mmm structures Fe 16B2\n(\u0014= 1:1) and Fe 4Hf2(\u0014= 3:0). Many other Fe-based and Co-based binaries\nhave been calculated and uploaded following this methodology.\nFigure 9: High-throughput calculation of \frst anisotropy constant K 1as a function of po-\nlarization\u00160Msfor some theoretical uniaxial Fe-based binaries (Fe-X where X=Al, B, P\nand Hf) with \u0001 HF<0. The dashed lines correspond to magnetic hardness parameter\n\u0014=p\nK1=(\u00160M2s) for values \u0014=1 and 0.1. Hard magnetic materials ( \u0014>1) can be used to\nmake e\u000ecient magnets of any shape. Black symbols stand for some known hard phases [2].\n4.2.2. Tetragonally distortion of cubic phases\nEnlightened by the work of Burkert et. al. on tetragonal FeCo alloys, [60] we\nhave performed high throughput calculations on the tetragonally distorted cubic\nmagnetic materials to screen for candidates as permanent magnets, as shown in\nFig. 10. All the binary and ternary compounds with cubic structures including\none of Cr, Mn, Fe, Co, and Ni from the Materials Project database [18, 19] are\n19considered, where 1% compressive strain is applied to get the tetragonally dis-\ntorted structures under the constant volume approximation. There are quite a\nfew compounds showing large magnetocrystalline anisotropy induced by the im-\nposed tetragonal distortions. For instance, Mn 3Pt shows an anisotropy as large\nasjK1j=6.8 MJ/m3, and Mn 2RhPt with an anisotropy of jK1j=2.8 MJ/m3.\nConcerning the sign of K1, it is noted that compressive and tensile strain will\nlead to opposite magnetocrystalline anisotropies as the original cubic materials\nare isotropic to the \frst order due to the high symmetry. An interesting ques-\ntion is how such tetragonal distortions can be stabilized. We suspect that light\ninterstitial atoms such as H, B, C, and N will lead to substantial tetragonal\ndistortions, as recently investigated in Fe [61] and FeCo [62, 63] alloys. Sys-\ntematic calculations have been performed on compounds with the Cu 3Au (as\nfor Mn 3Pt) and Heusler (as for Mn 2RhPt and Fe 2CoGa) structures with light\ninterstitials, and the results will be reported elsewhere.\n4.2.3. Modi\fed magnetic materials from \frst principles\nAnother interesting computational high-throughput strategy to \fnd promis-\ning PMs is to try di\u000berent dopants on known magnetic structures in order to tune\nand improve their properties. For instance, the Novamag database provides data\nof \frst principles calculations for the RE-free modi\fed Fe 3Sn compound [64].\nAmong all the Fe-based intermetallic compounds the Fe 3Sn phase is the most\nattractive, due to the highest concentration of Fe and therefore high magnetic\nmoment. However, as it has been shown experimentally, the magnetocrystalline\nanisotropy of this compound is planar [65], while the uniaxial anisotropy is a\nrequirement for a good PM. We have studied the in\ruence of dopants on the\nmagnetocrystalline anisotropy of Fe 3Sn (see Fig. 11 a). On the Sn sublattice\nwe considered 25 at.% of impurities, such as M = Si, P, Ga, Ge, As, Se, In, Sb,\nTe, Pb, and Bi. On the Fe sublattice Mn was added as a structure stabilizer.\nThe high temperature phase of Fe 3Sn has a hexagonal crystal structure with\nspace group P6 3/mmc (#194). The 1 \u00021\u00022 supercell of Fe 3Sn comprising of 12\nFe and 4 Sn atoms, one of which was substituted by a dopant (see Fig. 11 b) was\n20Figure 10: High-throughput calculation of the \frst anisotropy constant K 1as a function\nof magnetization for stable binary and ternary magnetic materials. The blue squares mark\nthe known experimental compounds, and the green circles denote three promising candidates\n(Mn 3Pt, Mn 2RhPt, and Fe 2CoGa) identi\fed as permanent magnets.\nconsidered. In the case of doping with Mn, 6 or 4 Fe atoms were substituted.\nStructures were relaxed using VASP [54, 55, 56] within the projector augmented\nwave (PAW) method [66]. The electronic exchange and correlation e\u000bects were\ntreated by the generalized gradient approximation (GGA) in the Perdew, Burke,\nand Ernzerhof (PBE) form [67]. For the calculation of the magnetic properties\nthe full-potential linear mu\u000en-tin orbital (FP-LMTO) method implemented\nin the Relativistic Spin Polarized toolkit (RSPt code) [68, 69] was used. We\nperformed integration over the Brillouin zone, using the tetrahedron method\nwith Bl ochl's correction [70]. The k-point convergence of the MAE for the\nchosen supercell size was found when increasing the Monkhorst-Pack mesh [71]\nto 24\u000224\u000224 used in all calculations.\nThe calculated MAEs were plotted as a function of the number of valence\n21Figure 11: MAE and crystal structure of the Fe yMn3\u0000ySnxM1\u0000xsystem, y=1.5, 2, and 3;\nx= 0.5, 0.75, and 1. a) The magnetocrystalline anisotropy energy (MAE) calculated from\n\frst principles as a function of the number of valence electrons, N, per formula unit. b) The\n1\u00021\u00022 Fe 3Sn hexagonal cell with one impurity atom on the tin sublattice. Fe atoms are\nshown with brown spheres, Sn atoms with grey and M (M = Si, P, Ga, Ge, As, Se, In, Sb, Te,\nPb and Bi) impurity atom is shown with the blue sphere. Violet spheres represent the Mn\natoms on the Fe sublattice.\nelectrons per formula unit for doping atoms on the Sn and Fe sublattices, see\nFig. 11. Positive numbers correspond to the uniaxial MAE. The addition of\ncertain elements, such as Sb, As, and Te allowed one to change MAE from planar\nto uniaxial, however the absolute value was rather small. The maximal uniaxial\nMAE found (for As and Sb dopants), is obtained for Group VA elements of the\nperiodic table.\nCalculated MAE of system with increased Sb content, i.e. Fe 3Sn0:5Sb0:5\nwith 28.5 valence electrons per formula unit, was found to be close to the value\nfor Fe 3Sn0:75Te0:25with the same number of valence electrons (see opaque circle\nin Fig.11). Therefore, for this system the MAE depends more on the number\nof valence electrons rather than on the dopant. It was shown experimentally\n[64] that the Fe 3Sn0:5Sb0:5system is unstable unless doped with Mn. How-\never, adding Mn to Fe 3Sn0:75Sb0:25system changes anisotropy back to planar.\n22Though the increase of Mn content leads to the increase of absolute value of\nMAE.\n4.3. Exchange integrals and Curie temperature\nAnother important intrinsic property of magnetic materials is the exchange\nenergy, which is typically described by the classical Heisenberg Hamiltonian,\nEex=\u00001\n2X\ni;jJijsi\u0001sj; (3)\nwhereJijare the exchange parameters and siis the unit vector along the mag-\nnetic moment of atom i-th. The factor 1 =2 takes care of the double counting\nin the summation. This is the convention adopted in the database, but note\nthere are many works and codes where Eq. 3 is de\fned without the factor\n1=2, so in these cases the Jijparameters are 2 times smaller than those from\nEq. 3. A warning message is displayed in the database's \feld \"exchange info\"\nwhen a di\u000berent convention is used in order to avoid any misunderstanding.\nFromJijone can determine the magnetic order (ferromagnet, antiferromagnet,\netc), roughly estimate the T Cusing mean-\feld approximation (MFA) or random\nphase approximation (RPA), and build atomistic spin dynamics (ASD) models\n[72, 73, 74, 75, 76, 77] to calculate the temperature dependence of magnetic\nproperties (magnetization, domain wall width, exchange sti\u000bness, etc).\n4.3.1. Rare-Earth-free compounds\nFirst principles calculations of J ijhave been calculated and uploaded to the\ndatabase for some promising Rare-Earth free PMs such as MnAl \u001c-phase [78]\nor L1 0FeNi. For instance, Fig. 12a shows the estimated exchange integrals\nof L1 0FeNi as a function of the interatomic distance using frozen magnon\ncalculations implemented in FLEUR code [73, 79, 80]. Note FLEUR code follows\nthe de\fnition given by Eq. 3 that includes factor 1 =2. We employed the spin\nspiral formalism with 1960 k-points in the irreducible Brillouin zone (IBZ),\nplane-wave cut-o\u000b of k max= 4.1 a.u.\u00001and 463 q-points in the IBZ. We see\nthat Fe-Fe intralayer exchange interaction is stronger than interlayer one. We\n23also observe that Fe-Ni and Ni-Ni exchange interactions are quite weak, in\nfact only \frst nearest neighbour interaction might play a relevant role in the\nspin dynamics. The Curie temperature given by ASD simulations using these\nexchange parameters is T C=800 K, which is in good agreement with experiment\n[81] (see Fig. 12b). The details of this ASD model and calculation's settings can\nbe found in the database. Note a better agreement of magnetization in the all\ntemperature range can be achieved by rescaling the temperature in the classical\nASD simulations [82] or including quantum e\u000bects [83, 84]. Additionally, the\ndatabase contains data about exchange sti\u000bness and domain wall width given by\nASD models, which can be useful for micromagnetics simulations (see Section\n4.4.2) within a multiscale modelling approach [85].\nFigure 12: (a) Estimated exchange integrals of L1 0FeNi as a function of the interatomic\ndistance. Inset shows a diagram of the Fe-Fe exchange parameters up to the sixth nearest\nneighbour (labelled by J ii=1,...,6). (b) Total magnetization versus temperature of L1 0FeNi\nwith no external magnetic \feld (red line) and external magnetic \feld \u00160H=5T (green line),\nobtained by ASD simulations. Blue circles correspond to experimental data of magnetization\nwith external magnetic \feld \u00160H=5T reported in Ref. [81].\n4.3.2. Rare-Earth-lean 1:12 compounds\nExchange parameters for the known SmFe 10V2system, as well as for the\nnewly stabilized SmFe 11V [86] were obtained using Lichtenstein's method [87,\n88], as implemented in FP-LMTO code RSPt [89]. Note code RSPt de\fnes Eq.\n3 without factor 1 =2, so the following results are based on the convention used\nin this code. The electronic exchange and correlation e\u000bects were treated by the\n24generalized gradient approximation (GGA) in the Perdew, Burke, and Ernzer-\nhof (PBE) form [67]. The k-point mesh equal to 18 \u000218\u000224 was used in all\ncalculations. In Fig. 13 the calculated J ijs are shown for pairs of atoms ijwith\ni\fxed to V, Fe, and Sm, and all corresponding js at distances R ijfrom the \frst\ncoordination shell up to 0.6 of the lattice constant. As one may see from Fig.\n13, the strongest exchange interactions are between Fe atoms that fall o\u000b with\nthe distance and are close to zero starting at a distance of approximately 0.35 of\nthe lattice constant. They are positive, i.e. favor the ferromagnetic alignment\nof magnetic moments on the Fe atoms, and therefore play a dominating role in\nthe paramagnetic-to-ferromagnetic transition and de\fne to a large degree the\ntransition temperature (T C). The behavior of the Fe-Fe interactions is qualita-\ntively rather similar in both SmFe 10V2and SmFe 11V compounds, although the\nhighest values di\u000ber by up to 50 % being larger for SmFe 11V. It is also notice-\nable that all the V-V as well as all the types of interactions involving Sm are\nrather weak. The Fe-V interaction at the \frst coordination shells is, however,\nrelatively large, though around factor 2 weaker than the strongest Fe-Fe inter-\nactions in SmFe 10V2. In both cases they favor the antiferromagnetic alignment\nof magnetic moments on the Fe and V atoms. In SmFe 10V2, where there are\nthe Fe-V pairs in the \frst coordination shell, the antiferromagnetic exchange\ninteraction between Fe and V is strong. However, it is well counteracted by the\nstrong Fe-Fe interactions. In the MFA, the T Ccan be obtained as follows (not\nincluding factor 1 =2 in the Heisenberg Hamiltonian):\nTMFA\nC =2J0\n3kB: (4)\nwherekBis the Boltzmann constant. Here T Cis proportional to the on-site\nexchange parameter J 0whereJ0= \u0006 iJ0i, i.e. the sum over all coordination\nshells. T Cs are rather similar in both compounds and well above the room\ntemperature. Note that we only took into account Fe interactions for T Cand\npossible contributions from Sm 4 fare not included in the approximation since\n4fare treated as spin-polarized core.\n25Figure 13: Inter-site exchange parameters J ijbetween atoms iandjseparated by the distance\nRijof SmFe 10V2(a) and SmFe 11V (b) compounds. The J ijs are shown for all R ijs involving\ni=V, Fe, and Sm (shown by large black, small \flled blue, and medium red circles, respectively).\nastands for the lattice constant. In the inset the mean-\feld estimation of T Cis given; x stands\nfor the V content, i.e. 2 in SmFe 10V2and 1 in SmFe 11V.\n4.4. Extrinsic magnetic properties\n4.4.1. Stoner-Wohlfarth model\nOnce the saturation magnetization ( MS) and magnetocrystalline anisotropy\nconstants ( K1andK2) are calculated, it is possible to roughly estimate hys-\nteresis loop properties as coercivity, remanence and maximum energy product\nby means of the Stoner-Wohlfarth model in the limit of coherent rotation [5]\nE\nV=K1sin2\u0012+K2sin4\u0012+\u00160\n4(1\u00003D)M2\nSsin2\u0012\u0000\u00160MSHcos\u0012; (5)\nwhereEis the energy, Vis the volume of the material, \u0012is the angle between\nmagnetization and the easy axis (along z-axis), \u00160is the vacuum permeability,\nDis demagnetization factor (which can be a value between 0 and 1) and His\nthe external magnetic \feld applied along the easy axis. The \frst two terms de-\nscribe the magnetocrystalline energy (discussed in Section 4.2), the third term is\n26the shape anisotropy created by the demagnetizing \feld of the magnet and the\nlast term is the Zeeman energy. In this model the remanence equals to satura-\ntion magnetization (perfectly rectangular hysteresis loop). Here, we de\fne the\nanisotropy \feld as the minimum \feld needed to overcome the magnetocrytalline\nenergy barrier (to reverse magnetization in the absence of shape anisotropy)\nHK=2(K1+K2)\n\u00160MS; (6)\nNote that there are other possible de\fnitions [5]. Including the shape anisotropy,\nthe minimum \feld to reverse the magnetization is the coercivity\nHC=2(K1+K2)\n\u00160MS+1\n2(1\u00003D)M2\nS: (7)\nThis model describes well nano-size magnetic materials. Taking into account\nnucleation processes, one arrives to Brown relation\nHC\u00152K1\n\u00160MS\u0000DM S: (8)\nThe value of coercivity corresponding to uniform magnetization reversal by\ncoherent rotation; generally, it is much lower than the anisotropy \feld, at\nbest about 25% of H K. On the other hand, the energy product reads [6]\n(BH) =\u00160D(1\u0000D)M2\nS, which is maximum for D= 1=2, that is,\n(BH)max=1\n4\u00160M2\nS: (9)\nNote this value can only be reached if HC>M S=2 (hardness parameter \u0014>1=2)\n[6]. The database is being updated with values of magnetic extrinsic properties\n(remanence, coercivity and maximum energy product) given by the Stoner-\nWohlfarth model (see Fig. 14) for structures with easy axis, obtained in the\nhigh-throughput calculation of MAE (Section 4.2.1).\n4.4.2. Micromagnetics\nIn real magnets, to approach theoretical maximum energy product and co-\nercivity it is essential to have an optimized microstructure. The intergranular\nstructure between the grains plays a signi\fcant role determining the magnetic\n27Figure 14: (a) (top) Perfectly rectangular M(H) loop and (bottom) B(H) loop for obtaining\nmaximum energy product. (b) High-throughput calculation of theoretical maximum energy\nproduct (BH) max and maximum coercivity ( HC;max =HK) given by the Stoner-Wohlfarth\nmodel for the Fe-based binaries showed in Fig. 9 (with \u0014>1=2).\nproperties, especially if the grain diameter is in the nanometer scale. Hard mag-\nnetic phases may be distorted locally near grain boundaries (GBs), because of\nthe di\u000berent lattice constant of the grain boundary phase. The distorted lattice\nmay give rise to locally di\u000berent intrinsic magnetic properties. Moreover, surface\ndefects located at GB are sources of strong local demagnetizing \felds, which act\nas nucleation centres. On the other hand, defects close to the GB can pin do-\nmain walls during the magnetization's reversal process. Nucleation and pinning\nmechanisms strongly a\u000bect extrinsic properties of hard magnetic phases, as the\ncoercivity [7]. Additionally, intergranular phases modify the exchange coupling\nbehavior between the hard magnetic grains. For instance, nonmagnetic phases\neliminate the direct exchange interaction between the hard magnetic grains.\nTypically, it leads to an increase of the coercive \feld.\nNovamag database contains theoretical results of extrinsic properties cal-\nculated with micromagnetic simulations considering optimized microstructures,\nconstructed with the open-source 3D polycrystal generator tool Neper [90], for\nsome of the magnets studied in Novamag project as Fe 3Sn0:75Sb0:25, L1 0FeNi,\n28Figure 15: (Left) Magnets structure with 3 nm thin grain boundary phase. (Mid) For both\nmaterials demagnetization curves M over H ext(Right) the corresponding B over H curve.\nMnAl\u001c-phase, etc [91]. In Fig. 15 we present demagnetization curves of such\nMnAl and L1 0FeNi magnets highlighting the in\ruence of di\u000berent grain bound-\nary magnetizations. As green solid line we show the results of M S;gb= 0.52\u0001MS\nand in blue dotted M S;gb= 0.04\u0001MSwhere M Sis the main phase's saturation\nmagnetization. The structure used for simulating the magnets consists of 20\nequi-axed grains with an average grain size of 46 nm separated by a 3 nm thin\nlayer denoted as the GB phase. With increased magnetization inside the GB\nphase the coercivity drops by 22% and 16% for MnAl and L1 0FeNi respectively.\nIn Fig. 15 on the right hand side we show the desheared BH loop using a shape\nfactor of 1/3. Computed energy density products for L1 0FeNi range from 230\nkJ/m3to 300 kJ/m3whereas for MnAl ranges from 80 kJ/m3to 100 kJ/m3for\nhigh and low saturation magnetization inside the GB, respectively.\n4.5. Synthesis and characterization of hard magnetic phase alloys\nThe comptutational high-throughput techniques described in previous sec-\ntions can only approximately estimate the \fnal properties and perfomance of a\n29Sample Space Group a(\u0017A) b(\u0017A) c(\u0017A)\nFe3Sn #194, P6 3/mmc 5.4621(5) 5.4621(5) 4.3490(6)\nFe2:25Mn0:75Sn0:75Sb0:25 #194, P6 3/mmc 5.4858(7) 5.4858(7) 4.3721(6)\nFe2:5Mn0:5Sn0:75Sb0:25 #194, P6 3/mmc 5.5551(1) 5.5551(1) 4.4398(1)\nFe2Mn1Sn0:75Sb0:25 #194, P6 3/mmc 5.5000(4) 5.5000(4) 4.3829(6)\nFe1:5Mn1:5Sn0:75Sb0:25 #194, P6 3/mmc 5.5338(1) 5.5338(1) 4.4270(2)\nFe1:5Mn1:5Sn0:9Sb0:1 #194, P6 3/mmc 5.5545(3) 5.5545(3) 4.4453(4)\nTable 2: Lattice parameters of the Fe 3\u0000xMnxSn1\u0000ySbyalloys, obtained by Rietveld re\fne-\nments of the X-Ray Di\u000braction patterns.\nreal bulk PM. On the other hand, there are also several high-throughput exper-\nimental screening techniques as reactive crucible [92] or thin \flm combinatorial\nsynthesis [93]. However, they allow for a limited number of characterizations\nand are not representatives of the material in bulk, i.e. of the magnet. There-\nfore, whether the goal is the manufacture of a magnet, it is necessary to give\nthe processing recipes of the material in bulk for a possible upscaling at an\nindustrial level.\nIn order to provide this kind of valuable information, the database is be-\ning extended with experimental results obtained in Novamag project, such as\nsynthesis routes and the structural and magnetic characterization for CRM-\nfree Fe 3Sn, FeNi and MnAl alloys [94], or the RE-lean ThMn 12-type tetragonal\nstructure magnetic materials. For instance, Table 2 shows the lattice parame-\nters obtained by Rietveld re\fnements of the X-Ray Di\u000braction patterns for a set\nof Fe 3\u0000xMnxSn1\u0000ySbyalloys synthesized by solid-state reaction in the attempt\nto tune Fe 3Sn alloys [64, 95], while Fig. 16 shows a statistical summary of the\nanisotropy \feld and M Sobtained in di\u000berent alloys [96, 97, 98, 99, 100]. We\nsee that the alloys can be classi\fed by their magnetic properties. As it is well\nknown, the intrinsic properties of the magnetic alloys suitable for magnets must\nhave high anisotropy \feld and high M S(dark grey lined zone) as SmFe 11V,\nSmFe 11Mo or SmFe 9Co2Ti. Other alloys, which have high M S, can be further\nprocessed in order to increase their anisotropy (light grey zone), as it is the case\n30of Fe 3Sn.\nFigure 16: Measured values of anisotropy \feld and saturation magnetization for some exper-\nimental magnetic samples included in the Novamag database.\n5. Possible strategies to exploit promising theoretical novel phases in\nthe Novamag database\nMGI has created a new scenario with many theoretical phases that could\nbe very appealing for di\u000berent technological applications. However, one of the\nmain challenges in MGI is to \fnd experimental routes to synthesize most promis-\ning theoretical materials given by computational high-throughput approaches.\nFrom a theoretical point of view, determining the stability of a predicted phase\nis a hard task that requires an accurate calculation of the full phase diagram at\n\fnite temperature (free energy) [101, 102]. In this section, inspired by isostruc-\ntural tie-lines [103, 104], we discuss about some general strategies to design ex-\nperimental procedures for exploring most promising theoretical materials found\nin the Novamag database, fully based on their crystallographic data, by doping\n31known stable phases. The main idea is to think about the predicted phases by\nAGA in a more \rexible manner, it means, as a hint or trend. Hence, making a\nmagnet with some crystal and atomic similarities to these structures might be\nenough to get a good PM. Bearing this in mind, let's consider a given unknown\nFe-based binary Fe xMygenerated by AGA that ful\flls all necessary criteria\nfor being a high-performance PM. First step is to search in literature (Hand-\nbook of crystallographic data or databases like Materials Project) for a known\nexperimental stable binary phase that contains M element but no Fe (no Fe-\nbased binary) A' x0My0or contains Fe but no M (Fe-based) Fe x0A\"y0with the\nsame crystallographic prototype structure as the theoretical Fe xMyand also\nsimilar stoichiometry. Next, these known stable phases (A' x0My0or Fe x0A\"y0)\nare used as starting point to explore either the ternary line (A' 1\u0000zFez)x0My0or\nFex0(A\"1\u0000zMz)y0, that is, doping with Fe or M, respectively. In this way, it is\npossible to experimentally explore and approach the neighborhood of promising\ntheoretical phase Fe xMy, where the compounds in this region with the same\ncrystal structure might have similar properties as Fe xMy. Additionally, since\nthis procedure is based on the same prototype structure as Fe xMyit could also\ntrigger the formation of this phase locally during the synthesis. Note that the\nroute using a no Fe-based starting point might be more challenging since one\nneeds to dope the ternary with high content of Fe in order to make it ferro-\nmagnetic, especially if A is not Co or Ni. Fig. 17 shows a diagram explaining\nthe above described strategy. Alternately, the selection of optimal substrates\nfor epitaxial growth of these novel theoretical phases could be a way to stabilize\nthem too [105].\nThe above described strategy can be reversed. For instance, let's consider\nwell-known phase Fe 3Sn (P6 3/mmc, space group 194) that exhibits an easy\nplane MAE. As indicated in Section 4.2.3, in this system it is necesseary to\n\fnd a way to switch MAE from easy plane to easy axis in order to make it\nsuitable for PM applications. To this end, we can use the search advanced tool\nof Novamag database[51] with the following \flters: (i) space group=194, (ii)\nnegative formation enthalpy \u0001H F<0, (iii)\u00160MS>1T, (iv) K 1>1 MJ/m3,\n32Figure 17: Proposed general strategy to experimentally explore the crystal phase space neigh-\nborhood of promising theoretical Fe-based binaries found in the database.\n(v) binary alloy, and (vi) Fe content equal to 75 at.%. Doing so, it found two\npromising phases Fe 3A with A=Ta,Ti that ful\fll these requirements. Hence,\nit suggests to explore the ternary line Fe 3Sn1\u0000xAx. A preliminary theoretical\nanalysis using DFT calculations, following the steps described in Section 4.2.1,\nwith software VASP [54, 55, 56] (PAW method [66] and GGA-PBE exchange-\ncorrelation type [67]) for 1x1x2 supercells (16 atoms) of Fe 3Sn1\u0000xAx(A=Ti,Ta)\nshows that replacing 50% of Sn by Ti or Ta in the Fe 3Sn phase might be enough\nto switch MAE from easy plane to easy axis, see Fig. 18. Obviously, a more\ndetailed analysis of the stability and MAE of these systems are required in order\nto clarify the role of these dopants on the Fe3Sn phase, see Section 4.2.3 [64].\nFinally, concerning the cubic phases in which a tetragonal distortion can\ninduce a large MAE (see Section 4.2.2), we think that adding light interstitial\natoms such as H, B, C, and N in these compounds might be a likely way to\nachieve it in practice [61, 62, 63].\n33Figure 18: Strategy to make use of Novamag database for tuning Fe 3Sn phase.\n6. Conclusions\nIn summary, based on the philosophy and standards of MGI, we have devel-\noped a comprehensive open database of magnetic material parameters that is\nbeing updated with many signi\fcant theoretical and experimental results from\nNovamag project. In the design process, e\u000borts have been made to use free soft-\nware, standard vocabulary and metadata \fles that might facilitate interoper-\nability with other existing databases. Additionally, its FOSS license maximizes\nopenness and minimizes barriers to software use, dissemination, and follow-on\ninnovation.\nUsers worldwide can access to crystallographic and magnetic information\nof many theoretical structures calculated by DFT combined with an AGA.\nPresently, it is also being updated with the results of a high-throughput cal-\nculation of MAE for a large set of uniaxial magnetic structures predicted with\nAGA, as well as for tetragonally distorted cubic phases and tuned known mag-\nnetic materials. In order to illustrate these methods, we have reported some\ninteresting phases with high MAE and M S, such as FeAl, Fe 16B2and Fe 4Hf2,\ntetragonally distorted Mn 3Pt and Mn 2RhPt, and doped Fe 3Sn.\nExamples of available multiscale theoretical and experimental data about\nother relevant magnetic properties (exchange integrals, T C, anisotropy \feld,\ncoercivity, maximum energy product, etc.) were also shown, which can be help-\nful for the design of new promising high-performance RE-free/lean PMs. Aiming\n34to link the computational high-throughput results to experiment, we presented\nsome possible strategies to exploit most promising theoretical materials found\nin the database. Further results from the consortium of Novamag project are\nexpected to be uploaded into the database in the near future, as well as possible\nnew features and improvements.\nAcknowledgements\nThis work was supported by the European Horizon 2020 Framework Pro-\ngramme for Research and Innovation (2014-2020) under Grant Agreement No.\n686056, NOVAMAG. The authors P. N. and S. A. thankfully acknowledge\nthe computer resources at Finisterrae and the technical support provided by\nFundaci\u0013 on P\u0013 ublica Galega Centro Tecnol\u0013 oxico de Supercomputaci\u0013 on de Galicia\n(RES-QCM-2018-3-0009). O. Y. V., H. C. H. and O. E. thankfully acknowl-\nedge the computer resources provided by the Swedish National Infrastructure\nfor Computing (SNIC) at PDC and NSC centers. O. E. acknowledges support\nfrom STandUP, eSSENCE, the Swedish Research Council and the Knut and\nAlice Wallenberg foundation (KAW) and the foundation for strategic research\n(SSF).\nReferences\n[1] O. Gut\reisch, M. 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ACS Applied Materials &Interfaces ,\n8(20):13086{13093, 2016.\n46"    },    {        "title": "1902.05384v2.Vectorial_observation_of_the_spin_Seebeck_effect_in_epitaxial_NiFe__2_O__4__thin_films_with_various_magnetic_anisotropy_contributions.pdf",        "content": "Vectorial observation of the spin Seebeck effect in epitaxial NiFe 2O4thin films\nwith various magnetic anisotropy contributions\nZhong Li,1, 2,\u0003Jan Krieft,3,\u0003Amit Vikram Singh,1Sudhir Regmi,1, 2Ankur Rastogi,1\nAbhishek Srivastava,1, 2Zbigniew Galazka,4Tim Mewes,1, 2Arunava Gupta,1, †and Timo Kuschel3, ‡\n1Center for Materials for Information Technology, The University of Alabama, Tuscaloosa, Alabama 35487, USA\n2Department of Physics & Astronomy, The University of Alabama, Tuscaloosa, Alabama 35487, USA\n3Center for Spinelectronic Materials and Devices, Department of Physics,\nBielefeld University, Universitätsstraße 25, 33615 Bielefeld, Germany\n4Leibniz-Institut für Kristallzüchtung, Max-Born-Str. 2, 12489 Berlin, Germany\n(Dated: May 14, 2019)\nWe have developed a vectorial type of measurement for the spin Seebeck effect (SSE) in epitaxial NiFe 2O4\nthin films which have been grown by pulsed laser deposition on MgGa 2O4(MGO) with (001) and (011) orien-\ntation as well as CoGa 2O4(011) (CGO), thus varying the lattice mismatch and crystal orientation. We confirm\nthat a large lattice mismatch leads to strain anisotropy in addition to the magnetocrystalline anisotropy in the\nthin films using vibrating sample magnetometry and ferromagnetic resonance measurements. Moreover, we\nshow that the existence of a magnetic strain anisotropy in NiFe 2O4thin films significantly impacts the shape\nand magnitude of the magnetic-field-dependent SSE voltage loops. We further demonstrate that bidirectional\nfield-dependent SSE voltage curves can be utilized to reveal the complete magnetization reversal process, which\nestablishes a vectorial magnetometry technique based on a spin caloric effect.\nThe discovery of the spin Seebeck effect (SSE), the genera-\ntion of a spin current from a thermal gradient, has attracted\na lot of interest in the field of spin caloritronics in recent\nyears [1–6]. Pt/YIG (yttrium iron garnet, Y 3Fe5O12) is one of\nthe most widely studied material systems in spin caloritronics\n[7] and magnon spintronics [8] regarding the SSE, coherent\nspin pumping [9], spin Hall magnetoresistance (SMR) [10]\nand nonlocal magnon spin transport [11]. This material sys-\ntem has also been used to introduce the longitudinal spin See-\nbeck effect (LSSE), the generation of a spin current parallel to\na temperature gradient that is usually aligned out-of-plane [2].\nThe spin current is converted to charge current in the Pt layer\nvia the inverse spin Hall effect (ISHE) [12]. In order to opti-\nmize the performance in LSSE experiments, a number of im-\nprovements have been investigated, including studies on the\ninfluence of YIG film thickness, temperature, applied mag-\nnetic field and interface between the spin-current-detecting\nmaterial and YIG layers [13–17]. The spin Hall conductiv-\nity of the spin-current detector changes systematically in re-\nsponse to the number of delectrons in the 4 dand 5 dtransition\nmetals [18]. Besides Pt, another material of this group with an\neffective spin-charge conversion and thus a sufficiently high\nspin-Hall angle is Pd [19], exhibiting highly transparent non-\nmagnetic/ferromagnetic material e.g. Pd/YIG interfaces [20]\nwith strong potential for spintronic applications [21].\nNFO (nickel ferrite, NiFe 2O4) thin films, which have po-\ntential applications in high-frequency microwave and spin-\ntronics devices [22–25], have also been used in LSSE stud-\nies [4, 5, 26, 27]. Recent studies for NFO on lattice-matched\nsubstrates, such as MgGa 2O4, report strongly improved mag-\nnetic properties which are comparable to YIG [28], as well\nas enhanced SSE and improved spin transport characteris-\ntics [29]. The choice of substrate material and crystal cut\ncan be used to taylor the magnetic anisotropy in the thin NFO\nfilm in order to study the impact of the anisotropy type onthe SSE. Using a four-contact device, we present in this let-\nter a vectorial magnetometry technique based on LSSE that is\nsuitable to study magnetic anisotropies and magnetization re-\nversal processes. We demonstrate the technique on NFO thin\nfilms grown on lattice-matched substrates with different crys-\ntal cuts and, thus, study different magnetic anisotropy combi-\nnations.\nWithin this Letter, the thermal generation of spin currents\nhas been realized in NFO thin films deposited on MgGa 2O4\n(MGO) with (001) and (011) orientation as well as CoGa 2O4\n(011) (CGO), thus varying the lattice mismatch. Pd has been\nused as spin-current detector material. Using x-ray diffrac-\ntion (XRD), we show that the lattice mismatches of NFO\nfilms on MGO substrates are larger than those on CGO sub-\nstrates. A larger lattice mismatch leads to a higher mag-\nnetic strain anisotropy in NFO//MGO thin films, as confirmed\nvia vibrating sample magnetometry (VSM) and ferromagnetic\nresonance (FMR) measurements. The vector detection tech-\nnique has been used to study the influence of magnetic strain\nanisotropy in thin films on the SSE response. We find that the\nshapes of SSE voltage curves detected by two orthogonally\naligned voltage probes measured as a function of the mag-\nnetic field strength and its orientation can vary significantly\nfrom each other due to the effect of magnetic strain anisotropy.\nWhile first attempts in this direction focused on detecting indi-\nvidual magnetic components separately [30], we demonstrate\nour simultaneous measurement method as a tool to study the\nmagnetization reversal process using the SSE signals detected\nby the two perpendicularly aligned voltage probes.\nAll films were grown using pulsed laser deposition (PLD)\nand structurally characterized using a Philips X0Pert diffrac-\ntometer with a Cu-K asource, which helped to quantify the\nlattice mismatches for the different substrates (Supplementary\nMaterial (SM) II). Magnetization hysteresis loops of the sam-\nples were measured by VSM in a PPMSrDynaCoolTMsys-arXiv:1902.05384v2  [cond-mat.mes-hall]  12 May 20192\nCu block atTCu block atT+ΔT\nPd/NFO\nRotational stageNPeltier modules\nS(a) (b) 360°\nSpacer\nSubstrate H MΦγ\nFIG. 1. (Color online) (a) LSSE setup. The sample is placed between\ntwo Cu blocks and a temperature gradient ÑTis applied. (b) The\ngeometry for four-point vectorial LSSE measurements and definition\nof the external magnetic field and magnetization angles \u001eand\r.\ntem. In the setup for LSSE measurements (Fig. 1(a)), the sam-\nple was placed between two Cu blocks. To carry out vectorial\nLSSE measurements (Fig. 1(b)), four Al wires were bonded\non four points of the Pd layer so that they are located on two\northogonally aligned axes. The two contacts of each axis were\nconnected to a separate nanovoltmeter (more details in SM I).\nMagnetization measurements were performed by VSM on\n\u0018450 nm thick NFO films grown on (011)- and (001)-oriented\nMGO and (011)-oriented CGO substrates to examine their\nmagnetic in-plane anisotropy characteristics. While the re-\nsults for NFO//MGO (011) and NFO//CGO (011) samples\nare presented in Figs. 2(a) and (b), we refer to the SM III\nfor NFO//MGO (001). For the NFO//MGO (011) thin film,\nwe observe a sharp switching of the magnetization when the\nexternal magnetic field is applied along the [01 ¯1] direction\n(Fig. 2(a)). With the external magnetic field applied along the\n[100] direction, we observe magnetic hard axis type switching\nbehavior with an anisotropy field of \u00181500 Oe. In addition,\nwe measured the SSE voltage signal VLSSE of the sample along\nthe two perpendicular directions (Fig. 1(b)). In the first config-\nuration, the VLSSE signal is measured along the [100] direction\n(Fig. 2(c)), while the VLSSE signal is measured along the [01 ¯1]\ndirection in the second configuration (Fig. 2(e)).\nIn the first configuration (Fig. 2(c)), when the magnetic\nfield is applied along the [01 ¯1] direction, the magnetization\nMof the NFO film is also aligned in the same direction.\nWhen the magnetic field direction changes polarity, the mag-\nnetization of the film Malso switches into the opposite di-\nrection. This results in a sharp switching in the VLSSE signal\n(\u001e=90\u000ein Fig. 2(c)) and it is comparable to the correspond-\ning magnetization measurement when the magnetic field is in\nthe [01 ¯1] direction (Fig. 2(a)). In the next step, we changed\nthe angle\u001eof the external magnetic field with respect to the x\naxis (insert in Fig. 2(c)) in the range from 0\u000eto 90\u000e. In satura-\ntion, the magnetization of the NFO film Mis almost aligned\nalong the direction of the external magnetic field for all \u001ean-\ngles. The voltage generation ( VLSSE =ELSSE\u0001d) due to the\nISHE is characterized by the projection of magnetization onto\nthe [01 ¯1] direction. The electric field ELSSE generated by the\nSSE is given by\nELSSEµJs\u0002\u001b; (1)\nFIG. 2. (Color online) (a),(b) Normalized in-plane magnetization\nversus magnetic field for NFO//MGO (011) and NFO//CGO (011)\n(The insets are substrate orientation schematic and close-up figures).\nThe magnetization is measured with an external in-plane magnetic\nfield applied in three different magnetic field direction (0\u000e[100], 45\u000e\nand 90\u000e[01¯1]). (c)-(f) LSSE measurements at various angles \u001efor\n(c),(e) NFO//MGO (011) and (d),(f) NFO//CGO (011) with voltage\nmeasured (c),(d) along the [100] (X) direction and (e),(f) along the\n[01¯1] (Y) direction ( DT=20 K).\nwhereJsis the thermally induced spin current which is par-\nallel to the ÑTinzdirection, and \u001bis the spin polarization\nvector which is aligned along M. Thus, from Eq. (1) we can\nconclude that\nVxµ\u001byµMy;Vyµ\u001bxµMx: (2)\nWith increasing the angle \u001ebetween the external magnetic\nfield and the [100] direction along the xaxis, the saturation\nvoltage Vxincreases in correspondence with a factor of sin \u001e\ndue to the cross product of the ISHE [12]. When the projec-\ntion of the magnetization in the [01 ¯1] direction Myincreases\n(decreases) due to the increase (decrease) of the external field,\nthe measured voltage signal of the ISHE also increases (de-\ncreases). At zero magnetic field the magnetization can rotate\ncompletely into the magnetic easy axis or partially (or fully)\nswitch into another magnetic easy axis reducing the projection\nMyand the corresponding voltage Vx. When increasing the ex-\nternal field into opposite direction the voltage usually changes\nsign and the coherent rotation of the magnetization out of\nthe magnetic easy axis along the [01 ¯1] direction (\u001e<90\u000e)\nis accompanied by a decrease of the absolute voltage value\n(Fig. 2(c)). For \u001e= 0\u000e, the saturation voltage is nearly zero as3\nwe expect from the ISHE. The small residual voltage signal in\nsaturation could be explained by a slight misalignment of volt-\nage contacts along the [100] direction or with an alignment of\nthe magnetization that is not fully saturated.\nIn the second configuration (Fig. 2(e)), the voltage contacts\nalong the [01 ¯1] (Y) direction are used. When the external\nmagnetic field is applied along the [100] direction to com-\nplete the typical LSSE configuration (i.e. \u001e= 0\u000e), we observe\na similar voltage value in magnetic saturation as compared\nto the previous configuration in Fig. 2(c)). When the exter-\nnal magnetic field decreases, the LSSE voltage signal does\nnot show a sharp switching, but follows the magnetization\nmeasurement in the [100] direction with low remanence (in\nFig. 2(a)), which significantly differs from that in the first\nconfiguration. While the projection of the magnetization onto\nthe [100] direction Mxincreases (decreases) monotonically\nuntil the magnetization switches, the LSSE voltage Vyalso\nincreases (decreases). This is comparable to the first configu-\nration in Fig. 2(c), detecting the same switching events while\nbeing sensitive to orthogonal projections of the magnetization\nvector. Therefore, the LSSE measurements provide a promis-\ning alternative compared to established optical measurement\nmethods to determine both in-plane components of the mag-\nnetization during field reversal.\nFigure 2(b) shows the VSM results for the NFO//CGO\n(011) thin film as a comparison with the NFO//MGO (011)\nsample. The magnetization curves for different external field\ndirections still indicate magnetic easy axis behavior in [01 ¯1]\ndirection as well as magnetic hard axis characteristics in\n[100] direction. However, the differences between easy-axis\nand hard-axis loops are not that pronounced as seen for the\nNFO//MGO (011) sample. We also performed SSE measure-\nments on this sample. As shown in Figs. 2(d) and 2(f), we find\nthat the SSE measurements along these two perpendicular di-\nrections are very similar, which is consistent with the previous\nmagnetization results in Fig. 2(b).\nWe have further carried out in-plane angular-dependent\nFMR measurements that identify the presence of a fourfold\ncombined with a strong uniaxial anisotropy for NFO//MGO\n(011) thin films. The results are shown in Fig. 3(a), with over-\nall in-plane magnetic easy axis along the [01 ¯1] direction and\nthe magnetic hard axis along the [100] direction. On the other\nFIG. 3. (Color online) In-plane angular dependence of reso-\nnance field, Hres, at 20 GHz for (a) NFO//MGO (011) and (b)\nNFO//CGO (011).hand, for NFO//CGO (011) thin films, the fourfold anisotropy\nis dominant and only a weak uniaxial anisotropy is present,\nresulting in a biaxial anisotropy with a more and less hard\naxis in 0\u000eand 90\u000edirection, as shown in Fig. 3(b). Thus,\nwe conclude that in addition to the anisotropy landscape mod-\nified by the crystal cut a strain anisotropy affects the over-\nall strength and direction of magnetic easy and hard axes.\nWhile we observe this anisotropy modification with various\nextent for NFO//MGO (011) and NFO//CGO (011) samples,\nthe NFO//MGO (001) control sample exhibits a pure four-fold\nmagnetocrystalline anisotropy. For the FMR results of this\nsample and a deeper fit analysis of the FMR measurements\npresented here see SM VI.\nThe reversal process of the magnetization vector depends\non the magnetic anisotropy and the direction of the external\nmagnetic field. The data used to display the LSSE hystere-\nsis loops (in Figs. 2(c) - 2(f)) can now be used in order to\nreconstruct the magnetization reversal process, utilizing both\nprojections of the in-plane magnetization vector on the [100]\nand [01 ¯1] directions from the vector LSSE measurement ( Mx\nandMy). From Eq. (2), we can assume that\nVx=Ax\u0001My;Vy=Ay\u0001Mx; (3)\nwhere AxandAyare material and setup parameters. A detailed\nanalysis of the voltage VSSEanisotropy is presented in SM IV .\nThe LSSE saturation voltages VxandVyin relation to\nthe azimuthal angle of the magnetization show no signifi-\ncant anisotropy yet in principle the magnetic transport and\nthe magnitude of the generated SSE voltages can depend on\nthe relative orientation to the NFO crystal axes. Notably in\nSMR measurements a current direction anisotropy can be ob-\nserved in NFO thin films where the precise origin of the cur-\nrent anisotropy is still up for debate [31]. In this SMR and\nin our SSE experiment the spin current is always generated\nin out-of-plane direction propagating along the same crystal-\nlographic axis and is therefore independent from the voltage\ndetection direction. Still, an anisotropy may originate from\nanisotropic SHE or spin mixing conductance as discussed by\nAlthammer et al. [31]. However, as presented in Fig. 2(c-\nf) the amplitude of VSSEat maximum field measured paral-\nlel to [100] direction is comparable to the voltage measured\nalong the [01 ¯1] direction for both NFO//MGO (011) as well\nas NFO//CGO (011). This is a strong indication that there is\nno significant anisotropy of the ISHE e.g. by strain similar\nto the reported immunity to electrostrain of the spin-current\ntransport and spin-charge conversion in Pt/YIG by Wang et\nal.due to robust magnon diffusion length and spin-mixing\nconductance at the interface [32].\nSubsequently, we can use the assumption Ax\u0019Ayto calcu-\nlate other quantities like the magnetization M,\nM=p\n(Mx)2+ (My)2=q\n(Vy\nAy)2+ (Vx\nAx)2\u00191\nAx\u0001q\nV2y+V2x:\n(4)\nWe calculate the in-plane orientation of the magnetization\nvector based on the magnitude jVj=q\nVx2+Vy2, and the az-4\n0°30°60°90°\n120°\n150°\n180°\n210°\n240°\n270°300°330°05101520VSSE [µV]\nγ(a)ϕ= 0◦\n0°30°60°90°\n120°\n150°\n180°\n210°\n240°\n270°300°330°05101520VSSE [µV]\nγ(b)ϕ= 30◦\n0°30°60°90°\n120°\n150°\n180°\n210°\n240°\n270°300°330°05101520VSSE [µV]\nγ\n(c)ϕ= 60◦\n0°30°60°90°\n120°\n150°\n180°\n210°\n240°\n270°300°330°05101520VSSE [µV]\nγ(d)ϕ= 90◦\n0°30°60°90°\n120°\n150°\n180°\n210°\n240°\n270°300°330°05101520VSSE [µV]\nγ\n(e)ϕ= 120◦\n0°30°60°90°\n120°\n150°\n180°\n210°\n240°\n270°300°330°05101520VSSE [µV]\nγ(f)ϕ= 150◦\nFIG. 4. (Color online) The NFO//MGO (011) reversal processes of\nthe magnetization vector for angles of the external magnetic field\n(blue dotted line) \u001efrom 0\u000eto 150\u000e((a) - (f)) relative to the [100] in-\nplane direction (inset Fig. 2(a)). The magnitude of the magnetization\nvector given byjVjof the combined LSSE measurements in µV is\nplotted against the rotation angle \rof the magnetization vector. The\nmagnetic easy (green) and magnetic hard axes (orange) are marked\nby dashed lines.\nimuthal magnetization angle \rcan be expressed as\n\r=arctan (My\nMx) =arctan (Vx\nVy\u0001Ay\nAx)\u0019arctan (Vx\nVy):(5)\nThis can now be applied to create polar plots of the magne-\ntization vector length versus \r. The plot compilation shown\nin Fig. 4 visualizes the NFO//MGO (011) reversal processes\nof the magnetization vector for the selected external mag-\nnetic field directions \u001e, defined as shown in Fig. 1. These\npolar plots represent the progress of the resulting magnetiza-\ntion vector state for each magnetic field strength during the\nstepwise change of the external magnetic field direction with\nrespect to the sample orientation. This analysis is based on\nthe magnetization measurements shown in Fig. 2. A definedmagnetic domain state exists at the beginning of each rever-\nsal process, since the external magnetic field strength is suf-\nficient to mainly saturate the magnetization orientation along\nthe external magnetic field direction. The subplot Fig. 4(a) ex-\nemplifies a reversal process when the external magnetic field\nis applied along the magnetic hard axis of the NFO//MGO\n(011) sample ( \u001e=0\u000e) according to the geometry defined in\nFig. 1(b). A saturating external magnetic field yields a maxi-\nmum in Vy(from Eq. (2)) and represents the starting point of\nthis reversal process as seen on the right side of Fig. 4(a). The\napplication of an external field along or close to a magnetic\nhard axis induces domain splitting if the external magnetic\nfield is decreased as indicated by the reducing magnitude of\nthe magnetization vector. When the external magnetic field is\nfurther reduced (still applied along \u001e=0\u000e), the magnetic mo-\nments of those domains switch or rotate towards the magnetic\neasy axis along [01 ¯1] direction, indicated by the black arrow\nat the top of Fig. 4(a). The preferred direction is defined by\nthe slight misalignment of the external magnetic field rela-\ntive to the magnetic hard axis [100] direction of the thin film.\nWith an increasing opposite external field completing the first\nbranch of the LSSE hysteresis loop, the magnetization rotates\nback to the external field direction.\nHowever, if the external magnetic field is not applied\nclose to a magnetic hard axis the system essentially follows\nthe coherent rotation model without transforming into multi-\ndomains. Here, the length of the magnetization vector is con-\nstant from saturation to remanence. For an external magnetic\nfield along the strong magnetic easy axis direction ( \u001e=90\u000e,\nFig. 4(d)), we can observe a simple switching of the magne-\ntization direction by 180\u000e, as expected along a magnetic easy\naxis direction where the LSSE hysteresis in the standard mea-\nsurement geometry shows a very small coercive field and high\nremanence. We furthermore used this vector LSSE technique\nto visualize the magnetization reversal process of correspond-\ning samples showing a pure fourfold and a fourfold plus weak\ntwofold anisotropy. The vector LSSE measurement results for\nthose NFO//MGO (001) and NFO//CGO (011) samples can\nbe found in the SM V .\nThe vectorial magnetization technique based on SSE is us-\ning electrical detection compared to magnetooptic instruments\nextracting the in-plane magnetization components by differ-\nent combinations of magnetooptic Kerr effects (MOKE) [33–\n35]. In contrast to the magnetooptic techniques, our vecto-\nrial SSE approach is also applicable for magnetic materials\nthat are not amenable to magnetooptic detection due to a van-\nishingly small Kerr rotation especially with standard vector\nMOKE techniques in the visible range of light such as ferrites,\nYIG [36] and in particular NFO. This illustrates the need for a\nwider range of applicable techniques for the probe of magne-\ntization reversals in the expanding field of spin caloritronics,\nspin orbitronics, and beyond. Still, this technique offers room\nfor improvement and simplification to allow for a broader ap-\nplicability. For example, it is possible to implement lock-in\ntechniques or replace the bonding process by designing an in-\nsulating spacer material pre-coated with defined voltage con-5\ntacts to increase precision and practicability.\nIn conclusion, we have experimentally found that the lat-\ntice mismatch between NFO film deposited on isostructural\nspinel CGO (011) and MGO substrates of two different ori-\nentations ((001) and (011)) results in varying magnetic strain\nanisotropy as determined from VSM and FMR measurements.\nThe strain anisotropy significantly influences the shape of the\nLSSE voltage hysteresis loop measured as a function of the\nexternal magnetic field. Based on vector measurement of the\nLSSE, we show that the complete reversal process of the mag-\nnetization vector can be determined, demonstrating an alterna-\ntive to study the magnetization reversal process of thin films\nbased on SSE and ISHE voltage detection.\nThis work was supported by NSF ECCS Grant No. 1509875\nand the Deutsche Forschungsgemeinschaft (DFG) within the\npriority program Spin Caloric Transport (SPP 1538), Grant\nNo. KU 3271/1-1. The MGO and CGO substrates were pre-\npared from bulk crystals obtained by the Czochralski method\nat Leibniz-Institut für Kristallzüchtung, Berlin, Germany [28,\n37]. We finally thank Matthias Althammer and Weiwei Lin\nfor additional input during the revision process of this letter.\n\u0003These authors contributed equally to this work.\n†E-mail: agupta@mint.ua.edu\n‡E-mail: tkuschel@physik.uni-bielefeld.de\n[1] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. 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Status Solidi (a) 212,\n1455 (2015)."    },    {        "title": "1902.08700v1.Strongly_Enhanced_Gilbert_Damping_in_3d_Transition_Metal_Ferromagnet_Monolayers_in_Contact_with_Topological_Insulator_Bi2Se3.pdf",        "content": "1 \n Strongly Enhanced Gilbert Damping in 3 d Transition Metal \nFerromagnet Monolayers in Contact with Topological Insulator Bi 2Se3 \nY. S. Hou1, and R. Q. Wu1 \n1 Department of Physics and Astronomy, University of California, Irvine, California \n92697 -4575, USA  \n \nAbstract  \nEngineering Gilbert damping of ferromagnetic metal films is of great importance to \nexploit and design spintronic devices that are operated with an ultrahigh speed. Based on \nscattering theory of Gilbert damping, we extend the torque method originally  used in \nstudies of magnetocrystalline anisotropy to theoretically determine Gilbert dampings of \nferromagnetic metals. This method is utilized to investigate Gilbert dampings of 3 d \ntransition metal ferromagnet iron, cobalt and nickel monolayers that are co ntacted by the \nprototypical topological insulator Bi 2Se3. Amazingly, we find that their Gilbert dampings \nare strongly enhanced by about one order in magnitude, compared with dampings of their \nbulks and free -standing monolayers, owing to the strong spin -orbit coupling of Bi 2Se3. \nOur work provides an attractive route to tailoring Gilbert damping of ferromagnetic \nmetallic films by putting them in contact with topological insulators.  \n  \n \n \n \n \nEmail: wur@uci.edu   \n \n \n \n \n \n 2 \n I. INTRODUCTION  \nIn ferromagnets, the time -evolution of their magnetization M can be described by the \nLandau -Lifshitz -Gilbert (LLG) equation [1-3]  \n1Meff\nSdd\ndt dt   MMM H M\n, \nwhere \n0B g  \n  is the gyromagnetic ratio, and \nMSM  is the saturation \nmagnetization. The first term describes the precession motion of magnetization M about \nthe effective magnetic field, Heff, which  includes contributions from external field, \nmagnetic anisotropy, exchange, dipole -dipole and Dzyaloshinskii -Moriya  interactions [3]. \nThe second term represents the decay of magnetization prece ssion with a dimensionless \nparameter \n , known as the Gilbert damping [4-8]. Gilbert  damping is known to be \nimportant for the performance of various spintronic devices such as hard drives, magnetic \nrandom access memories, spin filters, and magnetic sensors [3, 9, 10]. For example, \nGilbert damping in the free layer of reader head in a magnetic hard drive determines its \nresponse speed and signal -to-noise ratio [11, 12]. The bandwidth, insertion loss , and \nresponse time of a magnetic thin film microwave device also critically depend on the \nvalue of \n  in the film [13]. \n \nThe rapid developm ent of spintronic technologies calls for the ability of tuning Gilbert  \ndamping in a wide range. Several approaches have been proposed for the engineering of \nGilbert damping in ferromagnetic (FM) thin films, by using non -magnetic or rare earth \ndopants, addi ng differ ent seed layers for growth, or adjusting composition ratios in the \ncase of alloy films [9, 14-16]. In par ticular, tuning \n  via contact with other materials \nsuch as heavy metals, topological insulators (TIs), van der Waals monolayers or magnetic \ninsulators is promising as the selection of material combinations is essentially unlimited.  \nSome of these materials may have fundamentally different damping mechanism and offer \nopportunity for studies of new phenomena such as spin -orbit torque, spin -charge \nconversion, and thermal -spin-behavior [17, 18].  \n \nIn this work, we systematically investigate the effect of Bi 2Se3 (BS), a prototypical TI, on \nthe Gilbert damping of 3d transitio n metal (TM) Fe, Co and Ni monolayers (MLs) as they 3 \n are in contacted with each other. We find that the Gilbert dampings in the TM/TI \ncombinations are enhanced by about an order of magnitude than their counterparts in \nbulk Fe, Co and Ni as well as in the fr ee-standing TM MLs.  This drastic enhancement \ncan be attributed to the strong spin -orbit coupling (SOC) of the TI substrate and might \nalso be related to its topological nature . Our work introduces an appealing way to \nengineer Gilbert dampings of FM metal fi lms by using the peculiar physical properties of \nTIs.  \n \nII. COMPUTATIONAL DETAILS  \nOur density functional theory (DFT) calculations are carried out using the Vienna Ab-\ninitio  Simulation Package (VASP) at the level of the generalized gradien t approximation \n[19-22]. We treat Bi -6s6p, Se -4s4p, Fe -3d4s, Co -3d4s and Ni -3d4s as valence electrons \nand employ the projector -augmented wave pseudopotentials to d escribe core -valence \ninteractions [23, 24].  The energy cutoff of plane -wave expansion is 450 eV [22]. The BS \nsubstrate is simulated by five quintuple layers ( QLs), with an in -plane lattice constant of \naBS = 4.164 Å and a vacuum space of 13 Å between slabs along the normal axis. For the \ncomputational convenience, we put Fe, Co and Ni MLs on both sides of the BS slab. For \nthe structural optimization of the BS/TM slabs, a 6× 6× 1 Gamma -centered k -point grid is \nused, and the positions of all atoms except those of the three central BS QLs are fully \nrelaxed with a criterion that the force on each atom is less than 0.01 eV/Å. The van der \nWaals (vdW) correction in the form of the nonlocal vdW functional (optB86b -vdW) [25, \n26] is included in all calculations.  \n \nThe Gilbert dampings are determined by extending the torque method that we developed \nfor the study of magnetocrystalline anisotropy [27, 28]. To ensure the numerical \nconvergence, we use very dense Gamma -centered k -point grids and, furthermore, large \nnumbers of unoccupied bands. For example, the first Bri llouin zone of BS/Fe is sampled \nby a 37× 37× 1 Gamma -centered k -point grid, and the number of bands for the second -\nvariation step is set to 396, twice of the number (188)  of the total valence electrons. More \ncomputational details are given in Appendix A. Mag netocrystalline anisotropy energies \nare determined by computing total energies with different magnetic orientations [29].  4 \n  \nIII. TORQUE METHOD OF DETERMINING GILBERT DAMPING  \nAccording to the scattering theory of Gilbert damping [30, 31], the energy dissipation \nrate of the electroni c system with a Hamiltonian, H(t), is determined by  \n dis 2i j j i F i F j\nijE E E E Euu\n         \nHHuu\n. \nHere, EF is the Fermi level and \nu  is the deviation of  normalized magnetic moment away \nfrom its equilibrium, i.e., \n0m m u  with \n00 MsM m . On the other hand, the time \nderivative of the magnetic energy in the LLG equation is [32] \n mag 3S\neffM dEdt\n MH\n mm\n. \nBy taking \ndis magEE\n  , one obtains the Gilbert damping as:  \n4i j j i F i F j\nij SE E E EM u u\n        \nHH\n. \nNote that, to obtain Eq. (4), we use \n mu  since the eq uilibrium normalized \nmagnetization m0 is a constant. In practical numerical calculations, \nFEE  is \ntypically substituted by the Lorentzian function \n 22\n0 0.5 0.5 L          . \nThe half maximum parameter, \n1 , is adjusted to reflect different scattering rates of \nelectron -hole pairs created by the precession of magnetization M [10]. This procedure  \nhas been already used in several ab initio  calculations for Gilbert dampings of metallic \nsystems [8, 9, 32-35], where the electronic responses play the major role for energy \ndissipation .  \n \nIn this work, we focus on the primary Gilbert damping in FM metals that arises from \nSOC [10, 36-38]. There are two important effects in a uniform precession of \nmagnetization M, when SOC is taken into consideration. The first is the F ermi surface \nbreathing as M rotates, i.e., some occupied states shift to above the Fermi level and some \nunoccupied states shift to below the Fermi level. The second is the transition between \ndifferent states across the Fermi level as the precession can be viewed as a perturbation to 5 \n the system. These two effects generate electron -hole pairs near the Fermi level and their \nrelaxation through lattice scattering leads to the Gilbert damping.  \n \nNow we demonstrate how to obtain the Gilbert damping due to SOC thro ugh extending \nour previous torque method [27]. The general Hamiltonian in Eq. (4) can be replaced by \n SOC j j jr\n H l s\n [4, 27] where  the index j refers to atoms, and \njji lr and s \nare orbital and spin operators, respectively. This is in the same spirit for the determination \nof the magnetocrystal line anisotropy [27], for which  our torque meth od is recognized as a \npowerful tool in the framework of spin -density theory [27]. When m points at the \ndirection of \n , , ,x y zm m m n , the term \nls  in HSOC is written as follows:   \n22\n2211cos sin sin22\n1sin sin cos 52 2 2\n1sin cos sin2 2 2ii\nz\nii\nz\nii\nzs l l e l e\ns l l e l e\ns l l e l e\n\n  \n\n\n\n\n  \n\n     \n   \n  \nn ls\n \nTo obtain the derivatives of H in Eq. (4), we assume that the magnitude of M is constant \nas its direction changes [36]. The processes of getting angular derivatives of H are \nstraightforward and the results are given by Eq. (A1) -(A5) in Appendix B.  \n \nIV. RESULTS AND DISCUSSION  \nIn this section, we first show that our approach of determining  Gilbert damping works \nwell for  FM metals such as 3d TM Fe, Co and Ni bulks. Following that, we demonstrate \nthe strongly enhanced Gilbert dampings of Fe, Co and Ni MLs due to the contact with BS \nand then discuss the underlying physical mechanism of these enhancements.  \n \nA. Gilbe rt dampings of 3d TM Fe, Co and Ni bulks  \nGilbert dampings of 3d TM bcc Fe, hcp Co and fcc Ni bulks calculated by means of our \nextended torque method are consistent with previous theoretical results [10]. As shown in \nFig. 1, the intraband contributions decrease whereas the interband contributions increase \nas the scattering rate \n  increases. The minimum values of  \n have the same magnitude 6 \n as those in Ref. [10] for all three metals, showing the applicability of our approach for the \ndetermination of Gilbert dampings of FM metals.  \n \n \nFigure 1 (color online) Gilbert dampings of (a) bcc Fe, (b) hcp Co and (c) fcc Ni bulks. Black \ncurves give the total Gilbert damping. Red and blue curves give the intraband and interband \ncontributi ons to the total Gilbert damping, respectively.  \n \nB. Strongly enhanced Gilbert dampings of Fe, Co and Ni MLs in contact with BS  \nWe now investigate the magnetic properties of heterostructures of BS and Fe, Co and Ni \nMLs. BS/Fe is taken as an example and its atom arrangement is shown in Fig. 2a. From \nthe spatial distribution of charge density difference \nBS+Fe-ML BS Fe-ML        in Fig. \n2b, we see that there is fairly obvious charge transfer between Fe and the topmost Se \natoms. By taking  the average of \n  in the xy plane, we find that charge transfer mainly \ntakes place near the interface (Fig.2c). Furthermore, the charge transfer induces non -\nnegligible magnetization  in the topmost QL of BS (Fig. 2b). Similar charge  transfers and \ninduced magnetization are also found in BS/Co and BS/Ni (Fig. A1 and Fig. A2 in \n7 \n Appendix C). These suggest that interfacial interactions between BS and 3 d TMs are very \nstrong.  Note that BS/Fe and BS/Co have in -plane easy axes whereas the BS/ Ni has an \nout-of-plane one.     \n \n \nFigure 2 (color online) (a) Top view of atom arrangement  in BS/Fe. (b) Charge density difference \n\n near the interface in BS/Fe. Numbers give the induced magnetic moments (in units of \nB ) in \nthe top most  QL BS. Color bar indicates the weight of negative (blue) and positive (red) charge \ndensity differences. (c) Planer -averaged charge density difference \n in BS/Fe. In (a), (b), (c), \ndark green, light gra y and red balls represent Fe, Se and Bi atoms, respectively.  \n \nFig. 3a and 3b show the \n  dependent Gilbert dampings of BS/Fe, BS/Co and BS/Ni. It is \nstriking that Gilbert dampings of BS/Fe, BS/Co and BS/Ni are enhanced by about one  or \ntwo order s in magnitude  from the counterparts of Fe, Co and Ni bulks as well as their \nfree-standing MLs, depending on the choice of scattering rate in the range from 0. 001 to \n1.0 eV.  Similar to Fe, Co and Ni bulks, the intraband contributions monotonically  \ndecrease while the interband contributions increase as the scattering rate \n  gets larger \n(Fig. A3 in Appendix D).  Note that our calculations indicate that there is no obvious \ndifference between the Gilbert dampings of BS/Fe when f ive- and six -QL BS slabs are \nused (Fig. A4 in Appendix E). This is consistent with the experimental observation that \nthe interaction  between the top and bottom topological surface states is negligible  in BS \nthicker than five QLs [39].  \n \n8 \n  \nFigure 3 (color online) Scattering rate \n  dependent Gilbert dampings of (a) Fe ML, bcc Fe bulk, \nBS/Fe and PbSe/Fe, (b) Co ML, hcp Co bulk, BS/Co, Ni ML, fcc Ni bulk and BS/Ni. (c) \nDependence of the Gilbert dampin g of BS/Fe on the scaled SOC \nBS  of BS in the range from \nzero (\nBS0 ) to full strength (\nBS1 ). Solid lines show the fitting of Gilbert damping \nBS/Fe  \nto Eq. (6). The inse t shows Gilbert damping comparisons between BS/Fe at \nBS0 , bcc Fe bulk \nand Fe ML.  \n \nAs is well -known, TIs are characterized by their strong SOC and topologically nontrivial \nsurface states. An important issue is how they affect the Gilbert damping s in BS/TM \nsystems. Using BS/Fe as an example, we artificially tune the SOC parameter \nBS of BS \nfrom zero to full strength and fit the Gilbert damping \nBS/Fe  in powers of \nBS  as \n2\nBS/Fe 2 BS BS/Fe BS 0 (6)       \n. \nAs shown in Fig. 3c, we obtain two interesting results: (I) when \nBS  is zero, the \ncalculated residual Gilbert damping \nBS/Fe BS 0    is comparable to Gilbert dampings of \nbcc Fe bulk and Fe free -standin g ML (see the inset in Fig. 3c) ; (II) Gilbert damping \nBS/Fe\n increases almost linearly with \n2\nBS  , simi lar to previous results [36]. These reveal \nthat the strong SOC of BS is crucial for the enhancement of Gilbert damping.    \n  \nTo gain insight i nto how the strong SOC of BS affects the damping of BS/Fe, we explore \nthe k-dependent contributions to Gilbert damping, \nBS/Fe . As shown in Fig. 4a , many \nbands near the Fermi  level show strong intermixing between Fe and BS orbitals (mar ked \nby black arrows ). Accordingly, these k-points have large contributions to the Gilbert \n9 \n damping  (marked by red arrows in Fig. 4b). However, if the hybridized states are far \naway from the Fermi level, they make almost zero contribution to the Gilbert damp ing. \nTherefore, we conclude that only hybridizations at or close to Fermi level have dominant \ninfluence on the Gilbert damping. This is understandable, since energy differences EF-Ei \nand EF-Ej are important in the Lorentzian functions in Eq. (4).   \n \n \nFigu re 4 (color online) (a) DFT+SOC calculated band structure of BS/Fe. Color bar indicates \nthe weights of BS (red) and Fe ML (blue). Black dashed line indicates the Fermi level. (b) k -\ndependent contributions to Gilbert damping \nBS/Fe at sc attering rate \n26meV . Inset shows \nthe first Brillouin zone and high -symmetry k -points \n , \nK and \n .  \n \nIt appears that there is no direct link between the topologic al nature of BS and the strong \nenhancement of Gilbert damping. The main contributions to Gilbert damping are not \nfrom the vicinity around the \n -point,  where the topological nature of BS manifests.  \nBesides,  BS should undergo a topol ogical phase transition from trivial to topological as \nits SOC \nBS  increases [40]. If the topological nature of BS dictates the e nhancement of \nGilbert damping, one should expect a kink in the \nBS  curve at this phase transition \npoint but this is obviously absent i n Fig. 3c.  \n \n10 \n To dig deeper into this interesting issue, we replace the topologically nontrivial BS with a \ntopologically trivial insulator PbSe, because the latter has a nearly the same SOC as the \nformer. As shown in Fig. 3a, the Gilbert damping of PbSe/Fe is noticeably smaller than \nthat of BS/Fe, although both are significantly enhanced from the values of \n  of Fe bulk \nand Fe free -standing ML. Taking the similar SOC and surface geometry between BS and \nPbSe  (Fig. A5 in Appendix F) , the large difference between the Gilbert dampings of \nBS/Fe and PbSe/Fe suggests that the topological nature of BS still has an influence on \nGilbert damping. One possibility is that the BS surface is metallic with the presence of \nthe time -reversal protected t opological surface states and hence the interfacial \nhybridization is stronger.      \n \n \nFigure 5 (color online) Comparisons between Gilbert damping \n  of BS/Fe at \n26meV  and \n(a) total DOS, (b) Fe projected DOS and (c) BS projected DOS. Red arrows and light cyan \nrectangles highlight the energy windows where Gilbert damping \n  and the total DOS and Fe \nPDOS have a strong correlation . In (a), (b) and (c), all DOS are in units of state per eV and \nFermi level EF indicated by the vertical green lines is set to be zero.  \n \nA previous study of Fe, Co and Ni bulks suggested a strong correlation between Gilbert \ndamping and total density of states (DOS) around the Fermi level [36]. To attest if this is \n11 \n applicable here, we show the total DOS and Gilbert damping \nBS/Fe  of BS/Fe as a \nfunction of the Fermi level based on the rigid band a pproximation. As shown in Fig. 5a, \nGilbert damping \nBS/Fe and the total DOS behave rather differently in most energy \nregions. From the Fe projected DOS (PDOS) and BS projected PDOS (Fig. 5b and 5c), \nwe see a better  correlation between G ilbert damping \nBS/Fe and Fe -projected DOS, \nespecially in regions highlighted by the cyan rectangles . We perceive that although the \n\n-DOS correlation might work for simple systems, it doesn’t hold when hybridiza tion and \nSOC are complicated as the effective SOC strength may vary from band to band.  \n \nV. SUMMARY  \nIn summary, we extend our previous torque method from determining magnetocrystalline \nanisotropy energies [27, 28] to calculating Gilbert da mping of FM metals and apply this  \nnew approach to Fe, Co and Ni MLs in contact with TI  BS. Remarkably, the presence of \nthe TI BS substrate causes order of magnitude enhancements in their Gilbert dampings. \nOur studies demonstrate such strong enhancement is mainly due to the strong SOC of TI \nBS substrate . The topological nature of BS may also play a role by facilitati ng the \ninterfacial hybridiz ation and leaving more states around the Fermi level . Although \nalloying with heavy elements also enhances Gilbert dampings  [32], the use of  TIs pushes \nthe enhancement into a much wide r range. Our work thus establishes an attractive way \nfor tuning the Gilbert damping of FM metallic films, especially in the ultrathin reg ime.  \n \nACKNOWLEDGMENTS  \nWe thank Prof. A. H. MacDonald and Q. Niu at University Texas, Austin, for insightful \ndiscussions. We also thank Prof. M. Z. Wu at Colorado State University and Prof. J. Shi \nat University of California, Riverside for sharing their ex perimental data before \npublication. Work was supported by DOE -BES (Grant No. DE -FG02 -05ER46237). \nDensity functional theory calculations were performed on parallel computers at NERSC \nsupercomputer centers.   \n \n 12 \n Appendix A: Details of Gilbert damping calcula tions  \nTo compare Gilbert dampings of Fe, Co and Ni free -standing MLs with BS/Fe, BS/Co, \nand BS/Ni, we use \n33  supercells containing three atoms and set their lattice \nconstants to 4.164 Å, same as that of the BS substrate. This means that the lattice \nconstant of their primitive unit cell (containing one atom) is fixed at 2.40 Å. The relaxed \nlattice constants of Fe (2.42 Å), Co (2.35 Å) and Ni (2.36 Å) free -standing MLs are close \nto this value.   \nSystems  a (Å)  b (Å) c (Å) k-point grid            \nFe bulk  2.931  2.931  2.931  35× 35× 35  16 36 2.25 \n Co bulk  2.491  2.491  4.044  37× 37× 23  18 40 2.22 \nNi bulk  3.520  3.520  3.520  31× 31× 31  40 80 2.00 \nFe ML  4.164  4.164  -- 38× 38× 1  24 56 2.33 \nCo ML  4.164  4.164  -- 37× 37× 1  27 64 2.37    \nNi ML  4.164  4.164  -- 39× 39× 1  30 72 2.40 \nBS/Fe  4.164  4.164  -- 37× 37× 1  188 396 2.11 \nBS/Co  4.164  4.164  -- 37× 37× 1  194 408 2.10 \nBS/Ni  4.164  4.164  -- 37× 37× 1  200 432 2.16 \nPbSe/Fe  4.265  4.265  -- 37× 37× 1  174 376 2.16 \n \nTable A1. Here are details of Gilbert damping calculations of all systems that are studied \nin this work.     is abbreviated for the number of  valence electrons and     stands for the \nnumber of total bands.   is the ratio between     and    , namely,          . Note that \nfive QLs of BS are used in calculations for BS/Fe, B S/Co and BS/Ni.   \n \n \n \n \n \nAppendix B: Derivatives of SOC Hamiltonian HSOC with respect to the \nsmall deviation \nu  of magnetic moments  \nBased on the SOC Hamiltonian HSOC in Eq. (5) in the main text, derivatives of the term \nls\n against the polar angle \n  and azimuth angle \n  are    13 \n \n11sin cos cos22\n1 1 1cos sin sin A1 ,2 2 2\n1 1 1cos sin sin2 2 2ii\nnz\nii\nz\nii\nzs l l e l e\ns l l e l e\ns l l e l e\n\n  \n  \n  \n\n\n  \n\n       \n   \n  \nls \nand \n \n  \n 22\n22110 sin sin22\n10 sin cos A2 .2 2 2\n10 cos sin2 2 2ii\nn\nii\niis i l e i l e\ns i l e i l e\ns i l e i l e\n\n\n\n\n\n\n  \n\n         \n     \n    \nls\n \nNote that magnetization M is assumed to have a constant magnitude when it precesses , so \nwe have \n0SOC SOC    H M H m . When the normalized magnetization  m points \nalong the direction of \n , , ,x y zm m m n , we have: \nsin cosxm , \nsin sinym  \nand \ncoszm . Taking \n0m m u  and the chain rule together, we obtain derivatives of \nSOC Hamiltonian HSOC with respect to the small deviation  of magnetic moments as \nfollows:   \nsincos cos A3 ,sinSOC SOC SOC SOC SOC\nx x x x x\nSOC SOCu m m m m\n\n                   \nH H H H H m\nm\nHH\n \ncoscos sin A4 ,sinSOC SOC SOC SOC SOC\ny y y y y\nSOC SOCu m m m m\n\n                   \nH H H H H m\nm\nHH\n \nand  \nu14 \n \n sin A5 .SOC SOC SOC SOC SOC\nz z z z z\nSOCu m m m m\n\n                 \nH H H H H m\nm\nH \nCombining Eq. (5) and Eq. (A1 -A6), we can easily obtain the final formulas of \nderivatives of SOC Hamiltonian HSOC of magnetization m. \n \n \n \n \nAppendix C: Charge transfers and induced magnetic moments in BS/Fe, \nBS/Co and BS/Ni  \n \nFigure A1 (color online) Planar -averaged char ge difference \nBS TM ML BS TM-ML         \n(TM = Fe, Co and Ni) of  (a) BS/Fe, (b) BS/Co and (c) BS/Ni . The atoms positions are given along \nthe z axis.  \n \n15 \n  \nFigure A2 (Color online) Charge density difference \nBS TM ML BS TM ML          (TM = Fe, \nCo and Ni) nea r the interface betwee n the TM monolayer and the top most  QL BS of (a) BS/Fe, (b) \nBS/Co and (c) BS/Ni. The color bar shows the weights of the negative (blue) and positive (red) \ncharge density differences. Numbers give the induced magnetic moments (in units  of \nB ) in the \ntopmost  QL BS. Bi and Se atoms are shown by the purple and light green balls, respectively.  \n \n \nAppendix D: Contributions of intraband and interband to the Gilbert \ndampings of BS/Fe, BS/Co and BS/Ni  \n \nFigure A3 (color  online) Calculated Gilbert dampings of (a) BS/Fe, (b) BS/Co and (c) BS/Ni. \nBlack curves give the total damping. Red and blue curves give the intraband and interband \ncontributions, respectively.  \n16 \n Appendix E: Gilbert dampings of BS/Fe with five - and six -QLs of BS slabs  \n8  \nFigure A4 (color online). Gilbert dampings of BS/Fe with five (red) and six (black) QLs of BS \nslabs. In the calculations of the Gilbert damping of BS/Fe with six QLs of BS, we use a 39 ×39×1 \nGamma -centered k -point grid, and the number of the  total bands is 448 which is twice \nlarger than the number of the total valence electrons (216).    \n \nAppendix F: Structural c omparisons between BS/Fe and PbSe/Fe  \n \n17 \n Figure A5 (color online) (a) Top view and (c) side view of atom arrangement  in BS/Fe. (b) Top  \nview and (d) side view of atom arrangement  in PbSe/Fe. In (a) and (c), the xyz -coordinates are \nshown by the red arrows. In (b) and (d), the rectangles with blue dashed lines highlight the most \ntop QL BS in BS/Fe which is similar to the Pb and Se atom laye rs in PbSe/Fe. 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Wang  \nBeijing National Laboratory for Condensed Matter Physics  \nInstitute of Phy sics \nChinese Academy of Sciences  \nBeijing 100190, China  \nE-mail: fwwang@iphy.ac.cn  \n \nDr. S. L. Zhang, Prof. T. Hesjedal  \nDepartment of Physics, Clarendon Laboratory  \nUniversity of Oxford  \nOxford OX1 3PU, United Kingdom  \n \nDr. D. Alba Venero  \nISIS \nSTFC Rutherford Appleton Laboratory  \nChilton, Didcot OX11 0QX, United Kingdom  \n \nDr. J. S. White  \nLaboratory for Neutron Scattering and Imaging  \nPaul Scherrer Institute  \nCH-5232 Villigen, Switzerland  \n \nDr. R. Cubitt  \nInstitut  Laue -Langevin  \n6 rue Jules Horowitz, 38042, Grenoble, France  \n \nDr. Q. Z. Huang  \nNIST Center for Neutron Research  \nNational Institute of Standards and Technology  \n100 Bureau Drive, Gaithersburg, MD 20899, United States  \n \nDr. J. Chen  \nChina Spallation Neutron Sour ce \nDongguan 523808, China  \n \nProf. G. van der Laan  \nMagnetic Spectroscopy Group  \nDiamond Light Source  \nDidcot OX11 0DE, United Kingdom  \n \nX. Y. Li, H. Li, Prof. W. H. Wang, Prof. F. W. Wang  \nSchool of Physical Sciences  \nUniversity of Chinese Academy of Sciences       \n2 \n \nBeijing 101408, China  \n \nProf. L. H. He, Prof. W. H. Wang, Prof. F. W. Wang  \nSongshan Lake Materials Laboratory  \nDongguan 523808, China  \n \nKeywords: Biskyrmion, Disorder , Small angle neutron scattering, MnNiGa  \n \nA biskyrmion consists of two bound, topologically stable skyrmion spin textures. These \ncoffee -bean -shaped objects have been observed in real -space in thin plates using Lorentz \ntransmission electron microscopy (LTEM). From LTEM imaging alone, it is not clea r \nwhether biskyrmions are surface -confined objects, or, analogously to skyrmions in non -\ncentrosymmetric helimagnets, three -dimensional tube -like structures in bulk sample. Here, we \ninvestigate the biskyrmion form factor in single - and polycrystalline MnNiG a samples using \nsmall angle neutron scattering (SANS). We find that biskyrmions are not long -range ordered, \nnot even in single -crystals . Surprisingly all of the disordered biskyrmions have their in -plane \nsymmetry axis aligned along certain directions, gove rned by the magnetocrystalline \nanisotropy. This anisotropic nature of biskyrmions may be further exploited to encode \ninformation.  \n \n      \n3 \n \nSystems  consisting  of elastically  coupled  arrangements of  particles  or quasiparticles  are a \ncommon,  yet intriguing  many -body  problem  in physics.  First,  the elementary unit  has to be \nidentified,  and its effective  potential  has to be determined.  Next,  the interactions  among  these  \nparticles  will lead to characteristic  dynamics  of the ensemble. Depending on the nature of the \nparticles, an order parameter can be assigned to describe a phase transition.[1] For example, in \na crystallization process, atoms are usually treated as isotr opic particles, and the ordering is \nevaluated by a scalar structure factor. Nevertheless, in  many  occasions,  these  atoms  carry  \nmagnetic  moments that are anisotropic, and which contribute an extra degree of  freedom.  \nConsequently,  an additional  order  parameter (magnetization)  emerges  to describe  how these  \nmoments order, appearing as a vector field.[2] Moreover, the elementary units can have \ninternal degrees of freedom, which have to be taken into account to understand the emergence  \nof order  on all length scales.  \nMagnetic skyrmions are field -like solutions that  carry nontrivial topological properties, and \nwhich can be regarded  as quasiparticles  in continuum  field theory.[3,4] Discovered in \nnoncentrosymmetric helimagnets[5,6] and later found in magnetic monolayers and \nmultilayers,[7–9] frustrated magnet,[10,11] as well  as in polar magnets,[12] skyrmions are axial -\nsymmetric and localized magnetization configurations with d ifferent point -group symmetries \nof either Dn, T, O (Bloch -type) or Cnv (Né el -type).[13] They can be written as m = (mx, my, mz) = \n(sin Θ cos Ψ, sin Θ sin Ψ, cos Θ) in polar coordinates ( ρ, ψ), where Θ = Θ( ρ) is the radial \nprofile function; and Ψ = ψ + χ, where phase χ defines the helicity.[3,4] These two types of \nskyrmions  nearly always have isotropic  inter-skyrmion  potentials,[14] as evidenced by the long -\nrange ordered, hexagonal close -packed skyrmion  lattice  arrangement.[15] Therefore,  the \nordering of the skyrmions has scalar nature, justifying rigid -body modeling using molecular \ndynamics approaches.[16] A major modification of this rigid -body model has to be made to \ninclude the description of so -called antiskyrmions which  are composed of D2d skyrmions.[17]      \n4 \n \nThese antiskyrmions are anisotropic, thus their inte raction potential becomes twofold \nsymmetric, breaking O(2) symmetry.  The anisotropic  nature  of antiskyrmions may therefore  \nalso influence  their lattice  order.[17] \nSuch anisotropic properties are most pronounced in a  biskyrmion system in which the \nelementary quasiparticles are molecule -like bound skyrmion pairs with C2v symmetry, as \nobserved in LTEM experiments in lamella of La 2−2xSr1+2xMn 2O7 (x = 0.315),[18] MnNiGa,[19–\n21] Cr11Ge19,[22] and FeGd amorphous films.[23] In the stereographic projection representation, a \nbiskyrmion molecule can be written as a complex  number  Ω:[24] \nΩ(𝑑,𝛹,𝜆,𝜂)=𝑚𝑥+𝑖𝑚𝑦\n1+𝑚𝑧=Ω𝜒Ω−𝜒                                    (1) \nwhere Ω χ and Ω −χ are two axial -symmetric skyrmions with  opposite  helicity  angle,  χ and −χ. \nThe exact  biskyrmion structure is further determined by the skyrmion -skyrmion bond distance \nd, the azimuthal rotation angle Ψ, the polarity  λ (the magnetization  direction  of the core of Ω χ), \nand a shape factor η. η is the ratio be tween  the long and short  axes when  distorting  an \notherwise  rotation -symmetric skyrmion into an elliptical shape. Figure 1  illustrates the key \nparameters that describe a C2v biskyrmion  and their corresponding simulated magnetic form \nfactor, respectively. From the illustration, it is clear that the principle C2 axis defines a \ndirectional axis of the biskyrmion motif.  In other  words,  unlike  for isotropic  skyrmions,  \nbiskyrmion  molecules  have an internal  degree  of freedom where the in -plane direction Ψ can in \nprinciple take arbitrary  values.  Consequently,  when  considering  the assembly of biskyrmions \ninto ordered lattices, the biskyrmion directional order also has to be taken into account, in \naddition to their relative positions. For example, one can envision a hexagonally  ordered  \nbiskyrmion  lattice,  in which  the individual  biskyrmion  axes can orient  in different  ways,  e.g., \nparallel,  antiparallel,  or perpendicular  to each other.  \nSo far, the experimental characterization of biskyrmions  was mainly  based  on LTEM  \nstudies,[18–23] which provided microscopic insight into their real -space order in quasi -two-     \n5 \n \ndimensional  thin plate  samples. An in -plane magnetization of such a biskyrmion configuration \nis shown in Figure 1g . In most  of these  materials systems,[18,19,23,25] the energy density \nfunctional ω, and the total energy E, can be written in the micromagnetic framework as:  \n𝜔 =𝐴(∇𝐦)2− 𝐾𝑢𝑚𝑧2−𝐁∙𝐦\n𝐸= ∫𝜔𝑑3𝑟+𝐸dipolar                                                  (2)                         \nwhere A is the exchange stiffness constant, Ku is the uniaxial  anisotropy,  B is the external  \nmagnetic  field,  and Edipolar  is the non -local dipole -dipole  interaction  energy.  \nA stripe domain state ( B = 0) is observed experimentally and modeling shows its origin to \nbe stemming from the competition between the uniaxial anisotropy and demagnetization \nenergy,[4] with arbitrary directions of the stripe domains,  and with different domain forming \non large length scales.[25] When applying  a magnetic  field,  biskyrmions  nucleate  directly  from \nthe stripes.[18,19,22,23] Therefore, a biskyrmion lattice can be observed locally, and the lattice \norientation is locked, given by the orientation of the stripes from which they evolved. This \nlocking is observed in La 2−2xSr1+2xMn 2O7 (x = 0.315),[18] MnNiGa,[19] and FeGd thin films.[23] \nIn Cr 11Ge19, in which  the Dzyaloshinskii -Moriya interaction also plays a role, the biskyrmion  \nlattice  seems  to form  in the same  way.[22] \nNevertheless, even  in the absence of an ordered biskyrmion  lattice phase , the in-plane  direction \nof the biskyrmions (characterized by azimuthal angle Ψ) is aligned along certain directions in \nMnNiGa,[19–21] independent  of the pinch -off direction of the initial stripe  phase.[18] However, \nthe physical origin that determines this direction has not been reported. Furthermore, it still \nremains an open question whether biskyrmions are a purely surface -related phenomenon (as \nonly thin lamella with non -bulk-like properties had been investigated i n LTEM),[26,27 ] or \nwhether biskyrmions extend into the bulk, analogous to regular skyrmions.[28] On the one \nhand, it seems intuitive to assume that biskyrmions behave like pairs of skyrmions, inheriting \ntheir fundamental topological  properties. On the other hand , their deeply distorted      \n6 \n \nmagnetization distribution could also lea d to strongly surface -confined, bobber -like minimum \nenergy objects.[26,27] Here, we perform SANS on bulk crystal MnNiGa (the crystal information \nshown in Figure 1h, 1i, and 1j) to provide the answers to  these  two key questions, and which \nhave  remained  elusive  so far. \nFigure 2 a shows the typical SANS geometry used in our experiments and Figure 2b , 2f, \n2j, and 2n show the Ewald sphere representation for the SANS geometry for  differ ent \nmagnetic  reciprocal  space  configurations. First, a polycrystalline sample is measu red using the \nSANS2d instrument at ISIS with ki ⊥ B geometry to demonstrate the bulk effect of \nbiskyrmions, and the data shown in panels Figure 2d , 2h, 2l, and 2p. Then, a single -\ncrystalline sample is measured using the D11 instrument at ILL with ki || B geometry to study \nthe form factor of the biskyrmion. For the single -crystalline sample, the incident neutron \nbeam along ki is parallel to z (Figure 2a) , while the magnetic field is applied along the c-axis \nof the hexagonal magnet MnNiGa (Figure 2m).  \nThe magnetic  scattering cross  section  for SANS  experiments  can be written as:[29] \nσ(𝐪)∝𝐌⊥(𝐪)∙𝐌⊥∗(𝐪)                                               (3) \nwhere M⊥ is the magnetic structure factor that is projected onto the plane perpendicular to the  \nscattering wavevector q, and  𝐌⊥∗ is the complex conjugate of M⊥. The SANS pattern can thus \nbe simulated by inserting different magnetic structures into Equation (3). At zero field, the \nbulk ground state is expected to be the stripe  domain  phase,  or a labyrinth  domain  state.[4,19] \nThus, in reciprocal space, the magnetic scattering lies on a sphere of radius of qm as shown by \nthe yellow sphere in Figure 2b , where qm = 2π/LD, and LD is the periodicity of the stripe \nmodulation. The simulated SANS pattern in this case is shown in Figure 2c , from which a \nring-like pattern is expected, reflecting the random orientation and populations of the stripe \ndomains. The scattering observed in the corresponding experimental data shown in Figure 2d  \nis in agreement with the simulation. The radius of the ring is about 0.002 Å-1, corresponding      \n7 \n \nto the real -space stripe domain pitch LD of 300 nm, while our previous LTEM results showed \nthis value from 180 to 230 nm.[19–21] \nFigure 2e  shows the ki ⊥ B geometry. Upon applying a magnetic field, three possible \nmagnetic phases can evolve. The first one is the  biskyrmion  phase,  as observed  in LTEM  \nstudies.[18–23] This state will only have magnetic  correlations in the plane perpendicular to B, \nas illustrated in Figure 2f, giving rise to two spots with ±qx in the SANS pattern. The second \nphase is the single -domain cone phase, for which the propagation wavevector is along B,[5,14] \nas shown in Figure 2i . Consequentl y, one would expect two peaks with ±qy (Figure 2j). The  \nthird possible state is the field -polarized state, i.e., the ferromagnetically aligned state, which \nleads to a strong peak at the origin of reciprocal space.  It is therefore  possible  to \nunambiguously  distinguish  between  these  three  phases  in the ki ⊥ B geometry.  \nIn our experiments, when applying a finite field, the ring -like pattern gradually transforms \ninto a twofold -symmetric scattering pattern, as shown in Figure 2h . The magnetic  correlation \nlength (2 π/(0.0016 Å-1) ≈ 390 nm) is comparable to that in our previou s LTEM measured \nbiskyrmion results (from 300 to  500 nm).[19] As discussed above, this is a clear indication that \nthe stripe domains continuously evolve into the biskyrmion state. As SANS is a bulk probe, \nbeing relatively insensitive to surface effects, it can be concluded that biskyrmions stack \nalong the y-directio n, probably forming a tube -like three -dimensional structure (Figure 2e) \nanalogous to regular skyrmions .[28] By further increasing the field from 0.38 to 0.5 T, the  \ntwofold -symmetric  spots  become  weaker  (Figure  2l), suggesting that biskyrmions become \nmore w eakly  correlated and/or exist at lower density in this phase . Moreover,  the two spots  \nmove  closer  to each other,  which means that the inter-biskyrmion spacing increases with \nincreasing field. The biskyrmion total scattering inten sity, as well as the absolute value of the \nbiskyrmion scattering wavevector as  a function of magnetic field, is shown in Figure 2p . It is \nworth noting that throughout the phase diagram, we do not observe a conical phase.       \n8 \n \nNext, we investigate the ki || B || [001] geometry for a single -crystalline sample, which \nreveals the structure within the biskyrmion plane. When studying  chiral  skyrmions  in \nhelimagnets  using  the same geometry, the skyrmions form a long -range ordered hexagonal \nlattice, leading to six -fold-symmetr ic SANS peaks.[5] In SANS, the  magnetic  scattering factor  \nFm(q) = fmotif(q) ∗ flattice(q) is the convo lution  of the form  factor  of the biskyrmion  motif,  \nfmotif(q), and their lattice order flattice(q).[29] If the biskyrmions are not forming a periodic \nlattice, e.g., as in the disordered skyrmion phase hosted in the geometrically frustrated spin \nliquid material Co 7Zn7Mn 6,[11] flattice(q) smears  out into an isotropic  distribution  in the qx-qy \nplane, while the contribution of fmotif(q) to Fm(q) becomes more pronounced. Therefore, the \nnon-lattice  state offers  a unique  opportunity  to study  the form factor of the biskyrmion  motif.[30] \nFigure  2o shows  the experimental  biskyrmion  SANS pattern obtained at 0.4 T. First, no \ndiffraction peaks  are observed, suggesting that in MnNiGa bulk crystals the biskyrmions are \nnot long -range ordered in the hexagonal plane . Note that in LTEM imaging, owing to the \nsmall field -of-view,  biskyrmions may form a distorted hexagonal lattice,[18–23] which  is not in \ncontradiction  to the SANS data. In  such a disordered  state,  a possible  scenario  is that individual  \nbiskyrmions  have a random  in-plane  direction  due to spontaneous symmetry breaking. \nConsequently, there  would be no preferred direction of the biskyrmions when averaging over \nall quasiparticles. In this scenario, the scattering intensity is equally distributed forming a  \nring. \nInterestingly, the experimental scattering pattern (Figure  2o) is anisotropic  and has the \nshape  of an astroid (a hypocycloid with four cusps) . Therefore, it can be concluded that even \nthough the  biskyrmions  are not long-range  ordered,  surprisingly their  in-plane  directions  are \naligned,  and locked  along  certain directions, consistent with the LTEM measured results under \nsmall field -of-view.[19–21] As we will show  the particular  anisotropy in the scattering  pattern  \ndirectly reflects the form factor of the biskyrmion  motif.       \n9 \n \nFigure 3a-c shows the experimental SANS patterns for different magnetic fields in the ki || \nB geometry. Be low 0.4 T, where the biskyrmion phase forms, the astroid  shape remains \nalmost unchanged, while the  intensity varies with applied field. Most importantly, the \nfourfold -symmetric SANS pattern remains always locked along a direction, such that two of \nthe four corners are along the crystallographic [100 ] directions, as shown in Figure 3a  and \n3b. This findin g points directly towards an intimate  relationship  between  the in-plane  \nbiskyrmion  direction and the magnetocrystalline anisotropy.  The remanent field of the \nsuperconducting magnet, which is present during cool -down of the sample, may have \na similar effect  on the magnetic state as the field -cooling scenario reported by Peng et \nal.,[20] thereby explaining the ~0.03 T data. Figure 3d  shows the magn etic field \ndependence of the magnetization measured under H || [001]. Further increasing the field \nto saturation at 1.1 T leads to a smearing out of the  fourfold -symmetric  pattern,  suggesting  the \ntransformation  into the field-polarized  state. The single -crystalline sample shows  two easy -\nmagnetization axes which are separated by 90◦ with respect to each other within the ab-plane, \nas obtained  by the magnetic measurements (Figure 3e). We compared the results for the \npolycrystalline and the single -crystalline sample by plotting the reduced 1 -dimensional data as \nshown in supporting information Figure S1 . The main differences between field -polarized \nstate and biskyrmion state appear from Q ≈ 0.001 to 0.01 Å-1 for both data.  \nIn order to quantitatively study the biskyrmion motif structure, numerical simulations of \nthe SANS intensity were performed by inserting  Equation (1) into Equation (3), and by using  \nvarious  combinations  of structural  biskyrmion  parameters. By fitting the simulated form \nfactors to the observed  astroid -shaped  experimental  pattern,  the bond distance  d, shape  factor  \nη, as well as the in-plane  biskyrmion direction is obtained for the different biskyrmion \nconfigurations. Figure 4 a shows the form factor of a typical biskyrmion with d = 50 nm, η = \n2, and Ψ = 18 5◦, representing  a biskyrmion non-lattice  state (shown as inset , where brown      \n10 \n \narrows represent the direction of individual biskyrmions). It is clear that the form factor of the \nmotif exhibits twofold symmetry, while the symmetry axis of the SANS pattern coincides \nwith the C2 axis of the biskyrmion  configuration  in real space.  \nHowever, there are more ways in which the alignment direction of the biskyrmions can be \naffected by the symmetry of the host crystal. The first is the threefold degeneracy of the \nbiskyrmion order, as there are three  equivalent [100 ] crystalline axes within the ab-plane.  \nConse quently, a sixfold -symmetric form factor is expected,  as shown in Figure 4c , which \ndoes not agree with the  experimental data. Therefore, the only crystalline order that is \ncompatible with the astroid -shaped diffuse  scattering pattern is the formation of 90◦ separated \ntwo easy magnetization axes. The biskyrmion direction  is locked  along  the easy-magnetization \naxis direction,  giving rise to two possible directionally ordered states that are separated by 90◦. \nAs a result, the overall form  factor appears as a fourfold -symmetric pattern, as shown in \nFigure 4 d, and consistent with the in -plane magnetic anisotropy shown in Figure 3e . \nHowever, such a f ourfold -symmetric pattern does not have the shape of an astroid . The \nparameter that changes the shape of the boundary from convex to concave  is the bond  distance  \nd, as it is the key parameter that governs the detailed in -plane anisotropy described by the form \nfactor. As shown in Figure 4e , by using an appropriate d, the simulated contrast fits the \nexperimentally obtained pattern well. Lastly, we show that Ψ, the in -plane direction of the \nbiskyrmions, directly correlates with the azimuthal rotation of the SANS pattern, see Figure  \n4e and 4f. \nNext, we discuss the possible energy terms that could govern  such an ‘order  within  disorder’  \nphenomenon  in MnNiGa bulk crystals.  Here, order within disorder relates to the fact that \ndespite the lack of positional order of the biskyrmion lattice, there is directional order, i.e., the \nbiskyrmion axes all point in the same direction. Indeed, the formation of biskyrmions  has not \nbeen  accurately  captured  in the framework of micromagnetism, indicating that such       \n11 \n \nquasiparticles may exist as a metastable state. A simple modification of the model is to \nintroduce trigonal magnetocrystalline anisotropy  ωcry into Equation  (2), which  breaks  the O(2) \nsymmetry. However, such a term should also have an effect on the anisotropy of the stripe \ndomain as well. In our  SANS  experiments,  the stripe  domain  populates  random  orientations,  as \nevidenced  by the ring-like scattering pattern, excluding the possibility that the \nmagnetocrystalline energy is a first -order  term. Therefore, the magnetocrystalline  anisotropy  \ndoes not play an important role  in determining their  positional  arrangement,  which  is different  \nfrom  the case of Cr 11Ge19.[22] Nevertheless, as the biskyrmions  form  at elevated  fields,  their \npolar  axes may be affected by higher -order terms from the  trigonal magnetocrystalline \nanisotropy, such that the C2 axis is weakly locked along [100 ]  directions with threefold  \ndegeneracy. The next term to consider is the  biskyrmion -biskyrmion  interaction,  which  is \ndominated  by the effective force F(Ri − Rj) = −∇V, where Ri (j) are the neighboring \nbiskyrmion positions, and V is the  effective potential  energy  that results  from  Equation  (2). It \nis thus clear that  V (r) is highly  anisotropic  due to the C2v-shaped  biskyrmion  configuration.  \nTherefore,  the threefold  degenerate directional order would tend to lock in one of the three  \n[100 ]  directions  to minimize  the total energy  of the system. Moreover, as F(Ri − Rj) will also \nnot be strong,  since the biskyrmions do not display long -range order and form a lattice  state.  \nIn summary,  we have determined  the structural  parameters of the biskyrmion state in \nMnNiGa bulk crystals. First, biskyrmions do not assemble into a long -range -ordered lattice.  \nInstead, their correlation length always remains comparable to the periodicity of the stripe \ndomains,  and increases with increasing magnetic field. Second, although biskyrmions do not \norder  into periodic lattices on a large length scale, they are in -plane  rotationally ordered, with \nthe rotation direction being locked to certain crystallographic directions.  These polar \nmolecular properties  of the biskyrmion  system  suggest  an intricate energy balance in MnNiGa, \nwhich is fundamentally different  from  chiral  skyrmion  systems. Such anisotropic nature,      \n12 \n \nrepresenting an extra degree of freedom for this type of skyrmions, may be exploited as \ninformation carrier. In this case, the two binary states are encoded within the same topological \nstate, which reduces the energy consumption that is required for the state switching, \ndemonstrating advanced device scheme for skyrmion -based memories.  \n \nExperimental Section  \n    Sample preparation: Polycrystalline MnNiGa samples were synthesized by the growth \nmethod described in detail in reference [1 9], and single -crystalline samples were grown using \nthe optical floating zone technique, using a c-axis oriented seed crystal. The growth direction \nis along the c-axis direction . The polycrystalline sample used for SANS measurement s is a \ndisc with a diameter of 13 mm and thickness of 1 mm. The single -crystalline sample used for \nSANS measurements is a cube measuring 2.5 × 2.5 × 2.5 mm3, which was oriented using a \nLaue and an x -ray diffractometer. The data is shown in the main text in  Figure 1h . The Curie \ntemperature of the MnNiGa sample is ~350 K (as reported in reference [1 9]).  \n    Magnetic measurements: Single -crystalline sample magnetization data were measured by \napplying a magnetic field of 0.3 T at 215 K in a Quantum Design Magnetic Property \nMeasurement System (MPMS). The applied magnetic field is perpendicular to the c-axis and \nthe sample is rotated 360 ° around the c-axis in increments of 1 °. The data is shown in the \nmain text in Figure 3e . The magnetic field dependence of the magnetization measured \nfor H || [001] at 180, 200, 220, and 250 K is shown in the main text in Figur e 3d. The \ncharacteristic critical fields were measured by superconducting quantum interference device \n(SQUID) magnetometry for the polycrystalline sample, and shown to be qualitatively \nconsistent those reported in our previous LTEM study.[19] Note  that the critical  fields  defining  \nthe phase  boundaries differ between thin film lamella and bulk samples due to \ndemagnetization effects.       \n13 \n \n    Neutron diffraction measurement s: Neutron powder -diffraction m easurements were \nperformed at 400 K on the high -resolution powder diffractometer BT1 at the National \nInstitute of Standards and Technology (NIST), USA. A Cu(311) monochromator was used to \nproduce a monochromatic neutron beam of wavelength 1.5397 Å. The pro gram FullProf was \nused for the Rietveld refinement of the crystal structures of the compound. The \ncrystallographic structure of MnNiGa has a layered Ni 2In-type centrosymmetric  hexangular \nstructure with  space group P63/mmc . The data is shown in the main text in Figure 1j . Other \n300 K powder neutron diffraction patterns were obtained from the general purpose powder \ndiffrctometer (GPPD)[31] (90°  bank) at the China Spallation Neutron Source (CSNS) in \nDongguan, China. Compared with the paramagnetic neutron diffractio n data (Figure S2), we \nfound no magnetic satellites in the 300 K data which would be characteristic for helical and /or \nconical magnetic structures. The reason for the absence of the helical magnetic peaks is that \nthe helical periodicity is too large to be observed by neutron diffraction.   \n    SANS measurements:  SANS measurements on polycrystalline samples were performed \nusing the SANS2d instrument[32,33] at the ISIS Neutron and Muon Source, Didcot,  United \nKingdom. The neutron beam was collimated over a length of 12 m before reaching the sample, \nand the scattered neutrons were collected by a 2 -dimensional, position -sensitive multidetector \nplaced 12 m behind the sample. The applied magnetic field is perpendicular to the neutron \nbeam direction. The sample is measured at 2 15 K, which is the temperature at which the  \nmaximum topological Hall effect was measured[19] by neutrons with wavelengths from 1.75 Å \nto 12.5 Å with beam stop . The data reduction and analysis was done using the software \npackage MantidPlot.[34] The field -polarized state data has been used as a background. The \ndata are shown in the main text in Figure 2d , 2h, 2l, and 2p. SANS measurements on the \nsingle -crystalline sample were performed using the D11 instrument[35] at the Institut Laue -\nLangevin (ILL), Grenoble, France. The neutron beam was collimated over a length of 37 m \nbefore reaching the sample, and the scattered neutrons were collected by a 2 -dimensional,      \n14 \n \nposition -sensitive mul ti-detector placed 39 m behind the sample. The neutron wavelength was \nselected to be 8 Å. The circular aperture at the sample had a diameter of 7 mm. The sample is \nplaced in a Gd jacket and coated with a layer of Gd 2O3 to reduce the reflection by the surfa ce \nof the sample. The applied magnetic field was parallel to the neutron beam and approximately \nparallel to the sample [001] axis. The y-axis rocking scans covered a range of up to ± 10 ° with \na SANS measurement performed every 1 ° and the x-axis rocking scan s covered a range of up \nto ± 3 ° with a SANS measurement performed every 1 ° , without beam stop (for an illustration \nof the measurement geometry see Figure 2a) . The single -crystalline sample SANS patterns \nshown in the main text were obtained by summing over a ll rocking scans. The sample was \ncooled to 215 K in zero field and the rocking scans were subsequently measured at different \napplied magnetic fields, respectively.  The data reduction and analysis was done using the \nsoftware  package GRASP.[36] The data are shown in the main text in Figure 2o  and Figure 3a , \n3b, and 3c. Initial single -crystalline sample SANS measurements were perforemd on SA NSI \nat the Swiss Spallation Neutron Source, Paul Scherrer Institute (PSI), Villigen, Switzerland. \nThe in -plane symmetry axis of the disordered biskyrmion system is not  quite  dependent on \nthe temperature and  sample  magnetic field history.  We compared the re sults for the \npolycrystalline and the single -crystalline sample in the reduced 1 -dimensional data plot shown \nin Figure S1. There is more scattering intensity for the low field biskyrmion state compared \nto the high field polarized state in the Q region from 0.001 to 0.02 Å-1 for the polycrystalline \nsample, and from 0.0008 to 0.01 Å-1 for the single -crystalline sample.  \n \nSupporting Information  \nSupporting Information is available from the Wiley Online Library or from the author.  \n \nAcknowledgements  \nX. Y. L. and F. W. W. acknowledge financial support by the National Natural Science \nFoundation of China (No. 11675255) and the Na tional Key R&D Program by MOST of \nChina (No. 2016YFA0401503). S. L. Z. and T. H. acknowledge financial support by EPSRC \n(EP/N032128/1). W. H. W. acknowledge financial support by the National Key R&D \nProgram by MOST of China (No. 2017YFA0303202) and the Key  Research Program of the      \n15 \n \nChinese Academy of Sciences, KJZD -SW-M01 . J. S. W. acknowledge financial support by \nthe Swiss National Science Foundation (SNSF) via the Sinergia network \"NanoSkyrmionics\" \n(Grant No. CRSII5 -171003). The neutron scattering experimen ts were carried out at the \nISIS/STFC facility, which is sponsored by the Newton -China fund (proposal RB1620128). \nThis work is based partly on experiments performed at the Swiss spallation neutron source \nSINQ, Paul Scherrer Institute, Villigen, Switzerland.  \n \nReceived: ((will be filled in by the editorial staff))  \nRevised: ((will be filled in by the editorial staff))  \nPublished online: ((will be filled in by the editorial staff))  \n \nReferences  \n[1] L. D. Landau, Zh. Eks . Teor. Fiz.  1937 , 7, 19. \n[2] N. D. Mermin, Rev. Mod. Phys.  1979 , 51, 591.  \n[3] B. Lebech, J. Bernhard, T. Freltoft, J. Phys. Condens. Matter  1989 , 1, 6105.  \n[4] N. Nagaosa, Y. Tokura, Nat. Nanotechnol.  2013 , 8, 899.  \n[5] S. Muhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. 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Assoc. Equip.  2014 , 764, 156.  \n[35] X. Y. Li, R. Cubitt, and J. S. Whi te, Wide Temperature Range Biskyrmion Magnetic \nNanodomains in MnNiGa Study by Small Angle Neutron Scattering , 2018 , Institut \nLaue -Langevin (ILL) doi:10.5291/ILL -DATA.5 -42-460. \n[36] C. Dewhurst, GRASP, 2018 , https://www.ill.eu/en/users/support -labs-\ninfrastr ucture/software -scientific -tools/grasp/ . \n      \n19 \n \n \nFigure 1 . a-c) Structural parameters describing a C2v biskyrmion. The polarities are \nthe same for all three plots, namely λ = 1. Two Dn skyrmions are brought together in \na ferromagnetic background with an inter -skyrmion (bond) distance  d. Starting from \nthe structure shown in a), d is further decreased, while the biskyrmion shape \ndeforms into an ellipse ( η = 2). The principle symmetry axis is labeled by the \nblack  arrow,  which  gives  also the direction  of the biskyrmion ( Ψ = 30◦). d-f) The (a -\nc) corresponding simulated magnetic form factor (where | fM|2 is |fmotif(q)|2), \nrespectively. g) An in -plane magnetization of the biskyrmion configuration. h) The \nMnNiGa single -crystalline sample is checked by single -crystal x -ray diffractometer. \nThe data shows the sample has hexagonal lattice structure. i,j) The refinement result \nof high -resolutio n neutron powder diffraction data which demonstrated the structure \nof MnNiGa is a centrosymmetric hexangular structure. The Mn atoms located at 2a \nsite, Ni atoms located at 2d and Ga atoms located at 2c with a space group of \nP63/mmc .  \n     \n20 \n \n \nFigure 2 . a) Coordinate system. b,f,j,n) Ewald sphere representations for the various \nexperimental SANS configurations. c) Simulated and d) experimental SANS pattern in zero \nfield showing a ring -like contrast. e) Illustration of the 3D magnetization distribution o f the \ndisordered biskyrmion phase  in an applied out -of-plane field of 0.38 T. g,h) Simulated and \nexperimental SANS pattern for the corresponding state. i ) Assumed conical state in an applied \nout-of-plane field of 0.5 T, and k), corresponding simulated SANS contrast. l) The \nexperimental SANS data do not support the existence of the conical state. m) Biskyrmion \narrangement for ki || B, and o), corresponding experimental SANS pattern. p) The absolute \nvalue of the biskyrmion scattering wavevector, as well as the associated total scattering \nintensity as a function of the magnetic field for ki ⊥ B geometry. The panels shown in \n(d,h,l,p) are measured on the SANS2d  instrument at ISIS for the polycrystalline sample, and \nthe panel shown in (o) is measured on the D11 instrument at ILL for the single -crystalline \nsample (see experimental section for more details) . \n     \n21 \n \n \nFigure 3 . Evolution of the experimental SANS pattern in  the ki || B configuration for \napplied fields of a) ~0.03 T (remanent field of the superconducting magnet), b) 0.4 T, \nand c) 1.1 T. Note that the color scale is logarithmic to make weak features visible. d) \nMagnetic field dependence of the magnetization me asured under H || [001] at 180, \n200, 220, and 250 K, respectively. e) The single -crystalline sample shows two easy -\nmagnetization axes which are separated by 90 ◦ with respect to each other within the \nab-plane as obtained by the magnetic measurements.  \n \n   \n     \n22 \n \n \nFigure 4 . SANS contrast simulation for (laterally) disordered biskyrmion lattices. a) Single \ndomain oriented at Ψ = 185 °, and b) Ψ = 95°, and c) three domains separated by 120 ° in-plane \nrotation. Two domains (90 ° rotated) for d) d = 100 nm and e) 50 nm. f) Same as in (e) with an \nin-plane rotation of ∆Ψ = 45 °. For all simulations, η = 2. \n \n \n     \n23 \n \nSupporting Information  \n \n \nOriented Three -Dimensional Magnetic Biskyrmion in MnNiGa Bulk Crystals  \n \nXiyang Li, Shilei Zhang, Hang Li, Diego Alba Ve nero, Jonathan S White, Robert Cubitt, \nQingzhen Huang, Jie Chen, Lunhua He , Gerrit van der Laan, Wenhong Wang, Thorsten \nHesjedal, and Fangwei Wang*  \n \n \n \n \nFigure S1 : Reduced 1-dimensional data for the polycrystalline sample measured on \nSANS2d@ISIS and for th e single -crystalline sample measured on D11@ILL. The scattering \nintensity for the low field biskyrmion state is larger compared to the high -field polarized state.  \n \n     \n24 \n \n \nFigure S2 : Comparison of neutron diffraction data obtained at 300 K and 400 K. The Curie \ntemperature of the sample is ~350 K (as reported in reference [1 9]). Note that no magnetic \nsatellites associated with the helical/conical phase are measured at 300 K as the helical period \nis too large to be detected by neutron diffraction. The green lines represent the Bragg peak \npositions of the crystalline structure.  \n"    },    {        "title": "1903.00105v1.Effect_of_atomic_ordering_on_the_magnetic_anisotropy_of_single_crystal_Ni80Fe20.pdf",        "content": "arXiv:1903.00105v1  [cond-mat.mtrl-sci]  28 Feb 2019Effect of atomic ordering on the magnetic anisotropy of singl e crystal\nNi80Fe20\nMovaffaq Kateb,1,a)Jon Tomas Gudmundsson,1,2,b)and Snorri Ingvarsson1,c)\n1)Science Institute, University of Iceland, Dunhaga 3, IS-10 7 Reykjavik, Iceland\n2)Department of Space and Plasma Physics, School of Electrica l Engineering and Computer Science,\nKTH–Royal Institute of Technology, SE-100 44, Stockholm, S weden\n(Dated: 4 March 2019)\nWe investigatethe effect ofatomicorderingon the magnetic anisotr opyofNi 80Fe20at.% (Py). To this end, Py\nfilms were grown epitaxially on MgO (001) using dc magnetron sputter ing (dcMS) and high power impulse\nmagnetron sputtering (HiPIMS). Aside from twin boundaries obser ved in the latter case, both methods\npresent high quality single crystals with cube-on-cube epitaxial rela tionship as verified by the polar mapping\nof important crystal planes. However, X-ray diffraction results in dicate higher order for the dcMS deposited\nfilm towards L12Ni3Fe superlattice. This difference can be understood by the very high deposition rate of\nHiPIMS duringeach pulse which suppressesadatom mobility and order ing. We show that the dcMS deposited\nfilm presents biaxial anisotropy while HiPIMS deposition gives well defin ed uniaxial anisotropy. Thus, higher\norder achieved in the dcMS deposition behaves as predicted by magn etocrystalline anisotropy i.e. easy axis\nalong the [111] direction that forced in the plane along the [110] direc tion due to shape anisotropy. The\nuniaxial behaviour in HiPIMS deposited film then can be explained by pair ordering or more recent localized\ncomposition non-uniformity theories. Further, we studied magnet oresistance of the films along the [100]\ndirections using an extended van der Pauw method. We find that the electrical resistivities of the dcMS\ndeposited film are lower than in their HiPIMS counterparts verifying t he higher order in the dcMS case.\nPACS numbers: 77.55.Px, 75.30.Gw, 75.70.-i, 75.50.Bb, 81.15.Cd, 81.15.K k, 87.64.Bx\nKeywords: HiPIMS, Epitaxy, NiFe, uniaxial anisotropy, order\nI. INTRODUCTION\nThe phenomenon of uniaxial anisotropy in permal-\nloy Ni 80Fe20at.% (Py) films and its correlation with\nmicrostructure has attracted considerable scientific and\nindustrial interest for decades. The proposed expla-\nnations for uniaxial anisotropy include oriented defects\nand oxides,1,2directional ordering of Fe/Ni atoms pairs,3\nshape anisotropyof an elongated ordered phase,4compo-\nsition variation between grains5and more recently, local-\nized composition non-uniformity.6No one of these can\naccount for all instances of uniaxial anisotropy in the Py\nsystem, andoneormorecouldcontributesimultaneously.\nEpitaxial, single crystal Py films have a perfect lat-\ntice, while the arrangement of Ni and Fe atoms may\ncontain varying degrees of order. Among the expla-\nnations above, only the pair ordering and the local-\nized composition non-uniformity are applicable in such\na case. Py has vanishingly small magnetocrystalline\nanisotropy and magnetostriction, and low coercivity, but\nextremely large magnetic permeability.7,8This makes it\na unique system in which to study e.g. induced uniaxial\nanisotropy. Single crystal Py films have been deposited\nepitaxially by numerous techniques, including thermal\nevaporation,7,9,10electron beam evaporation,11molecu-\nlar beam epitaxy (MBE),12–14ion beam sputtering,15,16\na)Electronic mail: mkk4@hi.is\nb)Electronic mail: tumi@hi.is\nc)Electronic mail: sthi@hi.isrf magnetron sputtering,17,18dc magnetron sputtering,19\nand pulsed laser deposition.20Most of these deposition\nmethods haveresulted in asingle crystalPy (001)film on\nMgO (001) with biaxial anisotropy in the plane, and the\neasy directions being [110] or [100]. Unfortunately, stud-\nies that focused on the growth of single crystal Py did\nnot discuss the effect of ordering at all. Most of these\nstudies used a low deposition rate, which normally re-\nsults in higher order. On the other hand, the study of\natomic orderis limited to annealinga quenched specimen\nat about 500◦C for a very long time.3,21–23\nSeveral groups have shown that an increase in atomic\norder results in deterioration of the anisotropy constant\nK1.3,22,24,25Uniaxial anisotropy can be induced in single\ncrystal Py by applying an in-situmagnetic field during\ndeposition,7,15,16orbypostannealing26inmagneticfield.\nIt has been shown that magnetic field induced anisotropy\nstrongly depends on the crystal orientation of the Py for\nboth deposition and annealing in a magnetic field7,27i.e.\nit is most efficient along the /an}bracketle{t111/an}bracketri}htdirection, less so along\nthe/an}bracketle{t110/an}bracketri}htdirection and least along the /an}bracketle{t100/an}bracketri}htdirection.\nAnother method of inducing uniaxial anisotropy is de-\npositionunderananglewith respecttothesubstratenor-\nmal. We have shown that deposition under a 35◦angle\nis more effective than applying a 70 Oe in-situmagnetic\nfield when depositing polycrystalline films.28,29We have\nalso demonstrated growth of polycrystalline Py films un-\nder an angle using high power impulse magnetron sput-\ntering (HiPIMS) and compared with films deposited by\nconventional dc magnetron sputtering (dcMS).30During\nHiPIMS deposition high power pulses of low frequency2\nand low duty cycle are applied to a magnetron target\nwhich results in highly ionized sputtered material.31The\nHiPIMS discharge provides a highly ionized flux of the\nmetallic species and the averaged ion energy is signifi-\ncantly higher in the HiPIMS discharge than in dcMS dis-\ncharge and this energetic metallic ions are created dur-\ning the active phase of the discharge pulse.32–34For both\nmethods, deposition under an angle with respect to the\nsubstrate induces very well-defined uniaxial anisotropy\nin the film. Schuhl et al.35showed that tilt deposition\nbreaksthe symmetrybetween twoin-plane easyaxes, ap-\npearing as a stepped easy axis magnetization loop along\nthe flux direction. However the method of inducing uni-\naxial anisotropy using tilt deposition of a single crystal\nPy has not been studied so far. In this work we demon-\nstrate the epitaxial growth of single crystal Py films on\nMgO (001) substrates, by HiPIMS and by dcMS both\ndeposited under an incident angle of 35◦. We study the\neffect of the two above mentioned sputtering methods,\nwhose adatom energy differs by order(s) of magnitude,\non the structure, order and magnetic anisotropy of the\nfilms. Itmightbetemptingtothinkthatthehighadatom\nenergy involved in HiPIMS would cause severe struc-\ntural damage, but there appear to be only very subtle\nstructural disparities, while the ordering and magnetic\nanisotropy, however, are vastly different.\nII. METHOD\nThe substrates were single-side polished single crys-\ntalline MgO (001) with surface roughness <5˚A, and\nwith lateral dimension of 10 ×10 mm2and 0.5 mm thick-\nness (Latech Scientific Supply Pte. Ltd.). The MgO sub-\nstrates were used as received without any cleaning but\nwere baked for an hour at 400◦C in vacuum for dehy-\ndration. The Py thin films were deposited in a custom\nbuiltultra-highvacuummagnetronsputterchamberwith\na base pressure of <5×10−7Pa. The deposition was\nperformed with argon of 99.999 % purity as the working\ngasat 0.33Pa pressureusing Ni 80Fe20target of75 mm in\ndiameter and 1.5 mm thickness. During deposition, the\nsubstrateswererotated360◦backandforthat ∼12.8rpm\nwith 300 ms stop in between. Further detail on our de-\nposition geometry can be found elsewhere.28–30\nFor dcMS deposition a dc power supply (MDX 500,\nAdvanced Energy) was used and the power was main-\ntained at 150 W. For HiPIMS deposition, the power was\nsupplied by a SPIK1000A pulser unit (Melec GmbH) op-\nerating in the unipolar negative mode at constant volt-\nage, which in turn was charged by a dc power supply\n(ADL GS30). The pulse length was 250 µs and the pulse\nrepetition frequency was 100 Hz. The average power\nduring HiPIMS deposition was maintain around 151 W.\nThe HiPIMS deposition parameters were recorded by a\nhome-made LabVIEW program communicating with the\nsetup through high speed data acquisition (National In-\nstruments).X-ray diffractometry (XRD) was carried out using a\nX’pert PRO PANalitical diffractometer (Cu K α, wave-\nlength 0.15406 nm) mounted with a hybrid monochro-\nmator/mirror on the incident side and a 0.27◦collimator\non the diffracted side. A line focus was used with a beam\nwidth of approximately 1 mm. The film thickness, den-\nsity and surface roughness was determined by low-angle\nX-ray reflectivity (XRR) measurements with an angular\nresolution of 0.005◦. The data from the XRR measure-\nments were fitted using a commercial X’pert reflectivity\nprogram.\nFor the (002) pole figure the θ−2θwas set to a cor-\nresponding peak obtained in the normal XRD. However,\nthe (111) and (022) peaks do not appear in the normal\nXRD. To this end, first a rough pole scan was done ac-\ncording to θ−2θfound in the literature. This roughly\ngivesthein-plane( φ) andout-of-plane( ψ)anglesofthose\nplanes with respect to the film surface. Then we scan\nθ−2θat the right φandψto find each (111) and (022)\npeak. Finally, a more precise pole scan is made again\nat the new θ−2θvalues. Obviously, the θ−2θvalues\nreported in the literature might be slightly different than\nfor our samples due to strain in the film and accuracy of\ncalibration. The ψwas calibrated using the (002) peak\nof MgO normal to the substrate. In a similar way, the\nnarrow MgO (200) peak in-plane of the substrate was\nutilized for calibration of φ.\nTo obtain hysteresis loops, we use a homebuilt high\nsensitivity magneto-optical Kerr effect (MOKE) looper\nwith HeNe laser light source. We used variable steps in\nthe magnetic field i.e. 0.1 Oe steps around transitions of\nthe easy direction, 0.5 Oe steps for the hard axis and\nbefore transitions of the easy direction and 1 Oe steps\nfor higher field at saturation.\nFor the anisotropic magnetoresistance (AMR) mea-\nsurements we utilized Price36extension to van der Pauw\n(vdP)37,38method. We have already shown that the vdP\nmeasurement is more reliable in the AMR measurements\nsince it is less geometry dependent compared to conven-\ntional Hall-bar method.29Originally, vdP was developed\nfor determining isotropic resistivity and ordinary Hall\nmobilityofuniformandcontinuousthin filmsofarbitrary\nshape and has been used extensively for semiconductor\ncharacterization. In the vdP method, four small contacts\nmustbeplacedonthesampleperimetere.g.asillustrated\nin Fig.1. The measured resistances should satisfy vdP\nequation38\nexp/parenleftbigg\n−πd\nρisoRAB,CD/parenrightbigg\n+exp/parenleftbigg\n−πd\nρisoRAD,CB/parenrightbigg\n= 1 (1)\nwhereρisois the isotropic resistivity and dis the film\nthickness. The resistance RAB,CDis measured by forcing\ncurrent through the path AB and picking up the voltage\nat the opposite side between CD and RAD,CBis simi-\nlarly defined. Note that Eq. ( 1) is independent of sample\nshape and distances between contacts. This behavior is\na direct result of conformal mapping i.e. a sample has\nbeen mapped upon a semi-infinite half-plane with con-3\ntactsalongthe edgein which the contactdistancescancel\nout.\nxyIx\nVxxAB\nDCb Iy VyyA B\nDCa\nFIG. 1. Schematic illustration of set of measurements in vdP\nmethod.\nIt has been shown that the vdP method can be ex-\ntended to the case of anisotropic films if two of three\nprinciple resistivity axis are in the film plane while the\n3rd is perpendicular. In that case, ρisoobtained from\nEq. (1) stands for the geometric mean of the in-plane\nprinciple resistivities i.e. ρiso=√ρ1ρ2.39,40In principle,\nthe vdP method is based on conformal mapping and thus\nit should remain valid if an anisotropic medium with lat-\neral dimensions of a×bmapped into an isotropic one\nwith new dimensions e.g. a×b′. In practice one can\nmake a rectangular sample with sample sides parallel to\nthe principle resistivities.36,39Assuming the principle re-\nsistivities are aligned with xandydirections in Fig. 1,\nthe principle resistivities ρxandρycan be obtained from\nPrice’s36extension, as\n/radicalbiggρx\nρy=−b\nπaln/parenleftbigg\ntanh/bracketleftbiggπdRAD,CB\n16ρiso/bracketrightbigg/parenrightbigg\n(2)\nwhereaandbarethe sidelengthsofarectangularsample\nandRAD,CBis resistance along the bsides as described\nabove.\nEq. (2) yields to the ratio of principle resistivities and\nthe individual values can subsequently be obtained by\nρx=ρiso/radicalbiggρx\nρy(3)\nand\nρy=ρiso/parenleftbigg/radicalbiggρx\nρy/parenrightbigg−1\n(4)\nFor the AMR measurement according to Bozorth’s41\nnotation one must measure resistivity with saturated\nmagnetization parallel ( ρ/bardbl) and perpendicular ( ρ⊥) to\ncurrent direction. The AMR ratio is given by42\nAMR =∆ρ\nρave=ρ/bardbl−ρ⊥\n1\n3ρ/bardbl+2\n3ρ⊥(5)Assuming the x-axis being the current direction, the\nAMR response can be presented as ρxwhich only de-\npends on the direction of saturated magnetization\nρx=ρ/bardbl+∆ρcos2φ (6)\nhereφstandsforanglebetweencurrent( x) andsaturated\nmagnetization direction.\nItisworthnotingthatEq.( 6)statesthattheresistivity\nis only dependent on φ, not on the initial magnetization\ndirection of the films. This is because ρ/bardblandρ⊥are\nmeasured at saturation magnetization where the entire\ndomains are assumed to be aligned to external magnetic\nfield.41\nIII. RESULTS AND DISCUSSION\nA. HiPIMS discharge waveforms\nThedischargecurrentandvoltagewaveformareachar-\nacteristic of the HiPIMS process, which provides impor-\ntantinformationonboththe instrument(the pulserunit)\nand the physics of ionization. Fig. 2shows the current\nand voltage waveformsof the HiPIMS dischargerecorded\nduring deposition. It can be seen that a nearly rectan-\ngular voltage pulse of 250 µs length was applied to the\ncathode target. The oscillations at the beginning and\nafter ending the voltage pulse initiate from internal in-\nductance of the power supply, which creates a resonant\ncircuit along with the capacitance of the cathode tar-\nget and the parasitic capacitance of the cables. There is\nalso a local minimum corresponding to the current rise\nat 80µs.\nThe discharge current is initiated about 70 µs into the\nvoltage pulse. The dischrge current peaks at 110 µs into\nthe pulse and then decays until it is cut-off. As described\nby Lundin et al.43the current waveforms can be divided\ninto three distinct regions. (I) A strong gas compression\nduetotherapidaccumulationofsputteredfluxasplasma\nignites which give rise to current to the peak value. This\nis followed by (II) rarefaction i.e. collision of sputter flux\nwith the working gas which results in heating and expan-\nsion of the working gas and consequently current decay.\nMore recently, it has been shown that the later mecha-\nnismisdominatingforrarefactionathigherpressuresand\nionization loss is dominating otherwise.44(III) A steady\nstate plasma till the end of the voltage pulse which gives\na relatively flat current plateau.\nIt has been shown that increased pressures can pro-\nlong current decay time to the end of pulse and eliminate\nplateau regions.43We believe this is highly unlikely the\ncase in Fig. 2. We have already shown that 0.33 Pa is in\nvicinity of minimum pressure i.e. lower pressures results\nin non-linear increase of delay time for current onset and\nincrease time from current onset to peak current which\nis nearly constant at higher pressures.30Thus pressure\nis low enough to capture the third stage of the current4\n0 50 100 150 200 250 300\ntime (s)0100200300400500Cathode voltage (V)\n-50510152025\nCatode current (A)\nFIG. 2. The discharge current and voltage waveforms at\n0.33 Pa with the pulse frequency of 100 Hz and pulse length\n250µs.\nevolution but, the short pulse length here does not allow\nit to appear.\nIt is worth mentinong that, although we have tried to\nmaintain the HiPIMS average power the same as dcMS\npower (at 150 W), the HiPIMS pulse voltage and peak\ncurrent (465 V and 25 A) are well above dcMS counter-\nparts (321 V and 463 mA).\nB. Microstructure\n1. XRR\nValues of film thickness, film density and surface\nroughness obtained from fitting of the measured XRR\ncurves are shown in Table I. The surface roughnesses ob-\ntained here are slightly lower than previously reported\nvalues using dcMS.19The mass density of HiPIMS de-\nposited film is higher than for the dcMS deposited coun-\nterpart. It is somewhat lower than for polycrystalline\nfilms deposited by dcMS and HiPIMS (8.7 g/cm3)30and\nthe bulk density of 8.73 g/cm3(Ref45(p. 2-6)). Ac-\ncounting for the epitaxial strain the densities here are\nwithin a reasonable range.\nTABLE I. Values of film thickness, film density and surface\nroughness obtained by fitting the XRR measurement results.\nGrowth Thickness Deposition Density Roughness\ntechnique (nm) rate ( ˚A/s) (g/cm3) (˚A)\nHiPIMS 45.74 0.97 8.38 6.75\ndcMS 37.50 1.50 8.32 6.33\nItisworthmentioningthat, duringeachHiPIMSpulse,\nthe deposition rate is much higher during the active dis-charge phase than for dsMS i.e. more than 50 times ac-\ncounting 250 µs pulse width and 100 Hz frequency.\n2. XRD\nPermalloy has a fcc structure while both metastable\nhcp14,17,46and bcc8,47,48Py phases have been reported\nin utrathin films. Fig. 3illustrates the symmetric θ−2θ\nXRD pattern of the epitaxial films obtained by both de-\nposition methods. In the out-of-plane XRD, fcc (002)\npeak is the only detectable Py peak. This indicates that\nthe (002) planes of Py are very well aligned to that of\nthe MgO substrate i.e. Py (001) /bardblMgO (001). Similar\nresults were obtained by measurement in-plane of epitax-\nial films (ψ= 90◦) along the [100] directions of MgO i.e.\nnormal to substrate edges. Furthermore, in-plane mea-\nsurementsalongthe /an}bracketle{t110/an}bracketri}htdirectionofMgO(substratedi-\nagonals) show (220) peaks from both the MgO substrate\nand the Py film. These indicate a orientation relation-\nship of Py [100] /bardblMgO [100] and Py [110] /bardblMgO [110]\ni.e. the [100] and [110] directions of Py are fully aligned\nwith those of the MgO substrate. Thus, in spite of the\nlarge lattice mismatch (15.84%), high quality single crys-\ntal Py (aPy= 3.548˚A) film can be established on a MgO\n(aMgO= 4.212˚A) substrate for both deposition tech-\nniques. Furthermore, we compared the in-plane peaks\nalong the /an}bracketle{t100/an}bracketri}htand/an}bracketle{t010/an}bracketri}htdirections (not shown here)\nand detected no difference in lattice parameter even with\na precise scan i.e. angular resolution 0.0001◦and 10 s\ncountingtime. Thismeansweobservedidenticalin-plane\nstrain along the [100] directions in both of the films.\nInthedcMScase,thePy(002)peakshowsaslightshift\ntowards higher angles in the normal XRD scan. This\nis accompanied by the shift of in-plane peaks towards\nsmaller angles. Thus, tensile strain at the film-substrate\ninterface generates slight compression normal to the film\nplane.49However, in the HiPIMS case, both the in-plane\nandout-of-plane peaks are shifted towards smaller an-\ngles. This would indicate tensile strain in all tree dimen-\nsion that is impossible. However, we attribute the shift\nof (002) peak in the HiPIMS case to departure from the\nL12Ni3Fe superlattice.50As pointed out by O’Handley\n(Ref51(p. 548)), the Ni 3Fe phase exists in either a dis-\nordered or well ordered structure. It has been shown\nthat an ordered Ni 3Fe phase can be detected as a shift\nof XRD peaks towards larger angles3,23,51and narrower\npeaks.21,23In addition, the intensity of XRD peaks is ex-\npected to increase with the higher order.51(p. 549) All\nthese conditions can be observed in our dcMS deposited\nfilm, indicating that it is more ordered than its HiPIMS\ncounterpart. The more disordered arrangement in the\nHiPIMS deposition can be attributed to the high depo-\nsition rate during each pulse which suppresses adatom\nmobility.5\n40 45 50 55 60 65 70 75 80\n2 (°)024681012 log intensity (arb. units)MgO {200}\nPy {200}\nMgO {220} Py {220}Ni3Fe Ni1.91Fe\nFIG. 3. The symmetric XRD pattern of the epitaxial films\ndeposited by HiPIMS (right) and dcMS (left). The vertical\ndashed lines show the peak position of bulk Py and MgO. The\ncurves are shifted manually for clarity.\n3. Pole figures\nFig.4illustrates pole figures for the main Py planes of\nour epitaxial films. In the {200}pole figure, there is an\nintense spot at ψ= 0 that verifies that the (002) plane\nis lying parallel to the substrate i.e. epitaxial relation-\nship of Py(001) /bardblMgO(001) for both dcMS and HiPIMS\ndeposited films. There is also a weak spot with four-\nfold symmetry at ψ= 90◦due to in-plane diffraction of\n{200}planes parallel to substrate edges in both films.\nThis indicates there are Py {100}planes parallel to the\nsubstrate edges i.e. Py[100] /bardblMgO[100]. The {220}pole\nfigures, also depict four-fold symmetry of {220}planes\natψangle of 45 and 90◦as expected from symmetry\nin a cubic single crystal for both films. In both of the\n{111}pole figures, there is a four-fold spot at φ= 45◦\nandψ= 54.74◦which is in agreement with the angle\nbetween (002) and {111}planes. However, compared to\nthe (002) spots the {111}and{220}planes are slightly\nelongated radially, along the ψaxis. This indicates a lat-\ntice constant expanded in-plane of the substrate for both\nfilms, in agreement with shift observed in the in-plane\nXRDs (cf. Fig. 3). The FWHM of the spots are always\nnarrowerfor the dcMS deposited epitaxial film indicating\nhigher order in this case.\nThe extra dots that appear in the {111}pole figure\nof the HiPIMS deposited film belong to twin boundaries\nas have also been reported for epitaxially deposited Cu\nusing thermal evaporation52and HiPIMS.53The exis-\ntence of twin boundaries in the Py is a signature of high\ndeposition rate which has been observed previously in\nevaporated9,54and electro-deposited5films and studied\nindetailusingTEM.54–56Itcanbeseenthatthesedotsat\n23◦also appear in the dcMS deposited film but with very\nFIG. 4. The pole figures obtained for Py {111},{200}and\n{220}planes of epitaxial films deposited by HiPIMS and\ndcMS. The height represents normalized log intensity in arb i-\ntrary units.\nsmall intensity. This indicates that the fraction of twin\nboundaries is much lower in the dcMS deposited film.\nIn addition, there are three spots with four-fold symme-\ntry in the {200}pole figure of the HiPIMS deposited film\nwhich do not appear in the dcMS counterpart. The three\ndot pattern in the {200}pole figure has been character-\nized as an auxiliary sign of twin boundaries in the film.53\nIt is worth noting that these extra dots in both {200}\nand{111}plane were characterized by a θ−2θscan (not\nshown here) to make sure they belong to the Py film.\nC. Magnetic properties\nFig.5compares the results of in-plane MOKE mea-\nsurementsalong the [100] and [110]directions ofboth the\nepitaxialfilms. Fig. 5(a-b)indicateabiaxialbehaviourin\nthedcMS deposited filmconsistingoftwoeasyaxesalong\nthe [110] directions with Hcof∼2 Oe. This is consistent\nwiththe /an}bracketle{t111/an}bracketri}htdirectionbeingtheeasydirectionofthePy\ncrystalandthemagnetizationbeingforcedin-planealong\nthe/an}bracketle{t110/an}bracketri}htdirections due to shape anisotropy.9,57Along\nthe [100]directionsthe MOKEresponseisrelativelyhard\ni.e. open hysteresis with a gradual saturation outside the\nhysteresis. The gradual saturation can be explained by\nan out-of-plane component of the magnetization.58In\npolycrystalline films, the out-of-plane element of magne-6\n-1-0.500.51(a) dcMS (b) dcMS\n-15 -10 -5 0 5 10 15\nH (Oe)-1-0.500.51Normilized intensity (arb. units)(c) HiPIMS\n-6 -4 -2 0 2 4 6(d) HiPIMS\nFIG. 5. The average hysteresis loops of the epitaxial films\nobtained by MOKE measurements along the [100] and [110]\ndirections of the epitaxial Py films.\ntization increases with increase in the film thickness59,60\nand it gives perpendicular anisotropy at trans-critical\nthicknesses.1,2,61In single crystal films, however, it ap-\npears that an out-of-plane component of the magnetiza-\ntion is generally the case.12,19,62\nFig.5(a) also shows that the /an}bracketle{t100/an}bracketri}htand/an}bracketle{t010/an}bracketri}htdirec-\ntions are not completely equivalent for our dcMS de-\nposited film. The /an}bracketle{t100/an}bracketri}htdirection presents larger coerciv-\nity(∼2Oe)andsaturatesat12Oebutthe /an}bracketle{t010/an}bracketri}htdirection\ngives∼1 Oe coercivity and saturates at 15 – 18 Oe. This\ndifference arises from the fact that /an}bracketle{t100/an}bracketri}htis the direction\nof sputter flux during the 300 ms stop time while revers-\ning the rotation. Such a short time is enough to define\nuniaxial anisotropy in the polycrystalline film using both\ndcMS and HiPIMS.30However, it appears that for the\nepitaxial film deposited by dcMS, our deposition geome-\ntry is not enough to induce uniaxial anisotropy along the\n[100] direction in agreement with the previous study of\nSchuhlet al.35.\nAs shown in Fig. 5(c-d) the HiPIMS deposited epi-\ntaxial film shows very well-defined uniaxial anisotropy\nindicated by a linear hard axis trace without hysteresis\nand slightly rounded easy axis loop along the [100] direc-\ntions. The anisotropy field ( Hk) of the HiPIMS epitaxial\nfilm is 3.5 Oe, i.e. much lower than the values observed\nfor polycrystallinefilms deposited by HiPIMS on Si/SiO 2\n(11 – 14.5 Oe).30However the coercivity ( Hc) of 1.8 Oe\nhere is very close to that of polycrystalline films i.e. 2\n– 2.7 Oe. We have shown that in polycrystalline films\ntheHcdepends on the film density and increases as the\nfilm density drops.30In principle the Hcof a film de-\npends on the domain boundary structure which has been\nprovedto be dependent on the film thickness.63However,\nsince the grain size changes with the film thickness, it is\na common mistake to correlate Hcwith the grain size.We have shown that for a range of film thicknesses (10\n– 250 nm) the grain size changes continuously while Hc\nonly changes with the domain wall transition i.e. N´ eel to\nBloch to cross-tie.28\nA question that might arise here is what makes\nthe HiPIMS deposited epitaxial film present uniaxial\nanisotropy. It has been shown by several groups that\nformation of ordered Ni 3Fe results in lower uniaxial\nanisotropy constant ( K1).3,22,24,25,27According to both\npairordering3andlocalizedcompositionnon-uniformity6\ntheories, uniaxial anisotropy is not expected for a highly\nsymmetric Ni 3Fe. While in the case ofHiPIMS deposited\nfilm, lower order results in uniaxial anisotropy.\nD. Transport properties\nFig.6shows the AMR response of epitaxial films to\nthe rotation of 24 Oe in-plane saturated magnetization.\nThis field is large enough to saturate both films in any\ndirection. The θhere stands for angle between applied\nmagnetic field and the /an}bracketle{t100/an}bracketri}htdirection of films and should\nnot to be confused with the φin Eq. (6) i.e. the angle be-\ntween current direction and magnetic field. The result of\nEq. (6) is also plotted for comparison as indicated by the\nblack line. Eventhough, the dcMS deposited film is thin-\nner than the HiPIMS counterpart, the resistivites in the\ndcMS case are all lower than the HiPIMS ones. This be-\nhaviour is in contradiction with the Fuchs model64which\npredicts lower resitivity for thicker films. It can be ex-\nplained in terms of higher Ni 3Fe order achieved in the\ndcMS deposited film. It has been shown previously that\nthe resistivity depends on the order and decreases upon\nincrease in Ni 3Fe order.22\nIt can be seen that the AMR response of the epitaxial\nfilm deposited with HiPIMS conformsbetter with Eq.( 6)\nthan its dcMS counterpart. In the dcMS case, the devia-\ntion from Eq. ( 6) occurs at about 45 – 85◦, 95 – 135◦and\nso on. Since the deviation is symmetric around 90◦(the\n/an}bracketle{t010/an}bracketri}htorientation) it is less likely associated with a pin-\nning mechanism of some domains. Presumably, the de-\nviation originates form switching some domains towards\nthe easy axis at 45 and 135, 225 and 315◦i.e. [110] ori-\nentations. This so-called qusi-static switching in single\ncrystal Py has been studied using torque measurements,\nas characteristics of biaxial anisotropy.9\nThe AMR values obtained by Eq. ( 5) along the /an}bracketle{t100/an}bracketri}ht\nand/an}bracketle{t010/an}bracketri}htdirectionsaresummarizedin Table II. We have\nrecently shown that in polycrystalline films the AMR re-\nsponse is different along the hard and easy axis of the\nfilm.29It appears that the AMR response is always lower\nalong the /an}bracketle{t100/an}bracketri}ht(direction of flux) in the epitaxial films.\nIt is also evident that higher order reduces resistivity and\nincreases AMR.7\nTABLE II. Summary of the AMR results of epitaxial films.\n(All resitivity values are in µΩ-cm unit.)\nDeposition Current ρ/bardblρ⊥∆ρ ρaveAMR\nmethod direction ( µΩ-cm) (%)\ndcMS /an}bracketle{t100/an}bracketri}ht22.19 22.80 0.39 23.06 1.70\ndcMS /an}bracketle{t010/an}bracketri}ht16.92 16.46 0.46 16.77 2.74\nHiPIMS /an}bracketle{t100/an}bracketri}ht37.46 37.91 0.45 37.76 1.19\nHiPIMS /an}bracketle{t010/an}bracketri}ht27.16 27.62 0.46 27.47 1.67\n22.422.622.8 (a) 100 dcMS  \n1717.217.4\n(b) 010 dcMS  \n3737.237.4 (c) 100 HiPIMS  \n0 45 90 135 180 225 270 315 360\n (°)27.627.828\n(d) 010 HiPIMS  (cm) \nFIG.6. The AMRobtainedbyresistivitymeasurementsalong\nthe [100] directions of Py films deposited by (a – b) dcMS and\n(c – d) HiPIMS during rotation of 24 Oe magnetic field. The\nθhere stands for angle of in-plane magnetization with the\n/an}bracketle{t100/an}bracketri}htdirection. The black lines indicate the result of fitting\nwith Eq. ( 6). The vertical dashed lines indicate the direction\nof easy axes.\nIV. SUMMARY\nIn summary, we havedeposited Ni 80Fe20(001)films by\nHiPIMSanddcMS.Wehavecharacterizedthemcarefully\nwith detailed X-ray measurements, finding only rather\nsubtle structural differences. The pole figures display a\nsignatureof twin boundaries(stacking faults) in the HiP-\nIMS deposited film and it appears to be slightly more\nstrained or disordered, regarding dispersion of Ni and\nFe atoms, than the dcMS deposited film. However, the\ndifferences in the magnetic properties of said films are\nvast. The dcMS deposited film has biaxial symmetry inthe plane, with easy directions [110] as one might expect\nfor a bulk fcc magnetic material (the /an}bracketle{t111/an}bracketri}htdirection is\nout of plane and shape anisotropy forces magnetization\ninto the plane of the film). The HiPIMS deposited film\nexhibits different magnetic symmetry, as it has uniaxial\nanisotropywith /an}bracketle{t100/an}bracketri}htastheeasydirection. Furthermore,\nthe film is magnetically soft and has an anisotropy field\nof only 3.5 Oe, which is lower than most results we have\nobtained for polycrystallinefilms. We attributed the uni-\naxial anisotropy to less ordered dispersion of Ni and Fe\nat the atomic level in the film deposited by HiPIMS due\nto high deposition rate of HiPIMS during the discharge\npulse.\nACKNOWLEDGMENTS\nThe authors would like to acknowledge helpful com-\nments and suggestions from Dr. Fridrik Magnus and\nDr. Arni S. Ingason on the structure characterization.\nThis work was partially supported by the Icelandic Re-\nsearch Fund (Rannis) Grant Nos. 196141, 130029 and\n120002023, and the Swedish Government Agency for In-\nnovation Systems (VINNOVA) contract No. 2014-04876.\n1Y. Sugita, H. Fujiwara, and T. Sato, Appl. Phys. Lett. 10, 229\n(1967).\n2H. Fujiwara and Y. Sugita, IEEE Trans. Magn. 4, 22 (1968).\n3S. Chikazumi, J. Phys. Soc. Jpn. 5, 327 (1950).\n4S. Kaya, Rev. Modern Phys. 25, 49 (1953).\n5J. R. Kench and S. B. Schuldt, J. Appl. Phys. 41, 3338 (1970).\n6D. C. M. Rodrigues, A. B. Klautau, A. Edstr¨ om, J. Rusz,\nL. Nordstr¨ om, M. Pereiro, B. Hj¨ orvarsson, and O. Eriksson ,\nPhys. Rev. B 97, 224402 (2018).\n7S. Chikazumi, J. Appl. Phys. 32, S81 (1961).\n8L. F. Yin, D. H. Wei, N. Lei, L. H. Zhou, C. S. Tian, G. S. Dong,\nX. F. Jin, L. P. Guo, Q. J. Jia, and R. Q. Wu, Phys. Rev. Lett.\n97, 067203 (2006).\n9A.Yelon, O.Voegeli, and E.Pugh,J. Appl.Phys. 36,101 (1965).\n10D. S. Lo, J. Appl. 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Kurebayashi1,†\n1London Centre for Nanotechnology, University College London, London WC1H 0AH, United Kingdom\n2Department of Physics, University of Science and Technology Beijing, Beijing 100083, China\n3Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551\n(Dated: August 27, 2019)\nWe study the magnetisation dynamics of a bulk single crystal Cr 2Ge2Te6(CGT), by means of\nbroadband ferromagnetic resonance (FMR), for temperatures from 60 K down to 2 K. We determine\nthe Kittel relations of the fundamental FMR mode as a function of frequency and static magnetic\n\feld for the magnetocrystalline easy - and hard - axis. The uniaxial magnetocrystalline anisotropy\nconstant is extracted and compared with the saturation magnetisation, when normalised with their\nlow temperature values. The ratios show a clear temperature dependence when plotted in the\nlogarithmic scale, which departs from the predicted Callen-Callen power law \ft of a straight line,\nwhere the scaling exponent n,Ku(T)/[Ms(T)=Ms(2 K)]n, contradicts the expected value of 3 for\nuniaxial anisotropy. Additionally, the spectroscopic g-factor for both the magnetic easy - and hard\n- axis exhibits a temperature dependence, with an inversion between 20 K and 30 K, suggesting\nan in\ruence by orbital angular momentum. Finally, we qualitatively discuss the observation of\nmulti-domain resonance phenomena in the FMR spectras, at magnetic \felds below the saturation\nmagnetisation.\nI. INTRODUCTION\nTwo-dimensional (2D) van der Waals (vdWs) single\ncrystals, belonging to the family of lamellar ternary\nchalcogenides (i.e :CGT and Cr 2Si2Te6) and chromium\nhalides (i.e :CrI3and CrBr 3), have recently attracted\na great deal of interest due to the presence of long\nrange magnetic order in the 2D limit1{3,6. The pres-\nence of ferromagnetism in the 2D state has a poten-\ntial to open new avenues in the \feld of spintronics\nleading to new magneto-optical and magneto-electric\napplications5,7. Recent study of gate-tunable room tem-\nperature ferromagnetism in layered 2D Fe 3GeTe 27and\nthe discovery of near room temperature ferromagnetism\nin the cleavable Fe 5GeTe 28have highlighted the signif-\nicance that these layered vdWs systems can have for\nspintronics devices with room temperature applications.\nThermal \ructuations in 2D systems at \fnite tempera-\ntures can restrain the long range magnetic order accord-\ning to the Mermin-Wagner theorem9, however due to the\npresence of large magnetocrystalline anisotropy in these\nlayered systems the magnetic order remains dominant\ndown to a few layers.\nChromium tellurogermanate, CGT, is a layered 2D fer-\nromagnetic semiconductor with vdWs coupling between\nthe adjacent layers. Bulk CGT has a hexagonal crystal\nstructure with the R \u00163 space group10. CGT has been a\nsubject of vast experimental studies over the past few\nyears. Gong et al. discovered the intrinsic ferromag-\nnetism in atomic bilayers of CGT using the magneto-\noptical technique and showed a signi\fcant control be-\ntween paramagnetic to ferromagnetic transition temper-\nature with very small magnetic \felds6. For CGT with\nonly few layers (\u00193:5 nm thickness of crystalline \rakes),\nWang et al. demonstrated the control of magnetism by an\nelectric \feld, showing a possibility for new applicationsin 2D vdWs magnets11. Liu et al. report an anisotropic\nmagnetocaloric e\u000bect associated with the critical be-\nhaviour of CGT and provide evidence of 2D Ising-like fer-\nromagnetism, which is preserved in few-layer devices12.\nThere still remains an ambiguity in the type of magnetic\ninteraction in CGT, as experimentally, Heisenberg-like\nferromagnetism is reported6, but other reports looking\nat the critical exponents in CGT predict 2D Ising-like\nferromagnetism13,14.\nCGT has been proposed as a potential substrate for\ntopological insulators in order to realise the quantum\nanomalous hall e\u000bect, and a large anomalous hall e\u000bect\nin the bilayer structure of Bi 2Te3and CGT has been\nalready observed15. To this point, there has been only\none brief report of determination of the uniaxial magnetic\nanisotropy in CGT by ferromagnetic resonance (FMR)16,\nwhile recent experiments rely on probing the sample by\nmeans of magneto-optical, magnetometry and transport\ntechniques. Understanding the magnetisation dynamics\nby FMR is bene\fcial as it exactly measures the mag-\nnetic ground state17i.e. uniform-mode excitations of\nspin-waves with k\u00190. The FMR experiment allows to\ndetermine the magnetic anisotropies in the system, the\nspectroscopic splitting g-factor, and addtionally it can\nprovide information on the relative orbital contribution\nto the magnetic moment18. Therefore, it is signi\fcant to\nbetter understand the magnetisation dynamics in bulk\nCGT in order to unfold the full potential of these layered\nvdWs systems in the \feld of spintronics.\nIn this article, we report on a broadband FMR\nstudy in CGT in the temperature range of 60 K -\n2 K, with the external magnetic \feld applied along\nthe in-plane (ab-plane) and out-of-plane (c-axis) orien-\ntations. The extracted value of the uniaxial magne-\ntocrystalline anisotropy constant, Ku, is found to be\ntemperature-dependent. We \fnd that the scaling of mag-arXiv:1903.00584v2  [cond-mat.mtrl-sci]  25 Aug 20192\nnetic anisotropy constant and saturation magnetisation\nas a function of temperature deviates from the theoreti-\ncal prediction by the Callen-Callen power law. The de-\ntermined g-factor in CGT is found to be anisotropic for\nthe di\u000berent crystallographic directions. Finally, we ob-\nserve a domain-mode resonance phenomenon below the\nsaturation \feld. This indicates a presence of multiple\ndomain structures in CGT.\nII. EXPERIMENTAL DETAILS\nCGT single-crystalline \rakes were synthesized by the\ndirect vapour transport (or \rux) method. High-purity\nelemental Cr (99.9999% in chips), Ge (99.9999% in crys-\ntals) and Te (99.9999% in beads) were pre-mixed at mo-\nlar ratio of 20:27:153 and sealed in the quartz ampule at\nhigh vacuum (\u001910\u00005Torr). The ampule was loaded in\nthe single-zone furnace, heated up to 1273 K with the\nrate of 2 K/min and left for 2 days. For a low-defect\ncrystal growth, the furnace was set to slow cooling with\nthe rate of 5 K/hour down to 673 K. Then, the furnace\nwas turned o\u000b for natural cooling. The layered single-\ncrystalline \rakes were separated from the \rux build-up\nand stored in the inert atmosphere of the glovebox. The\nbulk single crystals of CGT are not air sensitive and it\nis also the case for our sample whereas it is di\u000berent for\nfew layers of the CGT where the bilayer degrades after\n90 mins of exposure to air6.\nMagnetisation measurements were carried out at 2K\nand 60K with a vibrating sample magnetometer in a\nQuantum Design PPMS-14T, with measurements taken\nfor both in-plane and out-of-plane applied magnetic\n\felds. The Curie temperature of CGT is determined\nthrough heat capacity measurements carried out in the\nabsence of any magnetic \felds by thermal relaxation\ncalorimetry in the PPMS19.\nA standard broadband FMR technique is used for the\nstudy of magnetisation dynamics in CGT20{22. To em-\nploy a broadband oscillatory magnetic \feld to the sam-\nple, a silver plated copper co-planar waveguide (CPW)\non a Rogers 4003c PCB, with a center conductor width\nof 1 mm and a gap of 0.5 mm, is used. The impedance\nmatching of the microwave circuitary connnected to the\nCPW is important for consideration when using it for\nFMR experiments, and it is carefully matched to 50 \n23.\nThe PCB is mounted inside a cooper sample box and\nfeatures two SMP connectors, allowing the connection to\nthe microwave circuitry. The sample box is attached to\na probe, which is \ftted inside a closed cycle helium \row\ncryostat, with a temperature range of 2 K - 300 K. In\naddition, the cryostat is equipped with a rotation stage,\nallowing static magnetic \feld angle dependence measure-\nments. The static magnetic \feld is generated by an elec-\ntromagnet (Bruker ER073) which is able to produce a\nhighly homogeneous \feld at the sample location.\nWe perform broadband microwave transmission exper-\niments, using a vector network analyzer (VNA, HewlettPackard). The microwave transmission S 21is measured\nas a function of VNA frequency and static magnetic \feld.\nThe FMR signature is identi\fed by a strong microwave\nabsorption, hence a reduction of transmission.\nIII. RESULTS & DISCUSSION\nThe Curie temperature of the CGT sample is deter-\nmined to be Tc= 64:7\u00060:5 K via a characteristic peak\nin the heat capacity associated with magnetic ordering\n(Fig. 1). The cusp in the heat capacity data as a function\nof sample temperature corresponds to collinear magnetic\nordering in CGT, as reported elsewhere14.\nFIG. 1. Heat capacity of a CGT sample measured by thermal\nrelaxation calorimetry. The anomaly in the heat capacity\nassociated with the paramagnetic to ferromagnetic transition\nindicates a Curie temperature of 64.7 \u00060.5 K.\nThe magnetisation of the CGT sample is measured be-\nlow the Curie temperature for \felds along the in-plane\nand out-of-plane directions, yielding respective satura-\ntion magnetisations of 2.8 and 3.2 \u0016Bper Cr atom at 2\nK (Fig. 2).\nThe broadband FMR, for externally applied magnetic\n\felds along the in-plane and out-of-plane directions for\n60 K, 30 K, and 2 K are shown in Fig. 3. The inset shows\nan exemplary experimental resonance spectra. The reso-\nnance is described by the derivation of the dynamic sus-\nceptibility from the Landau-Lifshitz and Gilbert equa-\ntion, and hence it is analyzed by \ftting a linear combi-\nnation of symmetric and anti-symmetric Lorentzian func-\ntions in order to determine the resonance \feld, Hr(i.e.\npeak position)24. The frequency dependence of Hrex-\nhibits the typical characteristic of magnetisation dynam-\nics in a ferromagnet with easy and hard magnetic axis.\nThe easy axis in this case is along the c-axis (out-of-\nplane), which shows a linear frequency dependence with\na non-zero y-intercept corresponding to the anisotropy\n\feld. The ab-plane (in-plane) resonance spectra shows3\nFIG. 2. Magnetisation data for CGT measured via vibrating sample magnetometry carried out at temperatures from 2 K to 60\nK. (a) Magnetisation curves with the \feld applied along the in-plane direction. (b) Magnetisation curves with the \feld applied\nalong the out-of-plane direction. (c) Magnetisation with an applied \feld of 5 T as a function of temperature, lines connecting\ndata points have been added as a guide to the eye.\na non-linear frequency dependence with the x-intercept\nincreasing in value as the temperature is decreased from\n60 K to 2 K, exhibiting an increase in the uniaxial mag-\nnetocrystalline anisotropy. The dotted lines show the\nexpected resonance spectra of a uniform ferromagnetic\nresonance in single domain crystals. An important ob-\nservation in the extracted FMR Kittel relations, is the\ndi\u000berence in slope for in-plane and out-of-plane orienta-\ntions. Additionally, the slope for the two orientations\nshows a temperature dependence. This suggests that the\nspectroscopic g-factor is not isotropic for CGT (discussed\nin more detail below).\n0 0.2 0.4 0.6 0.8 1\n0H (T)051015202530Freq (GHz)\n0.2 0.4 0.6 0.8\n0H (T)0.850.90.951Signal (mV)Data\nFit\nFIG. 3. The inset shows the experimental resonance spec-\ntra with a Lorentzian line shape (black line). Main plot\nshows frequency dependence of resonance \feld at 60 K (red),\n30K (blue) and 2 K (black), along the in-plane (squares data\npoints) and out-of-plane (circle data points) orientations. The\nsolid lines represents the Kittel \ftting for both orientations\ngiving straight line \ft along the c-axis (\ftted with Eq. 8)\nand square-root dependence along ab-plane (\ftted with Eq.\n7) i.e. easy and hard axis, respectively. The dotted line along\nin-plane orientation show an expected spectrum of frequency\ndependence for a single domain case.\nIn the following we discuss the \ftting equations derivedby the Smit-Beljers-Suhl approach25,26, which are used to\n\ft the experimental data along the in-plane and out-of-\nplane directions in Fig. 3. The spatial distribution of the\nfree energy density, F, is probed by the FMR experiment.\nThe free energy density for the CGT system excluding\nexchange energy (in-plane dimensions of 2 x 1.5 mm and\nthickness of 0.3 mm, i.e. in the out-of-plane direction)\nreads\nF=\u0000M ·H+ 2\u0019(M ·n)2\u0000Ku(M ·z\nMs)2;(1)\nwhere HandMare the external magnetic \feld and\nmagnetisation vectors, respectively. The \frst term in Eq.\n1 is the Zeeman energy, the second term is the demag-\nnetisation energy and the third term is the uniaxial mag-\nnetic anisotropy term, with Ku, the magnetocrystalline\nanisotropy constant, Ms, the saturation magnetisation,\nandnandzare the unit vectors normal to the surface\nof the sample and orientated along the easy axis (c-axis),\nrespectively. HandMare then written as\nH=H0\n@sin(\u0012H) cos(\u001eH)\nsin(\u0012H) sin(\u001eH)\ncos(\u0012H)1\nA; (2)\nM=Ms0\n@sin(\u0012M) cos(\u001eM)\nsin(\u0012M) sin(\u001eM)\ncos(\u0012M)1\nA; (3)\nwhere\u0012H/M and\u001eH/M are the polar (accounted from\nnormal to the rectangular sample platelet ab-plane) and\nazimuthal (in-plane) angles, respectively. The double\nderivatives approach of the magnetic free energy with re-\nspect to the polar and azimuthal angles (i.e. substituting\nequations 1, 2 and 3 into Eq. 4) is then used to analyt-\nically calculate the resonance frequency25,26,!= 2\u0019f,\nas4\n\u0012!\n\r\u00132\n=1\nMssin(\u0012M)\u0014@2F\n@\u00122\nM@2F\n@\u001e2\nM\u0000\u0012@2F\n@\u0012M@\u001eM\u0013\u0015\n;(4)\nwhere\r, is the gyromagnetic ratio. The azimuthal an-\ngles in the CGT system (i.e. in equation 2 and 3) can\nbe set to\u001eH/M = 0, as the in-plane magnetocrystalline\nanisotropy energy is negligible. Solving for the perpen-\ndicular anisotropy case ( n=z, along the easy axis i.e.\nparallel to out of plane c-axis), one arrives at the general\nKittel relation for arbitrary polar angles of the external\nmagnetic \feld with respect to the easy axis of the mag-\nnetisation, which reads\n\u0012!\n\r\u00132\n= [Hrcos(\u0012H\u0000\u0012M)\u00004\u0019Me\u000bcos2(\u0012M)]\n\u0002[Hrcos(\u0012H\u0000\u0012M)\u00004\u0019Me\u000bcos(2\u0012M)];(5)\nwhereHris the ferromagnetic resonance \feld,\n4\u0019Me\u000b= 4\u0019Ms\u0000Huis the e\u000bective demagnetisation\n\feld andHu= 2Ku=\u00160Ms, the perpendicular anisotropy\n\feld, and since the hexagonal CGT lacks the C4symme-\ntry in the unit cell, the fourth order anisotropy constant\nis not considered in the \ftting of the FMR spectra. On\nresonance, the values for \u0012Mcan be determined by the\ncondition@F=@\u0012 M= 0 and is given by\nsin(2\u0012M) =2Hr\n4\u0019Ms\u0000Husin(\u0012M\u0000\u0012H): (6)\nThe Kittel equations for the in-plane ( k) and perpen-\ndicular (?) orientations can be determined by setting\n\u0012M=\u0012Hin a saturated state, and are given by\n!(k)=\rp\nHr(Hr+ 4\u0019Me\u000b); (7)\n!(?)=\r(Hr\u00004\u0019Me\u000b): (8)\nThe above equations are used to \ft the in-plane and\nout-of-plane FMR data in Fig. 3, with spectroscopic g-\nfactor (contained in \r=g\u0016B=~) andMe\u000bas \ftting pa-\nrameters. The extracted parameters from the Kittel re-\nlations are also used to calculate the dependence of the\nresonance frequency as a function of externally applied\nmagnetic \feld, and the angle-dependent resonance spec-\ntrum together with the calculated curves is shown in the\nsupplementary material27.\nThe temperature dependence of the out-of-plane uni-\naxial magnetocryatlline anisotropy constant, Ku, is\nshown in Fig. 4(a). This was determined from the least-\nsquare \ftting of equations 7 and 8 to the frequency-\ndependent FMR spectra. A positive sign convention\nis used, where Ku>0 favors perpendicular magnetic\nanisotropy. Kuis found to be in the range of (0 :77\u00003:95)\u0002105erg/cm3from 2 K to 60 K, which con\frms\nthat the magnetic easy axis in CGT is along the c-axis as\nseen by the magnetometry data and previous experimen-\ntal observations16,28. The reported magnetocrystalline\nanisotropy energies for other similar layered 2D magnets\nareKu= 1:46\u0002107erg/cm3at 4 K and Ku= 3:1\u0002106\nerg/cm3at 1.5 K, for Fe 3GeTe 229and CrI 330, respec-\ntively. CGT is found to be less anisotropic then these\nsystems whereas the ferromagnetic T cof CGT is close to\nthat of CrI 3(\u001861 K) and Fe 3GeTe 2possesses high tran-\nsition temperature ( \u0018223 K). At 2 K, the anisotropy\nenergy value corresponds to 0.24 meV per unit cell of\nCGT, which is six times smaller than the value, 1.4 meV\nper unit cell, obtained from the DFT calculations27. The\ndi\u000berence in the experimental and calculated magnetic\nanisotropy energy can arise as (i) the DFT calculation is\ndone at 0 K, (ii) the structure model used in the calcu-\nlation is considered to be a defect-free system (the unit\ncell considered for DFT calculations can be seen in the\nsupplementary material27).\nIn Fig.4(b), the reduced anisotropy constant,\nKu(T)=Ku(2 K), and the reduced magnetisation,\nMs(T)=Ms(2 K), at di\u000berent temperatures are shown for\nthe CGT single crystal. The theory of the temperature\ndependence of magnetic anisotropies was developed\nby Callen and Callen, and others32{34. It predicts\nthat the magnetocrystalline anisotropy constant as a\nfunction of temperature for uniaxial and cubic systems\ndistinctly di\u000bers depending on the crystal symmetry\nand on the degree of correlations between the direction\nof neighbouring spins. The famous Callen-Callen power\nlaw based on the single-ion anisotropy model relates\nthe temperature dependence of KutoMsin the low\ntemperature limit (T <<Tc) as\nKu(T)\nKu(0)=\u0014Ms(T)\nMs(0)\u0015n\n; (9)\nwhere the exponent, n=l(l+ 1)=2, depends on the\ncrystal symmetry and degree of correlations between ad-\njacent spins, and lbeing the order of spherical harmon-\nics and it describes the angular dependence of the local\nanisotropy. In the case of uniaxial anisotropy, the expo-\nnent is predicted to be n= 3 and it suggests a single-ion\norigin of magnetic anisotropy.\nAn unexpected behaviour of the scaling between the\ntemperature dependence of the anisotropy constant and\nmagnetisation is found in hexagonal CGT. This can be\nseen in Fig. 4(b), where the experimental data departs\nfrom the trend of a straight line (plotted in logarithmic\nscale) with a slope of n= 3, predicted by the Callen-\nCallen power law for uniaxial systems. In simple ferro-\nmagnetic systems such as ultrathin Fe \flms35, the Callen-\nCallen power law for uniaxial anisotropy has been experi-\nmentally veri\fed, across the whole temperature range be-\nlow T c, with an exponent n= 2:9, and it is considered as\na good model for ferromagnets with localised moments.\nIn the past, deviations from the expected scaling expo-5\n0 10 20 30 40 50 60 70\nT (K)00.511.522.533.54Ku ( 105 erg/cm3)Ku [FMR]\nKu [Magnetometry]\n(a)\n0 0.05 0.1 0.15 0.2\nLN [Ms (T)/Ms (2K)]00.511.52LN [Ku (T)/Ku (2K)]LN (Ratio) - FMR\nCallen - Callen, n=3 (power law)\nLN (Ratio) - Magnetometry\nStright line fit: y = n*x         n = 3\n(b)\nFIG. 4. (a) Temperature dependence of uniaxial magne-\ntocrystalline anisotropy constant, Ku, where the \flled points\nare extracted from the FMR data as discussed in the main\ntext and the non-\flled points are extracted from the mag-\nnetometry measurements31. (b) The ratio Ku(T)=Ku(2 K)\nis compared with the ratio [ Ms(T)=Ms(2 K)] in the absolute\nvalue of the logarithm scale. A solid line with the exponent,\nn = 3, is plotted to highlight that the experimental data does\nnot follow the Callen-Callen behaviour.\nnent ofn= 3 have been reported in complex systems con-\ntaining a non-magnetic material with a large spin-orbit\ncoupling, which can contribute to magnetic anisotropy\nwithout having a signi\fcant e\u000bect on other magnetic\nproperties36{38. In highly anisotropic barium ferrite sys-\ntems,Kuwas found to be linearly proportional to Ms,\nclearly inconsistent from the theoretical predictions39.\nThe departure from the Callen-Callen theory for these\nreported systems is suggested due to the violation of sim-\nple assumptions considered in the theory40, (i) magneti-\nsation origin in the material coming from single-ions with\nlocalised magnetic moments, (ii) spin-orbit coupling be-\ning regarded as a small perturbation to the exchange cou-\npling and (iii) temperature dependence of anisotropy and\nmagnetisation having the same origin. Recent \frst prin-\nciples calculations in CGT predict the interplay betweensingle-ion anisotropy and Kitaev interaction8. They are\nfound to be induced by the o\u000b-site spin-orbit coupling of\nthe heavy ligand, i.e. Te atoms. Another recent study42,\ninvestigating the origin of magnetic anisotropy in lay-\nered ferromagnetic Cr compounds, suggests that an ad-\nditional magnetic exchange anisotropy induced by the\nspin-orbit coupling from the ligand p-orbital through su-\nperexchange mechanism plays a crucial role in determin-\ning the magnetocrystalline anisotropy in CGT and other\nlayered systems such as CrI 3and CrSiTe 3. Hence, CGT\nis a complex magnetic system with no simple correlation\nbetween the thermal behaviour of anisotropy and mag-\nnetisation.\n0 10 20 30 40 50 60 70\nT (K)1.81.922.12.22.32.4g-factorH\nH|| c-axis\n|| ab-plane\nFIG. 5. Temperature dependence of the spectroscopic g-\nfactor with external magnetic \feld along the c-axis (easy-axis,\nred circles) and the ab-plane (hard-axis, blue squares). The\ndashed horizontal line (magenta) at g= 2 indicates a case of\npure spin magnetism where the orbital moment is completely\nquenched.\nThe temperature dependence of the spectroscopic g-\nfactor for the in-plane (ab-plane) and out-of-plane (c-\naxis) orientations, can be seen in Fig. 5. There is con-\ntrasting behaviour in the temperature dependence of g-\nfactor, where gincreases with decreasing temperature\nin the in-plane (hard-axis) direction, and it decreases\nin the out-of-plane (easy-axis) direction as the tempera-\nture decreases. At 2 K, gk= 2:18\u00060:02 (ab-plane) and\ng?= 2:10\u00060:01 (c-axis), indicating that the g-factor is\nanisotropic in CGT and shows a crossing between 20 K\nand 30 K. The g-factor is determined by extracting the\nproportionality of the gyromagnetic ratio in Eqs. (7) and\n(8), given by \r=g\u0016B=~, where\u0016Bis the Bohr magne-\nton and ~is the reduced Planck \rs constant. Determining\na value of gis important since one can \fnd the relative\nspin and orbital moments of a material by using the well-\nknown relation43\n\u0016L\n\u0016S=g\u00002\n2; (10)6\nwhere\u0016Lis the orbital moment per spin and \u0016S=\u0016B\nis the spin moment. The deviation of the spectrocopic\nfactor from the electron \rs g-factor of g= 2 suggests an\norbital contribution to the magnetisation44. The gvalue\nof the free electron case is considered in systems where\nthe orbital moment is assumed to be nearly completely\nquenched due to the high symmetry of the bulk crys-\ntals, and hence magnetism in such systems is described\nin terms of pure spin magnetism. The non-spherical\ncharge distribution in the d-shells can prevent the com-\nplete quenching of the orbital momentum45, i.e.Lz6= 0.\nAn orbital moment of 8 % and 5 % for \u0016L=\u0016Salong the in-\nplane and out-of-plane orientations, respectively, is found\nin CGT at 2 K. The small orbital moment anisotropy\ncould be explained by the presence of the spin-orbit in-\nteraction from the d-orbitals of the Cr atoms as well as\nthe o\u000b-site spin-orbit coupling of p-orbitals of Te atoms27,\ngiving ga tensor character i.e. dependence on the crys-\ntallographic directions. The g-factor extraction from the\nFMR experiment here only gives a relative insight into\nthe orbital moment contribution to the magnetism in\nCGT. One can determine the orbital moment, \u0016L, and\nits anisotropy from the x-ray magnetic circular dichroism\n(XMCD) experiments at synchrotron facilities in order to\nfurther improve the understanding of orbital magnetism\nin these layered van der Waals magnetic systems46{48.\nIn fact, the study referenced previously42looking into\nthe origin of magnetic anisotropy in layered 2D systems\nalso carries out an initial XMCD investigation of Cr L2,3-\nedge in CGT at 20 K. They report a sizable anisotropy\nin the orbital angular momentum (i.e. related to the\norbital moment), where Lc=\u00000:045,Lab=\u00000:052\nand \u0001L= 0:007. This work complements our results\nof the g-factor anisotropy obtained in the FMR experi-\nment, where a deviation from g= 2 con\frms presence of\norbital angular momentum in CGT.\nIn Fig. 6, the temperature evolution from 60 K to 2\nK of the frequency-dependent FMR spectrum in the in-\nplane (hard-axis) orientation is shown. The red arrows\nindicate the saturation \feld points for the CGT system,\nwhere a transition from the multi-domain magnetic states\nto a single domain state occurs in the sample49. At 60K,\nthere is no sign of domain-mode resonance but as the\ntemperature decreases, the strength of the perpendicular\nuniaxial magnetic anisotropy increases, and the multi-\nple domain-mode resonances (dotted white lines) appear.\nQualitatively, this shows the presence of multi-domain\nstructures in bulk CGT50. The FMR setup used in this\nexperiment is sensitive to observe the resonance of these\ndomain structures. The minimum feature of the reso-\nnance frequency (red arrow) increases with decreasing\ntemperature. This type of feature is markedly di\u000ber-\nent for samples with a single magnetic domain state51,\nwhere the frequency-dependent resonance spectra has a\nx-intercept along the hard axis of magnetisation.\nThe M-H curves in Fig (1a & 1b) show no hysteresis\nbehaviour, but this does not re\rect signatures of a single\ndomain system below the saturation state for CGT. The\nFreq (GHz) \n510 15 20 \n¹0H (T)0.2 0.4 0.6 0.8 1\nS21  (dB) 0\n-2 \n-4 \n-6 \n-8 \nFreq (GHz) \n510 15 20 Freq (GHz) \n510 15 20 \nFreq (GHz) \n510 15 20 \n¹0H (T)0.2 0.4 0.6 0.8 1S21  (dB) 0\n-2 \n-4 \n-6 \n-8 \nS21  (dB) 0\n-2 \n-4 \n-6 \n-8 \nS21  (dB) 0\n-2 \n-4 \n-6 \n-8 FIG. 6. Temperature evolution of the frequency-dependent\nferromagnetic resonance spectra (in-plane orientation) from\n60 K to 2 K. The red arrow indicates the saturation \feld point\nin the system where the system goes from multi-domain to\nsingle domain state. The dotted white lines do not represent\nany \fttings and are used only for guidance.\nremanent magnetisation at zero external magnetic \feld\ndoes not have any net magnetisation, indicating mag-\nnetic moment cancellation due to multi-domain states\nwith up-and-down \rux closures. This seems to be com-\nmon for layered van der Waals magnets i.e. exhibiting\na soft magnetic behaviour of a linear M-H curve be-\nfore saturation52,53. The experimental evidence of the\npresence of multiple magnetic domain structures in bulk\nCGT along the in-plane sample orientation has been con-\n\frmed recently using magnetic force microscopy54. The\nobserved domain structures were classi\fed with di\u000berent\nsymmetry types and showed an evolution from one sym-\nmetry type to another as function of both temperature\nand external magnetic \feld. This rea\u000erms that the fea-\ntures seen in the magnetisation dynamics experiment be-\nlow the saturation \feld (Fig. 6) corresponds to dynamics\nof multiple magnetic domain structures. Although, the\npresence of multi-domains are realised by the FMR ex-\nperiment, it is not possible to obtain the symmetry and\ntype of structures for these domains by a resonance ex-\nperiment alone. Hence, one has to be careful in extract-\ning any magnetic parameters in the bulk layered CGT\nsystem at low magnetic \felds as a small dynamic demag-\nnetising \feld can exist from the interaction of di\u000berent\nmagnetic domains with each other and also with that of\ndomain wall boundaries.\nIV. CONCLUSION\nIn summary, we studied the FMR behaviour in bulk\ntwo dimensional CGT. In particular, broadband FMR ex-\nperiments were performed within the temperature range\nof 60 K - 2 K with external magnetic \felds along the\nin-plane and out-of-plane orientations. Additionally, the\nangular dependence of the FMR spectra was examined.\nWe focused on the temperature dependence of the mag-7\nnetic anisotropy energy, spectroscopic g-factor and the\nmulti-domain resonance features (observed below the sat-\nuration \felds along the ab-plane). A uniaxial perpendic-\nular magnetic anisotropy is found with easy axis paral-\nlel to the c-axis (out-of-plane). The scaling of the nor-\nmalised magnetocrystalline anisotropy constant and sat-\nuration magnetisation showed a deviation from the the-\noretically predicted Callen-Callen power law. The pres-\nence of spin orbit coupling from d-orbitals of Cr atoms\nand an o\u000b-site spin orbit coupling from p-orbitals of the\nligand Te atoms can be a contributing factor to this con-\ntradicting behaviour of power law dependence of mag-\nnetic anisotropy to magnetisation. The obtained g-factor\nshowed an anisotropic response, i.e. a cystallographic de-\npendence, and at 2 K it was found to be gk= 2:18\u00060:02\n(ab-plane) and g?= 2:10\u00060:01 (c-axis). The deter-\nmined g-factor values are greater than 2, again, indicat-\ning that the orbital magnetism plays an important role\nin bulk CGT when considering magnetic interactions in\nthis system. A small anisotropy in the orbital magneticmoment relative to the spin moment is found, and a de-\ntailed x-ray magnetic circular dichroism study could give\nfurther insight into the orbital moment anisotropy. Fi-\nnally, the presence of multi-domain structures is qualita-\ntively con\frmed as domain-mode resonance phenomena\nare observed along the ab-plane (in-plane) orientation.\nACKNOWLEDGEMENTS\nThis research was supported by the Leverhulme\nTrust (grant No. RPG-2016-391), the Engineering and\nPhysical Sciences Research Council (EPSRC) through\nUNDEDD (EP/K025945/1) as well as by the Euro-\npean Union's Horizon 2020 research and innovation pro-\ngramme under Grant Agreement Nos. 688539 (MOS-\nQUITO) and 771493 (LOQO-MOTIONS). We would like\nto thank Prof. M. Farle and Dr. J. Baker for the helpful\ndiscussions.\n∗safe.khan.11@ucl.ac.uk\n†h.kurebayashi@ucl.ac.uk\n1B. Liu, Y. Zou, L. Zhang, S. Zhou, Z. Wang, W. Wang,\nZ. Qu, and Y. Zhang, Scienti\fc Reports 6, 33873 (2016).\n2Z. Wang, I. Guti\u0013 errez-Lezama, N. Ubrig, M. Kroner,\nM. Gibertini, T. Taniguchi, K. Watanabe, A. Imamo\u0015 glu,\nE. Giannini, and A. F. Morpurgo, Nature communications\n9, 2516 (2018).\n3N. Richter, D. 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Farle, Journal of\nmagnetism and magnetic materials 299, L1 (2006).\n52M. A. McGuire, G. Clark, K. Santosh, W. M. Chance,\nG. E. Jellison Jr, V. R. Cooper, X. Xu, and B. C. Sales,\nPhysical Review Materials 1, 014001 (2017).\n53M. A. McGuire, H. Dixit, V. R. Cooper, and B. C. Sales,\nChemistry of Materials 27, 612 (2015).\n54T. Guo, Z. Ma, Y. Hou, Z. Sheng, and Q. Lu, arXiv\npreprint arXiv:1803.06113 (2018).1\nSupplementary Material: A spin dynamics study in layered van der Waals single\ncrystal, Cr 2Ge2Te6\nV. DENSITY FUNCTIONAL THEORY CALCULATIONS\nA. Computational details\nThe \frst-principle calculations are performed using the projector augmented-wave (PAW)S1method as implemented\nin the Vienna ab initio simulation package (VASP)S2. The exchange-correction interaction is treated with the Perdew-\nBurke-Ernzerhof (PBE) form of the generalized gradient approximation(GGA)S3. A large plane-wave cut-o\u000b energy of\n500 eV is used throughout. The van der Waals correction of layered CrGeTe 3is included using the method of Grimme\n(DFT-D2)S4. The GGA+U methodS5is adopted to improve the description of on-site Coulomb interaction between\nCr 3d electrons. It was previously reported that the appropriate U value should be within the range of 0 :2<U<1:7\neVS6{S8. Here, test calculations were performed using U e\u000b= 0;0:5;1:0;1:5;2:0;2:5 and 3:0 eV, respectively, and we\nchoose U = 1 :5 eV, based on the fact that (i) the calculated lattice constants (a = 6 :901 and c = 20 :143\u0017A) are in\ngood agreement with the experimental data (a = 6 :820 and c = 20 :371\u0017A), as shown in Table 1; (ii) the intralayer\nmagnetic coupling of bulk CrGeTe3 is ferromagnetic, and the ferromagnetic state is more stable due to low magnetic\ncoupling energy, compared to that obtained using other values of U e\u000b(see Table 1). Both lattice parameter and\natomic position are optimized under the constraint of the symmetry until the energy and residual force on each atom\nconverge to less than 1 \u000210\u00006eV and 0:01 eV/ \u0017A, respectively. The magneto-crystalline anisotropy energy (MAE) is\nextracted by calculating the energy di\u000berence between states with all spins along the x direction [100] and along the\nz direction [001], respectively, in the bulk with spin-orbit coupling (SOC). We also calculated the total energy of bulk\nCrGeTe 3along the y direction [010] and found that its energy is identical to that along the x direction. Thus, in the\nfollowing, the [100] direction is used for the in-plane direction of magnetization.\nUe\u000b a error c error M Cr Mtot band gap E FM- EAFM\n(eV) ( \u0017A) (%) ( \u0017A) (%) ( \u0016B) (\u0016B) (eV) (eV)\n0 6.889 1.011 19.936 -2.136 3.066 18.00 0.073 -0.299\n0.5 6.890 1.021 20.031 -1.669 3.111 18.00 0.162 -0.292\n1.0 6.895 1.099 20.079 -1.433 3.184 18.00 0.147 -0.300\n1.5 6.901 1.192 20.143 -1.119 3.257 18.00 0.152 -0.303\n2.0 6.907 1.278 20.208 -0.801 3.329 18.00 0.12 -0.301\n2.5 6.914 1.376 20.253 -0.577 3.399 18.00 0.10 -0.297\n3.0 6.922 1.494 20.284 -0.429 3.466 18.00 0.068 -0.289\nExp.S9,S106.820 | 20.371 | 3.0 | 0.38 |\nTABLE I. Calculated lattice constants, magnetic moment, band gap and intralayer magnetic coupling energy (the energy\ndi\u000berence between the ferromagnetic (E FM) and anti-ferromagnetic (E AFM) con\fgurations) of bulk CrGeTe 3with di\u000berent U e\u000b\nvalues. M totand M Crstand for the magnetic moment of total and a Cr atom, respectively. The error refers to the di\u000berence\nbetween the calculated and experimental lattice constant of bulk CrGeTe 3. For comparison, the available experimental values\nare also listed at the last line of the table.\nB. Results & Discussion\nBulk CrGeTe 3has a hexagonal crystal structure with the R \u00163 space group, containing 30 atoms per unit cell. To\nexplore the magnetic ground state, we considered the ferromagnetic (FM) and anti-ferromagnetic (AFM) magnetiza-\ntion con\fgurations as shown in Fig. S1. The total energy calculations reveal that the FM state is the ground state,\nthe AFM con\fguration is 0.303 eV per unit cell higher in energy than the FM state. Therefore, we discuss only FM\nstate in the following. It is clear from the spin polarized charge densities of bulk CrGeTe 3(Fig. S1) that the magnetic\nmoment is contributed mainly by the Cr atoms. The calculated magnetic moment is about 3.26 \u0016B/Cr which is in\ngood agreement with the experimental value of about 3.0 \u0016BS10. It is expected since there are three unpaired electrons\nin the Cr3+ionic con\fguration. A small magnetic moment, -0.12 \u0016B/Te and 0.034 \u0016B/Ge, respectively, is also induced\non the Te and Ge atoms. The total magnetic moment of CrGeTe 3is 3\u0016B/formula unit which is in good agreement\nwith the measured magnetization in bulk CrGeTe 3S10. The ferromagnetic coupling in bulk CrGeTe 3is attributed to\nthe near 90\u000eCr-Te-Cr super-exchange interaction between half-\flled Cr-t 2gand empty Cr-e gstates through the Te-p2\norbitals, against the direct t 2g-t2gexchange interactions of Cr in the AFM state. This is also supported by our \fnding\nof magnetic moment of Te atoms pointing in opposite direction from the magnetic moment of Cr atoms.\nFIG. S1. (a) Side view of local structure of bulk CrGeTe 3, showing that one Cr atom is octahedrally coordinated by six Te\natoms. (b) and (d) Side and top view, respectively, of spin polarized charge densities of bulk CrGeTe 3in the FM con\fguration;\n(c) and (e), the same information in the AFM con\fguration. The yellow color and blue color represent the charge density of\nspin up and down, respectively. The isosurface is 0.058 e \u0017A\u00003.\nUsing the spin-orbit coupling implementation of density functional theory (DFT), we estimate the magneto-\ncrystalline anisotropy energy for ferromagnetic bulk CrGeTe 3. Figure ??shows the magnetic anisotropy energy\nsurface of bulk CrGeTe 3, the energy maxima are observed along the [001] direction, meaning that the c-axis is the\neasy magnetization direction of bulk CrGeTe 3which agrees with the experimental mensuration. The calculated MAE\ndepends on the choice of U e\u000bas shown in Fig. ??, and the MAE is 1.4 meV per unit cell at U e\u000b= 1:5 eV. In this\nwork, we did not consider the shape anisotropy caused by dipole-dipole interactions which is expected to be small\ncompared to the magneto-crystalline anisotropyS11.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s45/s49/s48/s49/s50/s51/s52/s53\n/s69\n/s49/s48/s48/s32/s45/s32/s69\n/s48/s48/s49\n/s49/s46/s52/s32/s109/s101/s86/s77/s65/s69/s32/s40/s109/s101/s86/s47/s117/s110/s105/s116/s32/s99/s101/s108/s108/s41\n/s72/s117/s98/s98/s97/s114/s100/s32 /s85\n/s101/s102/s102/s32/s40/s101/s86/s41\n(a)\n (b)\nFIG. S2. (a) Magneto-crystalline anisotropy energy as a function of Hubbard Ue\u000b in bulk of CrGeTe 3. Positive MAE value\nmeans that the out of plane magnetization is the easy axis. (b) The three-dimensional shape of MAE at U e\u000b= 1:5 eV shows\nthat the easy axis of magnetization is the c-axis, that is, [001] direction of the crystallographic cell. The MAE is 1.4 meV which\nagrees well with the other calculated values of 1.41 meVS7.3\nTo understand the mechanism of the magnetic anisotropy in bulk CrGeTe 3, we calculated its spin polarized total\nand projected density of state (DOS) and the results are shown in Fig. S3. It is clear from Fig. ??that bulk CrGeTe 3\nis a ferromagnetic semiconductor with a band gap of 0 :15 eV. The valence band ranging from \u00004:0 to\u00000:01 eV\nand conduction band ranging from 0 :15 to 4:0 eV, originated mainly from the Cr d orbitals and Te p orbitals with\nvery small contributions from Ge. Only the PDOS of Cr and Te are shown in Figs. ??and??. There are strong\norbital hybridizations between Cr d orbitals and Te p orbitals at the valence band maximum (VBM) and conduction\nband minimum (CBM), which may enhance the magnetic anisotropy due to strong spin-orbit coupling of Te. The\nenergy splitting of the Cr d states around the Fermi level contributes primarily to the crystal \feld splitting in their\noctahedrally coordination environment as shown in Fig. S1(a) (one Cr ion is octahedrally coordinated by six Te\natoms). Hence, the band gap in this material is very sensitive to the Hubbard U onsite Cr ions as shown in Table\n1. According to the second-order perturbation theoryS12, the orbitals close to the Fermi energy contribute the most\nto MAE. Hence, the increase in MAE with the U e\u000bvalue (see Fig. ??) could be due to the enhanced SOC between\nthe d orbitals induced by the decrease of band gap with the increase of U e\u000bvalue. The majority-spin d z2states of\nCr atom are almost fully occupied. This means that the contribution to the MAE from the SOC between unoccupied\nmajority-spin d z2states and occupied majority- or minority-spin d states can be neglected and hardly contribute\nto the MAE. In contrast, the d x2\u0000y2, dxy, dyzand d xzorbitals of Cr atoms are dominant in both valence band\nand conduction band, thus their contributions would be large for MAE. Moreover, due to the SOC matrix elements\n<dxzjHSOjdyz>and<dx2\u0000y2jHSOjdxy>contribute to the out-of-plane anisotropy, while <dx2\u0000y2jHSOjdyz>,<\ndxyjHSOjdxz>, and<dz2jHSOjdyz>favor an in-plane anisotropyS7,S13, it can be deduced that the SOC interaction\nfrom the d x2\u0000y2, dxy, dyzand d xzorbitals of Cr as well as p orbitals of Te atoms lead to the MAE that prefers the\nout-of-plane anisotropy in the bulk CrGeTe 3material.\n/s45/s52 /s45/s50 /s48 /s50 /s52/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48\n/s83/s112/s105/s110/s32/s100/s111/s119/s110/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s47/s101/s86/s41\n/s69/s32/s45/s32/s69\n/s70/s32/s40/s101/s86/s41/s32/s84/s111/s116/s97/s108\n/s32/s67/s114\n/s32/s84/s101\n/s32/s71/s101/s83/s112/s105/s110/s32/s117/s112\n(a)\n (b)\n/s45/s52 /s45/s50 /s48 /s50 /s52/s45/s52/s45/s50/s48/s50/s52\n/s69/s32/s45/s32/s69\n/s70/s32/s40/s101/s86/s41/s80/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s47/s101/s86/s41/s32/s112\n/s120/s44/s112\n/s121\n/s32/s112\n/s122/s84/s101\n(c)\nFIG. S3. GGA+U calculated total DOS of CrGeTe 3(a) and site PDOS of the Cr atoms (b) and the Te atoms (c). The inset\nin (b) shows an enlarged DOS near the Fermi level. The fermi level is set to zero (denote by blue dashed line).4\nVI. ANGULAR-DEPENDENT FMR\nThe angular dependence of the resonance frequency with respect to the externally applied magnetic \feld for 60 K,\n30 K and 2 K is shown in Fig. S4. The red line shows the calculated resonance frequency curve as function of \u0012H,\nwith ab-plane (hard axis) at \u0012H= 0\u000eand c-axis (easy axis) at \u0012H= 90\u000e. The resonance frequency is calculated by\nusing equations 1 - 6 in the main text together with the parameters (i.e. Kuand g-factor) extracted from the Kittel\nrelations \ftting along the in-plane and out-of-plane orientations. The calculated curves show a good agreement to\nexperimental resonance spectra with a small discrepancy at intermediate angles between in-plane and out-of-plane\ncon\fgurations. This is due to the fact that the spectroscopic g-factor is found to be anisotropic from the Kittel\nrelations and here, it is assumedS14asg2(\u0012H) =g?cos2(\u0012H) +gksin2(\u0012H), where the g?andgkvalues are given in\nthe main text.\n(a)\n(b)\n(c)\nFIG. S4. Dependence of the ferromagnetic resonance frequency as a function of the applied external magnetic \feld, \u0012H, at (a)\n2 K , (b) 30K and (c) 60 K. The red line shows the the calculated frequency dependence using the extracted parameters from\nthe Kittel \fttings in the main text.\n∗safe.khan.11@ucl.ac.uk\n†h.kurebayashi@ucl.ac.uk\n[S1] P. E. Bl ochl, Physical review B 50, 17953 (1994).\n[S2] G. Kresse et al. , Phys. Rev. B 54, 11169 (1996).\n[S3] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical review letters 77, 3865 (1996).\n[S4] S. Grimme, Journal of computational chemistry 27, 1787 (2006).\n[S5] A. Liechtenstein, V. Anisimov, and J. Zaanen, Physical Review B 52, R5467 (1995).\n[S6] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, et al. , Nature 546, 265 (2017).5\n[S7] Y. Fang, S. Wu, Z.-Z. Zhu, and G.-Y. Guo, Physical Review B 98, 125416 (2018).\n[S8] C. Xu, J. Feng, H. Xiang, and L. Bellaiche, npj Computational Materials 4, 57 (2018).\n[S9] Y. Li, W. Wang, W. Guo, C. Gu, H. Sun, L. He, J. Zhou, Z. Gu, Y. Nie, and X. Pan, Physical Review B 98, 125127\n(2018).\n[S10] Y. Sun, R. Xiao, G. Lin, R. Zhang, L. Ling, Z. Ma, X. Luo, W. Lu, Y. Sun, and Z. Sheng, Applied Physics Letters 112,\n072409 (2018).\n[S11] G. Daalderop, P. Kelly, and M. Schuurmans, Physical Review B 41, 11919 (1990).\n[S12] D.-S. Wang, R. Wu, and A. Freeman, Physical Review B 47, 14932 (1993).\n[S13] H. Takayama, K.-P. Bohnen, and P. Fulde, Physical Review B 14, 2287 (1976).\n[S14] J. Stankowski, S. Waplak, and W. Bednarski, Solid state communications 115, 489 (2000)."    },    {        "title": "1903.01789v1.Electronic_structure_and_magnetism_of_transition_metal_dihalides__bulk_to_monolayer.pdf",        "content": "Electronic structure and magnetism of transition metal dihalides: bulk to monolayer\nA. S. Botana1,\u0003and M. R. Norman1\n1Materials Science Division, Argonne National Laboratory, Argonne, IL 60439\nBased on \frst principles calculations, the evolution of the electronic and magnetic properties of\ntransition metal dihalides MX 2(M= V, Mn, Fe, Co, Ni; X = Cl, Br, I) is analyzed from the bulk to\nthe monolayer limit. A variety of magnetic ground states is obtained as a result of the competition\nbetween direct exchange and superexchange. The results predict that FeX 2, NiX 2, CoCl 2and CoBr 2\nmonolayers are ferromagnetic insulators with sizable magnetocrystalline anisotropies. This makes\nthem ideal candidates for robust ferromagnetism at the single layer level. Our results highlight the\nimportance of spin-orbit coupling to obtain the correct ground state.\nINTRODUCTION\nLong range magnetic order is a common phenom-\nena in three-dimensional materials but not in lower di-\nmensions. According to the Mermin-Wagner theorem,\nlong-range magnetic order is not possible in 2D for\nspin-rotational-invariant systems.[1] However, magnetic\nanisotropies (i.e., those that break spin-rotational sym-\nmetry) remove this restriction. Magnetic van der Waals\n(vdW) materials are good candidates given their \rex-\nibility that can allow for a tuning of their magnetic\nanisotropy, as well as for their ease of exfoliation. They\no\u000ber the possibility of obtaining a magnetic ground state\neven in the single-layer limit.\nAn example is CrI 3, a layered Ising ferromagnet (FM)\nin which the Cr ions lie on a honeycomb lattice. In 2017,\nit was shown that ferromagnetism does survive at the\nsingle-layer level with a transition temperature near that\nof the bulk material (61 vs 45 K).[2] Magnetism in 2D\nhas the potential to open up a number of technologi-\ncal opportunities such as sensing, information, and data\nstorage. 2D ferromagnets are particularly interesting for\nspintronic applications. Novel functionalities based on\nvan der Waals heterostructures, in which magnetism adds\na new ingredient, can also be anticipated. The poten-\ntial use of CrI 3and other materials for building devices\nand tuning their properties through gating has already\nstarted to be explored, marking the birth of a new era\nof magnetism.[3{6] However, pushing T cto higher values\nwill be necessary for real applications.\nIn this regard, there is a materials family related to\nCrI 3that holds equal promise: binary transition metal\ndihalides MX 2(M = transition metal, X = halogen: Cl,\nBr, I). They form low dimensional crystal structures com-\nposed of either one dimensional chains or two dimensional\nlayers.[7] The layered structure is shown in Fig. 1 where\nthe vdW gap between layers is apparent. In contrast to\ntrihalides, layered dihalides contain a triangular lattice\nof transition metal cations, and geometrical frustration\nin such a lattice is expected when the magnetic interac-\ntions are antiferromagnetic (AFM). This set of materials\nhence provides a rich playground for examining low di-\nmensional magnetism. In the bulk, the magnetism of\nFIG. 1. Left panel: Crystal structure of transition metal di-\nhalides (MX 2) in the 1-T phase showing the triangular lattice\nformed by the magnetic (M) atoms (in gray). Halide (X)\natoms are shown in green. Right panel: Top view of the ef-\nfective triangular ferromagnetic lattice formed by the metal\natoms for Fe, Co, and Ni dihalide monolayers.\nmost of these materials was analyzed decades ago.[7] At\nthe monolayer level, some theoretical e\u000bort has been de-\nvoted to them,[8{12] but the full trends in magnetism and\nelectronic structure and in particular the magnetocrys-\ntalline anisotropy energy (crucial to establish 2D long\nrange magnetic order) have not been completely studied\nfor all possible 3 dtransition metal and halide ions.\nHere, we present a systematic study of the electronic\nstructure and magnetism of MX 2compounds from the\nbulk to the monolayer limit by means of \frst principles\ncalculations. After a general overview of the magnetic\ntrends in bulk dihalides, we turn to results at the mono-\nlayer level, where a variety of magnetic states is found\nas a result of the competition between direct exchange\nand superexchange via the halogen pstates. We predict\nthat FeX 2, NiX 2, CoCl 2and CoBr 2monolayers are fer-\nromagnetic insulators with an easy axis normal to thearXiv:1903.01789v1  [cond-mat.str-el]  5 Mar 20192\nplanes and a sizable magnetocrystalline anisotropy mak-\ning them ideal candidates for robust 2D ferromagnetism.\nOur results also highlight the importance of considering\nboth the trigonal distortion of the MX 6octahedra as well\nas the spin-orbit coupling to obtain the correct ground\nstate.\nCOMPUTATIONAL METHODS\nOur electronic structure calculations were performed\nusing the all-electron, full potential code WIEN2k [13]\nbased on the augmented plane wave plus local orbitals\n(APW + lo) basis set. The Perdew-Burke-Ernzerhof ver-\nsion of the generalized gradient approximation (GGA)\n[14] was used for structural relaxation and optimization.\nThen, the LDA+ Uscheme within the fully localized limit\nwas applied to treat the strong correlations.[15] We have\nstudied the evolution of the electronic structure with in-\ncreasingU(U= 2-6 eV,J= 0.8 eV). For all materials,\nwe performed calculations in supercells of size 1 \u0002p\n3 to\nallow for the possibility of an in-plane antiferromagnetic\n`striped' con\fguration.\nFor the calculations, we converged using R mtKmax=\n7.0 with a \fne kmesh of 17\u000217\u00023 for the bulk materials\nand 19\u000219\u00021 for the monolayers. Mu\u000en-tin radii of 2.48\na.u. for Fe, 2.45 a.u. for V, 2.45 a.u. for Co, 2.50 a.u. for\nMn, 2.50 a.u. for Ni, 2.11 a.u. for Cl, 2.48 a.u. for Br,\nand 2.50 a.u. for I were employed.\nTo determine the magnetocrystalline anisotropy en-\nergy (MAE), we calculate how the direction of the spin\nof the metal atom a\u000bects the energy when spin-orbit cou-\npling (SOC) is included with the moment orientation ei-\nther in-plane or out-of-plane. SOC was introduced in\na second variational procedure.[16] The calculations of\nmagnetic anisotropy require careful convergence of the\ntotal energy. We found that a converging criterion for\nthe total energy to within 10\u00006eV yields stable results.\nSTRUCTURAL PROPERTIES\nAs mentioned above, most of the bulk transition metal\ndihalides have a natural layered structure that contains\ntriangular nets of cations in edge-sharing octahedral co-\nordination forming MX 2layers separated by van der\nWaals gaps (Fig. 1). The octahedral crystal \feld will\nsplit the 3dorbitals of the metal atoms into higher lying\neg(dx2\u0000y2,dz2) and lower lying t 2g(dxy,dxz,dyz) states.\nAll of the nearest-neighbor distances within the MX 6oc-\ntahedra are the same, but there is a trigonal distortion\nwhich further splits the t 2gmanifold. The consequences\nthis has for the electronic structure are explained below.\nMX 2compounds adopt either the trigonal CdI 2struc-\nture (so called 1-T with P\u00163m1 space group) or the rhom-\nbohedral CdCl 2(R\u00163m) one. These structures have dif-TABLE I. Calculated in-plane lattice parameters for mono-\nlayer MX 2within GGA.\na Cl Br I\nVX 2 3.62 3.81 4.08\nMnX 2 3.64 3.84 4.12\nFeX 2 3.49 3.69 3.98\nCoX 2 3.49 3.73 3.92\nNiX 2 3.45 3.64 3.92\nferent stackings along the c-axis. The Cr and Cu mate-\nrials are slightly di\u000berent in that their structure in the\nbulk is 1D-like and monoclinic, respectively, hence we\ndo not analyze them here. The structure of monolayer\ndihalides is analogous to that of the intensively stud-\nied transition metal dichalcogenides in which ferromag-\nnetism at the monolayer level has been anticipated by\nDFT calculations.[17] Regardless, based on the known\nbulk values, larger magnetic moments in 2D dihalides\ncan be expected.\nIt was previously found that all dihalides prefer the 1-T\ncrystal structure at the monolayer level.[10] The stabil-\nity of single-layer dihalides was also evaluated from their\nformation energy con\frming that not only are metal di-\nhalide monolayers stable, but also that they could poten-\ntially be exfoliated.[10, 18] Based on this, for the mono-\nlayer we will focus on 1-T structures only. The calculated\nstructural parameters of MX 2monolayers are summa-\nrized in Table I and agree with Ref. [10] and with the\nexperimental bulk data.\nDIHALIDES - BULK MAGNETISM\nWe analyze \frst the electronic structure and mag-\nnetism of the bulk materials, experimentally studied al-\nready, to test the validity of our predictions. All the ma-\nterials are insulators as shown by experiment and con-\n\frmed by our calculations (to open up a gap, a Uis\nrequired in some cases). As mentioned above, there are\ntwo competing magnetic interactions in the planes: direct\nexchange between transition metal cations, and superex-\nchange through the halogen anions. The magnetism of\nthese materials can be understood from the Goodenough-\nKanamori-Anderson rules.[19] In the case of a 90\u000eM-X-M\nbond angle, the e g-egexchange is always FM and weak,\nthe direct t 2g-t2goverlap can give rise to an AFM ex-\nchange, and depending on the particular orbital occupa-\ntion, the t 2g-t2gsuperexchange via halides can either be\nAFM or weakly FM. For the case of edge-sharing octa-\nhedra, the t 2gorbitals on neighboring sites, pointing be-\ntween the oxygens, are directed toward each other. The\nresultingd\u0000dhopping turns out to be very important\nfor early 3dmetals (i.e., V) and it can give rise to AFM\nexchange. As a note, the magnetic order found in most of3\nTABLE II. Magnetism of bulk MX 2within GGA. The second\ncolumn re\rects the nature of the ground state (GS) where\nAFS stands for an antiferromagnetic striped con\fguration\n(in-plane stripes coupled AFM along c), AF for FM planes\ncoupled AFM along c, FM for ferromagnetic, I for insulator,\nand HM for half-metal. * indicates that FeX 2become insula-\ntors after SOC and Uare included. The third column shows\nthe energy di\u000berence (per unit cell) between ferromagnetic\nand corresponding antiferromagnetic ordering. The fourth\ncolumn shows the magnetocrystalline anisotropy energy for\nthe magnetic moments pointing in-plane versus out-of-plane\n(negative values re\rect out-of-plane moments). The last col-\numn shows the calculated magnetic moment.\nGS \u0001E FM\u0000AFM (meV) MAE (meV) MM ( \u0016B)\nBulk\nVCl 2 AFS-I 55.49 0.04 2.60\nVBr 2 AFS-I 25.57 0.03 2.61\nVI2 AFS-I 10.13 0.62 2.62\nMnCl 2AFS-I 18.45 0.09 4.50\nMnBr 2AFS-I 12.27 0.03 4.52\nMnI 2 AFS-I 8.63 0.16 4.51\nFeCl 2AF-HM* 14.74 -1.09 3.47\nFeBr 2AF-HM* 5.44 -0.82 3.54\nFeI2AF-HM* 2.72 -1.05 3.40\nCoCl 2 AF-I 5.03 0.62 2.54\nCoBr 2AF-I 1.49 0.76 2.49\nCoI 2 AF-I 13.60 0.17 2.24\nNiCl 2 AF-I 4.62 0.50 1.46\nNiBr 2 AF-I 2.99 0.26 1.39\nNiI2 AF-I 27.33 0.30 1.25\nthese compounds either consists on ferromagnetic planes,\nstripes, or is helimagnetic.[7] The results of the density\nfunctional theory (DFT) calculations are shown in Table\nII. We discuss them below in order of increasing dcount.\nThree magnetic structures were explored: AFS (stripe)\norder, which was taken to be alternating rows of ferro-\nmagnetic spins stacked AFM along c, AF order taken to\nbe ferromagnetic planes stacked AFM along c, and FM\norder. These allow us to capture the physics of most of\nthe materials. However, the non-colinear nature of the\norder observed in some of the materials is beyond the\nscope of the present work.\nVX 2.In these materials, V is in a high spin d3con-\n\fguration (S = 3/2). Neutron powder di\u000braction studies\nfound that all three vanadium dihalides order antiferro-\nmagnetically with N\u0013 eel temperatures of 36.0, 29.5, and\n16.3 K for Cl, Br and I, respectively,[20] with the low\nvalues relative to the Weiss temperatures (437 K, 335 K,\n143 K) [21] being an indication of geometric frustration.\nOur AFS (stripe) calculations indeed con\frm that the\nmagnetic order involves antiferromagnetic in-plane cou-\npling, with the moments predicted to lie in-plane. The\nactual moment orientation for VCl 2and VBr 2is non-\ncolinear (120 degree orientation of the moments) as typ-\nical for triangular antiferromagnets, and moreover the\nmoments appear to be tilted out of the plane.[20, 22, 23].VI2is more complicated in that the 120 degree order oc-\ncurs \frst, and then slightly below this at 14.4 K a stripe\nphase (the same as simulated here) sets in, though the\nmoments are probably also tilted out of the plane.[20, 24]\nThe closeness of the two phase transitions indicates that\nthe free energy di\u000berence between the 120 degree order\nand the stripe phase is small.\nMnX 2.In these materials, Mn is in a high spin d5con-\n\fguration (S = 5/2). For this \flling, superexchange is\nAFM. The magnetic structure for MnCl 2has ferromag-\nnetic stripes of width two rows within the layers with\nantiferromagnetic coupling between neighboring stripes\nand between the layers.[25] There is evidence that the\nmoments lie in-plane for the Cl and Br materials,[26]\nthat our DFT calculations con\frm. For simplicity, we\napproximate this state by the one width row AFS (stripe)\norder for the purposes of Table II. MnI 2adopts a com-\nplicated helical magnetic structure at low T stemming\nfrom competition with longer range exchange.[27] The\ndevelopment of a ferroelectric polarization was recently\nreported, spurring interest in this compound as a multi-\nferroic material.[28]\nFeX 2.Divalent iron, d6, is expected to be in a high\nspin state, S = 2. Given the partially \flled t2glevels,\nan orbital moment may be expected as well. The mag-\nnetic structure of FeCl 2and FeBr 2shows ferromagnetic\norder within the layers and antiferromagnetic stacking\nalongc.[29] Our calculations do reproduce the observed\nmagnetic order. The iodide counterpart, though, adopts\nthe two-row width stripes as found in MnBr 2.[30] The\nmoments are along the caxis in all cases. Our obtained\nmagnetocrystalline anisotropy energies con\frm that the\nmoments lie out-of-plane.\nCoX 2.Co2+,d7, can be in a high spin (S = 3/2)\nor low spin (S = 1/2) state. Neutron di\u000braction results\nshow that the high spin state is preferred, at least for\nCoCl 2and CoBr 2, and our DFT calculations con\frm this\npicture.[29] Below their ordering temperatures, both of\nthese compounds adopt a magnetic structure with ferro-\nmagnetic alignment within each layer and antiferromag-\nnetic stacking. This is correctly reproduced by our calcu-\nlations, along with the fact that the magnetic moments lie\nin-plane. The magnetic behavior in CoI 2is more complex\n- it is a helimagnet with a spiral spin structure, indicating\nthe presence of longer range exchange.[31]\nNiX 2.Divalent nickel in these compounds has a d8con-\n\fguration, with low spin being S=0 and high spin S=1,\nthe later being favored. Both NiCl 2and NiBr 2show\na magnetic con\fguration with moments lying within\nthe plane that are ferromagnetically aligned within each\nlayer, with antiferromagnetic stacking.[32] Our DFT cal-\nculations are able to reproduce this. NiBr 2develops he-\nlimagnetic order below a lower second transition, that is\nseen as well below T Nin the iodide material. Like MnI 2\nand CoI 2described above, NiBr 2and NiI 2also develop\na ferroelectric polarization in their helimagnetic states.4\nFIG. 2. Atom-resolved density of states (DOS) and band structures for di\u000berent dihalide monolayers within GGA: FeCl 2,\nCoCl 2and NiCl 2. Majority and minority spin channels are represented by up and down arrows.\n[33, 34]\nThe trend in the magnetic moments across the MX 2\nseries is the expected one according to the spin states de-\nscribed above: starting with \u00183\u0016Bper metal atom for\nthe V-halides, increasing to \u00185\u0016Bfor the Mn-halides,\nand then gradually decreasing to \u00182\u0016Bfor the Ni-\nhalides (Table II). The halogens develop a magnetic mo-\nment of 0.16-0.20 \u0016Band hence show a signi\fcant spin-\npolarization.The magnetic moment on the metal atoms\ndecreases with the Z of the halide atom (Cl - Br - I). In\nthe Fe and Co compounds, an orbital moment \u00180.1 and\n0.2\u0016B, respectively, is found when SOC is included.\nDIHALIDES-MONOLAYER MAGNETISM\nAfter con\frming the correct magnetic ordering trends\ncan be reproduced in the bulk dihalides, we turn our at-\ntention to the monolayers. The obtained ground states\nare very similar to those obtained in the bulk, a reason-\nable outcome given the weak interlayer coupling in vdW\nmaterials. In a similar fashion to the above bulk de-\nscription, comparison has been established between fer-\nromagnetic, and antiferromagnetic (AFS stripe) ordering\nwithin the layers. FeX 2, CoX 2(X= Cl, Br) and NiX 2pre-\nfer a FM ground state, while VX 2, MnX 2and CoI 2are\n(striped) AFS (Table III). These preferences can be jus-\nti\fed based on the above described competition between\ndirect exchange and superexchange for di\u000berent \fllings.\nCoI 2breaks the trend among the Co compounds due to\nthe sensitivity of superexchange for M-X-M bond angles\nnear 90\u000eto the relative position of the anion pstates\n(Ref. [19]).\nWe will focus on the description of materials with a\nFM ground state at the monolayer level starting withTABLE III. Magnetism for monolayer MX 2within GGA. The\nsecond column re\rects the nature of the ground state where\nAFS stands for antiferromagnetic (stripe) in-plane order, FM\nfor ferromagnetic in-plane order, I for insulator, and HM for\nhalf-metal. * indicates that FeX 2become insulators after\nSOC andUare included. The third column shows the en-\nergy di\u000berence between ferromagnetic and antiferromagnetic\nin-plane order. The fourth column shows the magnetocrys-\ntalline anisotropy energy for the magnetic moments pointing\nin-plane versus out-of-plane (negative values re\rect out-of-\nplane moments). The last column shows the calculated mag-\nnetic moment.\nGS \u0001E FM\u0000AFS (meV) MAE (meV) MM ( \u0016B)\nMono\nVCl 2 AFS-I 61.47 -0.53 2.67\nVBr 2 AFS-I 31.90 -0.30 2.67\nVI2 AFS-I 10.70 -0.25 2.67\nMnCl 2AFS-I 30.00 -0.20 4.54\nMnBr 2AFS-I 15.36 -0.25 4.52\nMnI 2 AFS-I 12.92 -0.18 4.46\nFeCl 2FM-HM* -121.58 -0.89 3.57\nFeBr 2FM-HM* -67.66 -0.33 3.53\nFeI2FM-HM* -35.90 -0.59 3.45\nCoCl 2 FM-I -57.52 -0.69 2.54\nCoBr 2FM-I -10.33 -0.68 2.49\nCoI 2 AFS-I 14.15 -0.50 2.23\nNiCl 2 FM-I -38.76 -0.12 1.68\nNiBr 2 FM-I -32.70 -0.02 1.63\nNiI2 FM-I -33.73 -0.18 1.53\nthe Fe compounds. FeX 2monolayers were the focus of\nprevious studies in which they were predicted to be FM-\nhalf metals.[8] We have analyzed the electronic struc-\nture of these materials in further detail, in particular,\nthe evolution of the electronic structure with Uwithin\nthe LDA+Umethod and upon inclusion of SOC. Fig. 25\nshows the band structure and density of states of the\nhalf-metallic FeCl 2monolayer obtained within GGA. The\nelectronic structure is consistent with the description in\nRef. 8: Fe2+being HS, the majority spin channel dstates\nare completely occupied; in the minority spin channel,\nthere are two partially occupied t 2gbands and three un-\noccupieddbands (a \rat t 2gone, and above this, two\negones). The electronic structure for the minority spin\nchannel is identical to the bulk electronic structure in\nwhich FM layers are stacked in an AFM fashion (not\nshown). FeBr 2and FeI 2display the same basic electronic\nstructure around E F. The spin moment per unit cell is\n4\u0016Bwith most of it residing on the Fe site ( \u00183.5\u0016B).\nOne interesting question is the origin of the t 2gsplit-\nting into an e\u0003\ngdoublet and a higher-lying a 1gsinglet\n(Fig. 3).[35] This is due to the above mentioned trigo-\nnal distortion (compression along the [111] axis of the\noctahedra, i.e., the c-axis of the crystal). This distor-\ntion is quanti\fed by the angle \u0012(Fig. 3) that is larger\nthan the value for an undistorted octahedron \u00120= 54.74\u000e\n= arccos(1/p\n3). Additionally, one has to take into ac-\ncount the contribution to the crystal \feld of neighboring\ntransition metal atoms to the a 1g-egsplitting. The cor-\nresponding wavefunctions for these states can be written\nin the form:\nja1gi=1p\n3(jxyi+jxzi+jyzi);\nje\u0003\ng\u0006i=\u00061p\n3\u0010\njxyi+e\u00062\u0019i=3jxzi+e\u00072\u0019i=3jyzi\u0011\n(1)\nThe a 1gorbital has a very simple shape in local coor-\ndinates. It is analogous to a z2- egorbital with its zaxis\ndirected along the [111] axis (in this case, the c-axis of\nthe crystal). The other two t2gorbitals, denoted as e\u0003\ng,\nhave a more complicated shape. These are states with\njlz=\u00061i(with the quantization axis along c), the a 1g\nstate being thejlz= 0ione.\nThis clearly explains why, once SOC is included, the\ne\u0003\ngorbitals are split. The overall band structure is similar\nto that of Fig. 2 with an orbital moment of 0.10 \u0016Bbeing\ninduced on Fe. This splitting increases once an on-site\nUis included, with an insulating gap opening up once\nUexceeds 3 eV (Fig. 3). As a consequence, a larger\norbital moment (\u00180.6\u0016B) develops along the the trigonal\naxis, parallel to the spin moment. It should be noted\nthat regardless of the Uvalue, a gap is not opened up\nunless SOC is included and the e\u0003\ngstates are split. The\ndegeneracy lifting due to spin orbit coupling is similarly\nrequired to open a gap in FeX 2bulk materials, which are\nknown to be Mott insulators.\nThe electronic structure of monolayer CoCl 2within\nGGA (d7\flling, HS) is shown in Fig. 2. The electronic\nstructure for the bulk (with a magnetic order consisting\nof FM layers stacked AFM) is equivalent to that of the\nmonolayer. At the GGA level, a gap already opens up in\nFIG. 3. Band structures for di\u000berent dihalide monolayers\nwithin LDA+SOC+U (U= 4 eV): FeCl 2, CoCl 2and NiCl 2.\nAll of them are FM insulators. The top right panel shows\nthe magnetic (M) atom surrounded by a trigonally distorted\noctahedron of halides. Distortions are determined by the an-\ngle\u0012; the value cos \u00120= 1/p\n3 corresponds to an undistorted\noctahedron. The corresponding crystal \feld splitting of the\nd-orbitals of the magnetic atom is shown. The splitting of\nthe t 2glevels into a1gandeg\u0003is due to both the trigonal\ndistortion (TD) of the octahedron and the contribution from\nneighboring M atoms to the crystal \feld. SOC additionally\nsplits theeg\u0003doublet.\nspite of the partial ( d7) \flling. The reason is once again\nthe trigonal distortion of the octahedra that gives rise to\nthe t 2gsplitting into an a 1gsinglet and e\u0003\ngdoublet (the\n\rat band right above the Fermi level in the minority spin\nchannel has a 1gcharacter, the two e\u0003\ngare occupied). In a\nsimilar fashion to the Fe case, the total moment per unit\ncell is 3\u0016Bwith most of it residing on the Co site site\n(\u00182.5\u0016B). Once a Uis included, the gap between the\noccupied e\u0003\ngand unoccupied a 1gstates increases (Fig. 3).\nThe orbital moment derived for LDA+SOC+U (U= 4\neV) is 0.2\u0016B. This, unlike for the Fe case, is the same\nas forU=0, an expected result given the complete \flling\nof the e\u0003\ngdoublet.\nNiCl 2also is a FM insulator at the monolayer level.\nThe insulating character with Ni being d8(HS) is sim-\npler to understand. The band gap opens up between t 2g\noccupied and e gunoccupied minority spin bands. The\nderived magnetic moment agrees with this picture (Ta-\nble III). In this case, a Usimply increases the gap as\nexpected (Fig. 3).\nAs mentioned above, the character of the ferromag-\nnetic ordering in these 2D materials is determined by the\nmagnetocrystalline anisotropy energy. It is magnetic 2D\nmaterials with an easy axis that can have a ferromagnet-\nically ordered phase at \fnite temperature. All Fe and\nNi dihalides, as well as CoCl 2and CoBr 2, are predicted6\nTABLE IV. Magnetism for monolayer MX 2(continued). The\nsecond column re\rects the nature of the ground state where\nAFS stands for antiferromagnetic (stripe) order, FM for fer-\nromagnetic order, I for insulator, and HM for half-metal. *\nindicates that FeX 2become insulators after SOC and Uare\nincluded. The third column shows the value of the exchange\ninteraction. The fourth column is an estimate of the Curie\ntemperature.\nGS J(K) T c(K)\nMono\nFeCl 2 FM-HM* 44.1 160\nFeBr 2 FM-HM* 24.5 89\nFeI2 FM-HM* 13.0 47\nCoCl 2 FM-I 37.1 135\nCoBr 2 FM-I 6.7 24\nCoI 2 AFS-I { {\nNiCl 2 FM-I 56.2 205\nNiBr 2 FM-I 47.4 173\nNiI2 FM-I 48.9 178\nto have their easy axis along c. This makes them a very\npromising platform for stable and robust ferromagnetism.\nAs in Ref. 10, we estimate the strength of the magnetic\ninteractions from the energy di\u000berence between a FM and\nan AFM con\fguration (denoted as \u0001E). In our case, for\nthe AFS state, a given magnetic ion has four AFM and\ntwo FM near neighbors, and for the FM state, six FM\nnear neighbors. The same counting applies for next near\nneighbors. This leads to an e\u000bective Jof:\nJ=\u0001E\n8S2(2)\nwhereSis the spin per metal atom (2, 3/2 and 1 for\nFe, Co and Ni). The calculated values for materials\nwith a FM ground state are shown in Table IV. The\nderived values are similar to those in previous works in\nwhich pseudopotentials were used,[10] with predicted val-\nues for FeCl 2and NiX 2being particularly high, noting\nthat our estimates are higher than theirs since they as-\nsumed a factor of 12 rather than 8 in the denominator\nof Eq. (2). This J, which can be considered as a sum\nof the near-neighbor J1and next near-neighbor J2, is\nrelevant if longer range Js are not important (the next\nnext near-neighbor J3drops out of this energy di\u000ber-\nence). As a cautionary note, the values we estimate for\nJtypically exceed those extracted from inelastic neutron\nscattering.[36] But, they should give some idea of the\ntrends inJas M and X are varied.\nFor uniaxial anisotropy, the appropriate model is the\nIsing one. For a triangular 2D lattice, the critical temper-\nature is 3.641 J/kB.[37, 38] The T cs based on this, from\nour estimate of J, are shown in Table IV for each of the\nmonolayers. Given the above caveats, these should be\nconsidered as overestimates. Still, we would like to point\nout that in most cases, MX 2materials have N\u0013 eel temper-\natures that exceed those of their MX 3counterparts.[7]Moreover, it is entirely conceivable that the monolayer\nTccould exceed the bulk T N. This is particularly rel-\nevant for NiX 2, which have the highest T Nof the MX 2\nmaterials besides TiCl 2.[7] If the spins do convert from\nxy-like (in-plane) in the bulk to Ising-like (along c) in\nthe monolayer, as we predict, then the NiX 2monolayer\nCurie temperatures could indeed be high. As suggested\nin Ref. 10, T ccould also be increased by strain from a\nsuitable substrate.\nSUMMARY\nIn summary, motivated by recent experiments for CrI 3,\nwe have explored the evolution of the magnetism in tran-\nsition metal dihalides from the bulk to the monolayer\nlimit. FeX 2, NiX 2, CoCl 2and CoBr 2monolayers are pre-\ndicted to be ferromagnetic insulators with out-of-plane\nmoments and sizable magnetocrystalline anisotropies.\nOur results highlight the importance of considering the\nsymmetry lowering at the transition metal site due to\nthe trigonal distortion of the MX 6octahedron, as well as\nincluding the e\u000bect of spin-orbit coupling, to obtain the\ncorrect ground states. This work con\frms the potential\nfor stable and robust 2D ferromagnetism in transition\nmetal dihalide-based monolayers, and we hope this will\nstimulate experimental e\u000borts to realize them in the lab-\noratory.\nACKNOWLEDGMENTS\nThis work was supported by the U.S. Dept. of En-\nergy, O\u000ece of Science, Basic Energy Sciences, Mate-\nrials Sciences and Engineering Division. We acknowl-\nedge the computing resources provided on Blues, a high-\nperformance computing cluster operated by Argonne's\nLaboratory Computing Resource Center.\n\u0003antia.botana@asu.edu\n[1] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133\n(1966).\n[2] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-\nHerrero, and X. Xu, Nature 546, 270 (2017).\n[3] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao,\nW. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G.\nLouie, J. Xia, and X. 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Phys. 30, 615 (1967)."    },    {        "title": "1903.02258v1.Isolated_zero_field_sub_10_nm_skyrmions_in_ultrathin_Co_films.pdf",        "content": "1 \n Isolated  zero field sub-10 nm skyrmions in ultrathin Co films   \nSebastian Meyer1,*, Marco Perini2,*, Stephan von Malottki1, André Kubetzka2, Roland Wiesendanger2, \nKirsten von Bergmann2,#, and Stefan Heinze1  \n1Institute of Theoretical Physics and Astrophysics, Christian -Albrechts -Universität zu Kiel, \nLeibnizstrasse 15, 24098 Kiel, Germany  \n2Department of Physics, University of Hamburg, Jungiusstrasse 11, 20355 Hamburg, Germany  \n* These authors contributed equally to this work.  \n# e-mail: kbergman@physnet.uni -hamburg.de  \nDue to their exceptional topological  and dynamical properties magnetic skyrmions1 – localized \nstable spin structures on the nanomet re scale – show great promise for future spintronic \napplications2-4. To become technologically competitive , isolated skyrmions with diamet ers below \n10 nm that are stabl e at zero magnetic field and room temperature  are desired5. Despite finding  \nskyrmions in a wide spectrum of materials6-14, the quest for a material with these envisioned \nproperties  is still ongoing. Here we report zero field isolated s kyrmions with diameters smaller \nthan  5 nm coexisting with  1 nm thin domain walls in Rh/Co atomic bilayers on the Ir(111) surface. \nThese  spin structures  are characterized by  spin -polarized scanning tunne lling microscopy  and can \nalso be detected  using non -spin -polarized tips due to a large non -collinear magnetoresistance15. \nWe demonstrate  that sub-10 nm skyrmions  are stabilise d in the se ferromagnetic  Co films at zero \nfield due to strong frustration  of exchange interaction , together with Dzyaloshinskii -Moriya \ninteraction and a large magnetocrystalline anisotropy.   \nThe stabilisation  of isolated magnetic skyrmions  at zero magnetic field in a ferromagnetic material \ndue to the Dzyaloshinskii -Moriya interaction (DMI)16,17 has been predicted already more than 20 \nyears ago based on a micromagnetic model18. After the experimental discovery of magnetic \nskyrmions6-9 it has been proposed to use such individual skyrmions in novel storage and logic \ndevices2-5. This triggered further theoretical studies with a focus on isolated skyrmions confined in \nnanostructures19,20, in ultrathin films21, and in multilayers22. From an experimental point of view \nultrathin transition -metal films and transition -metal multilayers have proven to be particularly useful \nto find novel skyrmion systems since the magnetic interactions can be tuned via interface \ncomposition and structure9,11,13,23,24. Typically , magnetic skyrmions arise in applied magnetic fields: at \nroom temperature skyrmions with diameters between 30 and 400  nm10-14 have been realized in \nmultilayers, whereas at cryogenic temperatures isolated skyrmions  with diameters down to a few \nnanomet res have been observed9,25,26. In zero  field, remanent isolated skyrmions were obtained  \nrecently in a ferrimagnetic material with diameters down to 16 nm27, however , in ferromagnets so \nfar they have been realized only with diameters of at least 100 nm in confined structures12, in \nmultilayers14 or stabilized by interlayer exchange coupling28. \nFigure  1a and b  show spin -resolved  differential conductance ( ܫ݀/ܷ݀ )map s of typical Rh  island s on \nCo/Ir(111)  in the magnetic virgin state , measured at different bias voltages . The Co monolayer grows \nas stripes from the Ir step edges, dominantly pseudomorphic in fcc stacking29. The Rh monolayer  \nforms com pact islands on Co  which are either in hcp stacking (blue areas in Fig. 1b) or in fcc stacking  \n(see Supplementa ry Fig.  S1). On top of the Rh/Co there are a  few small second layer Rh islands . We \nobserve  small -scale lateral variations in the differential conductance of both  the Co and the Rh/Co  2 \n films , presumably due to  some intermixing of the two materials. In the following, we will focus on the \nmagnetic state of the Rh hcp/Co bilayer (see Supplementa ry Fig.  S2 for Rhfcc/Co).  \nThe Rhhcp/Co islands in the ܫ݀/ܷ݀ map s of Fig. 1a dominantly exhibit a two -stage contrast , which we \nattribute t o the spin -polarized contribution to the tunnel current, i.e.  the tunnel magnetoresistance \n(TMR) . In combination with experiments using an external magnetic field ܤ (see Fig.  2 and \nSupplementary Fig.  S3) we conclude that the Rhhcp/Co is out -of-plane ferromagnetic ( FM) and th e \ntwo-stage contrast represents  domains oriented in the two opposite  magnetisation  directions . The \ndomain walls (DWs) separating oppositely magnetised domains  surpri singly do not minimize their \nlength but instead follow meandering  paths. This suggests that the position of the DWs is dominated \nby a combination of  small DW energy , i.e.  only a small energy penalty for their exist ence, and an \ninhomogeneous potential  landscape  within the film  due to some intermixing.  \nIn Fig. 1a the ܫ݀/ܷ݀ signal on the DWs is dominated by the TMR contribution and the signal strength \nscales with the in -plane magnetisation  direction within the wall. This in turn is found to be correlated \nwith the local direction of the wall: all  brighter DWs are located at the right side  of darker domains  in \nFig. 1a. Thus  the magnetic structures in Rh hcp/Co/Ir(111) have a fixed  sense of magnetisation  \nrotation. It originate s from the DMI, which favo urs Néel -type DWs and skyrmions with unique \nrotational sense in thin film systems9,30,31. \nAt the bias voltage used i n Fig.  1b the ܫ݀/ܷ݀ signal on the DWs is dominated  by the tunnel non -\ncollinear magnetoresistance (NCMR) contribution.  This contrast is not related to the TMR but arises  \nfrom changes of the local electronic properties within the non -collinear spin texture15,32. For the \nmeasurement of Fig.  1b we find a h igher  NCMR -related  ܫ݀/ܷ݀ signal for sample positions where \nadjacent spins have a larger  cant ing, i.e. bright lines mark the position s of DWs . The strength of this \nNCMR contribution in first approximation scales with the local mean angle between adjacen t \nmagnetic moments15. This electronic effect can be  exploited to detect DWs and skyrmions also with a \nnon-magnetic metallic electrode  and is demonstrated here for the first time  in a Co film, a wi dely \nused magnetic material . \nSmall c ircular domains are found in both of the  two oppositely magnetized FM domains  of \nRhhcp/Co/Ir(111) , see boxes in  Fig. 1a and b . The experimental ܫ݀/ܷ݀ maps can be reproduced by \nscanning tunnelling microscopy ( STM ) simulations of skyrmions , see gr ey scale images in Fig.  1c and \nd, with different contributions of TMR and NCMR for the two different bias voltages  (see \nSuppl ementary Text S5  for details) . The diameter s of the two skyrmions are 4.3 nm and 3.5 nm, and \nthe compar ison between experimental and simulated line profiles  is reasonable ( Fig. 1c and d). In \nFig. 1e the derived out -of-plane magnetisation  components are plotted , which show the typical \ncontinuous magnetisation  rotation across a skyrmion.  It is quite remarkable, that these opposite \nmagnetic skyrmions appear in the virgin state  of our Rh/Co film  and that they do not collapse \nregardless of  their small diameter.  \nBeside the occurrence of isolated skyrmions  in the FM virgin  state  of Rh hcp/Co (Fig.  1, Supplementary \nFig. S3 and S4 ),  they can be  induce d by shrinking larger FM domains in opposite magnetic field  as \nseen in the measurements of Fig. 2a and b:  the closed l oop domain wall imaged bright in the ܫ݀/ܷ݀ \nmap of Fig.  2a due to the NCMR encloses a n isolated FM domain in zero field, which  shrinks in size \nupon application of an out -of-plane magnetic field (Fig.  2b). The ܫ݀/ܷ݀ signal with in the white \nrectangles is plotted versus  the lateral position across the magnetic objects  in Fig.  2d and e. The solid  \nlines show the simulated  NCMR signal for two straight  180° domain wall s (Fig.  2d) and a cut through 3 \n a magnetic skyrmion  (Fig.  2e) for a DW width  ݓ of 0.8  nm; the corresponding spin structures are \nshown in Fig.  2c, where the atomic magnetic moments are colo ured according to their out -of-plane \nmagnetisation components. The experimental data is reproduced well by the simulated  NCMR signal \nand t he deriv ed skyrmion diameter is 2.8 nm, see Fig.  2f, which corresponds to about 10 atomic \ndistances between opposite in -plane magnetisation s. We would like to emphasise  that there is no \nplateau in the centre  of this circular domain, instead the spins rotate continuously . \nTo understand  why in th ese Rh/Co film s small magnetic skyrmions are stable  at zero magnetic field , \nwe apply density functional theory (DFT)  (see methods).  Figure 3a shows the calculated energ y \ndispersion ܧ() of homogeneous  spin spirals  in Rh/Co/Ir(111)  along the high symmetry directions of \nthe two -dimensional Brillouin zone (2D -BZ) neglecting spin-orbit coupling ( SOC); to reveal  the role of \nthe Rh  overlayer  the system of uncovered Co/Ir (111) is  also shown . All Co films have a large FM \nnearest neighbour exchange constant , as evident from the large energy difference of the FM state at \n߁ത and the antiferromagnetic states at the BZ boundary . At small ||, i.e. for small angles between \nnearest neighbo ur moments,  Co/Ir(111) shows a rise of the energy with ݍଶ, as expected for a typi cal \nferromagnet ( Fig. 3b). In contrast , ܧ() is extremely flat for Rh/Co/Ir(111) for both Rh stackings  and \nto describe the dis persion , a  ݍସ term is require d (Fig.  3b). Such an  energy dispersion is characteristic \nfor strong exchange f rustration , where antiferromagnetic interactions beyond neares t neighbours \ncompete with  FM exchange between nearest neighbours  (see Supplementary Tables S11 and S12 for \nvalues) . This is  quite  unexpected  for Co  films , but, as we will show, it turns out to be benefi cial for the \nstabilisation  of small  magnetic skyrmions  in zero field . \nCalculations including  SOC show a strong out-of-plane  magnetocrystalline anisotropy energy for both \nRh/Co stacking s (with 1.2 and 1.6  meV/ Co atom , for fcc - and hcp -Rh stacking, respectively ). In \naddition , they reveal  an energy contribution to cycloidal spin spirals due to the DMI  (Fig.  3c) \nfavouring  a clockwise rotational sense . To explore  the resulting  spin structures we apply atomistic \nspin dy namics using the  parameters from DFT (see methods  and Supplementary Tables S1 1 and S1 2). \nFor Rh fcc/Co we find out -of-plane FM domains and clockwise rotating domain walls with a width of \n1.4 nm, see Fig.  4a. The domain walls are exceptionally thin because of the large magnetocrystalline \nanisotropy in combination with the flat spin spiral dispersion  for ||<0.2 ଶగ\n. The latter is \nresponsible for a n almost vanishing  energy cost for spin cantings between nearest neighbours of up \nto a lmost  20° (cf. Fig.  3c). The DW energy amounts to only 2.0  meV/nm, one order of magnitude \nsmaller than for Co/Ir(111) or Pt/Co/Ir(111)31. For Rh hcp/Co the DW energy  is negative  due to the spin \nspiral energy minimum of ܧ=−0.7 meV/Co -atom  with respect to the FM state . However, the DMI i s \ntypically overestimated in our  approach33 and intermixing  at the Rh/Co interface , as suggested by the \nSTM measurements , will lead  to an additional  reduction of the DMI  (cf. Ref. [34] and Supplementary \nFig. S7). Therefore, we reduce  the DMI in the simulations for Rh hcp/Co below the critical value  to \nobtain  a FM ground state as in  the exper iment  (see Supplementary  Table S1 2 for values) . The \nresulting DW width is 1.3 nm, see Fig.  4a, similar to the experimental value. With this reduced DMI \nthe DW energy is positive, and its  small  value  of only 4.4 meV/nm  agrees well with  the experimental \nobservation of meandering domain walls with a path dominated by the inhomogeneous potential \nlandscape due to intermixing.  \nIn agreement with the experimental findings our spin dynamics simulations show  small zero \nmagnetic field skyrmions w ithin the FM ground st ate of the Rh/Co film s (Fig.  4a). The small skyrmion \ndiameter of about  4 nm is the result of the combination of  flat energy dispersion of spin spirals close \nto the FM state (Fig.  3c) which reduces the energy cost of a fast spin rotation, and large 4 \n magnetocrystalline anisotropy , which enforces the fast spin rotation.  Note that all previously found \nnanometre -sized  isolated skyrmions in ultrathin films9,25,35 were induced by a magnet ic field out of a \nspin spiral ground state and do not exist in zero magnetic field.  \nTo gain further insight into the mechanism that stabilise s these small skyrmions at zero field , we \ncalculate the energy barrier using minimum energy path calculations (see methods). Typically, \nskyrmion annihilation into the FM state occurs via a collapse in which the skyrmion shrinks in size  \nand the energy barrier stems from  DMI (cf. Supplementary Fig . S8). For this  radial symmet ric collapse  \nmechanism t he nearest -neighbour exchange interaction lowers the barrier . However, frustrated \nexchange interactions can provide a contribution t hat increases  the energy barrier36. For Rhhcp/Co \nfilms the frustration is so strong that the total exchange contribution to the energy barrier even \nexceeds that of the DMI ( Supplementary Fig . S9). To lower the energy barrier , a different annihilation \nmechanism is preferred  at zero magnetic field  and Fig. 4b shows how the energy evolves along the \nminimum energy path starting from the skyrmion and ending in the FM state. At the saddle point  \n(SP) we obtain  a large energy  barrier of about 300  meV . The contribution due to exchange  is more \nthan three times larg er than that of the DMI . Decomposing the exchange contribution by the \nneighbouring shells ( see Fig. 4b right ) shows that the barrier unexpectedly results from the large FM \nnearest -neighbour exchange interaction  (ܬଵ).  \nOne can understand the individual  contributions to the energ y barrier by looking at the annihilation \nmechanism  (Fig. 4c): just before the SP (SP-1) the spin structure is a slightly oval shaped skyrmion  \nwith a similar size as the initial skyrmion . At the SP a singular point  is formed in the in -plane \nmagnetized region of the skyrmion . This is energetically very unfavourable in terms of nearest -\nneighbour FM exchange interaction  ܬଵ. In contrast, the energy cost due to DMI is much lower since \nonly a small part of the spins are not rotatin g with  the preferred sense. The formation of the  singular \npoint  transforms the skyrmion into a so -called chimera skyrmion37 (SP+1) with a vanishing topological \ncharge which easily collapses into the FM state . This novel annihilation mechanism is preferred over \nthe radial symmetric skyrmion collaps e at zero field  because it greatly  reduce s the  DMI energy \nbarrier  and t he stability of zero -field skyrmions in this system is a result of the frustrated  exchange  \ninteraction . \nOur work demonstrates that isolated skyrmions with a diameter of below 5 nm can be stabilise d \nwithout applied magnetic field in ultrathin ferromagnetic Co films  due to strong exchange frustration \ntogether with moderate  DMI and large magnetocrystalline anisotr opy.  We have shown that the \ndetection  of individual sub -10 nm skyrmion s via the NCMR effect  is possible also in Co films , meaning \nthat these  skyrmions  can be  directly detected in an  all-electrical read -out. We anticipate that  \nmultilayer s can be tailored to  transfe r these advantageous properties to structures suitable for \napplications  at room temperature .  \nMethods:  \nSample preparation. The Ir(111) single crystal surface was cleaned by cycles of annealing in oxygen \nwith partial pressures in the range of 10-7 mbar up to about 1800  K to remove C impurities. For each \nsample preparation the su rface was sputtered with Ar ions of about 800  eV with subsequent \nannealing  to around 1500  K for 6 0 seconds . The Co was deposited onto the substrate at elevated \ntemperatures to achieve step flow growth. The Rh was deposited after the sample had reached room \ntemperature. T ypical  deposition rates for Co and Rh are between 0.1 and 0.2 atomic layers per \nminute . Samples were tran sferred in vacuo  to a low temperature scanning tunne lling microscope 5 \n equipped with a Cr bulk tip for spin -resolved measurements. The Cr bulk tip was introduced into \nultra -high vacuum after etching and in-situ cleaning  was perfor med by field emission.  \nDensity functional t heory . We apply  DFT based on the full potential linearized augmented plane \nwave (FLAPW) method as implemented in the FLEUR  code  (www.flapw.de ). This all-electron method \nranks among the most accurate implementations of DFT. Computational details  for Co/Ir(111) can be \nfound in Ref. [ 32]. For Rh/Co/Ir(111) w e performed structural relaxations within the ferromagnetic \nstate using a symmetric film consisting of a Rh/Co bilayer on both sides of five layers of Ir(111) with \nthe theoretical  lattice constant (ܽ=3.82\tÅ)39. Relaxed interlayer distances are given in \nSupplementary Table S13. The muffin tin ( MT) radii were  ܴMT=2.31\ta.u. for Ir and Rh and \nܴMT=2.23\ta.u. for Co . The cutoff for the basis functions was ݇max=4.0\ta.u.-1. We used 240 k-\npoints in the irreducible wedge of the two-dimensional Brillouin  zone ( 2D-BZ) and the generalized \ngradient approximation40.  \nFor spin spiral  calculations41,42 with and without spin -orbit coupling and to obtain the \nmagnetocrystalline anisotropy energy , an asymmetric slab with a Rh/Co bilayer on nine layers of the \nIr(111) substrate was used. The energy dispersion  of flat spin spirals (see Supplementary Text S1 4 for \ndetails ) are calculated with a dense k-point mesh of 44×44 k-points  in the full  2D-BZ and  a basis \ncutoff of  ݇max=4.0\ta.u.-1 was used. Close to the ߁ത-point ( ||→0), we checked the convergence of \nthe energy dispersions with up to 100 ×100 k-points and  ݇max=4.3\ta.u.-1. These calculations were \nperformed in local density approximation43. \nSpin -dynamics simulations . In order to relax the spin structures of  the domain walls and the isolated \nskyrmions and to  calculate the ir energy differences  with respect to the FM state , we used the \nLandau -Lifshitz equation : \n ℏd\ndݐ=߲ℋ\n߲×−ߙ൬߲ℋ\n߲×൰× (1) \nwhere =ࡹ\nெ is the unit vector of the magnetic moment at atom site  ݅ ,ߙ is the damping parameter \nand ℋ is the Hamiltonian:  \n ℋ=−ܬ൫⋅൯−ࡰ൫×൯\n +ܭ(݉௭)ଶ\n (2) \nHere ܬ denotes the strength of the exchange interaction between spins on atom sites ݅ and ݆ and \nࡰ݆݅ is the vector characterizing their DM I. ܭ represents the strength of the uniaxial anisotropy.  \nFor the simulations presented in Fig. 4, we have used various  damping parameter s of ߙ∈\n[0.05,1]\tand a time step of 0.1\tfs. The values of the exchange constants, the DMI, and the \nmagnetocrystalline anisotropy energy are given in the Supplementary Tables S1 1 and S1 2. We used  a \nhexagonal lattice of 70×70 spins and a magnetic moment of 2.5\tߤB which corresponds to a \ncombined value of Rh (0.6 \tߤB), Co ( 1.8\tߤB) and the Ir interface layer ( 0.1\tߤB) in the FM state.  \nGeodesic nudged elastic band method . Using the relaxed structures of skyrmions from spin -\ndynamics simulations, we calculate the minim um energy paths (MEPs) for annihilation processes \nusing the geodesic nudged elastic band (GNEB) method44. Starting from a local energy minimum  \n(skyrmion ), a path is generated into the global energy minimum (FM state ). The path is systematically \nbrought  into the MEP, while it is divided into a discrete chain of states, the so -called images. The first \nimage corresponds  to the skyrmion  and the last image to  the FM state. After the relaxation of the 6 \n starting point, the effective field is calculated along a local tangent to the path at each image. 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Accurate spin -dependent electron liquid correlation energies \nfor local spin density calculations: a critical analysis.  Can. J. Phys. 58, 1200 (1980) . \n[43] Bessarab, P. F., Uzdin, V. M., & Jónsson, H. Method for finding mechanism and activation energy \nof magnetic transitions, applied to skyrmion and antivortex annihilation . Comp. Phys. Comm. 196, \n335-347 (2015).  \nAcknowledgements:  \nThis project has received funding from the European Union’s Horizon 2020 research and innovation \nprogramme under grant agreement No 665095 (FET -Open project MAGicSky).  K.v.B. and A.K \nacknowledge financial support from the Deutsche Forschu ngsgemeinschaft (DFG, German Research \nFoundation)  - 402843438;  - 408119516 . S.M., S.v.M., and S.H. thank the Norddeutscher Verbund für \nHoch - und Höchleistungsrechnen (HLRN) for providing computational resources. We thank Pavel F. \nBessarab  for valuable discussions.  \nAuthor contribution statement  \nS.H. devised the project. M.P. performed the measurements . M.P., K.v.B. , and A.K. analysed the \nexperimental data . S.M. performed the DFT calculations . S.M. and S.v.M. performed the spin \ndynamics and GNEB calculations. S.M ., S.v.M.,  and S.H. analysed the calculations. M.P. and S.M. \nprepared the figures. M.P., K.v.B., S.M.,  and S.H. wrote the manuscript. All authors discussed the \nresults and contributed to the manuscript.  \nCompeting financial interests  \nThe authors declare no competing financial interests.  \n 9 \n  \nFigure 1 |  Zero field skyrmions in Rh/Co/Ir(111).  a,b, Spin -resolved  differential conductance \nmap s of 0.4 atomic layers of hcp-stacked Rh on 0.5 atomic layers of Co on Ir(111)  measured \nat different bias voltages . Whereas in a the TMR contribution dominates and the FM \ndomains can be identified by their two -stage contrast, in b the NCMR contribution is strong \nand the domain walls appear as bright lines . The magnetic tip is identi cal for these  \nmeasurements and  it is sensitive to both the out -of-plane and an in-plane component of the \nsample’s magnetisation . Isolated skyrmions with opposite magnetisation  are indicated by \nthe boxes  (a: U = -250 mV ; b: U = -400 mV ; a,b: I = 800 pA, B = 0 T, T = 4.2 K, Cr bulk tip , fast \nscan axis is vertical ). c,d, dI/dU signal (dots) across the isolated skyrmions in a,b , together \nwith line profiles (solid lines) of STM simulations of two skyrmions (see Suppl ementary Text \nS5). e, Derived o ut-of-plane magnetisation  component mz across the two skyrmions.  \n \n \n \n \n10 \n  \nFigure 2 | Domain wall width and skyrmion diameter . a,b , dI/dU maps of an isolated \nmagnetic object in Rh hcp/Co/Ir(111)  at zero field and a t B = +1.5 T  measured with a non -spin-\npolarized tip ; because of the NCMR contribution the domain walls have a higher signal than \nthe FM domains ( U = -250 mV, I = 800 pA, T = 4 K, Cr bulk tip).  c, sketches of a domain wall \nand a skyrmion ( w = 0.8 nm ), the colour of the atomic magnetic moments indicates the out -\nof-plane magnetization component. d,e, dI/dU signal  (dots)  across the two domain walls and \nthe skyrmion  of a,b, positions are indicated by  the white rectangles ; the solid lines are \nprofiles of STM simulations of the domain wall and skyrmion shown in c. f: Corresponding \nout-of-plane magnetisation  component mz across the domain wall and skyrmion of c-e. \n \n \n \n \n \n \n \n11 \n  \nFigure 3 | Energy disp ersion of spin spiral s for ultrathin Co films.  a, Energy dispersion of  \nplanar  homogeneous spin spiral s for Co/Ir(111), Rh fcc/Co/Ir(111) and Rh hcp/Co/Ir(111)  along \nthe high symmetry directions ܯഥ−߁ത−ܭഥ of the two -dimensional Brillouin zone without \nspin-orbit coupling (SOC). Energies are given  with respect to the FM state . The symbols  \n(open grey circles for Co/Ir(111), red filled circles  for Rh fcc/Co/Ir(111) and green triangles for \nRhhcp/Co/Ir(111)) represent the DFT calculations while the solid lines are the fit s to the \nHeisenberg exchange int eraction beyond nearest neighbo urs (see Supplementary Text S1 4 \nfor details) . b, Zoom around the FM area ( ߁ത-point) of a, where dashed  lines represent a ݍଶ fit \nin the case of Co/Ir(111) and a  fit including a n additional  ݍସ term  in the case of \nRh/Co/Ir (111) . c, Zoom around the FM area ( ߁ത-point) of the energy dispersion for cycloidal \nspin spirals including the effect of spin -orbit coupling . The DMI leads to the local energy \nminima for clockwise rotating spin spirals along both high symmetry directions a nd the \nmagnetocrystalline anisotropy energy is responsible for the constant energy shift of the spin \nspirals with respect to the FM state.  \n \n12 \n  \nFigure 4 | Profile and stability of zero -field  skyrmions in Rh/Co/Ir(111).  a, Domain wall \nprofiles ( open symbols ) and zero field skyrmion profiles ( filled symbols ), i.e. z-component ݉௭ \nof the local magnetic moment,  obtained based on atomistic spin dynamics for \nRhhcp/Co/Ir(111)  (green curves) and Rh fcc/Co/Ir(111) (red curves) . The dashed and solid lines \nare fits to the standard domain wall and skyrmion profile8, respectively. Note that there are \nsmall deviations in both cases due to the exchange frustration. The insets show the domain \nwall and skyrmion spin structures on the two-dimensional atomic l attic e. b, Total energy and \nenergy contributions from the different interactions (exchange, DMI, magneto -crystalline \nanisotropy  energy  MAE ) at zero field versus the reaction coordinate along the minimum \nenergy path from the initial isolated skyrmion (S k) state to the final ferromagnetic (FM) state  \nfor Rhhcp/Co/Ir(111)  (the same annihilation mechanism occurs for fcc Rh stacking) . Energies \nare summed over all atoms of the simulation box and ar e given relative to the energy  of the \nisolated skyrmion state.  The saddle point (SP) is indicated . To the right:  exchange e nergy at \nthe saddle point , ܧୗ, resolved with  respect to the exchange interactions of different shell s: \nܬଵ..ଵ. c, Images along the minimum energy path in b right before the SP ( SP-1), at the SP , and \njust after the SP (SP-1). Each data point in b corresponds to one image.  For all images of the \nminimum energy path see Supplementary Fig . S10. Note that in c only a small part of the full \nsimulation box which contained (70×70) spins  is shown . \n \n \n"    },    {        "title": "1903.11442v1.Structural_and_magnetic_properties_of_GdCo___5_x__Ni__x_.pdf",        "content": "Structural and magnetic properties of GdCo 5\u0000xNix\nAmy L. Tedstone, Christopher E. Patrick, Santosh Kumar, Rachel S.\nEdwards, Martin R. Lees, Geetha Balakrishnan, and Julie B. Staunton\u0003\nDepartment of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom\n(Dated: March 28, 2019)\nGdCo 5may be considered as two sublattices|one of Gd and one of Co|whose magnetizations\nare in antiparallel alignment, forming a ferrimagnet. Substitution of nickel in the cobalt sublattice\nof GdCo 5has been investigated to gain insight into how the magnetic properties of this proto-\ntype rare-earth/transition-metal magnet are a\u000bected by changes in the transition metal sublattice.\nPolycrystalline samples of GdCo 5\u0000xNixfor 0\u0014x\u00145 were synthesized by arc melting. Structural\ncharacterization was carried out by powder x-ray di\u000braction and optical and scanning electron mi-\ncroscope imaging of metallographic slides, the latter revealing a low concentration of Gd 2(Co, Ni) 7\nlamellae for x\u00142:5. Compensation|i.e. the cancellation of the opposing Gd and transition metal\nmoments| is observed for 1 < x < 3 at a temperature which increases with Ni content; for larger\nx, no compensation is observed below 360 K. A peak in the coercivity is seen at x\u00191 at 10 K\ncoinciding with a minimum in the saturation magnetization. Density-functional theory calculations\nwithin the disordered local moment picture reproduce the dependence of the magnetization on Ni\ncontent and temperature. The calculations also show a peak in the magnetocrystalline anisotropy\nat similar Ni concentrations to the experimentally-observed coercivity maximum.\nI. INTRODUCTION\nThe RETM 5[(RE = rare earth; TM = transition\nmetal)] family of materials have widely ranging mag-\nnetic properties owing to the di\u000bering number of 4 felec-\ntrons found in the RE elements. These materials crys-\ntallize into a hexagonal lattice (the CaCu 5type struc-\nture, space group P6=mmm , Fig. 1) with a unit cell con-\nsisting of layers with a central RE atom surrounded by\nTM atoms in 2 cpositions, alternating with layers of TM\natoms in the 3 gpositions [1]. One pertinent example\nis SmCo 5, a permanent magnet which can be favored\nover Nd-Fe-B magnets for its superior high-temperature\nperformance (Curie temperature of around 1020 K [2],\nas opposed to approximately 580 K for Nd-Fe-B mag-\nnets [3, 4]). Another member of this family is GdCo 5,\nwhere the symmetry of the Gd 4 fshell causes crystal-\n\feld e\u000bects to vanish [5]. The absence of crystal-\feld ef-\nfects make GdCo 5a particularly useful system to study\nthe rare-earth/transition-metal interaction via both the-\nory and experiment.\nGdCo 5is ferrimagnetic. Starting from T= 0 K\nits magnetization increases with increasing temperature\nreaching a maximum at around 800 K [6, 7]. With a\nfurther increase in temperature, the spontaneous mag-\nnetization decreases to zero at the Curie temperature\n(1014 K) [8]. This unusual temperature dependence is a\nconsequence of the Gd moments disordering more rapidly\nwith temperature than the Co moments [9, 10].\nDoping a RETM 5material can change its magnetic\nproperties in a controlled manner [10{19]. Here, only\ndoping of the TM sublattice is considered. The e\u000bects\nof doping on coercive \feld and saturation magnetiza-\ntion have been studied for single crystals of GdCo 5\u0000xCux\n\u0003J.B.Staunton@warwick.ac.ukby Grechishkin et al. [12] and later by de Oliveira et\nal. [13]. The Cu reduces the TM sublattice magnetiza-\ntion. Both papers report a peak in coercivity at a compo-\nsition ofx\u00191:5. This is found to be the compensation\ncomposition of this intermetallic at room temperature,\nwhere the (Co,Cu) sublattice magnetization exactly can-\ncels (fully compensates) the Gd sublattice magnetization.\nA peak in coercivity was also found in YCo 5\u0000xNix[14]\nand for RECo 5\u0000xNix(RE = Sm, La, Y, Th and Ce) [15]\nfor certain compositions. It is interesting that RECo 5\ncompounds containing nonmagnetic REs such as Y and\nLa still exhibit a peak in coercivity, despite not hav-\ning a compensation composition. Buschow and Brouha\n(Ref. 14) suggested that the presence of narrow Bloch\nwalls in YCo 5\u0000xNixis primarily responsible for the high\ncoercivity observed for certain compositions.\nIn a previous work [10] we prepared polycrystalline\nsamples of GdCo 5\u0000xTMxand YCo 5\u0000xTMx(TM = Ni\nand Fe) with x\u00141 and single crystal GdCo 5and YCo 5,\nTM\n(2c)RE\nTM\n(3g)\nFIG. 1. Ball-and-stick model of the CaCu 5structure adopted\nby RETM 5compounds, showing the rare earth site (purple)\nand inequivalent 2 cand 3 gtransition metal sites (gray).arXiv:1903.11442v1  [cond-mat.mtrl-sci]  27 Mar 20192\nand compared the experimentally determined magnetic\nproperties of these samples with theoretical calculations\nmade using density-functional theory. An increase (de-\ncrease) in magnetization was observed for Fe (Ni) doping.\nThe calculations showed that substituting Ni onto the Co\nlattice led to Ni preferentially occupying the 2 csite, al-\nthough experimentally this may depend on the method\nof sample preparation. The doping site was not found to\nhave a large e\u000bect on magnetization, but did a\u000bect the\nCurie temperature, with a larger change for the 2 csite\ndoping. However, possible e\u000bects on the coercivity and\nmagnetocrystalline anisotropy were not explored in the\npaper, nor were concentrations with x>1.\nThe magnitude of the Ni moment in GdCo 5\u0000xNixis\nvery small [11] compared to that of Co and Gd (in\nGdCo 5, the Co moment is \u00191:6\u0016B/atom at both the\n2cand the 3gsites and the Gd moment is \u00197\u0016B[10]),\nhence the (Co, Ni) sublattice magnetization is expected\nto decrease with increasing Ni content. The fully sub-\nstituted material GdNi 5is ferrimagnetic, but the main\ncontribution to the magnetization comes from the ferro-\nmagnetic Gd sublattice, giving a Curie temperature of\n32 K [11, 20]. At absolute zero at a particular composi-\ntion for GdCo 5\u0000xNixthe (Co, Ni) sublattice magnetiza-\ntion will fully compensate the Gd sublattice magnetiza-\ntion. The compensation composition at absolute zero will\nful\fll the condition \u0016Gd\u0000\u0016Co(5\u0000x)\u0000\u0016Nix= 0. Taking\napproximate zero temperature values for the moments of\nCo, Ni, and Gd (1.6, 0.6, 7 \u0016B/f.u. [formula unit]), re-\nspectively, the compensation composition is x\u00181. At\nother compositions there may exist a \fnite compensa-\ntion temperature where the di\u000berent disordering of the\nGd and (Co, Ni) sublattice magnetizations again leads to\ncompensation.\nChuang et al. [16] replaced Co with Ni in GdCo 5\u0000xNix\nfor 0\u0014x\u00145 measuring magnetization versus tempera-\nture at 12 kOe for several compositions, focusing primar-\nily on temperatures from 300 to 1015 K, with the excep-\ntion of GdCo 2Ni3and GdCoNi 4for which measurements\nwere taken over the ranges of 77{1015 K and 77{300 K,\nrespectively. Therefore, for compositions with x<3, any\ncompensation point at temperatures lower than 300 K\nwould not have been observed. GdCo 5\u0000xNixhas been\ninvestigated by Liu et al . [17] forx\u00141:05 in order\nto determine the intersublattice RE/TM coupling con-\nstant. Magnetization compensation has also been stud-\nied in Gd(Co 4\u0000xNix)Al [18], where increased Ni content\nwas observed to increase the compensation temperature\nand in RE(Co 4\u0000xFexB) (RE = Gd and Dy) [19], where\nthe compensation temperature was reduced for increased\nFe content.\nIn this paper the magnetic behavior of polycrystalline\npowders and buttons of GdCo 5\u0000xNixis reported for tem-\nperatures from 5 to 360 K for x= 0, 0.5, 1, 1.25, 1.28,\n1.3, 1.5, 2, 2.5, 3, 3.5, 4, and 5. The compensation tem-\nperature, coercivity and magnetization at 70 kOe are\npresented as a function of x. This extends the previ-\nous work of Chuang et al. [16] to a temperature range inwhich the compensation point can be observed for x<3.\nThe behavior is then analyzed with the help of density\nfunctional theory calculations within the disordered lo-\ncal moment picture [21, 22], calculating the composition-\ndependent magnetization, coercivity, and magnetocrys-\ntalline anisotropy. The calculations provide microscopic\ninsight into the macroscopic quantities observed experi-\nmentally, demonstrating the utility of the joint computa-\ntional/experimental approach in understanding the be-\nhavior of RE/TM permanent magnets.\nThe rest of this paper is organized as follows. Sec-\ntion II describes the experimental and theoretical tech-\nniques used. Sections III and IV describes the results\nof the experiments and calculations, respectively. The\nconclusions and summary are presented in Section V.\nII. METHODOLOGY\nA. Experimental approach\nPolycrystalline samples of the series GdCo 5\u0000xNixwere\nsynthesized by arc melting the constituent elements on a\nwater cooled copper hearth under an argon atmosphere.\nThe starting elements (99% purity) were taken in the sto-\nichiometric ratios with 1% excess of Gd to compensate for\nlosses during melting. To ensure homogeneity, the ingots\nwere \ripped and remelted at least three times. Annealing\nfor 10 days at 950\u000eC was tried to improve phase homo-\ngeneity, as discussed by Buschow and den Broeder [23].\nHowever, analysis of the annealed samples (via the same\nmethods described below) showed no convincing evidence\nthat annealing promotes the 1:5 [Gd:(Co, Ni)] phase for-\nmation over the neighboring phases of 2:17 and 2:7, and\nso as-cast samples were used (provided they were found\nto be su\u000eciently phase pure, as described in the remain-\nder of this section).\nThe structures were characterized by powder x-ray\ndi\u000braction using a Panalytical Empryean di\u000bractometer\nwith CoK\u000bradiation. To con\frm the phase content,\nmetallographic slides were prepared from slices of ingots\nmounted in Epomet-F plastic and polished using pro-\ngressively \fner diamond suspensions. Optical microscopy\nand scanning electron microscopy (SEM) imaging of the\nslides were used to further examine the structure.\nMagnetization measurements were made as a function\nof temperature and applied \feld using a Quantum De-\nsign Magnetic Property Measurement System (MPMS)\nsuperconducting quantum interference device (SQUID)\nmagnetometer. Free to rotate powders were used to ob-\ntain the best estimate of saturation magnetization. The\nmagnetization vs temperature, M(T), data were taken in\na 10 kOe \feld with the temperature decreasing from 360\nto 10 K at a rate of 3 K/min. The magnetization versus\n\feld [(M(H))] data were taken at 10 K (5 K for x= 0,\n0.5, and 1) by \frst applying a 70 kOe \feld then collecting\nthe magnetization data at incrementally decreasing \felds\n(until 0 kOe). Polycrystalline buttons were \fxed to a3\nsample holder using GE varnish to measure coercivity in\nan Oxford Instruments vibrating sample magnetometer.\nThe buttons were \fxed in this manner to prevent sam-\nple rotation, and thus get a more accurate measurement\nof coercivity. The coercivities were determined from four\nquadrant hysteresis loops starting at 70 kOe, as the coer-\ncivity is known to depend on the initial applied \feld [24].\nB. Theoretical approach\nThe magnetic properties of Ni-doped GdCo 5were cal-\nculated at zero and \fnite temperature using density-\nfunctional theory within the disordered local moment\n(DFT-DLM) picture [21]. In this approach, both the\ntemperature-induced local moment disorder and the\ncompositional disorder from the Ni doping are modeled\nusing the coherent potential approximation (CPA) within\nthe Korringa-Kohn-Rostocker (KKR) multiple-scattering\nformulation of DFT [25]. A detailed description of this\napproach applied to Ni-doped GdCo 5is given in Ref. 10;\nhere the computational details speci\fc to this work are\ngiven.\nThe calculations were performed on the CaCu 5struc-\nture with lattice parameters \fxed at a= 4:979 \u0017A,\nc= 3:972\u0017A, which were measured for pristine GdCo 5\nat 300 K [26]. As discussed in Section IV the Ni dopants\nwere set to either occupy the 2 cand 3gcrystal sites with\nequal probability, or to preferentially sit at the 2 ccrys-\ntal sites. The KKR multiple-scattering equations were\nsolved within the atomic sphere approximation (ASA)\nwith Wigner-Seitz radii of (1.58, 1.39, 1.42) \u0017A at the\n(Gd, 2c, 3g) sites.\nThe KKR-CPA code Hutsepot [27] was used to gen-\nerate scalar-relativistic potentials for the magnetically-\nordered (ferrimagnetic) state, expanding the key quan-\ntities in an angular momentum basis up to a maximum\nquantum number l= 3. Exchange and correlation were\ntreated within the local spin-density approximation [28],\nwith the local self-interaction correction [29] also applied\nto the Gd-4 felectrons.\nThe scalar-relativistic potentials were then fed into\nour own code which solves the fully-relativistic scatter-\ning problem in the presence of magnetic disorder [22].\nFor selected Ni concentrations an orbital polarization\ncorrection (OPC) [30, 31] was included on the dscat-\ntering channels [32, 33]. The DFT-DLM Weiss \felds,\nwhich govern the temperature dependence of the calcu-\nlated quantities, were calculated self-consistently using\nan iterative procedure [10, 34]. The magnetocrystalline\nanisotropy constants were calculated using the torque for-\nmalism described in Ref. 22, using an adaptive reciprocal-\nspace sampling scheme to ensure numerical accuracy [35].\nTo calculate magnetization versus \feld curves the \frst-\nprinciples approach to calculate temperature-dependent\nmagnetization vs \feld (FPMVB) curves introduced in\nRef. 36 was used. A set of 28 DFT-DLM calculations\nare used to \ft the parameters contained in F2, which\n203 04 05 06 07 08 09 00100002000030000400005000060000 \nData \nFit \nDifferenceIntensity (counts)S\ncattering angle (2ϑ)FIG. 2. The data and re\fnement (black data and red line\nrespectively) for the powder x-ray di\u000braction carried out on\nGdCo 3Ni2. The blue line shows the di\u000berence plot between\nthe original data and the re\fnement. The green ticks indicate\nindexed peaks. The goodness of \ft parameter is 2.03.\nquantify exchange and magnetic anisotropy. The magne-\ntization for a given \feld and sample orientation is then\ndetermined by minimizing the free energy with respect\nto the angles between the Gd and the transition metal\nmagnetizations and the crystal axes. The approach does\nnot account for any canting between moments within the\ntransition metal sublattices, since this was previously cal-\nculated to be less than 0.1\u000efor GdCo 5[36].\nIII. RESULTS\nA. Structural characterization\nFigure 2 shows the powder x-ray di\u000braction pattern\nobtained from GdCo 3Ni2(x= 2, black data). The red\nline is the \ft obtained when a Rietveld re\fnement is car-\nried out using the TOPAS software [37]. The goodness\nof \ft parameter is 2.03 (a similar value was found from\n\ftting the di\u000braction patterns of all samples). The blue\nline is the di\u000berence plot between the observed data and\nthe \ft. The green ticks represent indexed peaks. All\nthe peaks in each di\u000braction pattern have been indexed\nusing the GdCo 5structure|space group P6=mmm |as\ndemonstrated here, suggesting the samples form as sin-\ngle phase materials to within the detection limits of the\ntechnique. The lattice parameters for all xare given in\nTable I, along with the lattice parameters found in the\nliterature. As expected from previous research, there is\na contraction in the abplane with increasing Ni content,\nand a less pronounced general contraction along the c\naxis.\nThe x-ray di\u000braction measurements show that the sam-4\nTABLE I. Lattice parameters of GdCo 5\u0000xNixobtained from\nRietveld re\fnement of the powder x-ray di\u000braction patterns.\nThe results reported previously in literature are included for\ncomparison.\nReported Reported\nx a (\u0017A) c(\u0017A) a(\u0017A) c(\u0017A)\n0 4.946(9) 3.999(7) 4.979a3.972a\n4.960b3.989b\n4.974c3.973c\n0:5 4.9680(4) 3.9790(3) - -\n1 4.9681(4) 3.9794(3) 4.959a3.977a\n1:5 4.957(1) 3.9790(5) - -\n2 4.9493(1) 3.9803(1) 4.948a3.980a\n2.5 4.9439(1) 3.9785(1) 4.94a3.979a\n3 4.9338(2) 3.9738(2) 4.932a3.967a\n3.5 4.9306(1) 3.9708(1) - -\n4 4.9245(1) 3.9704(1) 4.92a3.969a\n5 4.9139(1) 3.9683(1) 4.909a3.965a\n4.91d3.967d\n4.90e3.97e\naRef. 16\nbRef. 38\ncRef. 39\ndRef. 40\neRef. 41\nples are single phase, but this technique may miss small\npercentages of impurity phases. For this reason, opti-\ncal microscopy and SEM images were taken of metallo-\ngraphic slides; some example SEM images are shown in\nFig. 3. A decreasing quantity of lamellae of a secondary\nphase were observed with increasing x, untilx= 2.5, be-\nyond which no lamellae were observed. Energy-dispersive\nx-ray spectroscopy showed that the majority phase is the\n1:5 [Gd:(Co, Ni)] phase and the small lamellae are a 2:7\nphase. Table II shows the percentage of the 2:7 phase\nfound in the as-cast samples. Buschow and den Broeder\nnoted that a slight excess of Co during the arc-melting\npromotes the formation of the 2:7 phase within the 1:5\nmatrix [23]. GdNi 5forms congruently from the melt,\nand is stable down to, at least, room temperatures. On\nthe other hand, GdCo 5undergoes eutectic decomposi-\ntion at 775\u000eC into Gd 2Co7(2:7 Gd:Co) and Gd 2Co17\n(2:17 Gd:Co) [8] and so increasing Ni content improves\nthe stability of the 1:5 phase.\nThe relative unimportance of the secondary 2:7 phase\nin these quantities (2{7%) to the measurement of intrin-\nsic quantities such as the magnetization can be demon-\nstrated for the case of no Ni doping. The moment per\nformula unit of GdCo 5is 1.37\u0016B/f.u. [8] and of Gd 2Co7\nis 2.5\u0016B/f.u. [8]. Assuming no other impurities or other\nphases, taking 93% of the total powder to be GdCo 5and\n7% to be Gd 2Co7(i.e. the maximum amount of the\nimpurity phase estimated) the total moment becomesTABLE II. Percentage of the 2:7 phase in GdCo 5\u0000xNixfor\nx= 1, 1.25, 1.5, and 2 determined from optical and SEM\nimages.\nx 2:7 phase (%)\n1 5.2\n1.25 6.7\n1.5 5.7\n2 2.1\n1.45\u0016B/f.u. This is an \u00186% increase to pure GdCo 5;\nwhich is insigni\fcant compared to the almost 300% in-\ncrease in moment from GdCo 5to GdNi 5. On the other\nhand, it is possible that the presence of a secondary phase\na\u000bects extrinsic properties such as the coercivity, as in\nSm-Co or Nd-Fe-B magnets [42]. As no 2:7 phase was\nobserved in as-cast samples of x\u00153, these can be used\nas a comparison when studying trends in the measure-\nments.\nB. Compensation temperature\nMversusTcurves are shown in Fig. 4 for x= 1:5, 2,\nand 3 measured in an applied magnetic \feld of 10 kOe.\nA clear minimum occurs at progressively lower tempera-\ntures as Ni content is decreased. At this minimum the net\ntotal magnetization aligns with the bias \feld of 10 kOe.\nIn contrast, when the sample is warmed in the (small,\nnegative) trapped \feld of the magnetometer magnet, af-\nter cooling in 10 kOe, the magnetization changes sign at\nthe compensation temperature, as shown in Fig. 5 for\nx= 2.\nAs shown in Fig. 4, the magnetization of the powders\nsubject to the 10 kOe applied \feld does not go to zero,\neven at the compensation temperature. An explanation\nfor this behavior in terms of a change in the internal mag-\nnetic structure of the powder particles (from antiparallel\nto canted RE/TM moments) is given in Sec. IV D.\nFig. 6 shows the dependence of the compensation tem-\nperature with composition. A small hysteresis in com-\npensation temperature with increasing/decreasing tem-\nperature was noted in a 10 kOe \feld for x\u00142. No hys-\nteresis was found for these compositions in the trapped\n\feld of the magnet. The compensation temperature in-\ncreases for increasing nickel content, as expected. Com-\npensation is not observed for x\u00153:5 andx\u00141 in this\ntemperature range. In the case of low Ni doping, the\nCo sublattice magnetization, whilst reduced by the ad-\ndition of Ni, remains dominant over the Gd sublattice\nfor all measured temperatures. At high Ni doping, the\nCo sublattice magnetization is reduced so much that the\nGd sublattice magnetization dominates for all measured\ntemperatures.\nChuang et al. (Ref. 16) reported a compensation tem-\nperature for x= 3 of 380 K in a 12 kOe \feld, how-5\nFIG. 3. SEM images (secondary electron mode) of (a) x= 0,\n(b)x= 1:5, and (c) x= 2. The secondary 2:7 phase\ncan clearly be seen as lamellae which are lighter gray than\nthe surrounding 1:5 matrix. Fig. 3(c) is relatively clear of\nlamellae.\never here it is 323 K in a 10 kOe \feld. Earlier results\nused a large temperature step size, and hence some dis-\nagreement is to be expected. Measurements here were\nobtained on as-cast powder samples, whereas Ref. 16 re-\nports on annealed samples.\n05 01 001 502 002 503 003 500.00.51.01.52.02.5 x = 1.5 \nx = 2 \nx = 3Magnetization (µB/f.u.)T\nemperature (K)FIG. 4. Magnetization of samples x= 1:5 (green triangles);\nx= 2 (black squares) and x= 3 (red circles) versus\ntemperature in a 10 kOe \feld. A clear minimum can be seen\nat 157, 230, and 323 K respectively.\n05 01 001 502 002 503 00-0.40.00.40.81.2Magnetization (µB/f.u.)T\nemperature (K)\nFIG. 5. Magnetization versus temperature curve for x= 2\nin zero applied \feld with the temperature increasing from\n10 to 360 K after \feld cooling in a \feld of 10 kOe. The\nmagnetization changes sign at 235 K.\nC. Magnetization and coercivity\nFour-quadrant MversusHloops for buttons of com-\npositionx= 0:5, 1.3, 2.5, and 5 are shown in Fig. 7, with\nthe inset showing the low-\feld region. It is clear that the\nsamples do not reach full saturation at 70 kOe, which\nis the experimental limit. This is to be expected, since\nthe samples consist of a number of grains with randomly\norientedcaxes, such that both easy and hard axes are\nbeing probed.\nThe magnetization at 70 kOe and the coercive \feld ob-6\n1.21.41.61.82.02.22.42.62.83.03.2100150200250300350 \n10 kOe applied field \nTrapped field of magnet Compensation temperature (K)N\nickel content (x)\nFIG. 6. Temperature of minimum magnetization (compensa-\ntion temperature) in a 10 kOe \feld (black squares), and zero\n\feld (red circles) as a function of Ni concentration.\n-60- 40- 200 2 04 06 0-6-4-20246 x = 0.5  \nx = 1.3 \nx = 2.5  \nx = 5Magnetization (µB/f.u.)M\nagnetic field (kOe)-8-6-4-202468-202\nFIG. 7. Four-quadrant Mversus Hloops of buttons of\nGdCo 5\u0000xNixforx= 0:5, 1.3, 2.5, and 5, measured at 10 K\n(5 K for x= 0.5). The inset shows the low \feld region of\nthese Mversus Hloops.\ntained from the MversusHloops on powders/buttons as\ndiscussed above are shown in Fig. 8 as a function of com-\nposition. The magnetization has a minimum at x\u00181,\nwhich is the composition for which the sublattice mag-\nnetizations cancel maximally in the presence of a 70 kOe\napplied \feld at 10 K. This is consistent with the value\npredicted in section I and with the values obtained via\nthe theoretical approach, section IV. The coercive \feld\nhas a broad peak that corresponds with the minimum\nin magnetization. This behavior is consistent with that\nobserved for single crystals of Cu-doped GdCo 5, [12, 13]\nand other RECo 5\u0000xNixmaterials [14, 15]. A correspond-\n0.00.51.01.52.02.53.03.54.04.55.001234567Magnetization at 70 kOe (µB/f.u.)N\nickel content (x)  012345678C\noercive field (kOe)FIG. 8. Magnetization at 70 kOe (black squares) and coercive\n\feld (red triangles) as a function of Ni content, x, measured\nat 10 K (5 K for x= 0, 0.5 and 1). Errors were estimated\nfrom the precision of the magnetization measurements. Lines\nare a guide to the eye.\ning peak in magnetocrystalline anisotropy is found in the\ntheoretical calculations discussed in IV.\nIV. THEORY\nA. Zero temperature magnetization\nFigure 9 shows zero temperature DFT-DLM calcula-\ntions of the magnetization of GdCo 5\u0000xNixas a function\nof nickel content x. The antiferromagnetic coupling of\nthe RE and TM moments means that the total mo-\nment is obtained as the di\u000berence between these two\ncontributions. For GdCo 5, without the orbital polari-\nsation correction a total moment of 0.62 \u0016B/f.u. is cal-\nculated, which consists of a contribution from the Co\nsublattices of (2\u00021:65 + 3\u00021:61 = 8:13\u0016B) and from\nthe Gd atom of 7.49 \u0016B. The main e\u000bect of the OPC\nis to increase the orbital moments on each Co atom by\n\u00190:1\u0016B, giving an increased Co contribution to the mo-\nment of (2\u00021:79 + 3\u00021:73 = 8:77\u0016B) and total moment\nof 1.30\u0016B/f.u. The theoretical justi\fcation for includ-\ning the OPC is that it approximates the contribution\nto the exchange-correlation energy from the orbital cur-\nrent, which is missing in the local spin-density approx-\nimation [43]. Practically, previous work both on YCo 5\nand GdCo 5found that including the OPC improved the\nagreement of magnetic moments and magnetocrystalline\nanisotropy with experiment [31, 36, 44].\nConsidering the other limit of GdNi 5, the Ni sublat-\ntices give a much weaker contribution of (2 \u00020:22 + 3\u0002\n0:35 = 1:49\u0016B) (no OPC). The weaker TM magnetism\nleads to a smaller induced contribution to the Gd mo-7\n0.00.51.01.52.02.53.03.54.04.55.0-2-10123456N\nickel content (x)Magnetization (µB/f.u.) No OPC, Ni at 2c/3g \nNo OPC, Ni at 2c \nOPC, Ni at 2c/3gT\nM dominatedGd dominated\nFIG. 9. Calculated zero-temperature magnetization of\nGdCo 5\u0000xNix. A negative value of M(\\TM dominated\") im-\nplies that the total moment points in the same direction as\nthe transition metal (Co and Ni) sublattice and opposite to\nthe Gd sublattice, and vice versa for a positive value (\\Gd\ndominated\"). The di\u000berent symbols correspond to calcula-\ntions: without the OPC and with preferential Ni occupation\nat the 2 csites (red circles); without the OPC and with equal\nNi occupation at the 2 cand 3 gsites (black squares); with\nthe OPC and with equal Ni occupation at the 2 cand 3 gsites\n(blue triangles).\nment, whose total value is reduced to 7.27 \u0016B. The total\nGdNi 5moment is therefore 5.78 \u0016B/f.u. The OPC has\na much smaller e\u000bect on the Ni orbital moments com-\npared to Co, so that the OPC-calculated total moment\nis reduced only slightly, to 5.73 \u0016B/f.u.\nThe absolute value of the moment of GdNi 5exceeds\nthat of GdCo 5. The di\u000berence is that, in GdCo 5the total\nmoment points in the same direction as the transition\nmetal moments (TM dominated), whereas in GdNi 5the\ntotal moment points in the same direction as the Gd\nmoment (Gd dominated). In Fig. 9 the sign convention is\nadopted that Gd (TM) dominated systems have positive\n(negative) moments. The gradual addition of Ni weakens\nthe TM contribution, causing a compositionally-induced\ntransition from TM to Gd dominated magnetism with\nincreasingx.\nAt the concentration when the TM and Gd contri-\nbutions to the magnetization are equal, the moments\nare fully compensated and the total magnetic moment\nof GdCo 5\u0000xNixis zero. Since the OPC increases the\nTM moment whilst leaving the Gd moment largely un-\na\u000bected, this compensation concentration is di\u000berent for\ncalculations with and without the OPC. With or with-\nout the OPC, compensation occurs at x= 1:04 orx=\n0:54 respectively. Comparing the calculations with the\nexperimentally-estimated compensation concentration of\nx\u00191 also supports the use of the OPC for GdCo 5\u0000xNix.\nThe calculations discussed above were performed as-\n05 01 001 502 002 503 00-2.5-2.0-1.5-1.0-0.50.00.5Magnetization (µB/f.u.)T\nemperature (K) x = 0  \nx = 1.26Gd dominatedT\nM dominatedFIG. 10. Calculated magnetization versus temperature for\nx= 0 (blue triangles) and x= 1 :26 (pink half-\flled\nsquares). The sign convention (TM/Gd dominated) is as in\nFig. 9.\nsuming that the Ni atoms substitute onto the 2 cand\n3gsites with equal probability, and are shown as the\nblue triangles and black squares in Fig. 9 (with and with-\nout OPC, respectively). However, previous calculations\nfound it to be more energetically favorable for Ni to sub-\nstitute at the 2 csites [10]. Neutron di\u000braction exper-\niments on Ni-doped YCo 5also found this preferential\n2coccupation [45]. To investigate how this site pref-\nerence a\u000bects the magnetic properties, calculations were\nalso performed where the Ni atoms \fll the 2 csites \frst,\nwith the 3gsites only becoming occupied with Ni atoms\nforx>2. The moments calculated in this way (without\nthe OPC) are shown as the red circles in Fig. 9. The\nlocation of the Ni dopants does not have a large e\u000bect on\nthe calculated moment, yielding a maximum di\u000berence of\n0:16\u0016B/f.u. atx= 2. The compensation concentration\nis also una\u000bected. However, as discussed below, there\nis a more pronounced e\u000bect on the magnetocrystalline\nanisotropy from site-preferential doping.\nB. Temperature dependent magnetization\nWe now consider the magnetization at \fnite tem-\nperature, focusing on two cases: pristine GdCo 5and\nGdCo 3:74Ni1:26. The latter Ni concentration was selected\ndue to the interesting coercivity behavior observed exper-\nimentally for samples around this composition, as shown\nin Fig. 8. In these calculations the OPC was included,\nand the Ni dopants occupied the 2 csites only.\nFigure 10 shows the DFT-DLM magnetizations calcu-\nlated for the temperature range 0{300 K. As in Fig. 9,\npositive values correspond to the Gd moment having a\nlarger moment than the TM contribution. The magneti-8\nzation of both GdCo 5and GdCo 3:74Ni1:26becomes more\nnegative (TM-dominated) in this temperature range.\nThe change is e\u000bectively linear with temperature, with\na di\u000berence of 1.0 \u0016B/f.u. for GdCo 5and 0:8\u0016B/f.u. for\nGdCo 3:74Ni1:26between 0 and 300 K.\nThe origin of the change in magnetization is a faster\ndisordering of Gd moments compared to the TM as the\ntemperature is increased [10]. This disordering is quan-\nti\fed by the order parameters ( mGd;mTM), which vary\nbetween 1 at 0 K and zero at the Curie temperature.\nAt 300 K, ( mGd;mTM) = (0:75;0:91) in GdCo 5and\n(0.75,0.83) in GdCo 3:74Ni1:26. Therefore in both cases\nthe relative strength of the TM contribution compared\nto Gd has increased with increasing temperature, pro-\nducing a shift towards TM dominated magnetization.\nThe fact that mGd= 0:75 for both cases at 300 K\nshows that the introduction of Ni at the 2 csites has not\na\u000bected the rate of Gd disordering, consistent with re-\nsults obtained previously [10]. However, the presence\nof Ni does lead to a faster disordering of TM moments\n(mTM= 0:83 compared to 0.91), which is why the\nchange in magnetization between 0{300 K is smaller for\nGdCo 3:74Ni1:26than GdCo 5. Overall, this faster disor-\ndering reduces the Curie temperature, which is calculated\nto be 915 K for GdCo 5and 713 K for GdCo 3:74Ni1:26.\nThese values are consistent with the experiments of\nChuang et al. [16], who observed a Curie temperature\nof 1000 K for GdCo 5and 730 K for GdCo 3:75Ni1:25.\nAs shown in Fig. 10, at 140 K GdCo 3:74Ni1:26switches\nfrom Gd to TM dominated magnetization. This temper-\nature, where the antiparallel Gd and TM moments cancel\neach other, is the calculated compensation point of this\ncomposition, and agrees well with the experimental data\nshown in Fig. 6. In passing, we note that not includ-\ning the OPC shifts the magnetization to a more negative\nvalue by 0.55 \u0016B/f.u. at 0 K, and raises the compensation\ntemperature to\u0018300 K (not shown).\nC. Zero temperature magnetocrystalline\nanisotropy\nWe next consider the experimentally-observed varia-\ntion in coercive \feld with composition (Fig. 8). Arguably\nthe simplest model of coercivity is based on magneti-\nzation rotation (the Stoner-Wohlfarth [SW] model) [46]\nwhich gives a coercive \feld of 2 K=M for a ferromagnet of\nanisotropy Kand magnetization M. The same expres-\nsion is obtained for the nucleation of reverse domains\nwithin micromagnetic theory [47]. Postponing a discus-\nsion ofMto Sec. IV D, we \frst consider the magnetocrys-\ntalline anisotropy of GdCo 5\u0000xNix. At zero temperature,\nthe angular variation of the free energy was calculated,\nwhen the Gd and TM moments are held antiparallel to\neach other and rotated from being parallel to perpen-\ndicular to the crystallographic caxis. This variation is\nwell described by Ean(\u0012) =K1sin2\u0012+K2sin4\u0012, with\nK2\u001cK1. Fig. 11 shows K1as a function of Ni compo-\n0.00.51.01.52.02.53.03.54.04.55.00.00.51.01.52.02.53.03.5 \nRigid band \nNi at 2c / 3g \nNi at 2cAnisotropy constant K1 (meV / f.u.)N\nickel content (x)FIG. 11. Anisotropy energy corresponding to a rigid rotation\nof the antiparallel TM and Gd sublattices, calculated at zero\ntemperature, for GdCo 5\u0000xNix. The di\u000berent symbols corre-\nspond to preferential Ni substitution at the 2 csites (red cir-\ncles), equal substitution over the 2 c/3gsites (black squares),\nor a rigid band calculation on pristine x= 0 (blue triangles).\nsitionx. The dominant contribution to this anisotropy\nenergy is the TM sublattice, with a minor 5 dcontribution\nfrom Gd [36].\nAs for the zero temperature magnetization in Fig. 9,\nboth preferentially substituting the Ni at 2 csites (circles\nin Fig. 9) and equally distributing the Ni over the 2 cand\n3gsites (squares) was investigated. In both cases, adding\nNi increases K1compared to pristine GdCo 5. Further-\nmore, both cases show a peak in K1with Ni content.\nFor preferential 2 csubstitution this peak occurs for xbe-\ntween 1.5{2.0, while for equal 2 c/3gsubstitution the peak\nforxis between 2.5{3.0. The enhanced K1is much more\npronounced for preferential 2 csubstitution, becoming 3.5\ntimes larger compared to pristine GdCo 5atx= 1:5.\nIn these calculations, the Ni doping has been modeled\nusing the CPA. In a simpler rigid-band calculation, the\ne\u000bects of Ni-doping are simulated by shifting the Fermi\nlevel of pristine GdCo 5so that the integrated density of\nstates equals the number of electrons in the Ni-doped\nsystem. The rigid band calculations of K1are shown as\nthe blue triangles in Fig. 11. Here, the enhancement in\nK1withxis even greater than that found with the CPA.\nThe rigid band model does not provide a fully consistent\npicture of doping, e.g. with the value of K1atx= 5\nnot coinciding with K1calculated for GdNi 5. Nonethe-\nless, the rigid band data emphasizes how, as has been\npreviously discussed for YCo 5[31, 44], changing the oc-\ncupations of the bands located close to the Fermi level\ncan have large e\u000bects on the anisotropy.\nThe calculations in Fig. 11 were performed with-\nout the OPC. Calculations including the OPC show\nthe same variation with band \flling, but the values of\nK1are strongly enhanced, as observed previously for\nYCo 5[31, 44]. For instance, for x= 1:26 with prefer-9\nential 2cNi doping, values of 2.0 and 6.5 meV/f.u. for\nK1without and with the OPC, respectively, are found.\nD. Zero temperature coercivity\nThe previous section showed that increasing the Ni\ncontent causes a boost to the anisotropy energy of\nthe transition metal sublattice. Assuming Ni substi-\ntutes preferentially at 2 csites, the calculated peak in\nanisotropy and the experimentally measured peak in co-\nercivity are located at similar concentrations. This obser-\nvation may explain the increased coercivity with Ni dop-\ning of RECo 5compounds with nonmagnetic REs [14, 15].\nHowever, as shown in Fig. 8, the maximum in the coer-\ncive \feld for GdCo 5\u0000xNixcoincides with a minimum in\nmagnetization. Referring again to the micromagnetic ex-\npression for the coercive \feld of a ferromagnet of 2 K=M ,\nwe note that naively setting Mto zero at \fnite Kshould\nyield a divergent coercive \feld at the compensation point.\nThis divergence remains even when Kronm uller's prefac-\ntor\u000b[48] is introduced in order to account for microstruc-\ntural variation in K. Therefore the boost in coercivity in\nGdCo 5may simply result from compensation of the Gd\nand TM magnetic moments.\nHowever, GdCo 5\u0000xNixis a ferrimagnet, so it is by no\nmeans obvious that models based on the rotation of a\nsingle magnetization vector should apply. Extending the\nSW model for a ferrimagnet produces a two-sublattice\nmodel, which was investigated for positive applied \felds\nin Ref. 49. Crucially, the competition between the exter-\nnal \feld, the antiparallel exchange interaction and the\nmagnetocrystalline anisotropy can lead to canting be-\ntween the Gd and the TM sublattices when a magnetic\n\feld is applied.\nWe recently introduced a method of calculating mag-\nnetization versus \feld curves including this e\u000bect from\n\frst principles, which we applied to GdCo 5(x= 0) at\nlow [36] and high [50] magnetic \felds. In this approach,\nDFT-DLM calculations are used to parameterize the fol-\nlowing expression for the free energy F2.\nF2(\u0012Gd;\u0012TM) =K1;TMsin2\u0012TM\u0000\u00160M\u0001H\n+K2;TMsin4\u0012TM+K1;Gdsin2\u0012Gd\n+S(\u0012TM;\u0012Gd) +A^MGd\u0001^MTM (1)\nwithM=MGd+MTM. The \frst line of Eq. 1 resem-\nbles the free energy found in the Stoner-Wohlfarth model.\n\u0012idenotes the angle that the magnetization of sublat-\nticeimakes with the caxis,Kj;irepresents the various\nanisotropy constants, and Srepresents the anisotropy\nenergy due to dipolar interactions. Aquanti\fes the ex-\nchange interaction, which with a positive value favors an-\ntiferromagnetic alignment of the Gd and TM moments.\nFor a compensated magnet, MGd=MTMand in the\nabsence of an external \feld the magnetic moments are\nantiparallel. Naively we might therefore set M= 0 and,\nfrom inspection of equation 1, argue that the external\n-1500- 1000- 5000 5 001 0001 500-6-4-20246Magnetization (µΒ / f.u.)M\nagnetic field (kOe) x = 0  \nx = 1.26FIG. 12. Magnetization versus \feld curves calculated at zero\ntemperature for x= 0 (blue) and x= 1:26 (pink). The\narrows label the coercive \felds at which the magnetization\nswitches from positive to negative values during the downward\n\feld sweep (second quadrant).\n\feld can have no e\u000bect on the free energy or magne-\ntization, corresponding to in\fnite coercivity. However,\nM= 0 is only true as long as the moments remain an-\ntiparallel. If the antiparallel alignment breaks, the mag-\nnetic sublattices couple individually to the external \feld.\nFor instance, in the limit of extremely strong external\n\felds both sublattices align to the \feld, giving a resul-\ntant magnetization of MGd+MTM.\nWe note that this model provides an explanation for\nthe magnetization measurements of the powder in a\n10 kOe \feld as a function of temperature (Fig. 4). On\nfree-to-rotate samples, the critical \feld required to trig-\nger the transition from antiparallel to canted moments\nessentially scales as jMGd\u0000MTMj[50, 51]. Therefore, as\none approaches the compensation point in the (free-to-\nrotate) powder, the antiparallel alignment can be broken\nwith a small \feld, and a nonzero magnetic moment mea-\nsured.\nNow, considering \fxed samples, in Fig. 12, the results\nof minimizing F2along the full multi-quadrant magne-\ntization curve are shown, sweeping the \feld along the\nsequence 0!Hmax!\u0000Hmax!Hmax, for GdCo 5and\nGdCo 3:74Ni1:26. Here the OPC is included, preferential\nNi doping at the 2 csites is assumed (cf. Fig. 10), and the\ncalculations performed at zero temperature. The \feld is\napplied along the crystallographic caxis. The size of the\n\feld is not intended to match experimental results, as de-\nscribed below, but the relative changes between di\u000berent\ncompositions can be extracted.\nFocusing \frst on GdCo 5(blue line) forjHj<820 kOe,\nthe boxlike curve resembles that of a SW ferromagnet. At\n820 kOe, there is a transition from the rigid antiparallel\nalignment of Gd and Co moments to a canted con\fgura-\ntion, with the energy gain of the Gd moments aligning\nwith the magnetic \feld competing with the exchange and10\nanisotropy terms. This transition is reversible, such that\nthere is no hysteresis in the \frst quadrant. In the sec-\nond quadrant, at H=\u0000473 kOe (blue arrow in Fig. 12)\nthere is a discontinuous jump in the magnetization corre-\nsponding to a simultaneous 180\u000erotation of the Gd and\nCo moments. This jump is irreversible, so returning the\n\feld to zero now gives a negative magnetization, with\nthe majority of the Co moments now pointing opposite\nto the \feld until the symmetric jump at H= 473 kOe\noccurs.\nGdCo 3:74Ni1:26(x= 1:26, pink line) shows broadly\nthe same behavior, but the nature of the transitions\nthemselves are slightly di\u000berent. At the high-\feld tran-\nsition from antiparallel to canted moments at j\u00160Hj=\n1060 kOe, the moments rotate rapidly with \feld such\nthat there is a very sudden, but reversible, increase in\nmagnetization. However, the demagnetizing curve in the\nsecond quadrant shows a new feature, which is a continu-\nous and reversible decrease of magnetization in the region\n-835 kOe<\u0016 0H < -785 kOe. The magnetization passes\nthrough zero at -792 kOe, and becomes increasingly nega-\ntive, exceeding its zero-\feld magnitude at -800 kOe. This\nnew feature is a result of the system getting trapped in a\nmetastable energy minimum corresponding to canted Gd\nand TM moments. For jHj>835 kOe this minimum dis-\nappears, and the system undergoes an irreversible tran-\nsition back to antiparallel moments.\nEquating the coercive \felds with the magnitudes of\nthe applied \felds which produce zero magnetization in\nthe second quadrant, we extract values of 473 kOe for\nx= 0 and 792 kOe for x= 1:26. For now ignoring\nthe fact that these numbers are huge compared to ex-\nperiment, in terms of relative magnitudes an increase in\ncoercivity by a factor of 1.7 is observed at a Ni doping of\nx= 1:26. We note that this increase is relatively mod-\nest compared to a naive prediction based on assuming\nthat the coercivity was proportional to K1=M; sinceK1\nandMincrease/decrease by a factor of 3 respectively,\nwe might have expected a coercivity enhancement by a\nfactor of 9.\nThe calculations in Fig. 12 are illustrative, but can-\nnot be considered a realistic picture of macroscopic mag-\nnetization reversal. In reality, the nucleation of reverse\ndomains, e.g. at the edge of the sample, will facilitate\nmagnetization reversal at far lower \felds than found\nhere [1, 52]. The coercivity will then depend on how\nthe domain walls propagate through the sample, which\nis likely to be a\u000bected by the presence of the secondary\nphase [42]. We also note that the peak in coercivity ob-\nserved experimentally here was found for polycrystalline\nsamples. However, it is interesting that the small single\ncrystals of Cu-doped GdCo 5reported in Ref. 12 do show\nbox-like demagnetization curves such as the calculated\nones shown in Fig. 12.V. SUMMARY & CONCLUSIONS\nPolycrystalline samples of GdCo 5\u0000xNixforx= 0 to\n5 have been synthesized using an arc furnace. The pre-\ndominant formation of a single phase was con\frmed by\ncomparing powder x-ray di\u000braction patterns to the pat-\ntern measured for pure GdCo 5. Optical and SEM imag-\ning showed small ( <7%) amounts of a 2:7 phase in the 1:5\nmatrix for x\u00142:5, and no 2:7 phase at higher concen-\ntrations. No evidence was found to say with con\fdence\nthat the annealing improves (or indeed, a\u000bects at all) the\nmicrostructure and phase purity of the samples.\nThe magnetization of the samples measured at 70 kOe\nand 10 K initially decreases as the nickel content is in-\ncreased. At a composition of x\u00191, the (absolute) mag-\nnetization reaches a minimum and then increases with\nfurther Ni addition. This behavior is due to the Ni weak-\nening the magnetization of the transition metal sublat-\ntice, such that at low temperature, for x < 1 the net\nmagnetization points along the direction of the transition\nmetal moments, while for x > 1 the net magnetization\npoints along the direction of the Gd moments. Zero tem-\nperature DFT-DLM calculations \fnd the compensation\ncomposition, i.e. the point at which the transition metal\nand Gd sublattice magnetizations cancel each other to be\nx= 1:04, in good agreement with the experimentally-\nobserved minimum. The calculations found the magne-\ntization to be rather insensitive to the location of the\nNi dopants, which can occupy either 2 cor 3gcrystallo-\ngraphic sites.\nFor a Ni content of 1 \u0014x\u00143, compensation tem-\nperatures in the range 10{360 K were observed. The\ncompensation temperatures increase with increasing Ni\ndoping, and occur due to the faster disordering of the\nGd moments compared to the transition metal. Finite\ntemperature DFT-DLM calculations on pristine GdCo 5\nand GdCo 3:74Ni1:26demonstrate this behavior explicitly,\n\fnding a compensation temperature of 140 K for the lat-\nter compound which corresponds well to the experimen-\ntal measurements shown in Fig. 6.\nThe coercivity of polycrystalline buttons measured be-\nlow 10 K is found to have a maximum value at a compo-\nsitionx\u00191, coinciding with the minimum in magnetiza-\ntion. One might argue that such behavior is consistent\nwith the Stoner-Wohlfarth model and micromagnetics,\nwhere the coercive \feld is inversely proportional to the\nmagnetization ( Hc= 2K=M ). However, such a picture\nis based on the rotation of a single magnetic sublattice\nand neglects the possibility that magnetization reversal\nmight proceed via a canted arrangement of Gd and tran-\nsition metal moments. Magnetization versus \feld loops\ncalculated allowing for such canting do show an increase\nin coercivity from x= 0 tox= 1:26, but not by as\ngreat an amount as predicted by the Stoner-Wohlfarth\nmodel given the reduction in M. Apart from the reduc-\ntion in magnetization, the DFT-DLM calculations also\nfound an increase in the magnetocrystalline anisotropy\nof the transition-metal sublattice with Ni doping. In-11\ndeed, assuming preferential substitution at the 2 csites\nthe peak in anisotropy was found for a concentration of\nx= 1:5, reasonably close to the experimentally observed\ncoercivity maximum. Such an explanation for increased\ncoercivity, independent of phenomena related to compen-\nsation, would be consistent with measurements on doped\nRECo 5compounds with nonmagnetic RE, which also un-\ndergo peaks in coercivity despite having no compensation\npoints.\nHowever, apart from these intrinsic factors, it should\nalso be noted that the peak in coercivity also coincides\nwith the largest amount of secondary 2:7 phase (Ta-\nble II). Although the amount of secondary phase we\nobserve is small in terms of measuring intrinsic quan-\ntities, interfaces between the 1:5 and 2:7 phases could\ninhibit the motion of domain walls through the sample,\nincreasing the coercivity. Being able to better control the\nformation of the 2:7 phase would allow the magnitude of\nthis extrinsic e\u000bect to be tested.\nThe current study emphasizes the complementary roles\nplayed by experiments and theory. On one hand, the ex-\nperiments provide valuable input for developing the cal-\nculations, particularly in terms of validating the method-\nology. On the other hand, the calculations provide mi-\ncroscopic insight into macroscropic measurements. Here\nwe have shown that quantitative comparisons are possi-ble between intrinsic quantities such as magnetizations\nand compensation temperatures.\nHowever, whilst the calculations can give hints about\nextrinsic quantities such as the coercivity, in reality\na multiscale approach capable of describing e.g. mi-\ncrostructure and long range demagnetizing \felds is re-\nquired. Nonetheless, the \frst-principles calculations\n(as validated by experimental measurements of intrinsic\nquantities) can still play a fundamental role by provid-\ning the microscopic parameters required as input for such\nsimulations.\nACKNOWLEDGEMENTS\nThis work forms part of the PRETAMAG project,\nfunded by the UK Engineering and Physical Sciences\nResearch Council (EPSRC) Grant No. EP/M028941/1.\nCrystal growth work at Warwick was also supported\nby EPSRC Grant No. EP/M028771/1. We thank the\nWarwick Research Technology Platform (X-ray Di\u000brac-\ntion and Electron Microscopy) for their assistance. We\nacknowledge the Warwick Manufacturing Group and\nBuehler for assistance with slide production. We thank\nE. Mendive-Tapia and G. Marchant for useful discus-\nsions.\n[1] K. Kumar, J. Appl. Phys. 63, R13 (1988).\n[2] K. Strnat, G. Ho\u000ber, J. Olson, W. Ostertag, and J. J.\nBecker, J. Appl. Phys. 38, 1001 (1967).\n[3] M. Sagawa, S. Fujimura, N. Togawa, H. Yamamoto, and\nY. 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D’Annunzio”,  66100 Chieti, Italy  \n3 International Research Centre MagTop, Insti tute of Physics, Polish Academy of Sciences, Aleja Lotnik ów 32/46, PL -\n02668 Warsaw, Poland  \n4 Consiglio Nazionale delle Ricerche CNR -SPIN, c/o Univ. L’Aquila, 67100 L’Aquila, Italy  \n5 Consiglio Nazionale delle Ricerche CNR -SPIN, c/o Universitá  di Salerno - Via Giovanni Paolo II, 132 - 84084 - \nFisciano (SA), Italy   \n \n \nABSTRACT  \nWe report a combined theoretical  and experimental investigation of magnetic proximity and Hall transpor t in Pt/Cr  bilayers . \nDensity functional theory indicates that  an interfacial magnetization  can be  induced  in the Pt layer  and a strong \nmagneto crystalline anisotropy with an easy axis out of plane arises  in the antiferromagnet . A signal ascribed to the  anomalous \nHall effect is detected and associated to the  interface  between Pt and Cr layer s. We show that this effect originates  from the \ncombination of proximity -induced magnetization and a  nontrivial topology of the band structure  at the interface . \n \n \n  \n                                                 \n* Author to whom correspondence should be addressed : matteo.cantoni@polimi. it 2 \n TEXT  \nAnomalous Hall Effect (AHE) is typically associated to ferromagnetic metals because  of broken  time reversal symmetry  \nand spin -orbit couplin g (SOC)  [1].  Recent observations of large AHE originating from a non -vanishing Berry phase in \nantiferromagnets with non -collinear spin arrangement  [2,3]  brought great interest for possible applications in the emerging \nfield of antiferromagnet spintronics  [4]. It is however generally accepted  that ordinary antiferromagnet s with collinear spin \nconfiguration should not exhibit AHE due to vanishing  Berry phase.  \nIn this work, we show  how interfacing a collinear antiferromagnet (AFM) with a non -magnetic (NM) metallic layer can produce  \na detectable AHE , whose sign is controll able by field -cooling, in the NM/AFM system P t/Cr. We propose a mechanism  for \nthis phenomenon  based on Magnetic Proximity Effect (MPE)  in Pt [5] and non -trivial band topology  at the interface.  Chromium \nis the prototypical itinerant antiferromagnet, characterized by a spin -density -wave (SDW) magnetic ordering , involving  a \nperiodic modulation of the amplitude of antiferromagnetically coupled magnetic moments  [6]. Platinum  is a non -magnetic \nmetal in its bulk form,  on the verge of Stoner ferromagnetism . In thin film heterostructures, MPE  has been demonstrated  by \nX-ray Magnetic Circular Dichroism (XMCD)  in systems where Pt interfaces FM 3d metals  [7–10]. MPE has also been inv oked \nto explain some transport properties of Pt on insulat ing Yttrium Iron Garnet , [11] although this is still debated  [12]. More \nrecently , MPE in Pt has been exploited to detect the state of a neighboring magnetoelectric antiferromagnet Cr 2O3 by means of \nthe AHE  [13,14] , demonstrating the use of Pt as  valuable probe in the context of the emergin g AFM -based spintronics  [4]. \nBy combining  ab-initio calculations and electrical measur ements, we demonstrate that a spin -polarized interface forms  between \nCr and Pt . Density functional theory (DFT) predicts a net magnetic moment induced in Pt  by neighboring  Cr with out  of plane \nanisotropy . We show the electrical detection of  AHE  also in relatively thick metallic heterostructures , despite the interfacial \norigin  of such effect . Exploiting  the peculiar transport properties of Chromium related to SDW antiferromagnetism  [15,16] , we \ncan unambiguously associate the  presence of  AHE  to the AFM state of  Cr. Finally, we discuss how the presence of AHE and \nits temperature dependence can be justified by the com bined effect of interfacial magnetization a nd band topology , which could  \nbe in principle expected also in other NM/ AFM systems.  \nTo model  the Pt/Cr interface, w e perform ed first-principles density functional calculations within the Gra dient Generalized \nApproximation  by using the plane wave VASP  [17] package and the PBE for the exchange -correlation functional  [18] \nincluding SOC . The core and the valence electrons were treated with th e projector augmente d wave  method  [19] and a cutoff \nof 450 eV for the plane wave basis. In supercell calculation s we use 8 Cr layers and the lattice constant of Cr bulk (2.89 Å). \n(See Supplemental Material S.1 for details ).We assume that the spin density wave in Cr [20] will not affect the properties of \nthe Pt/Cr interface, due to the small deviation from commensurate antiferromagnetism. In this 3d -5d interface we neglect the \npossibility of complex magnetism  [21,22]  since it was shown that th e Dzyaloshinskii -Moriya interaction  is negligible  at the \nCr/Pt interface  [22].  \nBulk Pt present s a van H ove peak close to the Fermi le vel. In general t he interface hybridization  and the magnetic proximity  \ntends to favor  a Stoner -type ferromagnetic instability  [23], so that Pt undergoes a magnetic phase transition at interfaces. We \ncalculate the profile magnetization of the Pt (N)/Cr heterostructure in the collinear approximation as function of the number (N) \nof Pt layers.  The Pt interface  always couple s ferromagnetically with Cr , as for 3d ferromagnetic materials  [24]. From a \ntheoretical p oint of view, the magnetic proximity demonstrates that the spin polarization, arising into the antiferromagnet, 3 \n penetrates  the Pt layers . Although the spatial profile is non -monotonous, the sum over the Pt layers leads to a non -vanishing \ntotal magnetizatio n of the order of 0.3 μ B (See Supplemental Material S.2 for the Cr/Pt magnetic profile ).  \nThe magnetocrystalline anisotr opy (MCA) is calculated from first -principles (including SOC) as a function of the lattice \nconstant.  As illustrated  in Fig. 1, the MCA shows  a strong sensitivity to strain effects, highlighted by the non-monotonic  \nbehavior  and in agreement with what  previously found for 3d ferromagnets interfaced with 5d materials  [25]. For the lattice \nconstant of bulk Cr, a large MCA  with an out of plane easy axis is obtained . The contribution to the MCA mainly comes from \nthe first layer of Cr , as demonstrated by means of DFT evaluation  of density of states  (DOS ) (See Supplemental Material S.1 \nfor the Cr/Pt DOS).   \nEpitaxial Pt/Cr bilayers were grown by Molecular Beam Epitaxy  on MgO (001) substrates  to experimentally verify the presence \nof AHE . Cr thin films were deposited in ultra -high vacuum conditions with thickness ranging from 25 nm to 75 nm. Details on \nthe growth of the heterostru cture will be published elsewhere. Residual strain as measured from X -ray diffraction is less than \n0.4% even in the thinnest film , thus largely comparable to the bulk case. According to first -principles calculations, an MCA \nwith out  of plane easy axis is thus expected.  Finally,  a Pt overlayer  with thickness between 0.6 (4 monolayers)  and 3 nm was \ndeposited  on top . Fifty  microns  wide Greek  crosses  were then defined  on the samples  by optical lithography and ion milling . \nThe compensation of geometrical offsets in this geometry has been achi eved by subtracting measurements in which current \nand voltage contacts are exchanged  [26]. At the same tim e, the current reversal method allows to compensate for thermoelectric \noffsets  (See Supplemental Material S.3 for the measurements setup)  [27]. The measurement protocol goes as follows: by \ncooling the sample from above its Néel temperature in a magnetic field B = ± 0.4 T (field cooling) perpendicular to the surface \nplane , we set the magnetic state of Cr.  Subsequently, this state is read by exploiting the anomalous Hall resistance versus  \ntemperature , raised with a constant rate of 2  K/min.  A close -circuit He cryostat was us ed to this scope.   \nFigure  2 reports  the transport properties of a Pt(2)/Cr(50)  bilayer  (thickness in nm)  measured with a 20 mA probing current .  \nFigure 2 (a) shows the longitudinal resistivity  ρxx of the sample (empty dots ) and its first derivative with respect to the \ntemperature  (black line ) in the 20-330 K temperature ( T) range . Here, it is crucial to consider that t he formation of the SDW in \nCr determines the opening of a gap in the Fermi surface along the directio n of the wave vector  [6]. This result s in a reduction \nof the carrier  density,  which produces a  “kink” in the resistivity versus T curve . Following Ref.  [15] the N éel temperature TN \ncan be associated to the local maximum in dρ/dT. From Fig. 2(a) we find  TN = 290 K5 K, which is slightly  inferior to bulk Cr \nTN = 311 K  and comparable to other epitaxial thin films  [16]. \nThe same information can be extracted from the ordin ary Hall resistivity ρxy since it is related to the carrier density  n by the \nsimple relation coming from the Drude model 𝜌𝑥𝑦=−𝐵𝑛𝑒⁄ (e is the electron charge ), where 𝜌𝑥𝑦=𝑅𝑥𝑦𝑡 (t is the film \nthickness) . The temperature dependence of n (empty squares ) and its first derivative  (black line) with respect to T are shown in \nFig. 2(b). For the reasons given above , the increase of d n/dT below 290 K is consistent with the AFM phase transi tion of \nCr [16],  in agreement with  the value establishe d from longitudinal resistivity  measurements . \nFinally, we report the difference (ΔRxy) between the anomalous Hall resistances measured  at rema nence in states prepared by \nopposite field -coolings (B=0.4 T ), shown in F ig. 2(c). Measurements at remanence permits to exclude a transverse \ncontribution a rising from ordinary Hall  and to isolate the effect related to the memory of the system achieved through the \nmanipulation of the AF state. The small signal , in the order of hundreds of nano -ohms, can be reproducibly detected and is \nfound to be dependent only on the field-cooling direction . ΔRxy decreases with temperature, approaching zero in correspondence \nof the temperature T*= 290 K , that happens to be coincident with the previously determine d TN, considering the experimental \nerror bar.  4 \n Below T*, the application of magnetic field s does no  longer affect  the transverse resistance at remanence , thus excluding the \npossible influence of ferromagnetic contaminations  on the signal . On the contrary, the robustness against external fields can be \nexplained  by the negligible  magnetic susceptibility of Cr in the AFM phase . This strict correspondence between Cr magnetic \norder , established  from ordinary transport measurements, and the presence of an AHE clearly relates the origin of the latter to \nthe antiferromagnet state. We emphasize that conventional magnetometry does not allow to observe this phase transition. No \nchange in the magnetic signal is observed with Vibrating Sample Magnetometry (VSM) across the phase transition on an \nunpatterned Pt(3)/Cr(75) 5x5  mm2 sample . Considering the experimental error of 1 μemu , this sets the upper limit  to the average \nmagnetic moment in Cr and Pt to 0.0 01 μB/atom and  0.03 μB/atom, respectively. The latter is still compatible with the predicted \ninduced moment by DFT with a  mediated value  smaller than 0.05 μ B/atom.  \nTo understand if the effect originates within the Pt layer , a first hint comes from the comparison of samples with different Cr \nand Pt thicknesses  (Figure 3 ). ΔRxy at 20  K is almost identical for samples with 25 , 50 and 75  nm thick Cr films  and 3 nm thick \nPt overlayer , with a n average  value of 680 ± 120 nΩ. In the good -metal regime , the anomalous Hall resistance  is proportional \nto the magnetization of the sample and independent on the sample resist ivity  [13]. Being Δ Rxy not related to Cr thickness, a \nbulk contribution should be therefore excluded. Instead, when  a thinner Pt layer (0.6 nm, corresponding to 4 monolayer s) is \nconsidered,  a smaller ΔRxy = 275  ± 30 nΩ is found between states written with opposite field. We inci dentally note that  ΔRxy \nof the thinnest Cr sample  (25 nm)  approaches zero already below 200 K. This is consistent with the reported increase  of Cr \ntransition temperature with thickness in epitaxial  thin films  [15]. \nA second fact supporting  of the interfacial origin comes from the insertion of a gold spacer between Pt and Cr . In th is case, \nDFT calculations do not expect  proximity induced moment in Au/Cr, at variance with Pt/Cr systems  (See Supplemental \nMaterial  S.2 for the magnetic profiles) . This is in excellent agreement with experimental results:  as can be  seen in Fig. 3, the \nPt(3)/ Au(3)/Cr(50) sample does not show any detectable signal  compared to any of the Pt/Cr samples. We emphasize that t he \nsame Cr(50) film was used for the comparison with Au, that was  deposited on only half of the sample  with the help of  a \nmechanical shutter . Hence, the differen t signal  cannot be ascribed to Cr quality which is  confirmed  to be the same  by resistivity \nand ordinary Hall measurements (See Supplemental Material S.4 for the data) . Moreover, t he AHE suppression with the \ninterposition of  the diamagnetic Au interlayer allows to exclude a ny significant  contribution to the signal given by  spin-Hall \nmagnetoresistance  [28], since  any phenomena related to  spin-current generation in Pt  would result  unaffected  as the spacer \nthickness is much thinner than the spin diffusion length in Au (λ> 30 nm   [29]).  \nWhile the different signal coming from Pt/Cr and Au/Cr interfaces could be justified with the different proximity induced \nmagnetization, the temperature trend of the anomalous Hall resistan ce cannot be uniquely explained as due to this net magnetic \nmoment.  \nIn a conventional picture the transverse resistivity in a system  with magnetization M follow s the relation  [30,31]  \n𝜌𝑥𝑦=𝑅0𝐻+𝛼𝑀𝜌𝑥𝑥0+𝛽𝑀𝜌𝑥𝑥02+𝑏𝑀𝜌𝑥𝑥2                   (1) \nwith 𝛼,𝛽 and 𝑏 being the skew scattering, the side jump, and the intrinsic term, respectively, 𝑅0 the ordinary Hall coefficient , \n𝜌𝑥𝑥0 the zero -temperature residual resistivity , and 𝜌𝑥𝑥 the total longitudinal resistivity.  \nAccording to the scaling relation for 𝜌𝑥𝑦 , the temperature variation of the anomalous Hall resistivity is linked to the thermal \ndependence of the magnetization and of the longitudinal resistivity. It is worth pointing out that the amplitude of 𝜌𝑥𝑥 is only \ndoubled when moving from 50 K to ab out 300 K (See Fig. 2(a)). Such observation implies that it should be the magnetization \nto play the major role in setting the thermal dependence of the anomalous Hall resistivity. However, a closer inspection of t he 5 \n thermal profile of 𝜌𝑥𝑦 indicates few distinct marks (see Fig. 3). 𝜌𝑥𝑦 does not follow a Curie -Weiss model 𝜌𝑥𝑦∝1/(𝑇−𝑇𝑁) \nbut, on the contrary, exhibits a smooth switching close to the antiferromagnetic transition .  Moreover, the anomalous Hall \nresistance keeps increasing from 150 K to 50 K , in a region  where t he longitudinal resistivity has only minor variations  and the \nmagnetization due to the magnetic proximity should also be saturated because the sublattice antiferromagnetic magnetization \nhas reached its maximal value.  \nSince uncompensated magnetization alone fails to account for these observations , an additional topological contribution  has to \nbe considered. Concerning this term,  one possible scenario would point to the presence of a non -collinear antiferromagnetic \npattern developing at the Pt/Cr interface. Such possibility is however ruled out as the theoretical investigation reports a s trong \nmagnetic anisotropy  and the Dzyaloshi nskii -Moriya  coupling is small at the Cr/Pt interface  [22]. For this reason, we consider \nthe possibility of having a topological contribution due to a non -trivial Be rry curvature  of the  bands  arising from the \nmodification of the electronic structure due to the antiferromagnetic order and the magnetic proximity.   \nHence, we investigate  a multiband low -energy tight-binding  model which includes the bands  at the Fermi level having  Pt and \nCr character , the SOC  at the Cr site,  and a magnetic interaction  that takes into account the layer dependent magnetization close \nto the interface  both in the Pt and Cr regions . Then , we deter mine the intrinsic geometric contribution to the anomalous Hall \nconductance  (AHC) by evaluating the Berry curvature of the Bloch bands. The AHC is proportional to the anomalous Hall \nresistance experimentally measured. The analysis is performed for differen t spatial  profile s of the magnetization at the Pt side \nof the heterostructure with parallel or anti - parallel magnetic moments in neighbor ing layers (see inset of Fig. 4). Moreover,  in \norder to assess the role of an interface  a comparison is made by replacing Pt with Au.  Since the magnetization in each layer is \nuniform, the  temperature dependence simulation of the AH C in the heterostructure can be performed  by assuming a  Stoner  \nmodel  with 𝑀=𝑀0 [1−(𝑇\n𝑇𝑁)2\n]1/2\n, where  𝑀0  is the zero temperature magnetization. Concerning the layer dependent \n𝑀0 close to the interface , we take as reference the values obtained within the  ab-initio analysis.  \nRepresentative  results  for the AH C are shown in Figure 4 where we compare three cases: i) Cr/Pt interface with a sign \nmodulation of the magnetization in the Pt layers  (red curve ), ii) Cr/Pt with a uniform penetration of the magnetization in the \nfirst layers close to the interface  (black cu rve), iii) Cr/Au interface with weak  ferromagnetic proximity but higher electron \ndensity in the Au subsystem  (orange curve) . Due to the presence of non -trivial sources of Berry curvature, we find that an \noverall positive amplitude of the AHC  is generally o btained  assuming different types of profiles for the magnetization at the \nPt/Cr interface and within the Pt subsystem, accordingly with the indications from the ab -initio analysis.  On the other hand, \nfor the case of Au/Cr interface the AHC  is negative in the whole range of t emperatures below the magnetic transition \ntemperature. Although  we are not aiming at  achieving quantitative matching between theory and experiments , the outcomes  \nare robust and , remarkably,  due to its topological origin, the AHC is not much dependent on the character of the magnetic \nproximity . Furthermore , as a distinctive mark , the AH C conductance can be observed even in the case of complete \ncompensation of the total magnetization  for collin ear magnetic moments both in the Pt /Cr and Au /Cr bilayers .  \nRecently, it was show n that in a cubic environment partially occupied t 2g bands , in presence of strong SOC and breaking of the \ntime symmetry, due to ferromagnetism, can also generate AHE in layered system s [32]. We observe that , although we deal \nwith antiferromagnets,  similar electronic mechanism s are at work here for the Cr/Pt heterostructure  with two effective sources \nfor the AHE: the magnetic Pt and the Cr interface layer s. In bulk Cr, we have a small SOC and the topological contribution s to \nthe AHC of different antiferromagnetic layers substantially cancel each other. At the Cr/Pt interface , we have an imbalance of  \nthe Cr magnetism  and the presence of a large atomic SOC. These two ingredients generate the AHE  from the Cr interface layer . 6 \n On the other hand, gold is less effective in doing so because large onsite energy makes it hybridize less with interfacial Cr  \nlayers. Consistently,  from our DFT calculation s, gold does not show any intrinsic magnetic moment  (See Supplemental \nMaterial S.2 for the magnetic profile ). The thermal dependence of M the can also lead to a sign change of the AHE, depending \non the details of magnetic profile of Pt, see Fig. 4. \nIn conclusion, DFT  and electrical measurement s agree on  the existence of a spin -polarized Pt layer  at the interface w ith Cr  with \nout of plane magnetization . At the  interface, Cr and Pt influence each other. In closer detail, t he main effect of Cr on Pt is the \ninduced magnetization  profile . On the other hand, Pt induces a large MC A on the Cr side , favoring the out of  plane component  \nof the magnetization . Exploiting the transport properties related to SDW ordering we can demonstrate a correspondence \nbetween AHE coming from the Pt/Cr interface and AFM ordering in Cr. Finally, w e calculate the Berry c urvature of the spin \npolarized electronic structure  showing the presence of the topological contribution to the AHE which can justify the occurrence  \nand the temperature dependence of the AHE.  \nWhile the results reported in this paper highlight the specific role of the interface with Pt , we foresee  similar physical scenarios \nwith other NM metals reported to have some proximity magnetization like Pd  [33], Ir,W [10], or V [34], thus making Cr  a \nparadigmatic  testbed to study these interface effects. At the same time, we expect AHE coming from interfaces to be generally \nexploit able in a large variety of antiferromagnetic systems, either metallic or insulating, making Hall measurements  a powerful  \nprobe of antiferromagnetism both for fundamental studies and in view of applications.  \n \n \nACKNOWLEDGMENTS  \nWe thank R. Bertacco , X. Marti, D. Di Sante , M. Bragato, S. Achilli, and M. I. Trioni  and for fruitful discussion. This \nwork was partially performed at Polifab, the micro and nanofabrication facility of Politecnico di Milano. This work was \npartially funded by Fondazione Carip lo via the project Magister (Project No. 2013 -0726). The work is supported by the \nFoundation for Polish Science through the IRA Programme co -financed by EU within SG OP.  C. A. was supported by CNR -\nSPIN via the Seed Project CAMEO. W.B. acknowledges partial support by Narodowe Centrum Nauki (NCN, National Science \nCentre, Poland) Project No. 2016/23/B/ST3/00839. We acknowledge the CINECA award under the ISCRA initiative IsC43 \n\"C-MONAMI\" and IsC54 \"CAMEO \" Grant, for the availability of high-performance  computing resources and support.  \n \n  7 \n BIBLIOGRAPHY  \n[1] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).  \n[2] S. Nakatsuji, N. Kiyohara, and T. Higo,  Nature 527, 212 (2015).  \n[3] A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel, A. C. 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(a) resistivity ( empty dots ) and first \nderivative ( black line ). (b) number of carriers from ordinary Hall measurements ( empty squares ) and first derivative ( black \nline). (c) anomalous Hall resistance difference taken at two opposite  amplitude of the  magnetic field . The red line is a guide \nfor the eye.  \n  \n11 \n  \n \n \n \n \n \n \n \n \n \nFigure 3. Difference of the a nomalous Hall resistance  taken at two opposite values of the magnetic field  as a function of \ntemperature in Pt/Cr and Au/Cr  bilayers. Nominal thicknesses are expressed in nm. Error bars indicate the standard error on \nmultiple samples. The lines are  only a guide for the eye.  \n  \n12 \n  \n \n \n \n \n \n \n \n \n  \nFigure 4. Thermal dependence of the AHC  obtained by theoretically determining the Berry curvature of the d -bands for the \nCr/Pt and Cr/Au layered structures assuming various magnetic profiles in the Cr, Pt and A u interface layers . The proximized  \nmagnetization for Pt or Au has a smaller amplitude than that of Cr and it can have a ferromagnetic ( dotted black) or \nantiferr omagnetic ( dashed red) profile, as  schematically reported in the inset. The Cr/Au interface ( solid orange) corresponds \nto a physic al configuration for the magnetization with a small leaking of magnetization in the Au interface  layers.  \n  \n13 \n Supplemental material to “ Anomalous Hall effect in \nantiferromagnetic/non -magnetic interfaces”  \nM. Asa,1 C. Autieri2,3, R. Pazzocco1, C. Rinaldi1, W. Brzezicki3, A. Stroppa4, M. Cuoco5, \nS. Picozzi2 and M. Cantoni1 \n \n1 Department of Physics, Politec nico di Milano, Via G. Colombo 81, 20133, Milano, Italy  \n2 Consiglio Nazionale delle Ricerche CNR -SPIN, c/o Univ. “G. D’Annunzio”, 66100 Chieti, Italy  \n3 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotników 3 2/46, PL -\n02668 Warsaw, Poland  \n4 Consiglio Nazionale delle Ricerche CNR -SPIN, c/o Univ. L’Aquila, 67100 L’Aquila, Italy  \n5 Consiglio Nazionale delle Ricerche CNR -SPIN, c/o Universitá di Salerno - Via Giovanni Paolo II, 132 - 84084 - \nFisciano (SA), Italy  \n \nS.1: The DOS of the Cr/Pt for different spin orientation  \nIn order to study the surface, we add 20 Å of vacuum to settle a setup for accurate calculations. We optimize the internal de grees \nof freedom by minimizing the total energy to be less than 10-6 eV and the remaining forces to be less than 20 meV/Å. A \n16×16×1 k -point Monkhorst -Pack grid is used for the relaxation and the calculation of the magnetocrystalline anisotropy. In \naddition, we use a 20×20×1 k -point for the determination of the density of s tates (DOS) with a smearing of 0.05 eV and an \nenergy grid with a step of 0.1 meV. The SOC is included to calculate the magnetocrystalline anisotropy as a function of the \nlattice constant. We calculate the layer projected DOS for the MgO/Pt/Cr heterostructu re to make an interface of the Pt with \nthe Cr and the MgO. The plot of the difference between the DOS of the system with in plane and out of plane easy axes is \nreported in Fig.1.  \n \nFigure 2: Difference between the layer projected D OS for the system \ncalculated with the out of plane (001) and in plane (100) easy axes.  \n \n14 \n The width of the oscillation is proportional to the effect of the SOC in the different layers. Large oscillations are present  in any \nPt layer. The interface MgO does no t show any oscillations. The Cr layers show large oscillations for the interface layer and \nsmaller oscillations for the inner layers. Therefore, the Pt induces a large SOC on the interface layer and the contribution to the \nMCA mainly comes from this first layer of Cr interfaced with Pt.   \n \n \nS.2: The magnetic profile for the Pt/Cr and Au/Cr interfaces  \nThe entire profile of the Pt(N)/Cr and Au(N)/Cr magnetizations is reported in Fig. 2. The Cr magnetic moment is almost \nindependent from the number of the 5d layers. At the surface the magnetic moment is close to 2 µ B and the magnetization of \nthe inner layers o scillates. The main difference in the Cr layers between the two 5d systems is the different magnetization of \nthe interface that is larger for the Au interface.  \n \nA magnification of the magnetic profile of the Au/Cr heterostructure is shown in the Fig. 3. The interface gold layer presents a \nsmall magnetization of 0.06 µ B independently on the Au thickness, while no magnetization is present for the other layers. Since \nthis magnetization goes rapidly to zero far from the Cr layers, we propose that magnetization of the interface Au is not relat ed \nto the intrinsic magnetic moment of gold but it is simply due to the tails of the magnetization of the Cr atoms. The AHE \nmeasure d for the Au/Cr system can be entirely attributed to the Cr interface. This explains also the smaller value of the AHE \nthat arises from one single Au layer, while in the case of the Pt/Cr system many Pt layers contribute to the AHE. The differe nt \nsign of t he AHE between Pt/Cr and Au/Cr also suggests that the main contribution to two cases is different.  \nA magnification of the magnetic profile of the Pt/Cr heterostructure is shown in the Fig. 4. The total magnetic moment of the  \nPt region is -0.429, -0.147, -0.084, 0.336 and 0.315 µ B, respectively, for N=3,4,5,6,7. The total magnetic moment of the Au \nregion, instead, is basically always equal to the induced magnetization of 0.06 µ B . \n   \nFigure 2 : Entire magnetic profile for the Pt/Cr (a) and Au/Cr (b) heterostructure. The Cr 1 (Pt1, Au 1) is \nthe Cr (Pt, Au) interface layers. and the Cr i (Pti, Au i) is the i -th layer from the interface.  \n15 \n  \nFigure 3: Magnetic profile of the Au(N)/Cr interface. The Au 1 is the \nAu interface layers. The Au i is the i -th layer from the interface.  \n \nFigure 4: Magnetic profile of the Pt(N)/Cr interface. The Pt 1 is the \nPt interface layers. The Pt i is the i -th layer from the interface.  \n \n \nS.3: Electrical measurements in Pt/Cr  \nBoth longitudinal and transverse resistance measurements were performed as a function of temperature using a current -\nsource/nanovoltmeter pair Keithley 6221/2182A operating in Delta Mode. Essentially, the Delta Mode automatically triggers \nthe current source to alternate the signal polarit y, then triggers a nanovoltmeter reading at each polarity. This current reversal \ntechnique cancels out constant thermoelectric offsets, ensuring the results reflect the true value of the voltage. A relay sw itching \nmatrix was used to acquire both transverse  and longitudinal measurements at the same time. The two contact configurations \nrequired for geometrical offset subtraction [1] are pictured i n Fig. 5, while the Van der Pauw measurements of resistivity [2] \nwere performed using the two configurations showed in Fig. 6.  \n \n16 \n  \nFigure 5: Transverse resistivity measurement configuration in Hall crosses.  \n \n \nFigure 6: Van der Pauw configuration for the measurement of resistivity in Hall crosses.  \n \nRepresentative measurements of transverse resistance ( Rxy in the main text) in Pt(3)/Cr(75) are shown in Fig. 7. Here both \ngeome trical and thermal offset compensation techniques are applied. Nevertheless, a temperature -dependent offset of non -\nmagnetic origin, whose amplitude is often comparable to the small magnetic signal coming from the Pt/Cr interface, is observe d. \nThis uncompen sated offset is different for each sample and presumably comes from minor non -uniformities in the sample. This \nspurious signal is fully reproducible over multiple subsequent measurements and can be subtracted as for the data reported in  \nFig. 2(c) and Fig. 3 of the main text.   \n \n \n \n \n \n \n \n \n \n \n \nFigure 7: Raw measurement of transverse resistance measured in P(3nm)/Cr(75nm) \nfollowing field cooling at +0.4 T (black points) and at -0.4 T (red triangles)  \n \n \n17 \n  \nS.4: Comparison of transport in Pt/Cr and Au/Cr  \nIn Fig. 3 of the main text we compared Pt(3)/Au(3) and Pt(3) capping layers on top of the same Cr(50) thin film. To foster th e \nconclusion that the different transverse resistance signal at remanence is relate d only to the different interface, we present here \nthe transport properties of the whole heterostructure (largely dominated by the chromium layer) indicating in both cases a so lid \nantiferromagnetic behavior below 290 K.  \n \nFigure 8 reports the longitudinal r esistivity and its first derivative for Au -capped (left) and Pt -capped (right) chromium devices. \nIn both cases the local maximum in the first derivative observed at 290 K can be used as a reference for the antiferromagneti c \ntransition of Cr.  \nThe same signa ture of the gap opening arising from the spin density wave in Cr can be observed as well, independently on the \nnature of the interface, in the carrier density extrapolated from Hall measurements in applied magnetic field ( B = ±0.4 T) shown \nin Fig. 9.  \n \nFigure 8: Longitudinal resistivity and its first derivative in Pt(3)/Au(3)/Cr(50) (left) and Pt(3)/Cr(50) (right).  \n \n \nFigure 9 : Carrier density and its first derivative in Pt(3)/Au(3)/Cr(50) (left) and Pt(3)/Cr(50) (right).  \n \nReferences:  \n[1] P. Daniil and E. Cohen, J. Appl. Phys. 53, 8257 (1982)  \n[2] L. J. van der Pauw, Philips Technical Review (1958)  \n"    },    {        "title": "1904.03873v2.Anisotropic_magnetocaloric_effect_in_Fe___3_x__GeTe__2_.pdf",        "content": "arXiv:1904.03873v2  [cond-mat.str-el]  27 Aug 2019Anisotropic magnetocaloric effect in Fe 3−xGeTe2\nYu Liu1*, Jun Li1, Jing Tao1, Yimei Zhu1, and Cedomir Petrovic1*\n1Condensed Matter Physics and Materials Science Department , Brookhaven National Laboratory, Upton, New\nY ork 11973, USA\n*yuliu@bnl.gov and petrovic@bnl.gov\nABSTRACT\nWe present a comprehensive study on anisotropic magnetocal oric porperties of the van der Waals weak-itinerant ferroma gnet\nFe3−xGeTe2that features gate-tunable room-temperature ferromagnet ism in few-layer device. Intrinsic magnetocrystalline\nanisotropy is observed to be temperature-dependent and mos t likely favors the long-range magnetic order in thin Fe 3−xGeTe2\ncrsytal. The magnetic entropy change −∆SMalso reveals an anisotropic characteristic between H//ab andH//c , which\ncould be well scaled into a universal curve. The peak value −∆Smax\nM of 1.20 J kg−1K−1and the corresponding adiabatic\ntemperature change ∆Tadof 0.66 K are deduced from heat capacity with out-of-plane fie ld change of 5 T. By fitting of the\nfield-dependent parameters of −∆Smax\nMand the relative cooling power RCP , it gives −∆Smax\nM∝Hnwithn= 0.603(6) and\nRCP∝Hmwithm= 1.20(1) whenH//c . Given the high and tunable Tc, Fe3−xGeTe2crystals are of interest for fabricating\nthe heterostructure-based spintronics device.\nIntroduction\nIntrinsic long-range ferromagnetism recently achieved in two-dimensional-limit van der Waals (vdW) crystals opens u p great\npossibilities for both studying fundamental two-dimensio nal (2D) magnetism and engineering novel spintronic vdW het eros-\ntuctures.1–5Fe3GeTe2is a promising candidate since its Curie temperature ( Tc) in bulk is high and depends on the concen-\ntration of Fe atoms, ranging from 150 to 230 K.6–11Intrinsic magnetocrystalline anisotropy in few-layer cou nteracts thermal\nfluctuation and favors the 2D long-range ferromagnetism wit h a lower Tcof 130 K.5Most significantly, the Tccan be ionic-\ngate-tuned to room temperature in few-layers which is of hig h interest for electrically controlled magnetoelectronic devices.12\nThe layered Fe 3−xGeTe2displays a hexagonal structure belonging to the P6 3/mmc space group, where the 2D layers of\nFe3−xGe sandwiched between nets of Te ions are weakly connected by vdW bonding [Fig. 1(a)].6There are two inequivalent\nWyckoff positions of Fe atoms which are denoted as Fe1 and Fe2 . The Fe1-Fe1 dumbbells are situated in the centre of the\nhexagonal cell in the honeycomb lattice, composed of covale ntly bonded Fe2-Ge atoms. No Fe atoms occupy the interlayer\nspace and Fe vacancies only occur in the Fe2 sites.13Local atomic environment is also studied by the M¨ ossbauer a nd X-ray\nabsorption spectroscopies.14,15Partially filled Fe dorbitals results in an itinerant ferromagnetism in Fe 3−xGeTe2,16which\nexhibits exotic physical phenomena such as nontrivial anom alous Hall effect,17–19Kondo lattice behavior,20strong electron\ncorrelations,21and unusual magnetic domain structures.22,23A second-step satellite transition T∗is also observed just below\nTc, and is not fully understood.10,15\nHere we address the anisotropy in Fe 3−xGeTe2as well as the magnetocaloric effect investigated by heat ca pacity and dc\nmagnetization measurements. The magnetocrystalline anis otropy is observed to be temperature-dependent. The magnet ic en-\ntropy change ∆SM(T,H)also reveals an anisotropic characteristic and could be wel l scaled into a universal curve. Moreover,\nthe−∆Smax\nM follows the power law of Hnwithn= 0.603(6) , and the relative cooling power RCP depends on Hmwith\nm= 1.20(1) .\nMethods\nHigh quality Fe 3−xGeTe2single crystals were synthesized by the self-flux technique .14The element analysis was performed\nusing energy-dispersive X-ray spectroscopy (EDX) in a JEOL LSM-6500 scanning electron microscope (SEM). The selected\narea electron diffraction pattern was taken via a double abe rration-corrected JEOL-ARM200F operated at 200 kV . The dc\nmagnetization and heat capacity were measured in Quantum De sign MPMS-XL5 and PPMS-9 systems with the field up to 5\nT.\nResults and Discussion\nThe average stoichiometry of our flux-grown Fe 3−xGeTe2single crystals was determined by examination of multiple p oints.\nThe actual concentration is determined to be Fe 2.64(6)Ge0.87(4)Te2[Fig. 1(b)], and it is referred to as Fe 3−xGeTe2throughoutFigure 1. (Color online). (a) Crystal structure and (b) X-ray energy- dispersive spectrum of Fe 3−xGeTe2single crystal. Inset\nshows a photograph of Fe 3−xGeTe2single crystal on a 1 mm grid. (c) X-ray diffraction (XRD) pat tern of Fe 3−xGeTe2. Inset\nshows the electron diffraction pattern taken along the [001 ] zone axis direction. (d) Temperature dependence of the red uced\nmagnetization with out-of-plane field of Fe 3−xGeTe2fitted using spin-wave (SW) model and single-particle (SP) m odel.\nInset shows the temperature dependence of zero-field-cooli ng (ZFC) magnetization of Fe 3−xGeTe2measured at H= 1 T\napplied along the caxis.\nthis paper. The as-grown single crystals are mirror-like an d metallic platelets with the crystallographic caxis perpendicular to\nthe platelet surface with dimensions up to 10 millimeters [i nset in Fig. 1(b)]. In the 2 θX-ray diffraction pattern [Fig. 1(c)],\nonly the(00l)peaks are detected, confirming the crystal surface is normal to thecaxis. The corresponding electron diffraction\npattern [inset in Fig. 1(c)] also confirms the high quality of single crystals.\nFigure 1(d) presents the low temperature thermal demagneti zation analysis for Fe 3−xGeTe2with out-of-plane field us-\ning both spin-wave (SW) model and single-particle (SP) mode l. The temperature dependence of zero-field-cooling (ZFC)\nmagnetization M(T)for Fe3−xGeTe2measured in H= 1 T applied along the caxis is shown in the inset of Fig. 1(d).\nLocalized-moment spin-wave excitations can be described b y a Bloch equation:24–26\n∆M\nM(0)=M(0)−M(T)\nM(0)=AT3/2+BT5/2+..., (1)\nwhereAandBare the coefficients. The M(0)is the magnetization at 0 K, which is usually estimated from t he extrapolation\nofM(T). TheT3/2term stems from harmonic contribution and the T5/2term is a high-order contribution in spin-wave\ndispersion relation. In an itinerant magnetism, it is a resu lt of excitation of electrons from one subband to the other. T he\nsingle-particle excitation is:24\n∆M\nM(0)=M(0)−M(T)\nM(0)=CT3/2exp−∆\nkBT, (2)\nwhereC,∆andkBare fit coefficient, the energy gap between the Fermi level and the top of the full subband and the Boltzmann\nconstant, respectively. It can be seen that the SW model give s a better fit than the SP model up to 0.9 Tc[Fig. 1(d)], indicating\npossible localized moment, in agreement with the bad-metal lic resistivity of Fe 3−xGeTe2.15It is also understandable that\nthe SP model fails due to strong electron correlation in Fe 3−xGeTe2.21The fitting yields A= 8.4(7)×10−5K−3/2,B=\n1.24(5)×10−6K−5/2,C= 3.4(1)×10−4K−3/2and∆ = 3.9(4) meV .\nFigure 2(a) shows the temperature dependence of heat capaci tyCpfor Fe3−xGeTe2measured in zero-field and out-of-\nplane field of 2 and 5 T, respectively. The ferromagnetic orde r anomaly at Tc= 153 K in the absence of magnetic field is\ngradually suppressed in fields. The entropy S(T,H)can be determined by\nS(T,H) =/integraldisplayT\n0Cp(T,H)\nTdT. (3)\n2/8/s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48 /s49/s56/s48 /s50/s48/s48/s50/s50/s48/s50/s52/s48/s50/s54/s48/s50/s56/s48\n/s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48 /s49/s56/s48 /s50/s48/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50\n/s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48 /s49/s56/s48 /s50/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s32/s48/s32/s84\n/s32/s50/s32/s84\n/s32/s53/s32/s84/s72 /s32/s47/s47/s32 /s99\n/s84 /s32/s40/s75/s41/s67\n/s112/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41/s40/s97/s41/s32/s50/s32/s84\n/s32/s53/s32/s84/s40/s98/s41\n/s84 /s32/s40/s75/s41/s45 /s83\n/s77/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41/s40/s99/s41/s32/s50/s32/s84\n/s32/s53/s32/s84/s84\n/s97/s100/s32/s40/s75/s41\n/s84 /s32/s40/s75/s41\nFigure 2. (Color online). Temperature dependences of (a) the specific heatCp, (b) the magnetic entropy change −∆SM,\nand (c) the adiabatic temperature change ∆Tadfor Fe3−xGeTe2with out-of-plane field changes of 2 and 5 T.\nThe magnetic entropy change ∆SM(T,H)can be approximated as ∆SM(T,H) =SM(T,H)−SM(T,0). In addition,\nthe adiabatic temperature change ∆Tadcaused by the field change can be derived by ∆Tad(T,H) =T(S,H)−T(S,0)at\nconstant total entropy S(T,H). Figures 2(b) and 2(c) present the temperature dependence o f−∆SMand∆Tadestimated\nfrom heat capacity with out-of-plane field. They are asymmet ric and attain a peak around Tc. The maxima of −∆SMand\n∆Tadincrease with increasing field and reach the values of 1.20 J k g−1K−1and 0.66 K, respectively, with the field change of\n5 T. Since a large magnetic anisotropy is observed in Fe 3−xGeTe2, it is of interest to further calculate its anisotropic magn etic\nentropy change.\nFigures 3(a) and 3(b) present the magnetization isotherms w ith field up to 5 T applied in the abplane and along the caxis,\nrespectively, in temperature range from 100 to 200 K with a te mperature step of 4 K. The magnetic entropy change can be\nobtained from dc magnetization measurement as:27\n∆SM(T,H) =/integraldisplayH\n0/bracketleftbigg∂S(T,H)\n∂H/bracketrightbigg\nTdH. (4)\nWith the Maxwell’s relation/bracketleftBig\n∂S(T,H)\n∂H/bracketrightBig\nT=/bracketleftBig\n∂M(T,H)\n∂T/bracketrightBig\nH, it can be rewritten as:28\n∆SM(T,H) =/integraldisplayH\n0/bracketleftbigg∂M(T,H)\n∂T/bracketrightbigg\nHdH. (5)\nWhen the magnetization is measured at small temperature and field steps, ∆SM(T,H)is approximated:\n∆SM(T,H) =/integraltextH\n0M(T+∆T)dH−/integraltextH\n0M(T)dH\n∆T. (6)\nFigures 3(c) and 3(d) show the calculated −∆SM(T,H)as a function of temperature in various fields up to 5 T applied in\ntheabplane and along the caxis, respectively. All the −∆SM(T,H)curves feature a pronounced peak around Tc, similar\nto those obtained from heat capacity [Fig. 2(b)], and the pea k broadens asymmetrically on both sides with increase in fiel d.\nMoreover, the value of −∆SM(T,H)increases monotonically with increase in field; the peak −∆SMreaches 1.26 J kg−1\nK−1with in-plane field change and 1.44 J kg−1K−1with out-of-plane change of 5 T, respectively. We calculate d the rotating\nmagnetic entropy change ∆SR\nMas\n∆SR\nM(T,H) = ∆SM(T,Hc)−∆SM(T,Hab). (7)\nThe asymmetry of −∆SM(T,H)is more apparent in the temperature dependence of −∆SR\nM[Fig. 3(e)]. Furthermore, there\nis a slight shift of −∆SMmaximum towards higher temperature when the field varies fro m 1 to 5 T [Figs. 3(c) and 3(d)].\nThis shift of Tpeak excludes the mean field model but could be reproduced by the He isenberg model due to its discrepancy\nwithTc.29\nAround the second order phase transition,30the magnetic entropy maximum change is −∆Smax\nM=aHn,31whereais a\nconstant and nis32\nn(T,H) =dln|∆SM|/dln(H). (8)\n3/8/s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48 /s49/s56/s48 /s50/s48/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50\n/s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48 /s49/s56/s48 /s50/s48/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48 /s49/s56/s48 /s50/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s32/s49/s32/s84\n/s32/s50/s32/s84\n/s32/s51/s32/s84\n/s32/s52/s32/s84\n/s32/s53/s32/s84/s72 /s32/s47/s47/s32 /s97/s98\n/s84 /s32/s40/s75/s41/s45 /s83\n/s77/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41/s40/s99/s41/s72 /s32/s47/s47/s32 /s99\n/s40/s100/s41\n/s84 /s32/s40/s75/s41/s45 /s83\n/s77/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41/s72 /s32/s47/s47/s32 /s97/s98/s32 /s32/s72 /s32/s47/s47/s32 /s99/s40/s101/s41/s32/s49/s32/s84\n/s32/s50/s32/s84\n/s32/s51/s32/s84\n/s32/s52/s32/s84\n/s32/s53/s32/s84/s45 /s83/s32/s82 /s77/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41\n/s84 /s32/s40/s75/s41/s50/s48/s48/s32/s75/s77 /s32/s40/s49/s48/s52\n/s32/s101/s109/s117/s32/s109/s111/s108/s45/s49\n/s41\n/s72 /s32/s40/s84/s41/s40/s97/s41\n/s72 /s32/s47/s47/s32 /s97/s98/s49/s48/s48/s32/s75\n/s50/s48/s48/s32/s75/s49/s48/s48/s32/s75\n/s72 /s32/s47/s47/s32 /s99/s40/s98/s41/s77 /s32/s40/s49/s48/s52\n/s32/s101/s109/s117/s32/s109/s111/s108/s45/s49\n/s41\n/s72 /s32/s40/s84/s41\nFigure 3. (Color online). Initial isothermal magnetization curves f romT= 100 to 200 K with temperature step of T= 4 K\nmeasured with (a) in-plane and (b) out-of-plane fields. Temp erature dependence of magnetic entropy change −∆SM\nobtained with (c) in-plane and (d) out-of-plane field change s, and (e) the difference −∆SR\nM.\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50\n/s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48 /s49/s56/s48 /s50/s48/s48/s48/s46/s54/s48/s46/s57/s49/s46/s50/s49/s46/s53/s49/s46/s56\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s32/s49/s32/s84\n/s32/s50/s32/s84\n/s32/s51/s32/s84\n/s32/s52/s32/s84\n/s32/s53/s32/s84/s68 /s83\n/s77/s47/s68 /s83/s32/s109/s97/s120 /s77\n/s116/s40/s99/s41\n/s72 /s32/s47/s47/s32 /s99/s32/s49/s32/s84\n/s32/s50/s32/s84\n/s32/s51/s32/s84\n/s32/s52/s32/s84\n/s32/s53/s32/s84/s72 /s32/s47/s47/s32 /s99/s40/s100/s41/s45/s68 /s83\n/s77/s47/s72/s32/s110\n/s101 /s47/s72/s49/s47 /s40/s98/s32 /s43\n/s32/s103 /s41/s110 /s32/s61/s32/s49/s32/s43/s32/s40 /s98 /s32 /s45/s32/s49/s41/s47/s40 /s98 /s32 /s43 /s32/s103/s41 /s32/s45/s50 /s48 /s50/s48/s49\n/s72 /s32/s47/s47/s32 /s97/s98/s68 /s83\n/s77/s47/s68 /s83/s32/s109/s97/s120 /s77/s32\n/s116/s32/s49/s32/s84\n/s32/s50/s32/s84\n/s32/s51/s32/s84\n/s32/s52/s32/s84\n/s32/s53/s32/s84/s72 /s32/s47/s47/s32 /s99/s40/s97/s41/s110\n/s84/s32 /s40/s75/s41/s110 /s32/s61/s32/s48/s46/s54/s48/s51/s40/s54/s41/s45/s68 /s83/s32/s109/s97/s120 /s77\n/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41\n/s72/s32 /s40/s84/s41/s40/s98/s41\n/s72 /s32/s47/s47/s32 /s99\n/s32/s45/s68 /s83/s32/s109 /s97/s120\n/s77\n/s45/s68 /s83/s32/s109 /s97/s120\n/s77/s32/s61/s32 /s97/s72/s32/s110\n/s48/s52/s48/s56/s48/s49/s50/s48\n/s109 /s32/s61/s32/s49/s46/s50/s48/s40/s49/s41\n/s82/s67/s80 /s32/s40/s74/s32/s107/s103/s45/s49\n/s41\n/s32/s82/s67/s80\n/s82/s67/s80 /s32/s61/s32 /s98/s72/s32/s109 \nFigure 4. (Color online). (a) Temperature dependence of nin various fields. (b) Field dependence of the maximum\nmagnetic entropy change −∆Smax\nM and the relative cooling power RCP with power law fitting in re d solid lines. (c) The\nnormalized ∆SMas a function of the rescaled temperature twith out-of-plane field and in-plane field (inset). (d) Scali ng\nplot based on the critical exponents β= 0.372 and γ= 1.265.14\n4/8Figure 4(a) shows the temperature dependence of n(T)in various fields. All the n(T)curves follow an universal behavior.33\nAt low temperatures, nhas a value close to 1. At high temperatures, ntends to 2 as a consequence of the Curie-Weiss law. At\nT=Tc,nhas a minimum. Additionally, the exponent natTcis related to the critical exponents:30\nn(Tc) = 1+/parenleftbiggβ−1\nβ+γ/parenrightbigg\n= 1+1\nδ/parenleftbigg\n1−1\nβ/parenrightbigg\n, (9)\nwhereβ,γ, andδare the critical exponents related to the spontaneous magne tizationMsbelowTc, the inverse initial suscep-\ntibilityH/M aboveTc, and the isotherm M(H)atTc, respectively.\nRelative cooling power (RCP) could be used to estimate the co oling efficiency:34\nRCP=−∆Smax\nM×δTFWHM, (10)\nwhere−∆Smax\nM is the entropy change maximum around TcandδTFWHM is the width at half maximum. The RCP also\ndepends on the field as RCP=bHm, wherebis a constant and mis related to the critical exponent δ:\nm= 1+1\nδ. (11)\nFigure 4(b) presents the field-dependent −∆Smax\nM and RCP. The RCP is 113.3 J kg−1within field change of 5 T for\nFe3−xGeTe2. This is one half of those in manganites and much lower than in ferrites.35,36Fitting of the −∆Smax\nM and\nRCP gives n= 0.603(6) andm= 1.20(1) , which are close to the values estimated from the critical ex ponents (Table I).\nTechnique β γ δ n m\n−∆Smax\nM 0.603(6)\nRCP 1.20(1)\nMAP 0.374(1) 1.273(8) 4.404(12) 0.620(1) 1.227(1)\nKFP 0.372(4) 1.265(15) 4.401(6) 0.616(2) 1.227(1)\nCI 4.50(1) 1.222(1)\nTable 1. Critical exponents of Fe 3−xGeTe2.14The MAP, KFP and CI represent the modified Arrott plot, the Kou vel-Fisher\nplot and the critical isotherm, respectively.\nThe scaling of magnetocaloric data is constructed by normal izing all the −∆SMcurves against the maximum −∆Smax\nM,\nnamely,∆SM/∆Smax\nM by rescaling the temperature tbelow and above Tcas defined in:\nt−= (Tpeak−T)/(Tr1−Tpeak),T < T peak, (12)\nt+= (T−Tpeak)/(Tr2−Tpeak),T > T peak, (13)\nwhereTr1andTr2are the temperatures of two reference points corresponding to∆SM(Tr1,Tr2) =1\n2∆Smax\nM.37All the\n−∆SM(T,H)curves collapse onto a single curve regardless of temperatu re and field, as shown in Fig. 4(c). In the phase\ntransition region, the scaling analysis of −∆SMcan also be expressed as\n−∆SM\naM=Hnf(ε\nH1/∆), (14)\nwhereaM=T−1\ncAδ+1Bwith A and B representing the critical amplitudes as in Ms(T) =A(−ε)βandH=BMδ,\n∆ =β+γ, andf(x)is the scaling function.38If the critical exponents are appropriately chosen, the −∆SM(T)curves\nshould be rescaled into a single curve, consistent with norm alizing all the −∆SMcurves with two reference temperatures.\nBy using the values of β= 0.372 and γ= 1.265 obtained by the Kouvel-Fisher plot,14we have replotted the scaled −∆SM\nfor Fe3−xGeTe2[Fig. 4(d)]. The good overlap of the experimental data point s clearly indicates that the obtained critical\nexponents for Fe 3−xGeTe2are not only in agreement with the scaling hypothesis but als o intrinsic.\nThen we estimated the magnetocrystalline anisotropy of Fe 3−xGeTe2. By using the Stoner-Wolfarth model a value for the\nmagnetocrystalline anisotropy constant Kucan be estimated from the saturation regime in isothermal ma gnetization curves\n[Fig. 5(a)].39Within this model the magnetocrystalline anisotropy in the single domain state is related to the saturation\nmagnetic field Hsand the saturation moment Mswithµ0is the vacuum permeability:\n2Ku\nMs=µ0Hsat. (15)\n5/8/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48/s56/s48/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48/s49/s46/s48/s49/s46/s50/s49/s46/s52\n/s77\n/s115/s32/s40/s49/s48/s52\n/s32/s101/s109/s117/s32/s109/s111/s108/s45/s49\n/s41/s40/s98/s41\n/s40/s100/s41/s75\n/s117/s32/s40/s107/s74/s32/s109/s45/s51\n/s41/s49/s52/s48/s32/s75/s72 /s32/s47/s47/s32 /s97/s98/s40/s97/s41/s77/s32 /s40/s49/s48/s52\n/s32/s101/s109/s117/s32/s109/s111/s108/s45/s49\n/s41/s49/s48/s32/s75\n/s84/s32/s61/s32/s49/s48/s32/s75/s72\n/s115/s40/s84 /s41/s40/s99/s41\nFigure 5. (Color online). (a) Initial isothermal magnetization curv es fromT= 10 to 140 K with in-plane fields. Temperature\nevolution of (b) the saturation magnetization Ms, (c) the saturation field Hs, and (d) the anisotropy constant Ku.\nWhenH//ab , the anisotropy becomes maximal. We estimated the saturati on magnetization Msby using a linear fit of M(H)\nabove a magnetic field of 2.5 T with in-plane field [Fig. 5(b)], which monotonically decreases with increasing temperatur e.\nThen we determined the saturation field Hsas the intersection point of two-linear fits, one being a fit to the saturated regime\nat high fields and one being a fit of the unsaturated linear regi me at low fields. The value of Hsincreases at low temperature,\nwhich is possibly related to a spin reorientation transitio n,15and then decreases with increasing temperature [Fig. 5(c)] . Figure\n5(d) presents the temperature evolution of Kufor Fe3−xGeTe2, which can not be described by the l(l+1)/2power law.40,41\nThe value of Kufor Fe3−xGeTe2is about 69 kJ cm−3at 10 K, slightly increases to 78 kJ cm−3at 50 K, and then decrease\nwith increasing temperature, which are comparable to those for CrBr 3, but smaller than those for CrI 3.42The decrease of Ku\nwith increasing temperature is also observed in CrBr 3and CrI 3,42arising from a large number of local spin clusters.43,44In\na pure two-dimensional system, materials with isotropic sh ort-range exchange interactions can not magnetically orde r. The\nlong-range ferromagnetism in few-layers of Fe 3−xGeTe2could possibly be favored by the large magnetocrystalline a nisotropy.\nConclusion\nIn summary, we have investigated in detail the magnetocalor ic effect of Fe 3−xGeTe2single crystals. The large magnetocrys-\ntalline anisotropy is found to be temperature-dependent an d probably establishes the long-range ferromagnetism in fe w-layers\nof Fe3−xGeTe2. The magnetic entropy change −∆SMalso reveals an anisotropic characteristic and could be wel l scaled into\na universal curve independent on temperature and field. By fit ting of the field-dependent parameters of −∆Smax\nM and the\nrelative cooling power RCP, it gives −∆Smax\nM∝Hnwithn= 0.603(6) andRCP∝Hmwithm= 1.20(1) whenH//c .\nConsidering its tunable room-temperature ferromagnetism and hard magnetic properties in nanoflakes, further investi gation\non the size dependence of magnetocaloric effect is of intere st.\nReferences\n1.McGuire, M. A. et al. Coupling of Crystal Structure and Magnetism in the Layered, Ferromagnetic Insulator CrI 3.Chem.\nMater. 27, 612 (2015).\n2.Huang, B. et al. Layer-dependent ferromagnetism in a van der Waals crystal d own to the monolayer limit. Nature 546,\n270 (2017).\n3.Gong, C. et al. 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Temperature-dependent magnetic properties of FePt: Effec t spin Hamiltonian model. Europhys. Lett.\n69, 805 (2005).\n42.Richter, N. et al. Temperature-dependent magnetic anisotropy in the layered magnetic semiconductors CrI 3and CrBr 3.\nPhys. Rev. Mater. 2, 024004 (2018).\n43.Zener, C. Classical Theory of the Temperature Dependence of Magnetic Anisotropy Energy. Phys. Rev. 96, 1335 (1954).\n44.Carr, W. J. Temperature Dependence of Ferromagnetic Anisot ropy. J. Appl. Phys. 29, 436 (1958).\nAcknowledgements\nWe thank J. Warren for help with the scanning electron micros copy (SEM) measurement. Work at Brookhaven National\nLaboratory is supported by the US DOE, Contract No. DE-SC001 2704.\nAuthor contributions statement\nY .L. and C.P. designed this study and synthesized crystals; Y . L. performed magnetization and heat capacity measuremen ts.\nJ.L, J.T. and Y .Z. contributed TEM measurement. Y .L. and C.P . organized and wrote the paper with input from all collabora tors.\nThis manuscript reflects the contribution and ideas of all au thors.\nAdditional information\nCompeting interests: The authors declare no competing inte rests.\n8/8"    },    {        "title": "1904.04247v3.Magnetic_anisotropy_and_entropy_change_in_trigonal_Cr___0_62__Te.pdf",        "content": "arXiv:1904.04247v3  [cond-mat.mtrl-sci]  25 Nov 2019Magnetic anisotropy and entropy change in trigonal Cr 5Te8\nYu Liu,1Milinda Abeykoon,2Eli Stavitski,2Klaus Attenkofer,2and C. Petrovic1\n1Condensed Matter Physics and Materials Science Department ,\nBrookhaven National Laboratory, Upton, New York 11973, USA\n2National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA\n(Dated: November 27, 2019)\nWe present a comprehensive investigation on anisotropic ma gnetic and magnetocaloric properties\nof the quasi-two-dimensional weak itinerant ferromagnet t rigonal Cr 5Te8single crystals. Magnetic-\nanisotropy-induced satellite transition T∗is observed at low fields applied parallel to the abplane\nbelow theCurie temperature ( Tc∼230 K).T∗is featured byan anomalous magnetization downturn,\nsimilar to that in structurally related CrI 3, which possibly stems from an in-plane antiferromagnetic\nalignment induced by out-of-plane ferromagnetic spins can ting. Magnetocrystalline anisotropy is\nalso reflected in magnetic entropy change ∆ SM(T,H) and relative cooling power RCP. Given the\nhigh Curie temperature, Cr 5Te8crystals are materials of interest for nanofabrication in b asic science\nand applied technology.\nI. INTRODUCTION\nAchieving long-range magnetism at low dimensions as\nwell as high temperature is of great importance for the\ndevelopment of next-generation spintronics. As stated in\nthe Mermin-Wagner theorem,1materials with isotropic\nHeisenberginteractionscannot magneticallyorderin the\ntwo-dimensional (2D) limit. Strong magnetic anisotropy\nsuch as exchangeanisotropyand/or single-ionanisotropy\nmay removethis restriction. Recently, 2D long-rangefer-\nromagnetism was observed in van der Waals (vdW) CrI 3\nand Cr 2Ge2Te6,2–8however the Curie temperature Tcis\nsignificantly lowered when reducing to monolayer or few\nlayers. The corresponding Tcin monolayer and bilayer\nwas only ∼45 K for CrI 3and 25 K in Cr 2Ge2Te6, re-\nspectively. Therefore, high- Tcferromagnets with strong\nanisotropy are highly desirable.\nBinary chromium tellurides Cr 1−xTe are such promis-\ning candidates with high- Tcranging from 170 to 360 K\ndepending on different Cr vacancies.9–13The carriers of\nmagnetism in CrI 3and Cr 2Ge2Te6are Cr3+ions with\nspinS= 3/2 which are octahedrally coordinated by Te\nligands and form a honeycomb magnetic lattice in the ab\nplane. Cr 1−xTe crystallizesin a defective NiAs structure,\nshowing similarities in crystal structure, magnetic ion,\nand magnetic ordering. The mechanism of FM in this\nsystem could be attributed to the dominant FM superex-\nchange coupling between half-filled Cr t2gand empty eg\nstates via Te porbitals, againstthe AFM direct exchange\ninteractions of Cr t2gstates. The saturation magnetiza-\ntion in Cr 1−xTe is observed as 1 .7∼2.5µB, which is\nmuch smaller than the expected moment value of Cr us-\ning ionic model due to the spin canting, itinerant nature\nof thedelectrons and the existence of mixed valence\nCr.14The electronic structure of Cr 1−xTe has further\nbeen studied by photoemission spectroscopy,15showing\nthe importance of electron-correlationeffect. Here we fo-\ncusontrigonalCr 5Te8withx= 0.375,whichhasahigher\nTcof 230 K compared with monoclinic space group.16–20\nStrong magnetic anisotropy with out-of-plane easy caxis\nhas been confirmed.11Critical behavior analysis of trig-onal Cr 5Te8illustrates that the ferromagnetic state be-\nlowTcconformstothree-dimensional(3D) Isingmodel,19\nholding a high potential for establishing long-range mag-\nnetic order when thinned down to 2D limit. Anisotropic\nanomalous Hall effect (AHE) was recently observed in\ntrigonal Cr 5Te8.20,21Calculations of band structure is\ndesired to explain its origin, calling for precise structural\ninformation.\nMagnetocaloriceffect(MCE)intheFMvdWmaterials\nprovides important insight into the magnetic properties.\nThe magnetocrystalline anisotropy constant Kuis found\nto be largerfor Cr 2Si2Te6when compared to Cr 2Ge2Te6,\nresulting in larger rotational magnetic entropy change at\nTc.22CrI3also exhibits anisotropic −∆Smax\nMwith values\nof4.24and 2.68J kg−1K−1at 5 T for H∝bardblcandH∝bardblab,\nrespectively.23Intriguingly, there is an anisotropic mag-\nneticanomaly T∗justbelow Tcinlowfields;2,23theorigin\nofT∗is still unknown.\nIn this work we investigate the anisotropy of magnetic\nand magnetocaloric properties in trigonal Cr 5Te8single\ncrystal. A satellite transition T∗belowTcwas observed,\nat which the huge magnetic anisotropy emerges. We as-\nsign its origin to the interplay among magnetocrystalline\nanisotropy,temperatureandmagneticfield. Whereasthe\nanisotropic magnetic entropy change ∆ SMin Cr5Te8is\nsmall, the magnetocrystalline anisotropy constant Kuis\nfoundtobetemperature-dependentandcomparablewith\nCrI3and Cr 2Ge2Te6. As a result of high- Tcin Cr5Te8, it\nwould be of interest to design low-dimensional FM het-\nerostructures.\nII. EXPERIMENTAL DETAILS\nPlate-like single crystals of Cr 5Te8were fabricated by\nthe self-flux method and the surface is the abplane.19,20\nThe structure was characterized by powder x-ray diffrac-\ntion (XRD) in the transmissionmode at 28-D-1beamline\nof the National Synchrotron Light Source II (NSLS II) at\nBrookhaven National Laboratory (BNL). Data were col-\nlected using a 0.5mm2beam with wavelength λ∼0.16682\nTABLE I. Structural parameters for Cr 5Te8obtained from\nsynchrotron powder XRD at room temperature.\nChemical formula Cr 5Te8\nSpace grroup P¯3m1\na(˚A) 7.7951\nc(˚A) 11.9766\natom site x y z Occ. U iso(˚A2)\nCr Cr1 0 0 0 0.87 0.0019\nCr Cr2 0.4920 0.5080 0.2515 1 0.0019\nCr Cr3 0 0 0.2747 1 0.0019\nCr Cr4 0.5 0 0.5 0.264 0.0019\nTe Te1 0.3333 0.6667 0.1164 1 0.00468\nTe Te2 0.3333 0.6667 0.6217 1 0.00468\nTe Te3 0.1652 0.8348 0.3816 1 0.00468\nTe Te4 0.8285 0.1715 0.1301 1 0.00468\n˚A. A Perkin Elmer 2D detector (200 ×200 microns) was\nplaced orthogonal to the beam path 990 mm away from\nthe sample. The x-ray absorption spectroscopy measure-\nment was performed at 8-ID beamline of NSLS II (BNL)\nin fluorescence mode. The x-ray absorption near edge\nstructure (XANES) and the extended x-ray absorption\nfine structure (EXAFS) spectra were processed using the\nAthena software package. The extracted EXAFS signal,\nχ(k), wasweighedby k2toemphasizethe high-energyos-\ncillation and then Fourier-transformed in a krange from\n2 to 12˚A−1to analyze the data in Rspace. The magne-\ntization data as a function of temperature and field were\ncollected using Quantum Design MPMS-XL5 system in\ntemperature range 200 ∼280 K with a step of 4 K on\ncleavedcrystalsin ordertoremovesurfacecontamination\nof residual Te flux droplets on the surface.\nIII. RESULTS AND DISCUSSION\nRietveld powder diffraction analysis was carried out\non data obtained from the raw 2D diffraction data in-\ntegrated and converted to intensity versus Qusing the\nFit2d software where Q= 4πsinθ/λ is the magnitude\nof the scattering vector.24The refinement was performed\nusing GSAS-II modeling suite.25Figure 1(a) shows the\nrefinement result (goodness of fit ∼4%) of Cr 5Te8\n(0.885 by weight) and residual Te flux impurity (0.115\nby weight). The crystal structure of trigonal Cr 5Te8can\nbe well refined in the P¯3m1 space group with four crys-\ntallographically different sites for both Cr and Te atoms,\nleading to the formation of a five-layer superstructure\nof the CdI 2type.26The detailed structural parameters\nare summarized in Table I. The superstructure is com-\nposed ofthree types ofCr layerswith a stackingsequence\nabcba. Cr2 and Cr3 occupy the vacancy-free atomic lay-\ners, while Cr1 and Cr4 distribute in the metal-deficient\nlayers [inset in Fig. 1(b)]. The average Cr-Te bond dis-/s53/s57/s54/s48 /s53/s57/s56/s48 /s54/s48/s48/s48 /s54/s48/s50/s48 /s54/s48/s52/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s51 /s54 /s57 /s49/s50\n/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53/s48/s52/s56/s49/s50/s49/s54/s50/s48\n/s40/s97/s41/s32/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s97/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s40/s97/s46/s32/s117/s46/s41\n/s80/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s40/s98/s41/s32/s32/s32/s67/s114/s32/s75/s45/s101/s100/s103/s101\n/s107/s50\n/s99 /s40/s107/s41/s32/s40/s197/s45/s50\n/s41\n/s107/s32/s40/s197/s45/s49\n/s41\n/s40/s99/s41/s99 /s40/s82/s41 /s32/s40/s97/s46/s32/s117/s46/s41\n/s82/s32/s40/s197/s41\n/s32/s32/s32/s32\n/s32/s32/s32/s32/s32/s32/s32/s32\n/s67/s114\n/s53/s84/s101\n/s56\nFIG. 1. (Color online). (a) Refinement of synchrotron x-ray\ndiffraction data of Cr 5Te8: data (solid symbols), structural\nmodel (red solid line), difference curve (blue solid line, off set\nfor clarity). Vertical tickmarks denote reflections in the m ain\nphase (black, top row) and the secondary Te impurity (red,\nbottom row). (b) Normalized Cr K-edge XANES spectra and\n(c) Fourier transform magnitudes of EXAFS data of Cr 5Te8\ntakenat room temperature. The experimentaldataare shown\nas blue symbols alongside the model fit plotted as red line.\nThe inset in (c) shows the corresponding EXAFS oscillation\nwith the model fit.\ntancesin vacancy-freelayersrangefrom 2.701to 2.805 ˚A,\nmatch well with the reported values for Cr 2(Si,Ge) 2Te6\n(2.741 - 2.803 ˚A) and CrTe 3(2.700 - 2.755 ˚A).26How-\never, the Cr-Te bond distances in metal-deficient layers\nare shorter with values of 2.673 - 2.681 ˚A. The short-\nest Te-Te interatomic separations range from 3.664 to\n3.779˚A, significantly shorter than the sum of their an-\nionic radii 4.2 ˚A, indicative of weak bonding interactions3\n/s48/s50/s48/s52/s48\n/s48/s56/s49/s54\n/s48/s49/s50\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s50/s48/s52/s48\n/s48/s56/s49/s54\n/s48/s52/s56\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s45/s49/s48/s49\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s45/s49/s48/s49/s32/s90/s70/s67\n/s32/s70/s67/s40/s97/s41\n/s72/s32/s47/s47/s32/s99/s32/s97/s116/s32/s49/s48/s32/s107/s79/s101\n/s40/s98/s41\n/s72/s32/s47/s47/s32/s99/s32/s97/s116/s32/s49/s32/s107/s79/s101/s77/s32/s40/s101/s109/s117/s47/s103/s41/s40/s99/s41\n/s72/s32/s47/s47/s32/s99/s32/s97/s116/s32/s49/s48/s48/s32/s79/s101\n/s40/s100/s41\n/s72/s32/s47/s47/s32/s99/s32/s97/s116/s32/s49/s48/s32/s79/s101/s32/s90/s70/s67\n/s32/s70/s67/s40/s102/s41\n/s72/s32/s47/s47/s32/s97/s98/s32/s97/s116/s32/s49/s48/s32/s107/s79/s101\n/s40/s103/s41\n/s72/s32/s47/s47/s32/s97/s98/s32/s97/s116/s32/s49/s32/s107/s79/s101/s77/s32/s40/s101/s109/s117/s47/s103/s41/s40/s104/s41/s72/s32/s47/s47/s32/s97/s98/s32/s97/s116/s32/s49/s48/s48/s32/s79/s101\n/s40/s105/s41\n/s72/s32/s47/s47/s32/s97/s98/s32/s97/s116/s32/s49/s48/s32/s79/s101/s100/s77/s47/s100/s84\n/s84/s32/s40/s75/s41/s32/s49/s48/s32/s79/s101\n/s32/s49/s48/s48/s32/s79/s101\n/s32/s49/s32/s107/s79/s101\n/s32/s49/s48/s32/s107/s79/s101/s40/s101/s41\n/s72/s32/s47/s47/s32/s99/s32\n/s84\n/s99\n/s100/s77/s47/s100/s84\n/s84/s32/s40/s75/s41/s32/s49/s48/s32/s79/s101\n/s32/s49/s48/s48/s32/s79/s101\n/s32/s49/s32/s107/s79/s101\n/s32/s49/s48/s32/s107/s79/s101/s40/s106/s41\n/s72/s32/s47/s47/s32/s97/s98/s84/s42 \n/s84\n/s99\nFIG. 2. (Color online). (a)-(d) Temperature dependence of\nzero field cooling (ZFC) and field cooling (FC) magnetization\nof Cr5Te8measured at indicated fields applied along the c\naxis. (e) Temperature-dependent dM/dT derived from FC\ncurves with H/bardblc. (f)-(i) Temperature dependence of ZFC\nand FC magnetization of Cr 5Te8measured at indicated fields\napplied in the abplane. (j) Temperature-dependent dM/dT\nderived from FC curves with H/bardblab.\nwith a quasi-2D character.\nFigures 1(b) and 1(c) show the normalized Cr K-edge\nXANES spectra and Fourier transform magnitudes of\nEXAFS spectra of Cr 5Te8, respectively. The XANES\nspectra is close to that of Cr 2Te3with Cr3+state.27The\nprepeak feature is due to a direct quadrupole transition\nto unoccupied 3 dstates that are hybridized with Te p\norbitals. In the single-scattering approximation, the EX-\nAFS could be described by the following equation:28\nχ(k) =/summationdisplay\niNiS2\n0\nkR2\nifi(k,Ri)e−2Ri\nλe−2k2σ2\nisin[2kRi+δi(k)],\nwhereNiis the number of neighbouring atoms at a dis-\ntanceRifromthephotoabsorbingatom. S2\n0isthepassive\nelectrons reduction factor, fi(k,Ri) is the backscattering\namplitude, λis the photoelectron mean free path, δiis\nthe phase shift of the photoelectrons, and σ2\niis the cor-\nrelated Debye-Waller factor measuring the mean square\nrelative displacement of the photoabsorber-backscatter\npairs. The corrected main peak around R∼2.7˚A in/s45/s53/s48 /s45/s50/s53 /s48 /s50/s53 /s53/s48/s45/s50/s45/s49/s48/s49/s50/s45/s53/s48 /s45/s50/s53 /s48 /s50/s53 /s53/s48/s45/s50/s45/s49/s48/s49/s50\n/s49/s50/s48 /s49/s54/s48 /s50/s48/s48 /s50/s52/s48 /s50/s56/s48/s48/s51/s54/s57\n/s49/s50/s48 /s49/s54/s48 /s50/s48/s48 /s50/s52/s48 /s50/s56/s48/s48/s51/s54/s57\n/s72/s32/s47/s47/s32/s97/s98/s77/s32/s40 /s109\n/s66/s47/s67/s114/s41\n/s72/s32/s40/s107/s79/s101/s41/s40/s98/s41\n/s32/s53/s32/s75\n/s32/s49/s48/s48/s32/s75\n/s32/s50/s48/s48/s32/s75\n/s32/s51/s48/s48/s32/s75/s72/s32/s47/s47/s32/s99/s77/s32/s40 /s109\n/s66/s47/s67/s114/s41\n/s72/s32/s40/s107/s79/s101/s41/s32/s53/s32/s75\n/s32/s49/s48/s48/s32/s75\n/s32/s50/s48/s48/s32/s75\n/s32/s51/s48/s48/s32/s75/s40/s97/s41\n/s50/s51/s49/s32/s75/s109/s39/s32/s40/s49/s48/s45/s50\n/s32/s101/s109/s117/s47/s103/s41\n/s84/s32/s40/s75/s41/s32/s72/s32/s47/s47/s32/s97/s98 /s32\n/s32/s72/s32/s47/s47/s32/s99/s50/s51/s56/s32/s75\n/s40/s99/s41\n/s49/s57/s57/s32/s75/s50/s50/s54/s32/s75/s50/s51/s52/s32/s75/s109/s39/s39/s32/s40/s49/s48/s45/s51\n/s32/s101/s109/s117/s47/s103/s41\n/s84/s32/s40/s75/s41/s40/s100/s41\n/s32/s72/s32/s47/s47/s32/s97/s98 /s32\n/s32/s72/s32/s47/s47/s32/s99\nFIG. 3. (Color online). Magnetization as function of field at\nvarious temperatures for (a) H/bardblcand (b) H/bardblab, respec-\ntively. Ac susceptibility real part (c) m′(T) and (d) imagi-\nnary part m′′(T) as a function of temperature measured with\noscillated ac field of 3.8 Oe and frequency of 499 Hz applied\nin theabplane and along the caxis, respectively.\nFig. 1(b) corresponds to the Cr-Te bond distances [Cr2-\nTe1: 2.70(1) ˚A, Cr2-Te4: 2.74(2) ˚A, Cr4-Te2: 2.683(1)\n˚A] extracted from the model fit in the range 2 ˚A to 3˚A,\nonly slightly modified when compared with the average\nbond distances. The peaks in high R range are due to\nlongerCr-Cr (3.01 ˚A∼3.29˚A) and Te-Te (3.66 ˚A∼3.78\n˚A) bond distances and multiple scattering effects.\nTemperature dependence of zero field cooling (ZFC)\nand field cooling (FC) magnetization M(T) taken at var-\nious magnetic fields for Cr 5Te8is presented in Fig. 2.\nWe observe an interesting anisotropic magnetic response\nfor fields applied along the caxis and in the abplane,\nrespectively. In Fig. 2(a), for H∝bardblcat 10 kOe, a typical\nincrease in M(T) on cooling from high temperature cor-\nresponds well to the reported paramagnetic (PM) to FM\ntransition.16Anappreciablethermomagneticirreversibil-\nity occurs between the ZFC and FC curves in lower fields\n[Figs. 2(b)-2(d)], in the magnetically ordered state, in-\ndicating magnetocrystalline anisotropy. On the other\nhand, for H∝bardblab, ananomalouspeak featureis observed.\nThe ZFC and FC magnetization in fields ranging from 10\nkOe to 10 Oe are virtually overlapping in the whole tem-\nperature range [Figs. 2(f)-2(i)]. A similar magnetization\ndownturn below Tcis also seen for Cr 2(Si,Ge) 2Te6and\nCr(Br,I) 3,29,30at which the strong magnetic anisotropy\nemerges. Since the ZFC curvemight be influenced by the\ndomain wall motion, we summarize the field dependence\nofTc(the minimum of dM/dT) andT∗(the maximum\nofdM/dT) from FC magnetization. As shown in Figs.\n2(e) and 2(j), both TcandT∗are field-dependent but\nwith an opposite tendency, i.e., with increasing field the\nTcincreases whereas the T∗gradually decreases.4\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s56/s49/s54/s50/s52/s51/s50\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s56/s49/s54/s50/s52/s51/s50\n/s49/s54/s48 /s49/s56/s48 /s50/s48/s48 /s50/s50/s48 /s50/s52/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s72/s32/s40/s84/s41/s50/s56/s48/s32/s75/s77/s32/s40/s101/s109/s117/s47/s103/s41/s40/s97/s41\n/s50/s48/s48/s32/s75/s72/s32/s47/s47/s32/s97/s98 /s72/s32/s47/s47/s32/s99\n/s50/s56/s48/s32/s75/s50/s48/s48/s32/s75/s40/s98/s41/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s72/s32/s40/s84/s41\n/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50\n/s32/s77\n/s115/s47/s77\n/s115/s40/s49/s54/s48/s75/s41\n/s32/s91/s77\n/s115/s47/s77\n/s115/s40/s49/s54/s48/s75/s41/s93/s51\n/s32/s91/s77\n/s115/s47/s77\n/s115/s40/s49/s54/s48/s75/s41/s93/s49/s48\n/s82/s97/s116/s105/s111/s115/s75\n/s117/s32/s40/s107/s74/s47/s109/s51\n/s41\n/s84/s32/s40/s75/s41/s32/s75\n/s117/s40/s99/s41\nFIG. 4. (Color online). Typical initial isothermal magneti za-\ntion curves from T= 200 K to 280 K with temperature step\nof 4 K measured (a) in the abplane and (b) along the caxis,\nrespectively. (c) Temperature-dependent anisotropy cons tant\nKu(left axis) and the ratios of [ Ms/Ms(160K)]n(n+1)/2with\nn= 1, 2, and 4 (right axis).\nIsothermal magnetization M(H) forH∝bardblcexhibits a\nclear FM behavior at low temperatures and a linear PM\nbehavior with small slope at 300 K [Fig. 3(a)]. The mild\nhysteresisfor both orientationsindicatesthe behaviorex-\npected for a soft ferromagnet. The observed saturation\nmagnetization of Ms≈1.8µB/Cr forH∝bardblcat 5 K which\nis smaller when compared to the expected value for Cr\nfree ion (3 µB/Cr), indicating weak itinerant nature of\nCr or canted FM in Cr 5Te8.19The saturation field Hs\ncould be derived from the xcomponent of the intercept\nof two linear fits, one being a fit to the saturated regime\nat high fields and one being a fit of the unsaturated lin-\near regime at low fields. As we can see, the magneti-\nzation is much easier to saturate for H∝bardblc[Fig. 3(a)]\nthan for H∝bardblab[Fig. 3(b)], indicating that the caxis\nis the easy magnetization direction. The large dispar-\nity ofHsfor two orientations can be explained by easy\n(hard) natures of the domain wall motion along different\nc(ab) axes. Figures 3(c) and 3(d) present the tempera-\nture dependence of ZFC ac susceptibility measured with\noscillated ac field of 3.8 Oe and frequency f= 499 Hz.\nA pronounced broad peak is observed at Tcin the real\npartm′, i. e. at 238 K for H∝bardblaband at 231 K for\nH∝bardblc. An additional anomaly occurs just below Tc, as\nseen in an obvioushump in the imaginarypart m′′at 199\nK forH∝bardblc, further confirming the anisotropic magnetic\nresponse.\nUsing the Stoner-Wolfarth model a value for the mag-\nnetocrystalline anisotropy constant Kucan be estimated\nfrom the saturation regime in the isothermal magnetiza-\ntion curve. Figures 4(a) and 4(b) present the magneti-\nzation isotherms with field up to 5 T in the temperature\nrange from 200 to 280 K for both H∝bardblabandH∝bardblc./s50/s48/s48 /s50/s50/s48 /s50/s52/s48 /s50/s54/s48 /s50/s56/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48\n/s50/s48/s48 /s50/s50/s48 /s50/s52/s48 /s50/s54/s48 /s50/s56/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48\n/s48 /s50 /s52/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s50/s48/s48 /s50/s50/s48 /s50/s52/s48 /s50/s54/s48 /s50/s56/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s84/s32/s40/s75/s41/s53/s32/s84/s45 /s83\n/s77/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41\n/s40/s97/s41\n/s49/s32/s84/s72/s32/s47/s47/s32/s97/s98\n/s84/s32/s40/s75/s41/s72/s32/s47/s47/s32/s99/s40/s98/s41/s45 /s83\n/s77/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41/s32/s49/s32/s84\n/s32/s50/s32/s84\n/s32/s51/s32/s84\n/s32/s52/s32/s84\n/s32/s53/s32/s84/s72/s32/s40/s84/s41/s32/s49/s57/s56 /s32/s75 /s32 /s32/s50/s48/s54/s32/s75 /s32 /s32/s50/s49/s52/s32/s75\n/s32/s50/s50/s50/s32/s75 /s32 /s32/s50/s51/s48/s32/s75 /s32 /s32/s50/s51/s56/s32/s75\n/s40/s99/s41/s45 /s83/s82 /s77/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s32/s48/s46/s50/s32/s84 /s32 /s32/s49/s32/s84\n/s32/s48/s46/s52/s32/s84 /s32 /s32/s50/s32/s84\n/s32/s48/s46/s54/s32/s84 /s32 /s32/s51/s32/s84\n/s32/s48/s46/s56/s32/s84 /s32 /s32/s52/s32/s84\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s53/s32/s84/s40/s100/s41/s45 /s83/s82 /s77/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41\n/s72/s32/s40/s84/s41/s50/s48/s50/s32/s75/s32 /s50/s49/s48/s32/s75\n/s50/s49/s56/s32/s75/s32 /s50/s50/s54/s32/s75\n/s50/s51/s52/s32/s75/s32 /s50/s52/s50/s32/s75\nFIG. 5. (Color online). Temperature dependence of magnetic\nentropychange −∆SMobtainedfrommagnetization measure-\nment at various magnetic field changes (a) in the abplane and\n(b) along the caxis, respectively. Inset shows the field depen-\ndence of −∆SMat low temperature. The rotational magnetic\nentropy change −∆SR\nMas a function of (c) temperature and\n(d) field.\nWhenH∝bardblab, where the anisotropy becomes maximal,\nthe saturation field Hsis associated with the magne-\ntocrystalline anisotropy constant Kuand the saturation\nmagnetization Mswithµ0being the vacuum permeabil-\nity: 2Ku/Ms=µ0Hs.31Figure 4(c) exhibits the tem-\nperature dependence of derived Kufor Cr 5Te8, which is\nabout 94 kJ cm−3atT= 160 K. The Kuin Cr5Te8\nis larger than those for Cr 2(Si,Ge) 2Te6and CrBr 3,22,30\nwhich most likely results in a more apparent magnetiza-\ntion downturn anomaly in Cr 5Te8[Figs. 2(f)-2(i)]. Here\nwe propose that the origin of this downturn in the mag-\nnetization curve for H∝bardblabis a continuous reorientation\nof the magnetization direction as a result of an inter-\nplay between the magnetocrytalline anisotropy, field and\ntemperature. The magnetocrytalline anisotropy favors a\nmagnetization direction perpendicular to the abplane,\nwhich for H∝bardblabthe field wants to align the magnetiza-\ntion direction parallel to the field. Assuming an external\nfieldH1is higher than the anisotropyfield H∗at temper-\natureT1belowT∗, the magnetization vector is aligned\nalong the external field direction. By reducing the exter-\nnal field to a value H2belowH∗at fixedT1, a tilting of\nthe magnetization vector towards the easy caxis will be\nachieved. The tilting in turn leads to a reduction of the\nmagnetization component parallel to the field. This is\nlike the spin-flop mechanism that usually is observed in\nan AFM ordered state. Such feature is also presented in\nthe field-dependent magnetization in the hard abplane\n[Fig. 3(b)] at 5 and 100 K.\nFurthermore, the magnetocrytalline anisotropy Kufor\nCr5Te8is found to be temperature-dependent [Fig. 4(c)],5\n/s50/s48/s48 /s50/s50/s48 /s50/s52/s48 /s50/s54/s48 /s50/s56/s48/s48/s46/s54/s48/s46/s57/s49/s46/s50/s49/s46/s53/s49/s46/s56\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s45/s49/s46/s54 /s45/s48/s46/s56 /s48/s46/s48 /s48/s46/s56 /s49/s46/s54 /s50/s46/s52/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50\n/s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s49 /s50/s50/s48/s48/s50/s52/s48/s50/s56/s48\n/s45/s49 /s48 /s49/s48/s49/s112\n/s84/s32/s40/s75/s41/s32/s49/s32/s84 /s32\n/s32/s50/s32/s84 /s32\n/s32/s51/s32/s84 /s32\n/s32/s52/s32/s84\n/s32/s53/s32/s84/s40/s97/s41 /s72/s32/s47/s47/s32/s99/s72/s32/s47/s47/s32/s99\n/s113/s32/s61/s32/s49/s46/s49/s50/s40/s50/s41/s45/s68 /s83/s109/s97/s120 /s77\n/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41\n/s72/s32/s40/s84/s41/s32/s45/s68 /s83/s109/s97/s120\n/s77\n/s32/s45/s68 /s83/s109/s97/s120\n/s77/s32/s61/s32/s97/s72/s112/s40/s98/s41\n/s112/s32/s61/s32/s48/s46/s54/s54/s55/s40/s50/s41\n/s48/s56/s48/s49/s54/s48\n/s32/s82/s67/s80\n/s32/s82/s67/s80/s32/s61/s32/s98/s72/s113\n/s82/s67/s80/s32/s40/s74/s32/s107/s103/s45/s49\n/s41/s68 /s83\n/s77/s47/s68 /s83/s109/s97/s120 /s77\n/s113/s32/s49/s32/s84 /s32\n/s32/s50/s32/s84 /s32\n/s32/s51/s32/s84 /s32\n/s32/s52/s32/s84\n/s32/s53/s32/s84/s40/s99/s41\n/s72/s32/s47/s47/s32/s99/s72/s32/s47/s47/s32/s99/s32/s49/s32/s84 /s32\n/s32/s50/s32/s84 /s32\n/s32/s51/s32/s84 /s32\n/s32/s52/s32/s84\n/s32/s53/s32/s84/s45/s68 /s83\n/s77/s47/s72/s110\n/s101/s47/s72/s49/s47/s40/s98/s43/s103/s41/s40/s100/s41/s72/s32/s47/s47/s32/s99\n/s32/s84\n/s114/s49/s32\n/s32/s84\n/s114/s50/s32/s84\n/s114/s32/s40/s75/s41\n/s72/s49/s47 /s68\n/s32/s40 /s84/s49/s47 /s68\n/s41/s72/s32/s47/s47/s32/s97/s98/s68 /s83\n/s77/s47/s68 /s83/s109/s97/s120 /s77\n/s113\nFIG. 6. (Color online). (a) Temperature dependence of pin\nvarious fields. (b) Field dependence of the maximum mag-\nnetic entropy change −∆Smax\nMand the relative cooling power\nRCP with power law fitting in red solid lines. (c) The normal-\nized ∆SMas a function of the rescaled temperature θwith\nout-of-plane field and in-plane field (inset). (d) Scaling pl ot\nof ∆SMbased on the critical exponents β= 0.315 and γ=\n1.81.19Inset shows TrvsH1/∆with ∆ = β+γ.\ngradually increasing with decreasing temperature. This\ntendency arises solely from a large number of local spin\nclusters fluctuating randomly around the macroscopic\nmagnetization vector and activated by a nonzero ther-\nmal energy.32,33In a simple classical theory, ∝angbracketleftK(n)∝angbracketright ∝\nMn(n+1)/2\ns, where ∝angbracketleftK(n)∝angbracketrightis the anisotropy expecta-\ntion value for the nthpower angular function,32,33in\nthe case of uniaxial anisotropy n= 2 and of cubic\nanisotropy n= 4, leading to an exponent of 3 and 10, re-\nspectively. The temperature-dependent Ms/Ms(160K),\n[Ms/Ms(160K)]3, and [Ms/Ms(160K)]10are also plot-\nted. The comparison in Fig. 5(c) points to the predom-\ninant uniaxial anisotropy with some deviation to cubic\nanisotropy in Cr 5Te8. As discussed above, reducing the\nexternal field from H1passingH∗toH2atT1(< T∗)\nresults in a tilting of the magnetization vector towards\ntheeasycaxis. Theobservedtemperature-dependent Ku\ngives a reduction of magnetic anisotropy when increasing\ntemperature to T2(> T∗). Then the alignment along the\nmagneticeasyaxisbecomes lessfavorableupon warming,\nwhich leads to a stronger tilting of the magnetization\nvector towards the abplane and an increased experimen-\ntally observed ab-component magnetization. Therefore,\nthe spin configuration of Cr 5Te8belowTcis supposed to\nbe an out-of-plane FM order with in-plane AFM com-\nponents, calling for further neutron scattering measure-\nment. This canted FM configuration might arise from\nthe ordered Cr-vacancies induced modification of local\nstructure and indicates competing exchange interactionsexist in Cr 5Te8.\nTo further characterize the anisotropic magnetic prop-\nerties in Cr 5Te8, we investigate the anisotropic magnetic\nentropy change:\n∆SM(T,H) =/integraldisplayH\n0/parenleftbigg∂S\n∂H/parenrightbigg\nTdH=/integraldisplayH\n0/parenleftbigg∂M\n∂T/parenrightbigg\nHdH,\nwhere/parenleftbig∂S\n∂H/parenrightbig\nT=/parenleftbig∂M\n∂T/parenrightbig\nHis based on Maxwell’s relation.34\nIn the case of magnetization measured at small discrete\nfield and temperature intervals,\n∆SM(Ti,H) =/integraltextH\n0M(Ti,H)dH−/integraltextH\n0M(Ti+1,H)dH\nTi−Ti+1.\nFigures 5(a) and 5(b) show the calculated −∆SM(T,H)\nas a function of temperature in selected fields up to 5 T\nwith a step of 1 T applied in the abplane and along the c\naxis, respectively. All the −∆SM(T,H) curves feature a\nbroad peak in the vicinity of Tc. The value of −∆SMin-\ncreases monotonically with increasing field; the maximal\nvalue reaches 1.39 J kg−1K−1in theabplane and 1.73 J\nkg−1K−1along the caxis, respectively, with field change\nof 5 T. Most importantly, the values of −∆SMfor the\nabplane are negative at low temperatures in low fields,\nhowever, all the values are positive along the caxis. The\nfield dependence of −∆SM(T,H) at low temperature [in-\nset in Fig. 5(a)] shows a clear sign change for H∝bardblab.\nThis behavior is similar with that in CrI 3,23in which the\nmagnetic anisotropy originates mainly from the superex-\nchange interaction.35The calculated magnetocrytalline\nanisotropy decreases with increasing temperature [Fig.\n4(c)], whereas the magnetization may exhibit opposite\nbehavior. At low fields, the magnetization at higher tem-\nperature could be larger than that at lower temperature\n[Figs. 2(f)-(i) and 3(b)], which gives a negative −∆SM.\nTherotationalmagneticentropychange∆ SR\nMcanbe cal-\nculated as ∆ SR\nM(T,H) = ∆SM(T,Hc)−∆SM(T,Hab).\nAs shown in Fig. 5(c), the temperature dependence of\n−∆SR\nM(T,H) also features a peak at Tcwith the field\nchange of 0.2 T, however, the temperature of the max-\nimum−∆SR\nM(T,H) moves away from Tcto lower tem-\nperature with increasing field. When T≤Tc, the field\ndependence of −∆SR\nM(T,H)increasesrapidlyatlowfield\nand changes slowly at high field [Fig. 5(d)]. This pos-\nsibly reflects the anisotropy field in Cr 5Te8, which de-\ncreases with increasing temperature and gradually dis-\nappears when T≥Tc. The obtained −∆SMof Cr5Te8is\nsignificantly smaller than those of well-known magnetic\nrefrigerating materials, such as Gd 5Si2Ge2, LaF13−xSix,\nand MnP 1−xSix,36however, comparable with those of\nCr(Br,I) 3and Cr 2(Si,Ge) 2Te6.22,37,38\nFor a material displaying a second-order transition,39\nthe field-dependent maximum magnetic entropy change\nshould be −∆Smax\nM∝Hp.40Another important param-\neter is the relative cooling power RCP=−∆Smax\nM×\nδTFWHM.41The−∆Smax\nMis the maximum entropy\nchange near TcandδTFWHMis the full-width at half\nmaximum of −∆SM. The RCP also depends on the field6\nwithRCP∝Hq. Figure 6(a) displays the temperature\ndependence of p(T) in various out-of-plane fields. At low\ntemperatures, well below Tc,phas a value which tends\nto 1. On the other side, well above Tc,papproaches 2\nas a consequence of the Curie-Weiss law. At T=Tc,\npreaches a minimum. Figure 6(b) summarized the field\ndependence of −∆Smax\nMandRCP. The RCPis calculated\nas 131.2 J kg−1with out-of-plane field change of 5 T for\nCr5Te8, which is comparable with that of CrI 3(122.6 J\nkg−1at 5 T).23Fitting of the −∆Smax\nMand RCP gives\np= 0.667(2) and q= 1.12(2).\nScaling analysis of −∆SMcan be built by normalizing\nall the−∆SMcurves against the respective maximum\n−∆Smax\nM, namely, ∆ SM/∆Smax\nMbyrescalingthereduced\ntemperature θ±as defined in the following equations,42\nθ−= (Tpeak−T)/(Tr1−Tpeak),T < T peak,\nθ+= (T−Tpeak)/(Tr2−Tpeak),T > T peak,\nwhereTr1andTr2are the temperatures of two reference\npoints that corresponds to ∆ SM(Tr1,Tr2) =1\n2∆Smax\nM.\nFollowing this method, all the −∆SM(T,H) curves in\nvarious fields collapse into a single curve [Fig. 6(c) and\ninset]. The values of Tr1andTr2depend on H1/∆with\n∆ =β+γ[insetinFig. 6(d)].42Inthephasetransitionre-\ngion, the scaling analysis of −∆SMcan also be expressed\nas−∆SM/aM=Hnf(ε/H1/∆), whereaM=T−1\ncAδ+1B\nwith A and B representing the critical amplitudes as in\nMs(T) =A(−ε)βandH=BMδ, respectively, and f(x)\nis the scaling function.43If the critical exponents are\nappropriately chosen, the −∆SM(T) curves should be\nrescaled into a single curve, consistent with normalizing\nall the−∆SMcurves with two reference temperatures\n(Tr1andTr2). This is indeed seen in Fig. 6(d) by us-\ning the values of β= 0.315 and γ= 1.81,19confirming\nthat the critical exponents for Cr 5Te8are intrinsic and\naccurately estimated. As is well known, the critical ex-\nponentsβ= 0.25,γ= 1.75 and β= 0.325, γ= 1.24 areexpected for 2D and 3D Ising models, respectively. The\nspatial dimension extends from d= 2 tod= 3 passing\nthrough Tcconfirms that trigonal Cr 5Te8features quasi-\n2D character. The spin dimension of Ising-type n= 1\nimplies strong uniaxial magnetic anisotropy, which is of\nhigh interest for establishinglong-rangemagnetism down\nto 2D limit.44\nIV. CONCLUSIONS\nIn summary, we have studied in detail the anisotropy\nof magnetic and magnetocaloric properties of trigonal\nCr5Te8with a high- Tcof 230 K. A satellite transition T∗\nis confirmed below Tc, featuring an anomalous magneti-\nzation downturn when low field applying in the hard ab\nplane. A canted out-of-plane FM configuration with in-\nplane AFM components is supposed for Cr 5Te8, suggest-\ning the existence of competing exchange interactions. In-\ntrinsic magnetocrystalline anisotropy in trigonal Cr 5Te8\nis established and also reflected in anisotropic magnetic\nentropy change. 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We demonstrate a con-\ntrolled tuning of magnetocrystalline anisotropy in Bi-dop ed Y3Fe5O12(Bi:YIG) films of high crys-\ntalline quality using growth induced epitaxial strain on [1 11]-oriented Gd 3Ga5O12(GGG) substrate.\nWe optimize a growth protocol to get thick highly-strained e pitaxial films showing large magneto-\ncrystalline anisotropy, compare to thin films prepared usin g a different protocol. Ferromagnetic\nresonance measurements establish a linear dependence of th e out-of-plane uniaxial anisotropy on\nthe strain induced rhombohedral distortion of Bi:YIG latti ce. Interestingly, the enhancement in the\nmagnetoelastic constant due to an optimum substitution of Bi3+ions with strong spin orbit cou-\npling does not strongly affect the precessional damping ( ∼2×10−3). Large magneto-optical activity,\nreasonably low damping, large magnetocrystalline anisotr opy and large magnetoelastic coupling in\nBi:YIG are the properties that may help Bi:YIG emerge as a pos sible material for photo-magnonics\nand other spintronics applications.\nMagneto-crystalline anisotropy and Gilbert damping\nare the crucial parameters for a material to be used in\nvarious spin-based device applications[ 1–4]. The emerg-\ning field of spintronics promises dense and fast mem-\nory architectures, enabling huge data storage and fast\ninformation processing[ 5–14]. The spin current based\ndevices would be highly efficient with almost no ther-\nmal losses unlike charge-based electronics and could be\nused in energy harvesting by recycle of heat waste via\nspin-caloritronics[ 1,15–19]. The miniaturization of such\nconcept-device prototypes requires material media in a\nthin film form, where the magnetic properties can vary\nsignificantly due to different film thicknesses, growth in-\nducedstrains, crystallographicorientationandsubstrate-\nfilm interface reactions. It is essential to have a physi-\ncal parameter to tune the magnetic anisotropy in thin\nfilms while maintaining the precessional damping as-low-\nas possible. The strain produced in thin films due to\nsubstrate-filmlattice mismatchservesas atuning param-\neter for magnetic anisotropy and can be varied by chang-\ning the film thickness. The uniaxial magnetic anisotropy\nis the main contributing term in a thin film’s total mag-\nnetic anisotropy and as the anisotropy field in a ferro-\nmagnetic system has one-to-one mapping with the effec-\ntive magnetization, we tried to establish a relationship\nbetween magnetic damping and the strength of effective\nmagnetization for different ferromagnetic systems.\nIn Fig. 1, we compile results from existing litera-\nture on Gilbert damping ( α) and effective magnetiza-\ntion (4πMeff) of different ferromagnetic systems irre-\nspective of the growth (different growth conditions and\nmethods), physical form (thin films or bulk), thickness\n(in thin films), crystallinity (amorphous or polycrys-\ntalline or epitaxial), dopants and other factors (refer-\nences provided at the end of this paper). Region I is\n∗zakir@iitk.ac.in\nFIG. 1. (Color online) Relationship between effective magne -\ntization and Gilbert damping coefficient. Here, we compare\nsome of the interesting work from existing literature; Regi on I\nand II: Ferromagnetic insulators in the form of bulk, thin fil ms\n(polycrystalline and epitaxial); Region III: Conducting- oxides\nand; Region IV: Pure metals and metal-alloys. Different re-\ngions of interest have been shaded with different colors. Not e:\nReferences are provided at the end.\nthe most exploited one because pure-YIG possesses very\nlow-damping( ∼10−4)[1,20–24]. Theapplicationofspin-\norbittorqueinheavymetals(HM)[ 25–29]andtopological\ninsulators (TI)[ 30–32] capped ferrimagnetic garnet het-\nerostructures show potential to improve the efficiency of\nmagnetic manipulations as it will not shunt a charge cur-\nrentappliedtothecappedconductinglayer[ 33]. Beinganinsulating material, only electron’s spin degrees of free-\ndom is allowed, resulting in pure spin current, which is\nnot the case with conducting-oxides (Region III), metals\nand metal-alloys (Region IV). Besides having the ability\nto generate pure spin current, the magneto-optical prop-\nerties of YIG enhances in proportion to Bismuth (Bi)\nconcentration at Yttrium site[ 34–37]. Due to enhanced\nmagneto-optical activity in the UV, visible and IR re-\ngions along with low propagation loss, Bi:YIG is a po-\ntential candidate in microwave and optical applications\nsuch as miniaturization of magnetic field sensors[ 38–43]\nand reciprocal transmission devices like isolators and cir-\nculators, respectively[ 44–46]. It has been well established\nthat the Bi:YIG films with in-plane magnetization can\nserve as basic sensors for magneto-optical imaging of do-\nmain formation in magnetic materials, magnetic flux in\nsuperconductors, currents in microelectronic circuits and\nrecorded patterns in magnetic storage media[ 47–52]. It\nis suggestive that the growth parameters optimization\nis crucial to obtain films with in-plane magnetization\nand free from effective domain activity[ 37][47]. Ferri-\nmagneticinsulatorswithin-planeeasymagnetizationcan\nalso be used to realize spin superfluidity[ 53–57]. The\ncoherent condensation of magnons in spin superfluid-\nity offers a unique opportunity to realize long distance\ncoherent superfluid like transport of the spin current,\nunlike the transport carried by the incoherent thermal\nmagnons which decays exponentially[ 55]. Recently, cou-\npling of light and spin wave has been demonstrated by\nirradiating a ferrimagnetic insulator using spatially mod-\nulated light beam[ 10][58]. This coupling gives rise to a\nmagnonic crystal that shows the capability to be effi-\nciently reprogrammed on demand via heat. The cou-\npling of electromagnetic waves to wave-like excitations\nin solids (magnons) could also be helpful to reduce all\nthe lateral dimensions by ordersofmagnitude for on-chip\nmicrowave electronics with optically reconfigurable and\nmultifunctional characteristics. Doping pure YIG with\nBi improves its sensitivity towards light and makes it\npursuable for magneto-optical based device applications.\nBeing a novel material for possible photon-based device\napplications, it is essential to optimize and investigate\nthestaticanddynamicmagnetizationaspectsofthislight\nsensitive material medium (Region II). Bi:YIG films with\noverwhelmingly large magneto-photonic activity coupled\nwith improved magnetic properties will provide a mate-\nrial platform for newly emerging photo-magnonics field.\nThe importance of Bismuth substituted YIG as a po-\ntential material for light based magnonics applications,\nmotivated the studies reported here. In this study, we\ngrow high quality epitaxial Bi:YIG films on GGG(111)\ncrystals using two different growth protocols which allow\nus to achieve different strain-states induced by rhombo-\nhedral distortion due to film-substrate lattice mismatch.\nWe prepared two sets of samples, Set-A and Set-B. Set-\nA consists of thin Bi:YIG films with large magnetocrys-\ntalline anisotropy due to the large magnitude of strain,\nand, Set-B consists of thick Bi:YIG films with reason-ably large strain. Despite being thick, the films from\nSet-B show large magnitude of strain that leads to large\nvalue of magnetocrystalline anisotropy, for an example;\nthe magnitude of uniaxial magnetocrystalline anisotropy\nfield for a 100 nm thick film from set-B is larger than\na 37 nm thin film from Set-A. The Gilbert damping co-\nefficient increases slightly due to strong spin-orbit cou-\nplingandinhomogeneityproducedbyBismuthdoping( ∼\n2×10−3), but still orders of magnitude smaller compare\nto metallic films[ 59–61] and aresuitable formagnonics[ 8–\n12]spintronics[ 6][7][62][63]andcaloritronics[ 1][15–18]ap-\nplications. The magnetoelastic constant of Bi:YIG films\ncomes out to be larger than YIG films[ 2] due to Bi3+\nsubstitution which enhances the spin-orbit coupling and\nhence the magnetoelastic coupling.\nEpitaxial Bi:YIG films were grown on GGG(111) crys-\ntalsusingaKrFExcimerlaser(LambdaPhysikCOMPex\nPro,λ= 248 nm) of 20 ns pulse width. The laser was\nfired at a repetition frequency of10Hz on solid state syn-\nthesized Bi0.25Y2.75Fe5O12target, placed 50 mm away\nfrom the substrate. The substrates were in-situ annealed\nat 800◦C for 120 minutes to get atomically flat surfaces\nand then cooled down to 500◦C in 4.0×10−2mbar\noxygen pressure to deposit the films. The target was\nsufficiently preablated before actual deposition to get a\nsteady state target surface. We incorporated two routes\nto deposit these epitaxial films to obtain different strain-\nstates by changing the laser fluence at a fixed oxygen\nambient and growth temperature. For set-A, the fluence\nwas∼1 Jcm−2with a spot sizeof ∼10.0mm2and hence\nthe realized growth rate was ∼0.25˚A/s. For set-B, we\nalmost doubled the fluence ( ∼1.9 Jcm−2) by reducing\nthe spot size ( ∼5.4 mm2) to achieve an enhanced growth\nrate of∼0.45˚A/s. We deposited five films of thicknesses\n10.2, 18.1, 37.0, 92.5 and 200 nm using set-A growth pa-\nrameters, hereafter denoted as A1,A2,A3,A4andA5,\nrespectively. Another five films of thicknesses 18.7, 39.8,\n100, 150 and 200 nm were grown using growth protocol-\nB, hereafter denoted as B1,B2,B3,B4, andB5respec-\ntively. The growth rate and hence the thicknesses of dif-\nferent samples were pre-calibrated using Dektak stylus\nprofilometer. PANalyticalX’PertPROfourcirclediffrac-\ntometer equipped with Cu-K α1source (λ= 1.54059˚A)\nwas used to characterize the crystallinity and to quantify\nthe state-of-strain. Room temperature Vibrating Sample\nMagnetometry (VSM) measurement was performed us-\ning a Quantum Design Physical Property Measurement\nSystem (PPMS). For the dynamic magnetization mea-\nsurements, we used both commercial and a custom-made\nFMR setup. Angular dependent FMR measurements\nwere performed using Bruker EMX EPR spectrometer\nwith cavity mode frequency f ∼=9.60 GHz. Frequency\ndependent FMR measurements were performed by using\na broadband coplanar waveguide (CPW). The CPW as-\nsembly was housed in an external homogeneous DC mag-\nnetic field along with the superposition of a small and\nlow frequency AC field. This small modulation of mag-\n2/s32/s110/s109/s32/s110/s109/s40/s100 /s41\n/s40/s99 /s41/s83/s101/s116\n/s32 /s32/s110/s109\n/s32 /s32/s110/s109\n/s32 /s32/s32/s110/s109\n/s32 /s32/s32/s110/s109\n/s32 /s32/s32/s110/s109/s83/s101/s116\n/s32 /s32/s110/s109\n/s32 /s32/s110/s109\n/s32 /s32/s110/s109\n/s32 /s32/s110/s109\n/s32 /s110/s109/s32/s110/s109\n/s32/s110/s109\n/s32/s110/s109\n/s32/s110/s109\n/s32/s110/s109/s32/s110/s109\n/s32/s110/s109\n/s32/s110/s109/s32/s40/s68/s101/s103/s114/s101/s101/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121 /s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s41\n/s32/s40/s68/s101/s103/s114/s101/s101/s41/s32/s110/s109/s32/s110/s109\n/s32/s110/s109/s32/s110/s109/s32/s110/s109/s32/s110/s109\n/s40/s97 /s41/s40/s98 /s41\n/s32/s40/s68/s101/s103/s114/s101/s101/s41\nFIG. 2. (Color online) X-ray measurements on Bi:YIG films gro wn by two different protocols; Panel (a): X-ray reflectivity\nmeasurements with fitted data to calculate the thicknesses o f different Bi:YIG films. Panel (b): Intensity normalized ω(Omega)\nscan profiles with low values of FWHM defines good crystallini ty. Panel (c): Thickness dependence of a⊥and percentage strain\n([ab−a⊥]/ab%) in the Bi:YIG films from both the sets. Panel (d): X-ray Diffr actograms of Bi:YIG films with trails of Laue\noscillations suggest high epitaxy.\nnetic field is required to get differential of absorbed radio\nfrequency (RF) power which is measured by a Schottky\ndiode detector and a lock-in amplifier.\nFig.2summarizes the X-ray measurements on Bi:YIG\nfilms grown on (111) oriented GGG substrates. Panel\n(a) shows reflectivity measurements on all the samples\nexcept 100, 150 and 200 nm (pre-calibrated using pro-\nfilometry), as there were no visible thickness fringes due\nto larger thickness. Reflectivity data was fitted to cal-\nculate and standardize the profilometric pre-calibrated\nthickness and gives very low roughness ranging from\n0.25 to 0.39 nm. The panel (b) of Fig. 2shows in-\ntensity normalized ωscan profiles with low values of full\nwidth half maximum (FWHM) ranging between 0.0448\nto 0.0072◦, signifies high crystallinity. The panel (d) of\nFig.2shows X-ray diffraction patterns of all the Bi:YIG\nsamples where the pronounced trail of Laue oscillations\ncharacterizes smooth surfaces and sharp interfaces. The\nbulk lattice constant for Bi0.25Y2.75Fe5O12comes out to\nbe 12.389 ˚A and the corresponding 2 θpeak position is\nshown by a vertical bar beneath substrate peak. Thin\nfilm lattice constant ( a⊥) differs due to lattice mismatch\nbetween substrate and film (shown by vertical up ar-\nrows). This lattice mismatch causes rhombohedral dis-\ntortion in the films and hence contributes to diagonally\nstretched unit cells along the [111] growth direction. The\nstrain induced rhombohedral distortion in these epitax-\nial Bi:YIG films can be quantified using the parameter\nσ= (ab−a⊥)/ab= ∆a/ab, where,abis the bulk Bi:YIG\nlattice parameter and a⊥is the stretched film lattice pa-rameteralongthe[111]direction[ 2][3][64]. Forset-Asam-\nples, XRD patterns show strain relaxation as the thick-\nness increases from 10.2 to 200 nm (2 θvalue approaches\nthe bulk value), the strain-induced lattice distortion de-\ncreases from 1 .162% to almost ∼0.0%. Surprisingly, set-\nBsamples havingthicknesses18.7, 39.8, 100, 150and 200\nnm, show relatively high strain (1 .122% for 18.7 nm thin\nfilm and 0 .171% for 200 nm thick film). The variation of\na⊥and the lattice strain ( σ) w.r.t. to Bi:YIG film thick-\nness from both the sets are shown in Fig. 2panel (c). It\ncanbeseenthatthevalueof a⊥approachesbulkvaluefor\na film of thickness 200 nm from set-A, whereas, a 200 nm\nthick film from set-B possess elongated a⊥. Similarly, a\n200nmthickfilmfromset-Ashownegligiblelatticestrain\nbut a 200 nm thick film from set-B possesses reasonably\nlarge lattice strain. The 2-axis ωvs. 2θ−ωmaps are\nshown in Fig. 3. The top panel shows symmetric maps\nin the 444 direction of Bi:YIG films. Whereas, the bot-\ntom panel shows the 642 asymmetric direction maps. We\nshow (444) symmetric and (642) asymmetric 2-axis maps\nfor 10.2, 37.0 and 92.5 nm films from set-A, and, 100 and\n200 nm films from set-B. It can be clearly seen that the\n2θ−ωvalueforfilm (representedby+; redcolored)shifts\ntoward higher value as the film thickness increases and\napproaches to the GGG substrate spot (represented by\n×; black colored). The map of a 92 .5 nm thick film from\nset-A shows large relaxation compare to a 100 nm thick\nfilmfromset-B.Whichconfirmstheinferencedrawnfrom\nθ−2θXRD measurement. The laser ablation conditions\ngreatlyimpactthelatticeconstantofdepositedfilmsirre-\nspective of oxygen pressure and growth temperature. We\n3FIG. 3. (Color online) The ωvs. 2θ−ω, 2-axis maps in (444) symmetric and (642) asymmetric direct ions: Left panel shows\nmaps of 10.2 nm, 37.0 nm and 92.5 nm Bi:YIG films from set-A. Rig ht Panel shows maps of 100 nm and 200nm Bi:YIG films\nfrom set-B.\nobserve that the laser fluence plays an important role in\ntuning the lattice constant of the films. The set-A films\nprepared using slow growth rate ( ∼0.25˚A/s) with a\nlower laser fluence ( ∼1 Jcm−2) show less lattice expan-\nsion and complete relaxation with thickness increment.\nWhereas, the set-B films prepared using almost doubled\ngrowth rate ( ∼0.45˚A/s) due to higher laser fluence ( ∼\n1.9 Jcm−2) show tendency to possess reasonably large\nlattice expansion even for higher thicknesses (panel (c)\nand (d) of Fig. 2). The laser fluence (growth rate) is low\nin the case of Set-A Bi:YIG films, which gives sufficient\nsettle down time to the ablated plasma species and hence\nlead to strain relaxation. In contrast, the higherlaserflu-\nence (growthrate) in the case ofBi:YIG films from set-B,\ndoesn’t allow the ablated plasma species to settle down\nand get relaxed. Table Icontains XRD, magnetization\nand FMR derived parameters for both the sets of sam-\nples. The negative sign of σindicates the presence of\ncompressive strain which relaxes with increment in film\nthickness[ 2,64–66].\nRoom temperature in-plane magnetic hysteresis loops\nare measured using VSM on Quantum Design PPMS.\nIn-plane and out-of-plane magnetization loops for a 37.0nm thick film from set-A is shown in Fig. 4(b), where\ntheparamagneticbackgroundfromGGGwassubtracted.\nThe values of saturation magnetization (4 πMS) for sam-\nples from set-A and set-B ranges between 1720 ±100 to\n1407±25 Oe and 1608 ±17 to 1457 ±12 Oe, respectively.\nThe coercivity ( HC) of these samples are in the range\nof∼13 to 23 Oe. These values fall in the range of\nreported YIG magnetization data[ 20][64–68]. To probe\nthe static and dynamic magnetic properties of Bi:YIG\nepitaxial films, we performed angular and frequency de-\npendent FMR measurements on both the sets of sam-\nples. Generally, the magnetic garnet thin films with a\nhard axis in the [111] direction (i.e., In-plane easy axis),\npossesses extrinsic uniaxial magnetic and intrinsic mag-\nnetocrystalline cubic anisotropies. FMR can directly de-\nduce the magnetic anisotropies in a precise manner. The\ncoordinate system used for FMR study on (111) oriented\nepitaxial Bi:YIG films is shown in Fig. 4(a). The ori-\nentations of static magnetic field Hand magnetization\nvectorMwith reference to coordinates x:[2 11], y:[011]\nand z:[111] are described by the angles φH,θHandφM,\nθM, respectively. The total free energy per unit volume\nof the media for (111) oriented cubic garnet system has\nthe form[ 69][70],\nF=−HMS/bracketleftbigg\nsinθHsinθMcos(φH−φM)\n+cosθHcosθM/bracketrightbigg\n+2πM2\nScos2θM−Kucos2θM+K1\n12/parenleftbigg7sin4θM−8sin2θM+4−\n4√\n2sin3θMcosθMcos3φM/parenrightbigg\n+K2\n108/parenleftbig\n−24sin6θM+45sin4θM−24sin2θM+4−2√\n2sin3θMcosθM/parenleftbig\n5sin2θM−2/parenrightbig\ncos3φM+sin6θMcos6φM/parenrightbig\n(1)\n4/s40/s99/s41/s40/s97/s41\n/s45/s52/s48/s48 /s45/s50/s48/s48 /s48 /s50/s48/s48 /s52/s48/s48/s45/s49/s48/s49\n/s32 /s102/s105/s108/s109\n/s110/s109/s32 /s102/s105/s108/s109/s77/s47/s77\n/s83\n/s72 /s32 /s40 /s101 /s41/s40/s98/s41\n/s50/s48/s48/s48 /s52/s48/s48/s48 /s54/s48/s48/s48\n/s72 /s32/s40/s79/s101/s41/s100/s73\n/s70/s77/s82/s47/s100/s72/s32 /s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s72/s61 /s57/s48/s111\n/s32/s53/s48/s111\n/s32/s32/s32/s32/s32 /s51/s48/s111\n/s32/s32/s32 /s50/s48/s111\n/s32/s49/s48/s111\n/s32/s48/s111\n/s32/s110/s109\nFIG. 4. (Color online) Room temperature magnetization and o ut-of-plane angular dependence of resonance field for Bi:YI G\nfilms from both the sets. (a) Typical schematic of spherical c oordinate system for FMR measurements and analysis of [111]\noriented epitaxial Bi:YIG/GGG(111) samples. (b) Magnetic hysteresis loops measured in in-Plane (red) and out-of-pla ne\n(Green) configuration of a 37.0 nm thin film from set-A using VS M. (c) Representative FMR derivative spectra for a 39.8 nm\nBi:YIG film from set-B. Panel (d) picturizes out-of-plane an gular variation ( θH) of the resonance fields ( Hres) and fitted curves\nfor set-A. Inset: Energy minimization comparison for 10.2 n m and 200 nm thick Bi:YIG films from set-A. Panel (e) picturize s\nout-of-plane angular variation ( θH) of the resonance fields ( Hres) and fitted curves for set-B. Inset: Energy minimization\ncomparison of 18.7 nm and 200 nm thick Bi:YIG films from set-B.\nThe first term in Eq. ( 1) corresponds to the Zee-\nman energy, the second term to the demagnetization en-\nergy, the third term to the out-of-plane uniaxial mag-\nnetocrystalline anisotropy energy Kuand the last two\nterms are due to first and second order cubic magne-\ntocrystalline anisotropy energies, K1andK2, respec-\ntively. The total free energy equation was minimized\n(∂F/∂θ M≡∂F/∂φ M≡0) to obtain the equilibrium ori-\nentation of the magnetization vector M(H). The evalu-\nation of resonance frequency ( ωres) of uniform magneti-\nzation precessional mode at equilibrium condition can be\nmade using total free energy and is expressed as:[ 70–72]\nωres=γ\nMSsinθM/bracketleftBigg\n∂2F\n∂θ2\nM∂2F\n∂φ2\nM−/parenleftbigg∂2F\n∂θM∂φM/parenrightbigg2/bracketrightBigg1/2\n(2)\nhereγandMSdenote gyromagnetic ratio and satu-\nration magnetization, respectively. These coupled and\nindirectly defined functional equations were solved nu-merically to obtain the equilibrium angles at resonance\ncondition and fit the angular dependent resonance data\n(Hresvs.θH) to determine g-factor, Ku,Hu,H1,H2\nandEani(see Table I). Fig. 4(c) shows representative\nangular-FMR spectra of a 39.8 nm thick film from set-B\nat a microwave frequency of ∼9.6 GHz. The peak-to-\npeak difference of FMR derivative gives linewidth (∆ H)\nwhich decreasesasthe film thickness increases. The mea-\nsured in-plane ∆ Hvalues for set-A samples A1,A2,A3,\nA4, andA5at∼9.6 GHz are 154, 120, 93, 39, and 14\nOe, respectively. Similarly, for set-B samples B1,B2,\nB3,B4, andB5the in-plane ∆ Hvalues are 150, 105, 50,\n44, and 23 Oe, respectively. The energy minimization\ngoverned by the correspondence between θHandθMis\nshown in the inset of Fig. 4(d-e), where the equilibrium\nmagnetization angle θMwas estimated numerically. It\ncan be seen that energy minimization attains large cur-\nvature for thin Bi:YIG film from both the sets and hence\nlarge anisotropy compare to thick film from the respec-\ntive sets. Fig. 4(d) and (e) show θHdependence of Hres\n5TABLE I. XRD, M−Hand FMR derived parameters of Bi:YIG epitaxial films grown by two protocols. Set-A ( A1,A2,A3,\nA4, andA5) and set-B ( B1,B2,B3,B4, andB5) are separated by a solid horizontal line.\nThickness 2 θ a ⊥ ∆a/ab 4πMS g-factor Ku Hu H1 H2 Eani\n(VSM)\n(nm) (Degree) ˚A % (Oe) ( ×103erg/cc) (Oe) (Oe) (Oe) ( ×103erg/cc)\n10.2 (A1) 50.406 12.533 -1.162 1720 ±100 2.12 -125.40 ±8.23 -1831 ±227 -15.3 ±1.9 3.1 ±1.1 -126.24 ±8.24\n18.1 (A2) 50.766 12.450 -0.492 1432 ±63 2.09 -55.68 ±4.21 -977 ±117 -29.7 ±2.2 13.8 ±1.6 -56.59 ±4.21\n37.0 (A3) 50.919 12.41 -0.210 1482 ±37 2.03 -27.68 ±3.02 -469 ±63 -40.9 ±1.8 30.9 ±1.7 -28.27 ±3.01\n92.5 (A4) 51.011 12.394 -0.0484 1507 ±38 2.01 -7.43 ±2.71 -124 ±48 -12.1 ±0.9 52.1 ±2.0 -5.03 ±2.70\n200 (A5) 51.033 12.389 0.0 1407 ±25 2.00 -3.91 ±1.32 -70 ±25 -3.2 ±0.6 3.2 ±0.7 -3.91 ±1.31\n18.7 (B1) 50.425 12.528 -1.122 1582 ±38 2.13 -81.42 ±6.72 -1292 ±137 -57.3 ±1.9 6.7 ±1.0 -84.61 ±6.81\n39.8 (B2) 50.530 12.504 -0.928 1545 ±25 2.05 -62.12 ±5.33 -1010 ±103 -14.4 ±0.8 160.2 ±3.4 -53.16 ±5.42\n100 (B3) 50.747 12.454 -0.525 1520 ±25 2.04 -39.23 ±4.28 -648 ±82 -40.0 ±1.2 65.1 ±1.9 -37.71 ±4.37\n150 (B4) 50.807 12.440 -0.414 1608 ±17 2.03 -15.78 ±3.04 -246 ±50 -19.6 ±0.6 118.5 ±1.4 -9.45 ±3.11\n200 (B5) 50.939 12.410 -0.171 1457 ±12 2.01 -6.25 ±0.91 -108 ±17 -23.9 ±0.7 113.5 ±1.2 -1.06 ±0.96\nfor set-A and set-B samples, respectively. The fit using\nEqs. (1) and (2) agrees well with the measured data. All\nthe extracted parameters for both the sets of samples are\nshown in Table I, separated by a solid line.\nWe mainly focus on the out-of-plane uniaxial anisotropy\nfield (Hu) due to its large contribution to total magnetic\nanisotropy and systematic variation with film thickness\nor lattice strain. In contrast, we couldn’t witness a sys-\ntematic thickness or strain dependence of cubic first and\nsecond order anisotropy which are weak in magnitude.\nInterestingly, Hufor 10.2 nm thin and 200 nm thick\nfilms from set-A comes out to be -1831 ±227 Oe and -\n70±25 Oe, respectively, which provides a strain tuning\nover a range of more than 1700 Oe. It suggests that\nthe rhombohedral distortion induces substantial out-of-\nplaneuniaxialanisotropyviathemagnetostriction,which\ndecreases systematically with increase in the film thick-\nness. The g-factor for thin films is as large as 2.13,\ngreater than the spin-only value 2 .0. This corroborates\nthe existence of spin-orbit coupling that lead to strain-\ninduced anisotropy. However, the g-factor for thick films\nare smaller ( ∼2.0). This variation in ‘g’ possibly arises\ndue to different strain state, which may change the oc-\ncupation of orbitals and hence the magnitude of orbital\nangular momentum and spin-orbit coupling. The strain\ninduced variation of HuandEaniis picturized in Fig.\n5(a) and (b), respectively. It is clear from Fig. 5(a)\nand (b) that the magnitudes of HuandEaniincreases\nalmost linearly as the magnitude of rhombohedraldistor-\ntion increases. The enhancement in uniaxial anisotropy\nfield is due to the larger magnitude of growth induced\nstraininthe samplesfromset-Bascomparetoset-A.The\nsubstrate-film lattice mismatch causes lattice-distortion\nin deposited films which results in a definite strain-state.\nThe lattice distortion influences the magnetic properties.This magnetization-lattice coupling gives rise to strain-\ninduced out-of-plane uniaxial anisotropy field, Hu. The\nstrain induced by rhombohedral distortion in a cubic lat-\ntice relaxes as the film thickness increases and hence re-\nsults in very low or almost negligible strain, which ulti-\nmately makes the film isotropic, having properties sim-\nilar to bulk. The value of Hufor Bi:YIG films B1,B2,\nB3,B4, andB5from set-B are found to be -1292 ±137,\n-1010±103, -648 ±82, -246±50, and -108 ±17 Oe, respec-\ntively. It is important to note that the values of Hufor\nthicker films from set-B are larger compare to respective\nfilm thicknesses from set-A. If we compare the uniaxial\nanisotropy field of Bi:YIG films from both the sets of al-\nmost equal thicknesses, i.e., A2(18.1 nm) and B1(18.7\nnm), comes out to be -977 ±117 Oe, and -1292 ±137 Oe,\nrespectively. The uniaxial anisotropy field magnitude for\nset-B Bi:YIG film is almost 300 Oe larger compare to the\nvalue of set-A Bi:YIG film.\nThe magnetoelastic energy density for a strain depen-\ndent FMR measurement is given by FME=−σb[cos]2Θ,\nwherebis magnetoelastic constant and Θ is the angle\nbetween Mand strain direction[ 2][3]. ForMpointing\nin the [111] direction, the magnetoelastic energy den-\nsity has the form, FME=−σb. Fig.5(b) shows the\nlinear dependence and least-square fit of anisotropy en-\nergyEani=−1/2[MSHu] with different strain states of\nBi:YIG films from both the sets. The derived expressions\nfrom least-square fit in Fig. 5(b) for set-A and set-B are\nEani= (−3.46±1.06)×103+(10.74±0.51)×106[(ab−\na⊥)/ab] (erg/cc) and Eani= (12.58±3.59)×103+(7.71±\n1.18)×106[(ab−a⊥)/ab] (erg/cc), where the slope of\nthe lines give −b= (10.74±0.51)×106(erg/cc) and\n−b= (7.71±1.18)×106(erg/cc), respectively. The neg-\native sign of bimplies that the magnetic easy axis is par-\nallel to the compressed lattice plane; [111]. The magne-\n6/s32/s83/s101/s116\n/s32/s76/s105/s110/s101/s97/s114/s32/s70/s105/s116\n/s40/s97\n/s98/s45/s97 /s41/s47/s97\n/s98/s32/s40 /s41/s69\n/s97/s110/s105/s32/s40 /s32/s101/s114/s103/s47/s99/s109 /s41/s72\n/s117/s32/s40/s71/s41\nFIG. 5. (Color online) (a) Out-of-plane uniaxial anisotrop y\nfieldHuand (b) total anisotropy energy Eanias a function of\nthe rhombohedral distortion (( ab−a⊥))/ab% of the Bi:YIG\nfilms on GGG(111). Blue solid lines are the least-square fit\nto obtain magnetoelastic coupling constant. Dashed curves\nserve as a guide to the eye.\ntoelastic constant of Bi:YIG comes out to be larger than\nin pure-YIG film[ 2]. Pure YIG exhibits almost quenched\norbital momentum of half-filled dshell inFe3+electron\nconfiguration, leads to weak SOC and shows low magne-\ntoelastic coupling constant. The substitution of strong\nSOC ions such as Bi3+,Dy3+andTm3+etc. enhances\nthe spin-orbit coupling which results in improved mag-\nnetoelastic coupling. It suggests that the strain-tuning\ncould be very crucial to obtain large magnetocrystalline\nanisotropy even in thick ferrimagnetic-insulating films.\nGilbert damping coefficient αfor our Bi:YIG films has\nbeen calculated from frequency-dependent FMR mea-\nsurement between 7 and 12 GHz. The external mag-\nnetic field is swept at various fixed frequencies. Fig.\n6(a) and (b) show the frequency vs. Hresdata and\nits fit (corresponding colored solid curves) for set-A and\nset-B, respectively, using reduced form of Eqs. ( 1)\nand (2) in a limiting in-plane magnetic field geometry\n(θH= 90◦,φH= 0◦). The derived compact expression in\nasymptotic limit has the form (in-plane Kittel equation),\nωres=γ/radicalBig\nHres(Hres+4πMeff) (3)\nwiththeeffectivemagnetization4 πMeff= 4πMS−Hani,\nwhere,Haniis the anisotropy field parameterizes out-of-\nplane uniaxial and cubic anisotropies. It is clear from\nFig.6(a) and (b) that the data fits perfectly without\neven considering additional in-plane anisotropy contri-\nbutions. In Eq. ( 3) we do not consider a renormalization\nshift in the resonance frequency and a small shift in res-\nonance field which can arise by two-magnon scattering\nand a static dipole interaction between the ferrimagnetic\nfilm and the paramagnetic substrate, respectively, due to\nnegligiblysmallcontributions. Fig. 6(c)and(d)showin-\nplane frequency dependencies of linewidth (∆ H) for set-\nA and set-B films, respectively. The standard Landau-\n/s56 /s49/s48 /s49/s50/s48/s49/s48/s51/s48/s52/s48/s57/s48/s49/s50/s48/s49/s53/s48/s49/s56/s48\n/s56 /s49/s48 /s49/s50/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s57/s48/s49/s50/s48/s49/s53/s48/s49/s56/s48\n/s40/s100/s41/s40/s99/s41\n/s110/s109/s32/s32/s110/s109/s110/s109/s32/s32/s110/s109/s32/s110/s109/s40 /s101 /s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40/s71/s72/s122/s41/s32/s110/s109\n/s110/s109/s110/s109\n/s32/s32/s110/s109\n/s110/s109\nFIG. 6. (Color online) In-plane, frequency and thickness de -\npendent room temperature FMR measurements. Panel (a)\nand (b) represent frequency vs. resonance field plots for set -\nA and set-B, respectively. The fit to experimental data has\nbeen shown by corresponding coloured solid curves. Panel (c )\nand (d) represent frequency dependent linewidth variation for\nset-A and set-B, respectively. Black solid lines represent fit\nto the experimental data.\nLifshitz-Gilbert equation justifies the linear dependence\nof ∆Hwith frequency and used for straightforward de-\ntermination of the intrinsic Gilbert damping coefficient\n(α): ∆H= ∆H0+ (4πα/√\n3γ)fres, where ∆ H0is the\nextrinsic linewidth broadening due to magnetic inhomo-\ngeneities within the material. The extracted values of\n4πMeff,αand ∆H0for films from both the sets are\nshown in table II.\nFig.7(a)showsstraindependentvariationsof4 πMeff\nand 4πMS. The values of 4 πMefffor both the sets\nsystematically decreases with the increase in film thick-\nness but the values strongly depend on the state-of-the-\nstrain in the films. It can be seen that 4 πMeffis signif-\nicantly larger than the Bi:YIG saturation magnetization\ngenerated simple shape anisotropy i.e., 4 πMS, reveal-\ning the presence of a negative uniaxial anisotropy, sig-\n7TABLE II. Frequency and thickness dependent FMR derived\neffective magnetization, Gilbert damping coefficient and in-\nhomogeneous broadening of Bi:YIG epitaxial films grown by\ntwo different protocols. Set-A and set-B are separated by a\nsolid horizontal line.\nThickness 4 π Meff α(×10−3) ∆ H0\n(nm) (Oe) (Oe)\n10.2 (A1) 3482 ±65 18.3 ±1.3 84\n18.1 (A2) 2441 ±27 12.7 ±0.9 72\n37.0 (A3) 1970 ±8 6.9 ±0.7 67\n92.5 (A4) 1673 ±46 2.4 ±0.3 30\n200 (A5) 1510 ±2 2.0 ±0.1 5\n18.7 (B1) 2928 ±15 16.1 ±1.5 92\n39.8 (B2) 2437 ±2 9.6 ±0.6 68\n100 (B3) 2125 ±3 3.4 ±0.1 37\n150 (B4) 1787 ±4 3.2 ±0.1 31\n200 (B5) 1399 ±2 2.9 ±0.2 10\nnature of easy in-plane magnetization. The gap between\n4πMeffand 4πMSrepresents magnitude of anisotropy\nfieldHani= 4πMS−4πMeffwhich decreases with in-\ncrement in film thickness. The magnitude of Hanifor\n∼100 nm thick Bi:YIG film from set-B is larger than\nthat expected and comparable to ∼37 nm thin film\nfrom set-A, which is due to growth induced large strain.\nFig.7(b) shows magnetization (4 πMeff, 4πMS) depen-\ndence on uniaxial anisotropy field, where, the magne-\ntization decreases in proportion with the magnitude of\nuniaxial anisotropy field. Fig. 7(c) shows the variation\nofαwith respect to the film thickness from both the\nsets. Whereas, inset shows induced strain dependency\nofα. We notice that the value of αdecreases nonlin-\nearly as film thickness increases (or strain relaxes) and\nvice-versa. We include effective magnetization, uniax-\nial anisotropy field and damping data of YIG/GGG(111)\nfilms from literature by Bhoi et. al.[64] which also follow\nthe same trend. The lowest damping possessed by a 200\nnm thick film from set-A is (2 .0±0.1)×10−3with an\ninhomogeneous broadening of ∼6 Oe, whereas, a 200\nnm thick film from set-B shows slightly larger damp-\ning (2.9±0.2)×10−3with an inhomogeneous broad-\nening of ∼10 Oe but inherit reasonably large uniaxial\nanisotropyfield ( −108±17Oe) which is almosttwotimes\nlarger compare to former. Although, the damping in Bi\ndoped YIG enhances due to strong spin orbit coupling,\nstill it’s passably small compare to metallic systems[ 59–\n61]. As the values of αand|Hu|increases as a func-\ntion of the induced strain, we therefore plot αvs.Hu\ngraph (see Fig. 7(d)) to see the correlation between\nthe precessional damping and magnetic anisotropy. In\nour Bi:YIG thin film system, we observe a nonlinear re-\nlationship between αandHu, similar to YIG and can\nbe attributed to spin wave damping induced by incre-\nment in strain[ 64]. Rhombohedral distortion arising due\nto lattice mismatch between the film and the substrate\nleads to changein magnetic properties throughspin orbit\n/s32/s82/s101/s102\n/s72\n/s117/s32 /s40/s71 /s41/s40/s100 /s41\nFIG. 7. (Color online) Magnetization (4 πMSand 4πMeff)\ndependencies on (a) epitaxial strain and (b) Hu. Precessional\ndamping dependencies on (c) thickness (inset: on strain) an d\n(d)Hu. Panel (b) and (d) include YIG/GGG(111) data from\nref.[64]. Dashed curves serve as a guide to the eye.\ncoupling[ 3]. The inclusion of lattice distorted SOC along\nwith phonon-magnon scattering, two-magnon scattering\nor charge transfer relaxation may explain the thickness\ndependent enhancement of uniaxial anisotropy and re-\nduction of magnetic damping[ 2,3,33,60,61,64,73].\nIn summary, we have been able to grow high quality\nepitaxial Bi:YIG thin films on GGG(111) crystals as\nevidenced by prominent Laue oscillations in X-ray\ndiffraction pattern. A usual trend of the film lattice\nrelaxation and decrease in magnetic anisotropies as\nthe film thickness increases has been observed. Our\nstudy shows that strain can be a crucial parameter to\ntune the magnetocrystalline anisotropy. We optimize\na growth protocol to get thick epitaxial films with\nlarge lattice strain which allows us to achieve large\nmagneto-crystalline anisotropy. The Bi:YIG films grown\nusing higher laser fluence show large magneto-crystalline\nanisotropy compare to films of respective thicknesses\ngrown using lower laser fluence. We show that the\nincorporation of growth induced large strain in thick\nBi:YIG films can be helpful to improve the magnetic\nproperties. Out-of-plane uniaxial anisotropy varies\nlinearly with strain induced rhombohedral distortion\nof Bi:YIG lattice. Still, we are able to achieve fairly\nlow Gilbert damping ∼2×10−3with enhanced mag-\n8netoelastic coupling. Further, as Bismuth substitution\nenhances the magneto-optical responses enormously,\nthe coupling of large magnetocrystalline anisotropy,\nimproved magnetoelastic coupling and low damping\nwith strong magneto-optical activity in Bismuth sub-\nstituted YIG may provide unique opportunities for\nphoton-based-magnonics to develop efficient and low\nloss spintronics and caloritronics devices.\nACKNOWLEDGEMENTS:\nWe thank Prof. R. C. Budhani for fruitful discussion\nand Dr. Veena Singh for technical assistance during\nFMR measurements.Reference details of the relationship between\neffective magnetization and Gilbert damping\ncoefficient shown in Fig. 1. It was constructed using\nthe effective magnetization (saturation magnetization\nin few cases) and Gilbert damping coefficient values\nfrom various (Region I and II) ferro- and ferrimagnetic\ninsulators, (Region III) conducting oxides and (Region\nIV) pure metals and metal-alloys, as reported in previous\nstudies.\n9FIG. 8. Region - I: Hauser et al. [23], Chang et al. [21], Chang et al. [1], Bhoiet al. [64], Lucas et al. [74], Leet al. [75], Onbasli et\nal.[20], Liuet al. [73], Yanget al. [27], Gallagher et al. [76], Sunet al. [77], Jungfleish et al. [78], Howe et al. [22], Patiet al. [79],\nWuet al. [80], Jermain et al. [33], Budhani et al. [81], Nosach et al. [82], Yoshimoto et al. [83], Dubset al. [84], Pirroet al. [85],\nHeinrich et al. [86], Haertinger et al. [87], Fanget al. [88], Hariiet al. [89], Chang et al. [1].\nRegion - II: Iguchi et al. [90], Kehlberger et al. [91], Vasiliet al. [92], Siuet al. [93].\nRegion - III: Lee et al. [94], Emori et al. [95], Qinet al. [96], Qinet al. [97], Luoet al. [98].\nRegion - IV: Ando et al. [99], Guoet al. [61], Leeet al. [100], Tuet al. [101], Luet al. [102], Fermin et al. [103], Gong et\nal.[104], Ikedaet al. [105], Lindner et al. [106], Belmeguenai et al. [107], Heet al. [108], Kurebayashi et al. [109], Zhaoet al. 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Kalashnikova1\n1Io\u000be Institute, 194021 St. Petersburg, Russia\n2School of Physics and Astronomy, The University of Nottingham, NG7 2RD Nottingham, UK\n3Experimental Physics II, Technical University Dortmund, D44227, Dortmund, Germany\n(Dated: August 21, 2019)\nUsing a time-resolved optically-pumped scanning optical microscopy technique we demonstrate\nthe laser-driven excitation and propagation of spin waves in a 20-nm \flm of a ferromagnetic metallic\nalloy Galfenol epitaxially grown on a GaAs substrate. In contrast to previous all-optical studies\nof spin waves we employ laser-induced thermal changes of magnetocrystalline anisotropy as an\nexcitation mechanism. A tightly focused 70-fs laser pulse excites packets of magnetostatic surface\nwaves with an e\u00001-propagation length of 3.4 \u0016m, which is comparable with that of permalloy.\nAs a result, laser-driven magnetostatic spin waves are clearly detectable at distances in excess\nof 10\u0016m, which promotes epitaxial Galfenol \flms to the limited family of materials suitable for\nmagnonic devices. A pronounced in-plane magnetocrystalline anisotropy of the Galfenol \flm o\u000bers\nan additional degree of freedom for manipulating the spin waves' parameters. Reorientation of\nan in-plane external magnetic \feld relative to the crystallographic axes of the sample tunes the\nfrequency, amplitude and propagation length of the excited waves.\nI. INTRODUCTION\nIn magnonics coherent spin waves (SWs) are employed\nfor encoding, transferring, and processing information [1{\n3]. The use of SWs enables scaling of magnonic elements\ndown to the nanometer range owing to short wavelengths\nand the reduction of Joule heating associated with the\ncharge transfer in conventional electronics. Moreover, an\nextended functionality of magnonic devices is provided\nby the possibility to manipulate the amplitudes, phases,\nand wavevectors of SWs. Progress in the \feld of magnon-\nics relies on the development of approaches to generate\nand transfer SWs in a controllable manner. E\u000ecient\nconversion mechanisms between electrical/optical pulses\nand collective magnetic excitations are required to cou-\nple magnonic elements to electronic and photonic units\n[4{6]. For rapid growth of the \feld of magnonics, it is\nessential to extend the range of materials and structures\nsupporting long SWs propagation distances and enabling\ntheir control [7{11].\nOptical radiation allows the magnetic parameters of\nmaterials to be altered reversibly at various timescales\ndown to femtoseconds [12, 13]. This has led to the emer-\ngence of a photo-magnonics [14{20], where laser pulses\nare employed as a tool for both driving SWs and ma-\nnipulating their propagation. In particular, it has been\ndemonstrated that femtosecond laser pulses enable exci-\ntation of SWs with controlled wavevectors and propaga-\ntion directions [18, 21{23]. However, up to now the e\u000bects\nof short laser pulses employed to drive SWs have been\nlimited to ultrafast opto-magnetic phenomena [18, 21{\n25], ultrafast demagnetization [17, 26{29], and coherent\nenergy transfer from elastic waves to the magnon sub-\nsystem [24, 30{33]. These mechanisms place constraints\non the properties of the media, the laser pulse parame-ters, and the excitation geometries, while the excitation\nof propagating SWs by other ultrafast magnetic phenom-\nena [13] remains unexplored. Furthermore, the range of\nmaterials where the optical generation of SWs has been\nrealized is also very limited and, in fact, coincides with\nthe known suitable media for magnonics [3].\nIn this Article we examine the feasibility and advan-\ntages of excitation of propagating SWs in an anisotropic\nferromagnetic \flm by ultrafast laser-induced thermal\nchanges of the magnetocrystalline anisotropy. Using\ntime-resolved optically pumped scanning optical mi-\ncroscopy (TROPSOM) [17], we reveal propagating mag-\nnetostatic surface waves (MSSWs) excited by a femtosec-\nond laser pulse in a thin \flm of a ferromagnetic metal-\nlic alloy Galfenol (Fe 0:81Ga0:19) epitaxially grown on a\nGaAs substrate. We demonstrate that the propagat-\ning MSSWs packets are launched via the laser-induced\nthermal decrease of the magnetic anisotropy occurring\non a picosecond time scale and localized within the ex-\ncitation spot. Owing to this excitation mechanism, the\nMSSWs are excited in a simple geometry with an in-\nplane external magnetic \feld, and their characteristics\ncan be controlled by the orientation of the in-plane \feld\nwith respect to the magnetocrystalline anisotropy axes\nof the \flm. We show that the 20-nm thick Galfenol\n\flm supports an e\u00001-propagation length of MSSWs as\nlarge as 3.4 \u0016m, comparable to that of Permalloy { a\nmodel metallic material for magnonics [3]. Our results\npromote epitaxial Galfenol to the limited family of mate-\nrials for magnon-spintronics, recon\fgurable magnonics,\nand ultrafast photo-magnonics. Furthermore, the ultra-\nfast thermal changes of magnetic anisotropy as a driving\nmechanism for the excitation of propagating SWs can\nbe applied to a broad range of materials without spe-\nci\fc constraints imposed on their electronic and magnetic\nstructures [34].arXiv:1904.05171v2  [cond-mat.str-el]  20 Aug 20192\nThe Article is organized as follows. In Sec. II we de-\nscribe the FeGa/GaAs sample and the details of the\nTROPSOM experimental setup. In Sec. III we \frst\npresent the experimental results on laser-induced excita-\ntion of the magnetization precession and SWs (III A), dis-\ncuss the excitation mechanism (III B), the main parame-\nters and features of the MSSWs' propagation (III C), and\nreconstruct the MSSWs' dispersion from experimental\ndata (III D). This is followed by a theoretical analysis of\nthe MSSW dispersion relation speci\fc to the anisotropic\nFeGa \flm, calculations of the main MSSWs' parameters\nand their comparison to those obtained experimentally\n(III E). In Sec. IV we summarize our \fndings and dis-\ncuss their possible impact on the \feld of magnonics.\nII. EXPERIMENTAL DETAILS\nA. Sample\nFor our study, we chose a \flm of Fe 0:81Ga0:19with\na thickness d= 20 nm epitaxially grown on a 350- \u0016m\nthick (001)-GaAs substrate by magnetron sputtering, as\ndescribed elsewhere [35]. The Galfenol \flm was capped\nwith 3-nm thick Al and 120-nm thick SiO 2protective\nlayers. The back-side of the substrate was polished to\noptical quality. X-ray di\u000bractometry showed that the\nGalfenol \flm has a mosaic structure with grain sizes of\n\u001812 nm, and their crystallographic axes misorientation\nof\u00181.3 deg. It is well established that thin \flms of iron\nand iron-based alloys epitaxially grown on GaAs exhibit\nintrinsic cubic and substrate-induced in-plane uniaxial\nmagnetic anisotropies [36{39]. In particular, in Galfenol\n\flms on (001)-GaAs substrates the uniaxial anisotropy\naxis emerges along the [110] direction [35, 40].\nB. Experimental setup\nOptically-excited SWs were studied using the\nTROPSOM setup which enables detection of the spatial-\ntemporal evolution of the polar magneto-optical Kerr\nrotation \u0001\u0012kproportional to the transient changes of the\nout-of-plane magnetization component Mz[Fig. 1]. Here\nxyz is the laboratory frame with the z\u0000axis directed\nalong the sample normal and the x\u0000axis chosen to be\nalong the direction of an external DC magnetic \feld H.\nOptical pulses with nominal duration of 70 fs, central\nwavelength of 1050 nm, and 70 MHz repetition rate\ngenerated by the Yb-doped solid-state oscillator laser\nsystem were split into pump and probe parts. The\ncentral wavelength of the pump pulses was converted to\n525 nm using a \f-BaB 2O4crystal. The amplitude of the\npump pulses was periodically modulated at a frequency\nof 84 kHz using a photoelastic modulator placed between\ntwo crossed Glan-Taylor prisms. Two microscope\nobjectives were used to focus the pump and probe pulses\nonto the Galfenol \flm from the cap and the substrate\ny\nProbe\n1050 nm\nObjective lensxy-stage\nPump\n525 nm\nH || x ⊥ [001]\n[010]\n x[100]\nφ\nH[001]\nFeGa\nGaAs\nz[100][001]\n[010]\nx2\nx1x3, \nHM�m�z\nxk�FIG. 1. Experimental geometry. A DC magnetic \feld His\napplied along the xaxis, while the azimuthal orientation of\nthe sample 'can be varied. The relative pump-probe dis-\ntance is changed along either the x- ory-direction to detect\npropagation of BVMSWs or MSSWs modes, respectively. The\nblue-red pattern represents SW propagation in the xyplane\nschematically. Inset shows the x1x2x3reference frame linked\nto the crystallographic axes of the sample, and the angles\n'm;'; de\fning the in-plane orientations of the magnetiza-\ntionM, external DC magnetic \feld H, and the SW wavevec-\ntork, respectively.\nsides, respectively. The pump and probe radii \u001b, i.e.\nthe half-width of the beam's spatial pro\fle at which\nthe intensity drops bype, were measured to be 0 :8\u0016m\nusing the knife-edge method. The pump \ruence was\n3.5 mJ/cm2, and the probe \ruence was approximately\n20 times lower. The microobjective for the probe pulses\nwas \fxed. The microobjective for the pump pulses was\nmounted on the piezoelectric stage, which moves in the\nxyplane and controls the relative spatial displacements\n\u0001x;\u0001ybetween the pump and probe pulses. The\npump-probe temporal delay twas controlled by a delay\nline in the pump pulse optical path. In the experiments\nthe temporal dependences \u0001 \u0012k(t) of probe were obtained\nat various pump-probe displacements \u0001 x;\u0001y, and at\nvarious azimuthal orientations of the sample de\fned by\nthe angle'between the [100] crystallographic axis and\nthex-axis. The probe polarization rotation \u0001 \u0012k(t) was\ndetected using a conventional scheme with a Wollaston\nprism and a balanced photodetector. The signal from the\nphotodetector was registered using a lock-in technique\nwith the reference frequency of the pump amplitude\nmodulation. Thus, only the probe polarization rotation\nrelated to the pump-induced dynamics was detected.\nThe measurements of SWs propagation were performed\nat room temperature and H= 100 mT.3\n�x=0\n�y (�m)�x=0\n��1\n-1-11\n-11\n��y=0 �m�y=2 �m�y=4 �m�y=6 �m\n�x=0\n(a) (b) (c)\n(e) (d)\n��y=0 �m�y=2 �m\n�x=0\n(f)-11�\n�y=0-30�y (�m) �x (�m)6\n4\n2\n0\n-212\n10\n8\n6\n4\n2\n012\n10\n8\n6\n4\n2\n0\nFIG. 2. (a,b) Experimental spatio-temporal \u0001 y\u0000tmaps of \u0001\u0012Kobtained at '= 45\u000e(a) and'= 60\u000e(b) and \u0001x= 0,\ni.e when the pump-probe distance \u0001 yis scanned transversely to H, corresponding to the MSSW con\fguration. Arrows are\nguides to the eye showing the centers of the propagating MSSWs packets. (c) Experimental (symbols) temporal evolution of\n\u0001\u0012Kat di\u000berent pump-probe distances \u0001 y, at'= 45\u000eand \u0001x= 0. Solid lines: the \fts using Eqs. (1,2). (d) Experimental\nspatial-temporal \u0001 x\u0000tmap of \u0001\u0012Kobtained at '=\u000030\u000eand \u0001y= 0, i.e. when the pump-probe distance \u0001 xis scanned\nalong H, corresponding to the BVMSWs con\fguration. (e) Experimental (symbols) and calculated using Eq.(5) dependence\nof the precession frequency f0on the strength of the magnetic \feld Happlied at'= 30\u000e. (f) Calculated (symbols) temporal\nevolution of \u0001 \u0012Kat di\u000berent pump-probe distances \u0001 y, at'= 45\u000eand \u0001x= 0. Solid lines: the \fts using Eqs. (1,2).\nIII. EXPERIMENTAL RESULTS AND\nDISCUSSION\nA. Laser-induced excitation of propagating spin\nwaves\nFigures 2(a,b) show the spatial-temporal evolution of\n\u0001\u0012Kobtained by scanning the pump-probe time delay t\nwhen the pump and probe spots are shifted with respect\nto each other by 0 \u0014\u0001y\u001412\u0016m at \u0001x= 0, i.e. trans-\nversely to H. Experimental data for two orientations of\nthe sample '= 45\u000eand'= 60oare presented. The for-\nmer geometry corresponds to the \feld applied along the\n\flm hard axis [110]. Two types of pump-induced signal\n\u0001\u0012K(t) can be distinguished depending on whether the\npump and probe spots overlap spatially or not. We show\nthis in more detail in Fig. 2(c), where the cross-sections of\nthe spatial-temporal maps at various \u0001 yare presented.\nWhen the pump and probe spots overlap spatially, i.e.\nat \u0001y <p\n2\u001b, decaying oscillations of \u0001 \u0012K(t) are ob-\nserved. Examination of the dependence of the oscilla-\ntion frequency on the external \feld strength [Fig. 2(e)]\ncon\frms that they originate from the laser-induced pre-\ncession of the magnetization. Outside the pump-probe\nspatial overlap, i.e. at \u0001 y >p\n2\u001b, well-de\fned wave-\npackets are observed in the \u0001 \u0012K(t) signal. The tilts of\nthe signal maxima reveal a positive phase shift of thepropagating waves. Therefore, these wave-packets can\nbe con\fdently ascribed to laser-induced MSSWs propa-\ngating transversely to H.\nSpatial-temporal \u0001 x\u0000tmaps obtained at \u0001 y= 0\n(Fig. 2(d)) have revealed the presence of fast-decaying\nbackward volume magnetostatic waves (BVMSWs) with\nnegative phase shifts. Such a di\u000berence in the propaga-\ntion characters of laser-driven MSSWs and BVMSWs ap-\npears to be typical for thin metallic \flms [26{28]. Below\nwe focus our discussion on the excitation and propagation\nof the MSSWs.\nAs can be seen in Figs. 2(a,b), the parameters of the\nmagnetization precession at \u0001 y= 0 and of the MSSWs\nat \u0001y6= 0 vary with '. To quantify the azimuthal de-\npendences of the parameters, we \ftted temporal signals\n\u0001\u0012K(t) obtained at di\u000berent 'and \u0001ywith either of the\nfunctions:\n\u0001\u0012K(\u0001y= 0;t) =A0\nSWsin(2\u0019f0t\u0000\u001e0)e\u0000t=\u001c;(1)\n\u0001\u0012K(\u0001y;t) =ASW(\u0001y) sin(2\u0019ft\u0000\u001e)e\u0000(t\u0000t0)2\n2w2:(2)\nHereA0\nSW,ASW(\u0001y) are amplitudes of the precession at\n\u0001y= 0 and \u0001y>0, respectively; \u001c,f0andf,\u001e0and\u001e\nare the decay time, frequencies, and initial phases of the\nprecession; wandt0{ width and center position of the\nGaussian packet. Good agreement between the experi-\nmental data and the \ftted curves is reached [Fig. 2(c)],4\napart from a small discrepancy at the tails of the pre-\ncession and wavepackets, which is addressed below in\nSec. III E.\nB. Mechanism of MSSWs' excitation\nFigures 3 (a,b) show the azimuthal dependences f0(')\nandA0\nSW(') at \u0001y= 0. The former has a pronounced\n4-fold symmetry with a 2-fold distortion. It corresponds\nwell to the intrinsic cubic magnetic anisotropy de\fned by\nthe anisotropy parameter K1>0 with easy axes along\n[100] and [010] and the substrate-induced in-plane uniax-\nial magnetic anisotropy de\fned by the parameter Ku<0\nwith an easy axis along [110] expected for such \flms [38].\nA pronounced azimuthal dependence of the amplitude\nA0\nSW(') of the laser-driven precession [Fig. 3(b)] al-\nlows us to identify the excitation mechanism as an ultra-\nfast laser-induced thermal change of the e\u000bective mag-\nnetocrystalline anisotropy \feld demonstrated earlier for\na range of Galfenol \flms of various thicknesses [41, 42].\nIn brief, excitation of the precession stems from a rapid\ndecrease of the anisotropy parameters K1andKu[43{\n45] and of the saturation magnetization Ms[46] in re-\nsponse to the laser-induced increase of electronic and lat-\ntice temperatures. For the in-plane orientation of exter-\nnal magnetic \feld considered here, the amplitude of the\nexcited precession is a measure of the abrupt reorienta-\ntion of the total e\u000bective \feld in the sample plane due\nto the laser-induced changes of K1=MsandKu=Ms. At\nthe timescale longer than the electron-phonon thermal-\nization time\u00182 ps, the thermodynamic approximation\nfor the temperature dependence of anisotropy can be ap-\nplied [47]. Thus, the decrease of the parameters K1,Ku\nis expected to be stronger than that of the magnetization\nMs, and, therefore, the laser excitation yields an abrupt\ndecrease of the e\u000bective anisotropy \feld. Experimentally\nveri\fed independence of the \u0001 \u0012k(t) signal on the pump\npulse's polarization supports the thermal nature of the\nexcitation mechanism.\nIn the following we refer to the MSSWs excita-\ntion mechanism based on laser-induced thermal change\nof the e\u000bective magnetocrystalline anisotropy \feld\n\u0018(K1+Ku)=Msas the ultrafast laser-induced change\nof the magnetic anisotropy. This is to distinguish it from\nthe excitation of the MSSWs via ultrafast changes of the\nshape anisotropy related to the demagnetization solely\nand demonstrated in [17, 26{29, 48]. The latter mech-\nanism is deliberately excluded here by the chosen ex-\nperimental geometry with an in-plane external magnetic\n\feld. We can also exclude two other mechanisms of laser-\ninduced MSSWs' excitation reported earlier. Ultrafast\ninverse magneto-optical e\u000bects employed in [18, 21{25]\nare proven to be e\u000ecient in dielectric media mostly (for a\nreview see e.g. [49]), and are pump-polarization sensitive\nwhich was not the case in our experiments. Driving SWs\nby optically-excited elastic waves shown in [24, 30{33]\ncan, in turn, work in any material with su\u000eciently strong\n�\n�\n��FIG. 3. (a,b) Azimuthal dependences of the frequency f0\n(a) and amplitude A0\nSW(b) of the precession observed at\n\u0001y= \u0001x= 0. Symbols show experimental data; solid lines\nare the theoretical \fts. (c) Normalized amplitude of MSSWs\npacketsASW(\u0001y)=ASW(\u0001y= 0:5\u0016m) vs distance \u0001 yas ob-\ntained for several angles '; lines are the \fts using Eq. (3). (d)\nPropagation length Lpropof MSSWs packets vs 'as obtained\nfrom the \fts of the experimental (symbols) and using Eq. 10\n(solid line).\nmagnetoelastic coupling at any laser-pulse polarization.\nFurthermore, the magnetoelastic properties of Galfenol\nfavor such a mechanism. However, excitation of SWs by\nelastic waves becomes e\u000ecient only near the crossing of\ntheir dispersion curves [50], which is not realized in the\nstudied \flm in the range of the applied magnetic \felds\nused in the experiments (see details in Sec. III D). Never-\ntheless, in the experiments we have detected two acous-\ntic waves seen as two concentric ripples independently\npropagating slower than the SWs, in analogy with the\nobservations reported in [17].\nC. MSSWs' propagation length\nThe azimuthal dependence of the MSSW's amplitude\nASW(') at \u0001y= 0:5\u0016m, i.e. close to the edge of the\nexcitation spot, naturally resembles the one for A0\nSW.\nAt \u0001y >p\n2\u001bthe situation changes drastically, as the\namplitudes of the MSSWs packets outside the excitation\nspot are de\fned not only by the e\u000eciency of excitation at\nthe speci\fc \feld direction [Fig. 3(b)], but by the spatial\ndecay as well. To single out the latter contribution, we\nplot the spatial dependence of the normalized amplitudes\nof the MSSWs packets ASW(\u0001y)=ASW(\u0001y= 0:5\u0016m)\n[Fig. 3(c)]. The spatial decay is minimum at '=\u000645\u000e\nin the so-called \"hard-hard\" con\fguration, where the\nequilibrium magnetization and the MSSW's wavevector5\nare oriented along the two orthogonal hard axes. Thus,\nthe MSSWs packets propagate larger distances along the\nhard axes, despite small initial amplitudes de\fned by\nthe excitation mechanism. In contrast, no propagating\nMSSWs packets are observed if His directed close to\nthe easy axes ( '= 0\u000e;\u000690\u000e), and the small amplitude\nprecession is excited.\nThe experimental dependences ASW(\u0001y) can be well\n\ftted by a single exponential decay function [Fig. 3(c)]:\nASW(\u0001y)\u0018e\u0000\u0001y=Lprop; (3)\nwithLprop being the propagation length. This is an\nimportant parameter of SWs, which, in particular, de-\ntermines whether Galfenol is a suitable material for\nmagnonic applications. As can be seen in Fig. 3(d), Lprop\ndemonstrates a pronounced azimuthal dependence with\ntwo maxima corresponding to the geometries with Hap-\nplied along the hard axes. The largest Lprop= 3:4\u0016m is\nobserved when His aligned along the hardest anisotropy\naxis [1 10]. Importantly, this value is very close to the\npropagation length found for optically excited MSSWs\nin a 20-nm thick Permalloy \flm [26].\nD. Reconstruction of MSSW dispersion\nThe \u0001y\u0000tmaps presented in Figs. 2(a,b) allow us to\nreconstruct the dispersion relation f(ky) for the excited\nMSSWs using a 2-dimensional fast Fourier transform\n(2D-FFT) of the data outside the pump spot [24, 27, 32].\nFigure 4(a) shows the 2D-FFT result for '= 45\u000e, at\nwhich the MSSWs possess large Lprop (see Sec. III C).\nThe 2D-FFT gives an expected near-linear dispersion and\nshows that the wavenumbers kyof the MSSWs detected\nin the experiment reach about 3.5 rad/ \u0016m. As discussed\nin Ref. [27] the wavenumbers of the SWs detected in the\noptical pump-probe experiment are limited by both the\npump and probe spot sizes, which yields k\u001b= 1=\u001bfor\nthe wavenumbers at the 1 =pe-level (see Appendix for\ndetails). Given the spot size \u001bused in the experiment,\nthe corresponding value is k\u001b= 1:3 rad/\u0016m, and the ex-\nperimentally found largest value 3.5 rad/ \u0016m corresponds\napproximately to 3 k\u001b. The value also con\frms that the\nobserved MSSWs are not driven by propagating surface\nacoustic waves (SAWs) in the GaAs substrate. Indeed,\nthe SAWs velocity in a (001)-GaAs substrate is 2.7 km/s\n[51], which gives a maximum SAW frequency of 1.5 GHz\nat a wavenumber of 3.5 rad/ \u0016m. The value is well below\nthe frequencies of the excited MSSWs.\nIt is important to note that the laser-induced heat-\ning and the changes of magnetization and anisotropy are\nexpected to alter the precession frequency fwithin the\npump spot. These changes, indeed, can be clearly seen in\nf\u0000\u0001ymaps, where fis obtained as the FFT of \u0012K(t) at a\nspeci\fc position \u0001 y[Fig. 4(b,c)]. For Happlied along the\nhard axis ('= 45\u000e)fexperiences an increase inside the\nheated area. If His not aligned with the hard axis (e.g.\nat'= 30\u000e), the situation is opposite, and fis decreased\nH H'Hf\nheatednon-heatedH||h.a. H||e.a.Frequency f (GHz)\nky (rad/�m)0      2      4      620\n15\n10\n5\n0   = 45\nHf\nheatednon-heated(a)\n(d) (e)\na a\n(b)(c)\n�y(�m)0      4       8    12\n�y(�m)0         3         6    = 45\nFFT intensity (arb.units)\n-1        0        1 0                     1FFT intensity (arb.units)\n   = 30Frequency f (GHz)20\n15\n10\n5\n0FIG. 4. (a) Dispersion of MSSWs reconstructed via 2D FFT\n(real part) at '= 45\u000e. (b,c) FFT spectra at di\u000berent pump-\nprobe shifts \u0001 yat'= 45\u000eand 30\u000e. (d,e) Schematic repre-\nsentation of the frequency changes in the heated area when H\nis along a hard and easy anisotropy axis, respectively. Solid\nlines: dependences of f(H) for a non-heated \flm, dashed\nlines:f0(H) for a heated \flm. Vertical dashed lines repre-\nsent the magnitude of Hin the experiments.\ninside the pump spot. Both observations are in agree-\nment with the scenario of the magnetic anisotropy being\ndecreased abruptly within the laser-excited spot. Indeed,\nin the \"hard-hard\" con\fguration the laser-induced ul-\ntrafast heating partly suppresses the e\u000bective anisotropy\n\feldHaresulting in an increase of fatH > Ha, as\nschematically illustrated in Fig.4(d). The opposite sit-\nuation is expected in the \"easy-easy\" con\fguration [see\nFig.4(e)].\nIt should be noted that the laser-induced local decrease\nof the eigen frequency f0can lead to the formation of\na potential well for MSSWs, with eigen frequencies be-\nlow the spectrum of MSSWs outside this well, as demon-\nstrated in Ref. [52]. Then the escape of the MSSWs from\nthis well would be strongly suppressed. We do not ob-\nserve the formation of such traps as they are formed,\nwhen the diameter of the hot spot is several times larger\nthan the MSSW wavelength. In our pump-probe experi-\nments the shortest excited MSSW wavelength is compa-\nrable to the pump spot size. In other words, when the\npump spot size becomes comparable to the length scale\nde\fned by the ratio of the group velocity of a MSSW\ntof0, the precession of the magnetization propagates as\nMSSWs from the pump spot to the non-excited area [27].6\nE. Theoretical analysis\n1. Dispersion of magnetostatic waves in an anisotropic \flm\nIn order to obtain the expression for the spin wave\ndispersionf(k) we utilized the approach developed in\nRef. [53] and adapted it to the particular anisotropy of\nthe studied \flm. The basic expression for the \flm's free\nenergy density used in our calculations has the form:\n\u0001F=K1(m2\n1m2\n2+m2\n1m2\n3+m2\n2m2\n3)\n+Kum1m2\u0000\u00160Msm\u0001H;(4)\nwhere mis the unit vector in the magnetization direction,\nandmiis its projection on the crystallographic axis xi\n[see inset in Fig. 1]. The form of the in-plane anisotropy\ntermKum1m2was chosen after Ref. [38] to accommo-\ndate our system of an iron-based alloy grown on a (001)-\nGaAs substrate. Note that Eq. (4) does not contain the\nmagnetostatic energy term. In the derivation of the dis-\npersion relation the demagnetizing \feld is accounted for\nseparately by solving the Maxwell's equations. Apply-\ning the procedures of Ref. [53] to the free enery density\ngiven by Eq. 4, we obtained a transcendental equation\nconnecting fandk, which can be simpli\fed and written\nin a closed form in the limit kd\u001c1 relevant in our case\n[54], yielding the dispersion relation for magnetostatic\nwaves to be:\n!(k) = 2\u0019f(k) =\n=\rr\u0010\nB\u000b+\u00160Ms(1\u0000kd\n2)\u0011\u0010\nB\f+\u00160Mskd\n2sin2 \u0011\n;\n(5)\nwithB\u000bandB\fde\fned as\nB\u000b=\u00160Hcos('\u0000'm)\u0000Ku\nMssin(2'm)\n+2K1\nMs\u0014\n1\u00001\n2sin2(2'm)\u0015\n;\nB\f=\u00160Hcos('\u0000'm)\u00002Ku\nMssin(2'm)\n+2K1\nMscos(4'm);(6)\nwhere is the angle between the equilibrium mag-\nnetization and k[see inset in Fig. 1], \r= 1:76\u0002\n1011rad\u0001s\u00001\u0001T\u00001is the electron's gyromagnetic ratio,\nand'mis the angle between the magnetization and the\n[100] axis obtained from the equilibrium condition:\n\u00160Hsin('\u0000'm)\u0000K1\n2Mssin(4'm)\n\u0000Ku\nMscos(2'm) = 0:(7)The discrepancy between the solution of the exact\ntranscendental equation for f(k) and its approximation\n(5) is negligible at k < 5 rad/\u0016m andd= 20 nm. We\nnote that at k= 0, i.e. in the case of a homogeneous\nmagnetization precession, Eq. (5) gives exactly the same\ndependence f0(') as the analytical formula for ferromag-\nnetic resonance frequency in a thin anisotropic \flm [55].\nHaving obtained the explicit dispersion relation, we\nderived the expression for the precession ellipticity:\n\u000f=j\u0001Mxyj\njMzj=s\njB\u000b+\u00160Ms(1\u0000kd=2)j\njB\f+\u00160Ms(kd=2) sin2 j;(8)\nwhere \u0001MxyandMzare the transient changes of in-\nplane and out-of-plane components of the magnetization\nprecession, respectively. The damping factor as a func-\ntion of'andkcan be found using the relation Ref. [56]:\n\u000b=\u000b01\n\u00160\r@!(k)\n@H; (9)\nwhere\u000b0is the Gilbert damping of the precession in the\nexternal \feld Hatk= 0 without the account of demag-\nnetization e\u000bects and anisotropy. It is interesting to note\nthat Eq. (9) leads to an anisotropic damping with higher\nand lower values when the magnetization is aligned along\nhard and easy axes, respectively.\nOnce the Gilbert damping parameter is known, the\nformula for the propagation length Lpropcan be written\nin a straightforward way as:\nLprop=\u001cvgr=1\n\u000b!(k)@!(k)\n@k; (10)\nwhere\u001cis the relaxation time and vgris the group veloc-\nity of the MSSW.\n2. Analysis of the magnetization precession within the\nexcitation spot\nIn order to apply the expressions derived above for\na description of the experimentally observed magnetiza-\ntion dynamics, one needs to account for the timescales\nof the laser-induced changes of magnetic anisotropy and\nmagnetization. In metals a few picoseconds after excita-\ntion both magnetization and magnetic anisotropy possess\nslow relaxations which can be neglected on the timescales\nwhere the precession is observed. Thus, the temporal\npro\fle of the total e\u000bective \feld \u0018(K1+Ku)=Msis de-\nscribed below by the Heaviside function.\nBy analyzing the experimental dependences f0(') and\nA0\nSW(') within the pump spot, we extract the anisotropy\nparameters K1;Kuand saturation magnetization Msof\nthe excited and equilibrium \flm, using the following\nprocedure. First, the frequency f0within pump spot\nis de\fned by modi\fed parameters K1,Ku, andMsof\nthe laser-exited material. Fitting the experimental de-\npendencef0(') to Eq. (5) with k= 0 [solid line in7\nFig.3(a)] we get \u00160Ms= 1:56 T,K1= 2:8\u0002104J/m3\nandKu=\u00001\u0002104J/m3.\nHaving determined the parameters of the \flm within\nthe laser-excited area, we now can extract the laser-\ninduced changes of the anisotropy parameters K1and\nKuwith respect to their room temperature (RT) val-\nues by analyzing the azimuthal dependence A0\nSW(') of\nthe laser-excited precession [Fig.3(b)]. Indeed, two sets\nof the parameters, the initial ~K1,~Kuand ~Msat RT\nand the modi\fed K1,KuandMsof the laser-excited\n\flm, yield the di\u000berence between the e\u000bective \feld ori-\nentation at equilibrium and upon the excitation. This\ndi\u000berence de\fnes the amplitude of the in-plane compo-\nnent \u0001M0\nxyof the precession. The amplitude of the\nout-of-plane component \u0001 M0\nzcan be then found taking\ninto account the precession ellipticity (8). The \ftting\nofA0\nSW(')\u0018\u0001M0\nzthen gives the ratios ~K1=~Msand\n~Ku=~Ms. Using the experimental value of the RT satu-\nration magnetization \u00160~Ms= 1:7 T [57, 58], we obtain\nthe RT anisotropy parameters ~K1= 3:33\u0002104J/m3and\n~Ku=\u00001:03\u0002104J/m3, which are in a very good agree-\nment with the anisotropy parameters found earlier for a\n22-nm FeGa \flm on a (001) GaAs substrate [58]. We note\nthat the laser-induced change \u0001 K1=~K1\u0019\u000016% shows\nrather good agreement with the one reported earlier for a\n100-nm Galfenol \flm [41], while \u0001 Ms=~Ms\u0019\u00008% agrees\nwith the magnitude of the ultrafast demagnetization ob-\nserved in a thinner Galfenol \flm [42].\n3. Analysis of the laser-driven spin waves propagation\nIn order to explain the observed azimuthal depen-\ndenceLprop('), we used Eq. (10). The \ftting procedure\natk= 1rad/\u0016m gives the Gilbert damping parameter\n\u000b0= 0:017. As can be seen in Fig. 3(d), Eqs. (9,10)\ncorrectly predict the azimuthal dependence of MSSWs'\npropagation. In particular, the theory con\frms that the\ndi\u000berent propagation lengths for di\u000berent misorientations\nbetween the Hand MSSW's wavevector are dictated by\nthe anisotropy of the \flm. We note that the observed\nanisotropy of the propagation of the laser-driven MSSWs\npackets is in agreement with the one demonstrated in cu-\nbic iron \flms recently [59]. In both cases the largest prop-\nagation length is observed in the \"hard-hard\" con\fgura-\ntion. In contrast to the experiments with antennae [59],\nwhere the MSSWs' propagation is one-dimensional, laser-\ninduced MSSWs packets propagating along the y\u0000axis\nare spreading slightly along the x\u0000axis as well. Our re-\nsults show that this spreading does not change the main\nfeatures of the propagation of MSSWs packets.\nTheoretically obtained MSSWs' group velocities vgr\nare in the range of 5-9 km/s, and are in an agreement\nwith the experimental values of 4.5-13 km/s. These val-\nues of group velocities are typical for MSSWs in thin\nmetallic \flms [3].\nFinally, having obtained the MSSWs' parameters, in-cluding the anisotropic damping \u000b, we have calculated\nspatial-temporal \u0001 y\u0000tmaps at di\u000berent ', following\nthe procedure described in [18, 21, 24, 27]. In this ap-\nproach the dispersion relation (5) for MSSWs was used.\nWithin the area corresponding to the excitation spot the\nadditional e\u000bective \feld with the Heaviside temporal and\nGaussian spatial pro\fles were introduced to account for\nthe localized change of magnetic anisotropy triggering\nthe precession (see Appendix for details). Figure 2(f)\nshows exemplary crossections of the calculated \u0001 y\u0000t\nmaps demonstrating good agreement with the experi-\nmental data. The calculations show good agreement with\nexperimentally obtained precession and MSSWs pack-\nets [Fig. 2(c,f)]. Importantly, the calculations reveal the\nsame deviation of the MSSW packets' shapes from the\nGaussian ones as the experimentally observed signals\n\u0001\u0012K(t), evident in Figs. 2(c,f). The most prominent de-\nviation is observed at large time delays t, i.e. at the tails\nof the wavepackets, as was noted by other groups as well\n[26, 27, 60]. Our calculations based on the MSSWs' dis-\npersion relation show that this deviation originates from\nthe dispersion of SWs generated by a sudden change of\ne\u000bective \feld.\nIV. CONCLUSION\nIn conclusion, we have demonstrated laser-induced ex-\ncitation and propagation of magnetostatic spin waves in\na 20 nm thick epitaxial Galfenol \flm on a GaAs sub-\nstrate characterized by pronounced in-plane magnetic\nanisotropy. We have shown that ultrafast thermal mag-\nnetic anisotropy changes induced by tightly focused fem-\ntosecond laser pulses excite propagating MSSWs. The\nstrong in-plane anisotropy ( \u0018104J/m3) of the \flm en-\nables the laser-induced excitation of MSSWs in a sim-\nple geometry with an in-plane external magnetic \feld.\nThe anisotropy of the \flm provides the possibility to\ntune the frequency, amplitude and propagation length\nof the excited waves by changing the in-plane \feld ori-\nentation. We \fnd that the propagation length of the\nMSSWs in the studied \flm reaches 3.4 \u0016m, which, along\nwith other recent results on spin dynamics in Galfenol\n\flms [40, 42, 61], con\fdently promotes epitaxial Galfenol\nto the limited family of metallic materials for magnonics.\nFurthermore, epitaxial Galfenol is also known for hav-\ning a large magnetostriction constant [58] and so o\u000bers\nthe prospect of controlling the SWs propagation via a\nvoltage-induced strain when, for example, coupled to a\npiezoelectric substrate, as was earlier realized in YIG [11].\nIt is important to note that the laser-induced thermal\nmagnetocrystalline anisotropy change represents a novel\nfundamental process for the excitation of SWs, which\ncan be applied to a broad range of materials without\nlimitations on their electronic and magnetic structures.\nFinally, we note that introducing laser-induced ultrafast\nthermal changes of magnetic anisotropy as a tool to gen-\nerate SWs can have even broader impact on magnon-8\n(a) (b) (c)\n(d) (e) (f)M  (arb. units)z M  (arb. units)z\nFIG. 5. Calculated SWs signals in the xyplane at di\u000berent time delays tafter the excitation, 0.3 ns (a,d), 0.7 ns (b,e), and 1\nns (c,f) for the angles 'of -15\u000e(a-c) and -45\u000e(d-f).\nics. Since abrupt and local changes of the magnetic\nanisotropy by laser pulses yield strong local modi\fca-\ntions of the SWs' dispersion relation, they can be seen as\na pathway to realize an ultrafast optically-recon\fgurable\nmagnonic medium for e\u000ecient steering and conversion of\nSWs [15, 16, 62, 63].\nACKNOWLEDGMENTS\nThe authors thank L. V. Lutsev and I. V. Savochkin\nfor valuable discussions. The setup construction and\nexperiments were carried out under a support of RSF\n(grant No. 16-12-10485); theoretical analysis and calcu-\nlations were performed under a support of RFBR (grant\nNo. 18-02-00824). Collaboration between Io\u000be Institute\nand TU Dortmund is a part of the TRR 160 ICRC \"Co-\nherent manipulation of interacting spin excitations in tai-\nlored semiconductors\" supported jointly by RFBR (grant\nNo. 19-52-12065) and DFG, and of the COST action\nCA17123 Magnetofon.\nAPPENDIX: CALCULATION OF\nSPATIAL-TEMPORAL MAPS\nOF LASER-DRIVEN SPIN WAVES\nTo calculate the spatial-temporal dependences of\n\u0001\u0012K(r;t) corresponding to the experimental \u0001 y\u0000tmaps,we used the procedure described in Refs. [18, 21, 24, 27].\nThe probe pulses are re\rected normally, so the change of\ntheir polarization is a measure of the transient changes of\nthe out-of-plane component Mz. The normal component\nis given by an integration over all excited wavevectors\nk= (kx;ky) [27]:\nMz(r;t)\u0018Z\ndk(1=\u000f)h(k;!) sin[kr\u0000!(k)t]\n\u0002exp[\u0000\u000b!(k)t];(11)\nwhereh(k;!) is the Fourier transform of h(r;t) - an e\u000bec-\ntive \feld describing the e\u000bect of the laser pulse excitation\nof the sample, \u000fis the ellipticity (8), \u000bis the damping\nparameter (9).\nWe consider the case of thermal changes of anisotropy,\nwhich are approximated by the laser-induced e\u000bec-\ntive \feld introduced as the di\u000berence between the\ntotal e\u000bective \feld in the equilibrium and laser-\nexcited states h(r;t) =\u0000@\u0001F(~K1;~Ku;~Ms)=@M+\n@\u0001F(K1(r;t);Ku(r;t);Ms(r;t))=@M[45]. It is assumed\nthat a few picoseconds after excitation both the mag-\nnetization and magnetic anisotropy possess slow relax-\nation which can be neglected on the time scales of the\nmagnetization precession. Thus, the \feld h(r;t) may be\nrewritten as h(r;t)\u0018h(r)\u0002(t), where \u0002( t) is a Heav-\niside function. Hence, the Fourier transform h(k;!)\nat!6= 0 is weighted by i=![24]. Next, the pump\nbeam in our experiments has a spatial pro\fle close to9\nthe Gaussian one, i.e. h(r)\u0018exp(\u0000r2=2\u001b2). There-\nfore,h(k;!)\u0018exp(\u0000k2\u001b2=2). Thus,k0\n\u001b=p\n2=\u001bgives a\nwavenumber for the excited SW component which ampli-\ntude is at a level 1 =peof the component with the highest\namplitude. It is also important to take into account that\nthe probe spot radius in our experiment is \u001b, as well.\nAs a result, the measured wavevnumbers are de\fned by\nan e\u000bective width ofp\n2\u001b[27], and the corresponding\nwavenumber is k\u001b= 1=\u001b.\nUsing the procedure described above, the maps of\nMz(r;t) were calculated in the time interval 0{1500 ps for\ndi\u000berent'. 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The remarkable chemical stability and simple chemical\nformula makes metallic SrRuO\n3 quite attractive for use in epitaxial thin film\nheterostructures with other perovskite oxides when metallic layers are\ndesired.  Indeed, the nearly cubic SrRuO3 [177] is often preferred over the\nmore distorted CaRuO3 when making structures such as electrodes for\nferroelectrics or superconductor-normal metal-superconductor junctions.\nThe other striking feature of SrRuO3 is its ferromagnetism with a\nreasonably high transition temperature (163 K) and large saturation moment\n(> 1µB) [174, 175, 178, 179].  S rRuO3 is the only ferromagnetic perovskite oxide\nof a 4 d or 5 d transition metal.  Moreover, S rRuO3 has the largest saturation\nmoment known to arise from 4d  electrons, making it more related to the iron\ngroup ferromagnetic metals (Fe, Co and Ni) than to the weak itinerant\nelectron ferromagnets such as ZrZn2.  SrRuO3 also has very strong cubic\nmagnetic anisotropy, requiring magnetic fields in excess of 10 Tesla to saturate\nthe magnetization in the hard directions.  Such a strong anisotropy makes\nmeasuring even the simplest properties, such as saturation magnetization,\ndifficult.  It is the purpose of the present work to determine the magnetic\nproperties of SrRuO3 by measurements of magnetically-soft single crystals\nalong the easy magnetic direction.Abstract\nThe Magnetization of Single Crystal  SrRuO 3 is studied as a function of temperature\nalong different crystallographic directions. The magnetocrystalline anisotropy and \nbehavior near the critical transition temperature are analyzed in detail. The magnetization \nvs temperature is found to vary more like T2 rather than T3/2 expected for spin waves.This work was first published in a thesis in 1997, reference [1]G. Jeffrey Snyder\nStanford University, Stanford CA USA\ncurrent address Northwestern University, Evanston IL USACritical Behavior and Anisotropy in Single Crystal SrRuO3 2\nThere have been several studies of S rRuO3 in the past three decades,\nmostly on polycrystalline samples which are quite easy to prepare.  SrRuO3\nhas a very slightly distorted (GdFeO3 type) perovskite structure.  The\ndeviation from perfect cubic perovskite is so small that it has often been\nundetected.  The resistivity of polycrystalline, and epitaxial thin film SrRuO3\nshows a cusp in dρ/dT at the ferromagnetic Curie temperature TC.  This is\ncommonly observed for metallic ferromagnets and is attributed to spin\ndisorder scattering [125].  Reported TC's tend to vary from 150K to 165K.\nThe saturation magnetization of SrRuO3 has been both difficult to\nmeasure and interpret.  Low spin Ru4+ in an octahedral coordination has four\n4d electrons in the t2g triply degenerate state, giving two paired and two\nunpaired electrons.  Since the orbital component of angular momentum J\nwill be quenched, J = S = 1 is expected.  The measured Curie constant of the\nparamagnetic state is consistent with this model (expected:  µeff = √2gJ(J + 1) µB\n= 2.83 µB; measured = 2.67 µB [175]).  For a localized moment ferromagnet, the\nsaturation magnetization MS is predicted to be MS = gJ µB = 2.0 µB for SrRuO3.\nMeasured values of MS are much less.  Polycrystalline SrRuO3 reaches about\n0.85 µB [175, 178, 179] in low fields but co ntinues to increase in higher\nmagnetic fields.  At 125 kOe it was noted that M had reached 1.55 µB but had\nnot yet saturated [178].  Early neutron diffraction derived a moment of 1.4 ±\n0.4 µB [178] with no evidence for any antiferromagnetic order (spin canting).\nRecent theoretical investigations predict incomplete band splitting and a large\nmoment of about 1.6 µB [176, 180].  Early e xplanations for the low value of MS\nincluded spin canting, band magnetism, and incomplete alignment of the\nmagnetization due to magnetocrystalline anisotropy [178].  The present work\nshows that SrRuO3 has a large saturation moment of 1.6 µB as predicted by\nthese calculations.  \nSingle crystals of SrRuO3 can be grown from a SrCl2 flux [36].  The\nresistivity of such crystals is c onsistent with the results on polycrystalline3\nsamples.  Magnetization and magnetic torque measurements of single crystals\n[181-183] showed that S rRuO3 has a high cubic anisotropy with 〈110〉 (cubic\ncell) being the easy direction, with nearly square hysteresis loops.  Some\nprevious measurements repo rted for single crystals are que stionable, for\nexample, the measured value of MS = 1.1 µB.  In this work new magnetization\ndata is shown to clarify these points.\nExperimental\nSingle crystals of SrRuO3 were grown by slow cooling in a SrCl2 flux [36,\n184].  Polycrystalline S rRuO3 was prepared from stoichiometric quantities of\nSrCO3 and Ru metal repeatedly reacted at 1260 °C, and was used as the source\nmaterial for crystal growth.  The SrCl2 was dried in air at 110 °C.  A mixture\nwith approximate weight ratio 1:20 of SrRuO3:SrCl2 was melted in a platinum\ncrucible with lid at 1260 °C for 94 hrs.  The sample was cooled to 800 °C at 1°/hr\nand then to room temperature at ~40 °/hr.  Crystals of SrRuO3 less than 1 m m\nin diameter were found at the bottom of the crucible after removing the flux\nwith water.  Most crystals were cubo-octahedron shaped and grew with a\n3-fold symmetric axis (presumably [111]) perpendicular to the Pt surface.   The\n(cubic) crystal orientation was determined by the symmetry of the faces.  The\nactual or thorhombic symmetry [177] was confirmed by powder x-ray\ndiffraction and Transmission Electron Microscopy.  \nMagnetization in fields up to 70 kOe was measured using a Quantum\nDesign MPMSR2 SQUID magnetometer.  The samples were attached to a\nplastic straw with a small amount of Apiezon N vacuum grease.  Samples\nwere oriented on the straw visually using crystal faces which had obvious 2-,\n3- and 4-fold rotational symmetry.  At room temperature, the crystal can\nrotate in a large field to align a paramagnetic easy direction with the field.\nThis was used as a final adjustment when aligning along the easy 〈110〉Critical Behavior and Anisotropy in Single Crystal SrRuO3 4\ndirections.  At low temperatures, the grease solidifies so the sample cannot\nrotate.  \nThe accuracy of the magnetization measurement was estimated by\nmeasuring a sphere of yttrium iron garnet described [1].  For the measure-\nments reported here, the sample holder was fixed at the rotation angle \nwhich gave the maximum magnetization. Several crystals were\nmeasured.\nResults\nFigure 1 shows hysteresis loops at 5 K for SrRu O3 single crystal in various \ncrystallographic direction s and a polycrystalline pellet.  The  crystals have a low \ncoercive field ( ≈ 10 Oe) compared to that of the polycrystall ine pellet (3000 Oe).  \nBut more importantly, the crystals show very littl e hyste resis1.11.21.31.41.51.61.7\n0 1 02 03 04 05 06 07 0SrRuO3 Single Crystal\n[110]\n[1-10]\n[101]\n[111]\n[100]Magnetic Moment (µB/Ru)\nMagnetic Field (kOersted)√(2/3) MS\nMS/√2MS\n-2-1012\n-75 -50 -25 0 25 50 75PelletM (µ\nB/Ru)\nH (kOe)\nFigure 1.  Magnetization at 5 K of SrRuO3 single crystal  \nalong several crystallographic directions showing strong cubic\nbut not uniaxial magnetocrystalline \nanisotropy.  Inset shows\nthe full h\nysteresis loop of the single crystal data along with\nthat of a polycrystalline pellet for comparison.5\nwhile the pellet displays noticeable hysteresis even in fields greater than\n40 kOe.  \nThe rapid, linear approach to saturation (with respect to the applied field)\nfound in all directions can be attributed to demagnetization.  Until the sample\nbecomes fully magnetized, the demagnetization field Hd is equal to the\napplied field Ha resulting in an internal field of zero ( Hi = Ha - Hd).  From this\nslope ( M = Ha/4πN), one can calculate the demagnetization factor N and\ntherefore calculate the internal field.  The measured demagnetization factors\nN, (Hd = 4π NM) are 0.25, 0.28, 0.31, 0.49, 0.66 for the [110], [1-10], [101], [100] and\n[111] directions respectively.  These seem reasonable considering the shape ofthe crystals.  Since a uniaxial magnetocrystalline anisotropy will also give a\nlinear increase in M if H is applied along the hard direction, it is difficult to\ndistinguish it from the effect of the demagnetization field in this study.  Since\nthe demagnetization field can be as large as a thousand Oersted, one can only\nconclude that the uniaxial anisotropy field is less than a thousand Oersted,\nwhich is considerably less than that ( > 50 kOe) reported previously [182].\nIn the [110] direction S rRuO\n3 rapidly approaches saturation and then\nremains relatively constant (square hysteresis loop), as is expected for a\nmagnet with a magnetic field along the easy direction.  The crystal was also\nmeasured in the [1-10] and [101] directions which would be equivalent by\nsymmetry to the [110] if the crystals were cubic.  Since the magnetization is the\nsame along these 〈110〉 type directions, it can be concluded that the magnetic\nproperties of single crystal SrRuO3 are essentially cubic, i.e. only cubic\nmagnetocrystalline anisotropy is detected.  Beyond this initial saturation\nalong the easy 〈 110〉  directions, there is a small but measurable increase in the\nmagnetization which is linear in magnetic field and has a slope of 6 × 10-7\nµB/Oe.  Critical Behavior and Anisotropy in Single Crystal SrRuO3 6\nThe H = 0, T = 0 saturation moment MS found along the easy 〈110〉\ndirections are about 1.62 µB/Ru.  In the remnant state ( H reduced to zero), the\nmagnetization should lie along the nearest easy direction.  Thus the expected\nremnant (H  = 0) magnetization along the [100] or [111] direction is simply the\ncosine of the angle it makes with the closest 〈110〉 direction.  For 〈100〉 the\nexpected remnant magnetization is MS/√2, and for 〈 111〉  directions √(2/3)MS is\nexpected.  These are extremely close to the experimental values (Figu re 1).\nIn the other primary directions, there is a nonlinear approach to satur-\nation which is characteristic of materials with cubic anisotropy.  The magneto-\ncrystalline anisotropy constants K1 and K2 can be estimated from these\nmagnetization curves [185].  The magnetocrystalline anisotropy energy E  can\nbe defined in terms of the cosine of the angle M makes with the three crystal\naxes: αi = xi • M/M.  For a cubic material E = K1 (α12α22 + α22α32 + α32α12) +0.00.20.40.60.81.0\n0 5000 10000 15000 20000 25000 30000SrRuO3 Crystal Arrott PlotM2 (µ\nB2)\nH/M (Oe/µB)140K\n145K\n152K\n156K\n158K\n160K\n162K\n163K = Tc\n164K166K168K170K1234\n-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 T = 163K   H  ∝ MδLog(H)\nLog(M)δ = 4.2(1)\nFigure 2.  Arrott Plot of SrRuO3 single crystal along easy [110] \ndirection.  Inset, cri tical isotherm (T = 163K ≈ TC) on a log scale \nfit to Mδ ∝ H  with δ = 4.2.7\nK2 (α12α22α32) + … .  In SrRuO3 the magnetic easy axes are 〈110〉, which requires\nK1 < 0.  Similarly, 〈100〉 are the hard axes ( 〈111〉 are intermediate) which\nimplies 2.25|K1| < K2 < 9| K1|.  The [100] magnetization should intersect that\nof the easy axis [110] at Ha = -2 K1/MS.  An extrapolation of the [100] M vs. H\ncurve gives -2K1/MS ≈ 109 kOe.  Alternatively, the area between the [100] and\nthe [110] M  vs. H curves should be equal to - K1/4.  This method gives a value\nfor -2K1/MS ≈ 96 kOe.  The area method can also be used with the [111] curve\nto estimate 2K2/MS ≈ 540 kOe.\nIn the easy [110] direction, M  vs. H curves were measured for various tem-\nperatures and are shown in Figure 2 as M2 vs. H/M (Arrott plot).  The iso-\ntherms below TC should be approximately linear and intersect H/M = 0 at M0.0.00.20.40.60.81.0\n125 130 135 140 145 150 155 160 165SrRuO3     M0 ∝ (TC-T)βM0 (µB)\nTemperature (K)Average β = 0.36\nTC = 163.2(1) K\n-0.5-0.4-0.3-0.2-0.2-0.10.0\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Log(M0)\nLog(TC-T)β = 0.32\nT far from TC\nβ = 0.39\nT near TC\nFigure 3.  Zero field magnetization M0 of SrRuO3 single  \ncrystal along easy [110] direction.  Sol id line shows the fit to\nM0(T) ∝ (1 - T /TC)β with β  = 0.36.  Inset showing the same data\non a log plot.  The critical exponent β appears to change from\nHeisenberg-like β = 0.39 near TC to Ising-like β = 0.32 as T\ndecreases.Critical Behavior and Anisotropy in Single Crystal SrRuO3 8\nFrom these M0(T), shown in Figu re 3, the critica l exponent β ≈ 0.36 and TC = \n163.2 ± 0.2 K can be e stimated by fitting M0(T) ∝ (1 - T/TC)β in the critical region.  \nThe fit with a single value for β = 0.36 is poor considering th e apparent precision \nof the data.  Closer to TC, the data fit better with a larger β ≈ 0.39 which is close  to \nthat predicted in the 3-d Heisenber g model (β = 0.38). Farther fr om TC, the \nexponent is smaller, β ≈ 0.32, near that predicted by the Ising model (β = 0.33).  It \nis possibl e that this is due  to a crossover from Heisenbe rg to Ising behavior at T \ndecreases from TC.  A similar model  has been proposed to explain \nmeasurements of thin film SrRuO3 [73].\nThe T > TC isothe rms of Figure 2 should interse ct M2 = 0 at 1/χ(T, H = 0) = 1/\nχ0.  The critical exponent γ is then found from 1/χ0 ∝0510152025\n160 165 170 175 180 185 190 195SrRuO3 Crystal Inverse Susceptibility1/χ0 (emu-1 mol Gauss)\nTemperature (K)3.03.54.04.55.0\n0 0.5 1 1.5Log(1/χ\n0)\nLog(T-T\nC)χ0 ∝ (T-TC)-γ\nγ = 1.17\n050100150200\n150 200 250 300 350Pellet\nSingle Crystal1/χ0 (emu-1 mol Gauss)\nTemperature (K)\nFigure  4.  Zero field inverse susceptibility 1/χ0 of SrRuO3 single \ncrystal along easy [110] direction.  Solid line shows the\nfit to 1/χ0(T) ∝ (1 - T/TC)γ with γ = 1.17 and TC = 163.2 K.  The\ninset shows the same data on a log plot.  9\n(T/TC - 1)γ.  From these data (Figure 4) γ = 1.17 ± 0.02 is estimated.  The plot  of 1/\nχ vs. T should be close to linea r for high temperatures.\nThe M(H, T) data can be replotted using the scaling hypothesis [90] with\nβ = 0.36 and γ = 1.17.  According to the hypothesis, the magnetic equation of\nstate in the critical region depends only on the scaled variables H/|TC/T - 1|β+γ\nand M/|TC/T - 1|β.  A plot of th e scaled M2 and scaled H/M, shown in Figure 5, \nwill then ha ve only two curves: one  branch for the T < TC data and another for T \n> TC.  Not all the curve s fit on a single  line, this is due to the apparent change in \nβ as discussed above.\nThe critical isotherm ( T = TC) should obey the relation Mδ ∝ H, where\naccording to the scaling relation δ = γ/β + 1.  From the previously measured\nvalues of β  and γ , the exponent δ is therefore expected to be 4.3 ± 0.5.  This is\nin good agreement with the measured value (Figure  2) δ = 4.2 ± 0.2.0.01.02.03.04.05.06.0\n0 500 1000 1500 2000 2500SrRuO3 Crystal Scaled Arrott PlotM2 × |T/T\nC - 1|-2β (µ\nB2)\nH/M × |T/T\nC - 1|-γ (kOe/µ\nB)T < T\nC\nT > T\nCβ = 0.36\nγ = 1.17\nTC=163.2K\nFigure 5.  Scaled Arrott Pl ot of SrRuO3 single cryst al along easy \n[110] direction with β = 0.36 and γ =1.17.  Symbols are the same \nas those used in Figure  2.Critical Behavior and Anisotropy in Single Crystal SrRuO3 10\nThe temperature dependence of M along the easy direction is shown in\nFigure 6 for various applied fields.  The demagnetization field of 4πNM ≈  800 \nOe should be subtracted from the applied field to give the internal field.\nBetween 5 K and 100 K, and for internal fields from 0.2 kOe to 68 kOe the\nmagnetization is well approximated by M = M0(1 - ( T/Θ2)2) where M0 and Θ2\nare fitting parameters.  Other possible analyses are discussed below.\nDiscussion\nThe low temperature magnetization data (Figure 1) of bulk SrRuO3 can be \nexplained with 〈110〉 easy directions and a large cubic, but very little\nuniaxial, magnetocrystalline anisotropy.  The low coercivity of the crystals\n(~10 Oe), compared to the 3 kOe coercivity found in polycrystalline samples,1.301.351.401.451.501.551.601.65\n0 1 02 03 04 05 06 07 08 09 0 1 0 0SrRuO3 Single Crystal along easy direction [110]\n7T\n6T\n5T\n4T\n3T2T\n1T\n2kG\n1kGMagnetization (µ\nB/Ru)\nTemperature (K)1.31.41.51.61.7\n0 2000 4000 6000 8000 10000M (µB/Ru)\nT2 (K2)\nFigure 6.  Magnetization as a function of temperature of\nSrRuO3 single crystal along e asy [110] direction.  Inset shows \nthe approximate T2 dependence of the magnetization.  11\nprovides highly reversible and square hysteresis curves.  There is no\nindication of a magnetic multi-domain structure [183].  \nThe cubic mag netocrystalline anisotropy in SrRuO3 is very large, as\nreported previously [181, 183].  Typical values of K1 for cubic 3- d ferromagnets\nare 100 times smaller than that found for SrRuO3.  Such large values of K1\nusually refer to uniaxial anisotropy, for instance in hexagonal materials,\nwhich is a lower order effect ( K1 refers to the first non-zero anisotropy\nconstant).  This large anisotropy probably results from the strong spin-orbit\ncoupling of the heavy Ru atom, which also gives SrRuO3 a strong Kerr effect\n[186].\nFrom measurements along different 〈 110〉 directions, no indication of\nuniaxial magnetocrystalline anisotropy [182] is found, although it is allowed\nfrom the orthorhombic symmetry.  This might be expected since the\ncrystallographic unit cell lengths [177] vary b y only 0. 03%, and the angles by\n0.4% from the perfect cubic ones.  The distortion from cubic is primarily due\nt\no a r otation of the RuO6 octahedra, which alters the symmetry much more  \nthan the shape of the unit cell [177].  I n the crystals reported here, the\northorhombic cell was confirmed using TEM.  Because the unit cell is only\nslightly distorted, \nthe few reports claiming cubic [184] or tetragonal [182]\ncrystallographic \nsymmetry without supporting evidence, should be\nreevaluated in this context.  Significant uniaxial anisotropy in thin  films of \nSrRuO3 may result from growth induced anisotropy, as was found in films \nof magnetic garnets used in magnetic bubble technology [188].\n  Due to the large magnetocrystalline anisotropy, the saturation moment\nof SrRuO3 is difficult to measure.  In directions other than 〈110〉 the magnetic \nmoment does not saturate even in fields of several 10 kOe.  The strain and\nsmall particle \nsize of a polycrystalline sample apparently makes it even more\ndifficult to \nsaturate than the hard direction in a single crystal (Figure 1).\nPolycrystalline S\nrRuO3 has a remnant magnetization ( H = 0) of MS ≈ 0.85 µB,Critical Behavior and Anisotropy in Single Crystal SrRuO3 12\nwhich increases non-linearly past 1.55 µB (H = 125 kOe) [178].  Cl early a\nmagnetically soft single crystal with H along the easy direction is needed to\nmeasure MS.  In a previous experiment [181] on single crystal SrRuO3 with a\nsquare hysteresis loop, MS = 1.1 µB at H  = 0 was reported.  However, that value\nof MS is clearly too small since by 17 kOe it is smaller than the value for a\npolycrystalline sample [181].  Our measured value of MS = 1.6 µB (H = 0) is\nconsistent with the high field polycrystalline results.\nThe magnetic critical exponents measured here are in the range typically\nseen in large moment ferromagnets and expected theoretically for 3-\ndimensional ferromagnets.  Experimental values for the critical exponent β  in\nFe, Ni and YIG [92] are 0.37 ± 0.02, which are near the theoretical values (Ising\nβ = 0.33, Heisenberg β  = 0.36).  The related metallic ferromagnets La0.5Sr0.5CoO3\n[92] has β  = 0.361.  The apparent decrease in β as T decreases from TC, may be \ndue to the lar ge magnetocrystall ine anisotropy.  A similar ef fect has been seen in \nthin film SrRuO3, where it is suggested that the magne tocrystalline anisotropy \nindu ces a cr ossover from Heisenberg to Ising behavior.  The weak, itinerant \nelectron ferromagnet ZrZn2 has mean field critical expon ents β = 0.5 [93].  The \nmost prominent theory on itinerant electron ferromagnetism by Moriya [84 , 85] \npredicts a TC4/3 - T4/3 dependence on the magnetization,  which is essen tially β = \n1.\nThe critical exponents γ and δ are also typical for large moment\nferromagnets (Fe, Ni and YIG [92] γ = 1.2 ± 0.2) as opposed to those found for\nthe weak, itinerant electron ferromagnet ZrZn2 which has mean field critical\nexponents γ = 1.0 and δ = 3 [93].  The three dimensional Ising and Heisenberg\nmodels predict γ = 1.24, δ = 4.8 and γ = 1.39, δ = 4.8 respectively.  The critical\nexponents measured here also obey the scaling relation δ = 1 + γ /β.13\nA positi ve cur vatur e persists in the plot of 1/χ vs. T (Figure 7) even at\nhigher temperature where it should be linear for a Curie-Weiss ferromagnet.\nSuch a curvature can be caused by a temperature independent term in the\nsusceptibility χ = χConst + C/(T - Θ), where C/(T - Θ) is the Curie-Weiss\nsusceptibility and is always positive (T  > Θ ).  A χConst < 0 of about -4 × 10-4 emu\nG-1 mol-1 will provide the observed curvature, and has been independently\nobserved elsewhere [175].  The temperature independent term χConst, should\ncontain a positive contribution due to Pauli paramagnetism.  This can be\nestimated from measurements of the linear term of the specific heat [176],\ngiving χPauli ≈ +4 × 10-4 emu G-1 mol-1.  The Landau diamagnetism should be050100150200\n150 200 250 300 350SrRuO3 Polycrystal Inverse Susceptibility\nPellet\nSingle Crystal\nTemperature (K)1/χ0 (emu-1 mol Gauss)\nFigure 7.  Inverse magnetic susceptibility (1/χ = M/H) at H = 10 \nkOe of polycrystalline SrRuO3 compared to the singl e crystal \ndata from Figure A- 4 .  The solid line is the straight-line fit \nwith TC = 165K which demonstrates the slightly posit ive \ncurvature of the data.Critical Behavior and Anisotropy in Single Crystal SrRuO3 14\nnegative and for simple band structures is smaller than χPauli (for free electrons\nχLandau  = -χPauli/3).  The core diamagnetism can be estimated from tables of\nexperimental values [76] giving χCore = -0.7 × 10-4 emu G-1 mol-1.  The sum of\nthese theoretical estimates χConst = χPauli + χLandau  + χCore is, however, still\npositive while the experimental value appears to be negative.  In the critical\nregion, 1/χ  ∝ (T/TC - 1)γ with γ  = 1.17, provides a positive curvature in the plot\nof 1/χ vs. T.  Since the mean field exponent γ = 1 is expected to be valid far\nfrom TC, some other mechanism must provide the effective γ > 1 observed at\nthese higher temperatures.  \nMagnetic excitations which become thermally induced as the temperature\nis raised above T = 0 reduce the magnetization from the ground state value.\nThe exponential decrease predicted by the mean field model has some\nqualitative value but is never in good agreement with experiment.  Collective\nspin wave excitations and single particle (Stoner) excitations both decrease the\nmagnetization according to a power law M = MS(1 - (T /Θn)n) which is in accord\nwith experiments where n  ≈ 2 ± 1 and Θn is of the order TC.\nThe limiting low T, H = 0 behavior of collective, spin wave excitations\n(section \n3.2.2.2.8) predicts n  = 3/2 and for SrRuO3, Θ3/2 ≈ 2.42TC = 400 K.  The  \nStoner \ntheory of single (k-space) particle excitations (section 3.2.2.2.2 of [1])\npredicts\n n = 2 and Θ2 ≈ 1.41 TC (= 230 K for SrRuO3).  As the temperature an d \nmagnetic field is raised, these models require further corrections.  For example, \nhigher order corrections such as an additional n  = 5/2 term in predicted in the \nspin wave theory.  For H ≠ 0, the low temperature spin-wave excitations \nbecome\n quenched which can have the effect of increasing the average value on \nn\n [97]. The generalized model of spin fluctuations in ferromagnets ([84] section15\n3.2.2.2.3 in [1]) includ es both types of interacting magnetic excitations and predicts\nn = 3/2 for H  = 0, T  = 0 but also n  ≈ 2 in calculations [86].\nExperimental results on metallic ferromagnets with substantial saturation\nmoments such as Fe, Ni [88, 189- 192] and La0.67Sr0.33MnO3 ([97], section 4.1.1 in [1])\ncan display n = 3/2 consistent with the spin wave stiffness determined by\nneutron di ffraction if significant corrections are included and only low\ntemperature data is analyzed.  In contrast “weak” i tinerant-electron-50050100150200\n0 25 50 75 100 125 150A parameter in fit to M = MS(1 - AT3/2 - BT2)T3/2 Coefficient (10-6/K3/2)\nMaximum Temperature of Fitting Range- 2 0 0 2 04 0 6 08 010152025Correlation of A and B \nT3/2 Coefficient (10-6/K3/2)T2 Coefficient (10-6/K2)\nFigure 8.  Variation of the T3/2 parameter in fitting the\nmagnetization data of single crystal SrRuO3 to M = MS (1 -\nAT3/2 - BT2) a s the fitting range is increased.  The upper inset\nshows the correlation of the A and B parameters.  In the\nregion where A i\ns relatively stable (around Tmax = 60 K), A \ndecreases as Tmax is lowered.  The symbols are the same as  \nthose used in Figure 9.Critical Behavior and Anisotropy in Single Crystal SrRuO3 16\nferromagnets such as ZrZn2, Ni3Al and Sc3In, show n = 2 [86, 193-195] over a\nwide temperature range.\nAs shown above (Figure 6) the magnetization of SrRuO3 single crystal  can be \nwell described by M = MS(1 - (T/Θn)n) with n  = 2 in the temperature range of \nreliable measurement, \n5 K to 100 K.  A fit with n = 3/2 in this temperature range \nis unsatisfactory.\nIf one assumes n = 2 arises from Stoner excitations and collective\nexcitations result in n = 3/2, then one may expect the variation of-1001020304050\n0 25 50 75 100 125 150B parameter in fit to M = M0(1 - AT3/2 - BT2)\n7T T2 term\n6T T2 term\n5T T2 term\n4T T2 term\n3T T2 term2T T2 term\n1T T2 term\n2kOe T2 term\n1kOe T2 termT2 Coefficient (10-6/K2)\nMaximum Temperature of Fitting Range\nFigure 9.  Variation of the T2 parameter in fitting the\nmagnetization data of single crystal SrRuO3 to M = MS (1 -\nAT3/2 - BT2) a s the fitting range is increased.  In the region\nwhere B is relatively stable (around Tmax = 60 K), B  increases as \nTmax is lowered.17\nmagnetization with temperature to fit with a combination of the n = 2 and\nn = 3/2 terms, M = MS (1 - AT3/2 - BT2) [150] at low temperatures.  Since an\nadditional parameter is included, an improvement of the fit does not\nnecessarily prove the significance of the added parameter.  Furthermore, the\nfunctions T3/2 and T2 are very similar making the fitting parameters highly\ncorrelated.  \nThe magnetization data for single crystal S rRuO3 were fit to a polynomial \nexpression M = MS (1 - AT3/2 - BT2) from T  = 2 K to T  = Tmax.  As expected, the A \nand B parameters are highly correlated (Figure A- 8) with ∆B ≈ 2.2 × 10-5 K-2 -\n0.2 K-1/2 ∆A.  The parameters A and B  can then be plotted as a function of Tmax\n(Figure 8 and Figure 9).  For very low Tmax (< 30 K) the fit is unstable since the \ndifference in the magnetization at T  <Tmax becomes comparable to precision of \nthe measurement.  Thus, the divergence of the fitting parameters at low Tmax is \nan artifact, and not physical.  At high temperatures, nearing the critical \ntemperature, \nthe polynomial expression is not expected to be valid, so the \ndivergence at high Tmax can also be ignored.  The  flat, linear region centered \naround Tmax = 60K, is presumably the region where the  fitting parameters may \nhave physi cal meaning.  From this region, A and B parameters can be extracted \nand extrapolated to the Tmax = 0 K values for comparison with the theory.  This \nanalysis was done for fitting the magnetization data both to M = MS (1 - AT3/2 - \nBT2) and M  = MS (1 - BT2). \nImplicit in the use of M  = MS (1 - AT3/2 - BT2) is the assumption that these\ntwo terms are theoretical low temperature expansions, i.e. they should\nbecome more accurate as T approaches 0 K.  As the temperature range for\nwhich the data is fit to this type of expression is decreased to lower\ntemperatures, the fitting parameters should be stable, or only be slightly\nvarying.  In principle, the true low temperature form should gradually\nbecome more dominant, at the expense of the other terms, as the fitting range\nis decreased to lower temperatures.Critical Behavior and Anisotropy in Single Crystal SrRuO3 18\nIn fitting to the data on SrRuO3  single crystal the B parameter (Figure 9) is \nrelatively stable and increases as lower temperature fitting ranges (Tmax) are \nused.  The A parameter (Figure 8) is not as stable and appears to decrease as\nTmax is \nlowered.  If this trend were to continue, then the T3/2 contribution \nwould be very small at low tempe ratures.  The removal of the AT3/2 term\nfurther stabilizes the B  parameter (Figure 10), and makes B more\nindependent of Tmax than with AT3/2 included.  Thus, from this analysis of the \nmagnetization data, there is little evidence for a large AT3/2 contribution to\nthe magnetization.  This is in contrast with measurements on thin film [186,\n187] which show T3/2 dominating over T2.\nThe two corrections to the spin wave T3/2 theory mentioned above (finite\nmagnetic field and higher order T(2n+1)/2 terms) may not adequately explain the\ndata presented here.  The applied magnetic field will suppress spin wave-1001020304050\n0 25 50 75 100 125 150B parameter in fit to M = M0(1 - BT2)T2 Coefficient (10-6/K2)\nMaximum Temperature of Fitting Range210215220225230235240\n01 23 45 67Θ\n2 (T\nmax = 60K)Θ2 (K)\nInternal Field (T)233K = √ 2T\nC\nexpected from\nsimple Stoner Theory\nΘ2 = B1/2\nFigure 10.  Variation of the T2 parameter in fitting the\nmagnetization data of single crystal SrRuO3 to M = MS (1 - BT2) \nas the fitting range is increased.  The parameter B for this fit is\nmore \nstable and constant than that shown in Figure 8.\nInset, variation of Θ2 in a magnetic field.19\nexcitations f or T < gµBH/kB,  resulting in an effective n > 3/2.  If this were a \nsignificant e ffect, the T3/2 fitting parameter A should decrease as H is increased\nparticularly as Tmax approaches 0 K.  Such a systematic decrease in A is not  \nobviously apparent (Figure 8).  One cannot however, exclude the\ndemagnetization or anisotropy field which are present even when the applied\nH = 0.  \nIf the true magnetization were a sum of T3/2 and T5/2 terms, M/MS =  \n(1 - (T /400 K)3/2 - (T/600 K)5/2 ) for instance, the data would fit well to M/MS = \n(1 - AT3/2 - BT2).  The fitting parameters A and B as a function of Tmax for this \nexample are shown in Figure 11.  The A parameter is relatively constant,\nincreasing slightly as Tmax approaches 0 K.  In the limit Tmax approaches 0 K,\nA = 125 × 10-6 K-3/2 or Θ3/2 = 400.3 K, which is very close to the true value of\n400 K.  The B fitting parameter is not nearly as stable and decreases as Tmax020406080100120140\n0.00.51.01.52.0\n0 2 04 06 08 0 1 0 0M = 1 -  (T/400 K)3/2 - (T/600 K)5/2\nfit with M = MS(1 - AT3/2 - BT2)\nA\nBAT3/2 Coefficient (10-6/K3/2)BT2 Coefficient (10-6/K2)\nMaximum Temperature (K) of Fitting Range\nFigure 11.  Variation of A and B fitting parameters in the\nhypothetical case where t he true magnetization is given by T3/2\nand T5/2 terms.Critical Behavior and Anisotropy in Single Crystal SrRuO3 20\napproaches 0 K.  Since the exponent of the BT2 fit (2) is smaller than 5/2,\nwhich is the exponent used in the example, a larger B  is needed to fit the data\nat higher temperatures.  Thus, a dramatically decreasing B as Tmax approaches\n0 K, is indicative of a fitting exponent (2 in this example) smaller than the\ntrue exponent (5/2 in this example).  \nThe data on S rRuO3 shows the opposite effect:  B is relatively constant\nwhile A decreases as Tmax approaches 0 K.  This decrease of A  indicates that\nthe exponent 3/2 is smaller than the true exponent, and therefore, the\naddition of higher order terms to the spin wave form of the magnetization\nwill not fully explain the data on SrRuO3.\n Positive B (and A) parameters can be converted to the Θn values used\nabove via Θ2 = B-1/2 and Θ3/2 = A-2/3.  The Θ2 values extracted from fits\ndescribed above are shown as a function of the internal magnetic field in\nFigure A- 10.  The values of Θ2(H) extrapolated to Tmax = 0 are 20 K less than\nthe corresponding Θ2(H) for Tmax = 60 K.  The single crystal Θ2 parameters\nincrease only slightly as the field is increased, about 2.6 K/Tesla.  These values\nfor Θ2 are consistent with mea surements of the field dependent heat capacity of \npolycrystalline SrRuO3 [1] via the Maxwell relation\n∂CH\n∂H\n\nT=T∂2M\n∂T2\n\n\nH.\nConclusion\nVarious magnetic properties of single crystal SrRuO3 along the magnetic\neasy direction have been measured.  The saturation moment, 1.61µB/Ru, is\nlarger than that reported previously for single crystal material but in accord\nwith experiments on polycrystalline material and theoretical calculations.\nThe crystals show extremely large cubic magnetocrystalline anisotropy while\nno substantial uniaxial magnetocrystalline anisotropy.  The critical exponent21\nβ changes from Heisenberg like 0.39 near TC to Ising like 0.32 further from TC. \nThe susceptibility exponent γ = 1.17 persists to 2 TC, wh ere it is expected to \ndecrease to 1.  The magnetization from 5 K to 100 K is well described by\nM = MS(1 - ( T/Θn)n) where n ≈ 2. The addition of a T3/2 term is not obviously  \nsignificant.\nAcknowledgement\nThis work was performed in the KGB lab, Applied Physics, Stanford \nUniversity 19 93-1997. I thank Ted Geballe, Lior Klein, and Kathryn Moler for \nuseful discussion s. The w ork at Stanford was supported in part by \nthe Air Force  Office of Scientific Research and the Stanford \nCenter for Materials Research under the NSF-MRL program. \nG.J.S. w ould like to acknowledge the generous support of the Hertz  \nfoundation.  This manuscript has not been updated for publication since June \n1997.22References\n[1]  G. Jeffrey Snyder, Ph.D. Thesis, Stanford University (1997).\n[36]  R. J. Bouchard and J. L. Gillson, Mat. Res. Bull. 7, 873 (1972).\n[73]  L. Klein, J. S. Dodge, C. \nH. Ahn, G. J. Snyder, T. H. Geballe, M. \nR. Beasley, and A. Kapitulnik, Phys. Rev. \nLett. 77, 2774 (1996).\n[76]  P. W. Selwood, Magnetochemistry (Interscience, New \nYork and \nLondon, 1956).\n[84]  T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism \n(SpringerVerlag, 1985).\n[85]  T. Moriya, in Metallic Magnetism, Topics in Current Physics, edited by H.\nCapellman (Springer, Berlin, 1987), Vol. 42, p. 15-55.\n[86]  G. G. Lonzarich and L. Taillefer, J. Phys. 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Gao1, and J. –M. Liu1,21 \n1Institute for Advanced Materials, South China Academy of Advanced Optoelectronics and \nGuangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, \nSouth China Normal University, Guangzhou 510006, China  \n2Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China   \n \n[Abstract] Searching  for new method s controlling antiferromagnetic (AF M) domain wall is \none of the most important issues for AFM spintronic device operation . In this work, we study \ntheoretically the domain wall motion of an AFM nanowire, driven by the axial anisotropy \ngradient generated by external electric field, allowing the electro control of AFM domain wall \nmotion in the merit of ultra -low energy loss . The domain wall velocity depending on the \nanisotropy gradient  magnitude  and intrinsic material properties is simulated based  on the \nLandau -Lifshitz -Gilbert eq uation and also deduced using  the energy dissipation theorem. It is \nfound that t he domain wall  moves at a nearly constant velocity  for small gradient , and \naccelerates for large gradient due to the enlarge d domain wall  width. The domain wall \nmobility  is independent of lattice dimension  and type s of domain wall , while it is enhanced by \nthe Dzyaloshinskii -Moriya interaction. In addition , the physical mechanism  for much faster  \nAFM wall dynamics than ferromagnetic wall dynamics is qualitatively explained.  This work \nunveils a promising strategy for controlling the AFM domain walls , benefiting to future AFM \nspintronic applications.  \n \nKeywords: antiferromagnetic dynamic s, domain wall, anisotropy  gradient   \n                                                   \n*D1Wen and Z Chen contributed equally to this work;  \n†E-mail: qinmh@scnu.edu.cn   I. Introduction  \nNowadays, the interest in antiferromagnets significantly increases due to the promising \napplication potential s for spintronics.1,2 Comparing with ferromagnets based storage devices, \nantiferromagnets based devices are more stable against magnetic field  perturbation s and could \nbe designed with high element densities without producing any stray field, attributing to zero \nnet magnetizati on and ultralow susceptibility of an antiferromagnetic (AFM) element.3-6 \nMoreover, AFM materials show fast magnetic dynamics ,7 including the spin wave modes with \nhigh frequencies result ing from their complex spin configurations , which  also favors them for \npromising potentials in future devices.8 One of the major challenges for these applications is \nhow to manipulate an AFM domain wall . A number of driv ing schemes  including spin waves,9 \nspin-orbit torques,10,11 asymmetric magnetic fields,12 and thermal gradients13-15 have been \nproposed. For example, a wall motion speed as high as ~30 km/s , driven by electric current \ninduced Néel spin -orbit torques , available in CuMnAs16 and Mn 2Au17, was predicted . This \nspeed is likely three orde rs of magnitude larger than that for ferromagnetic (FM)  domain wall \nmotion typically, although electric current driving would be highly energy -costing. \nFurthermore, it was revealed that the competition between entropic torque and Brownian \nforce under temperature gradient determines the wall motion  direction . Importantly, in these \ncontrol schemes, the AFM wall motion  remain s non-tilted and high wall mobility is expected \nsince no w alker breakdown18,19 which limits the motion of typical FM domain wall motion is \navailable in the AFM wall motion .  \nWhile t hese proposed schemes  are meaningful for future experiments and device design, \nseveral shortc omings can be found and they may be detrimental  in some cases . For instance , \nthe Néel  spin-orbit torque  is available only in some specific antiferromagnets with locally \nbroken inversion symmetry . This torque however drives  the neighboring domain walls to \napproa ch to and annihilate with each other, seriously hindering its applications. Some \nschemes, such as temperature gradient and spin wave driven wall motion , seem to be schemes \nof more theoretical sense  and will encounter difficulties practically . It is a lso noted that some \nof these schemes rely essentially on the electric current driven mechanism , which is highly \ndeficient for its high energy loss due to the Joule heating. Along this line, it is still on the way \nto find alternative scheme that is well -controlled, energy -saving, and high efficient for future AFM spintronic applications.  \nFirst, electric field control rather than electric current driving would be highly preferred if \none is concerned with the energy cost  which is detrimental for ultra -density memories . Indeed, \nvoltage control or say electro -control of magnetism has been an issue deserved for full \ncommitment. Fortunately, the electric field controlled magnetic anisotrop y has been \nexperimentally revealed in  magnetic heterostructures,20-23 and the anisotropy gradient can be \nobtained through elaborate structure design. Under such a gradient, the AFM domain wall \ntends to move towards the low anisotropy side in order to reduce  free energy. As a matter of \nfact, the anisotropy gradient has been proven to efficiently drive the skyrmions motion,24-26 \nand this scheme should be also utilized to control the AFM domain wall motion with \npreferred merit . However, so far no report on the dynamics of AFM domain wall under such \nanisotropy gradient is available, and this subject urgently deserves to be investigated \ntheoretically as the first step towards promoting the application process for AFM spintronics.  \nIn this work, we study the anisotropy gradient driven AFM domain wall motion  in a \nnanowire with the AFM domain structure . The dynamics and its dependence of the gradient \nmagnitude and intrinsic physical parameters are simulated based on the Landau  - Lifshitz  - \nGilbert (LLG) equation and also calculated based on the energy dissipation theorem. It is \npredicted that the domain wall does shift at a nearly constant speed  under  small  anisotropy  \ngradient, and can be accelerated under large  anisotropy gradient because of the wi dened \ndomain wall during the motion  stage . It is confirmed that the observed phenomena are \nindependent of the lattice dimension and wall type. Moreover, the Dzyaloshinskii -Moriya \n(DM) interaction is discussed specifically and its influence on spin dynamics is addressed , \nbenefiting  a lot to future materials design and dynamic manipulation . In addition , the physical \nmechanism of the AFM wall dynamics faster than FM wall dynamics is also qualitatively \nexplained.  \n \nII. Model and computational details  \nWithout l osing the generality, w e study a one-dimensional AFM model  with the isotropic \nHeisenberg exchanges between the nearest neighbors and the uniaxial anisotropy term :27 2\n,()zz\ni j i i\ni j iH J S S K S\n  ,            (1) \nwhere J > 0 is the AFM coupling constant, Si = i /s represent s the normalized magnetic \nmoment at site  i with three components Sx \ni, Sy \ni and Sz \ni. The second term in Eq. (1) is the \nanisotropy energy with the easy axis along the z-direction  (i.e. nanowire axis) , and the \nuniaxial  anisotropy constant Kz \ni at site i is described by Kz \ni = K0 + iaK where K0 is the \nanisotropy constant at the low anisotropy  end and K describes the anisotropy gradient \nmagnitude , a is the lattice constant .  \nThe AFM dynamics is investigated by solving the LLG equation  based on the atomistic \nspin model :28,29  \n 2()(1 )i\ni i i i\nsSS H S Ht   \n,          (2) \nwhere  is the gyromagnetic ratio,  = 0.00 2 is the Gilbert damping constant, Hi = − ∂H/∂Si is \nthe effective field. Unless stated elsewhere, the LLG  simulations are performed on a \none-dimensional nanowire lattice with 1  1  400 spins with open boundary conditions using \nthe fourth -order Runge -Kutta method with a time step t = 1.0  10−4 s/J. The local \nstaggered magnetization 2 n = m1  m2 is calculated to describe the spin dynamics where m1 \nand m2 are the magnetizations of the two sublattices. After sufficient relaxation of the domain \nstructure , the anisotropy gradient is applied to drive the domain wall motion , as schematically \ndepicted in Fig. 1 . \n \nFig.1. (color online) Illustration of a domain w all in antiferromagnetic nanowire under an anisotropy \ngradient.  \n \nIII. Resu lts and discussion  \nA. Domain wall motion  \n \nFig.2.  (color online) (a) Profile of the domain  wall at different times, and (b) the domain wall position and \nenergy as functions of time for ΔK = 6×10-6 J/a and K0 = 0.01 J. (c) A schematic depiction of torques acting \non domain  wall spins under the anisotropy gradient, and (d) the position of the domain  wall versus time for \nvarious ΔK for K0 = 0.01 J. \n \nWe first demonstrate the domain wall motion upon a finite anisotropy gradient K. It is \nclear to see no identifiable wall motion at K = 0. The motion trajectory at K0 = 0.01 J and K \n= 6  106 J/a, at five given times, are presented in Fig. 2(a) where the domain wall profiles are \nplotted. The domain wall motion in roughly steady state is clearly demonstrated as long as a \nfinite anisotropy gradient  is imposed . The wa ll always shifts from the high anis otropy side to \nthe low anisotropy side,  with a nearly constant speed . Moreover, the wall profile i s rather \nrobust and remains in its initial configuration during the motion. The simulated wall position \n(z) and wall energy  (EDW, the internal energy difference between the systems with and without \nthe wall) as functions of time t are shown in Fig. 2(b) . It is seen that the wall position z \ndecreases very slowly in the very beginning period and then the wall motion is accelerated up \nto a state with roughly constant  speed. The wall energy EDW however experiences serious \nfluctuations in the beginning period , and then reaches a state where EDW approximately \nlinear ly decreases with time t plus very weak fluctuations . It is thus clearly demonstrated that \nthe domain wall motion towards the low anisotropy side is a spontaneous process . In the \nbeginning period, the wall symmetry  is broken  slightly by introducing the anisotropy gradient, \nresulting in the distortion of the wall structure and serious fluctuation of EDW. Moreover , the \ntrajectory of n depends not only  on its instantaneous position but also on the spin dynamics .30 \nSpecifically , n tends to  continue its path even after reaching the equilibrium  position (along its \neffective field) , accounting for  the weak fluctuations of EDW after the wall structure was \nreadjusted .  \nAs a matter of fact, the wall motion can be qualitatively understood from the competition \namong various torques acting on the wall spins , as depicted in Fig. 2(c). In the absence of \nanisotropy gradient, the domain wall configuration is symmetric with respect to its central \nplane. Th is symmetry is broken if one considers a nonzero anisotropy  gradient. Specifically, \nfor the two spins  (SL and SR) neighboring the spin S0 exactly on the wall central plane , the \ndamping torque on SL, result ing from the anisotropy term , is smaller than that on SR, leading \nto the fact that  SL deviates more  from the easy axis than SR does. As a result, the damping \ntorque (d \nR) on S0 from the exchange interaction with S R is larger than the damping torque (d \nL) \nfrom SL, resulting in the net damping torque ~ (d \nR  d \nL) which efficiently drive s the wall \nmotion  toward  the low anisotropy  region . Undoubtedly, the net driv ing torque increases with \nincreas ing K, which significantly enhances the wall motion  speed , as clearly shown in Fig. \n2(d) where the wall position z as a function of time t at several K is plotted .  \nIn order to uncover  the under lying physics more clearly , we calculate the wall motion \nspeed  from the perspective of energy conservation. Considering  the wall motion as a \nconsequence of wall energy dissipation , one notes that the wall energy EDW = 2(2 JKc)1/2 \nwhere Kc is the anisotropy constant on spin S0,31 noting Kc > K0 if K > 0. The Rayleigh \ndissipation function  R = ṅ2dz/2 is introduced to describe the energy dissipation of the \nwhole lattice,32 where  = s/a is the density of staggered spin angular momentum per unit \ncell and ṅ represents the derivative with respect to time . Subsequently, according to the \nclassical mechanics , we obtain : \n2DWdE dt R\n,                (3) \nA necessary mathematical substitution  leads to the wall velocity  v (see Appendix for \ndetail s): 21/ 2\n2-,\nc sJaKKv    \n  \n\n,            (4) \nwhere  is the domain wall width.  \n \nFig.3.  (color online) (a) The position of the domain  wall as a function of time for various K0 for ΔK = \n4×10-6 J/a. The simulated (empty points) and calculated (solid lines) velocities as functions of (b)  ΔK for \nvarious K0 and  = 0.002, and (c)  for various Δ K for K0 = 0.01 J. (d) The simulated domain  wall position \nand velocity as functions of time for large ΔK = 0.0003  J/a.  \n \nEq. ( 4) suggests that the wall velocity  depends not only on anisotropy gradient K but \nalso anisotropy constant  Kc. It implies that a large anisotropy K makes the wall motion fast \nbut a larger Kc does the opposite. This result is confirmed from our s imulations. Fig. 3(a) \npresents the simulated wall position z as a function of t for various K0 at K = 4  10-6 J/a, \nnoting that a large K0 also implies a large Kc. Certainly, an enhanced anisotropy  is expected to \nsuppress the domain wall motion seriously, as shown in Fig. 3(a) and Fig. 3(b) . In Fig. 3(b), \nthe simulated speeds (empty dots) and calculated ones (solid lines)  from Eq. (4) are plotted \ntogether for a comparison, demonstrating the good consistence between the simulation and \nmodel on one hand and on other hand revealing the fact that the anisotropy magnitude ( K0 or \nKc) and anisotropy gradient (K) play the opposite roles in controlling the domain wall \nmotion, enabling an accelerated and decelerated wall motion respectively.  \nThe effect of the damping constant   on the wall velocity is also investigated, and the \ncorresponding results are shown in Fig. 3(c) which gives the simulated and calculated \nvelocities as functions of  for various ΔK for K0 = 0.01 J. It is noted that an enhanced \ndamping term always lowers the wall mobility, and the velocity decreases with the  increas ing \n. Furthermore, the wall width is simply consider ed to be a constant during the wall motion, \nwhich well describes the case of small ΔK. However, for large ΔK, the wall width is expected \nto increases with the wall motion toward the weak  anisotropy side, resulting in the additional \nacceleration of the wall motion. In order to better understand the acceleration behavior, we \ninvestigated the wall motion under huge anisotropy gradient  ΔK = 0.0003 J/a. The simulated \nwall position and velocity are presented in Fig. 3(d), which clearly shows that the velocity \nlinearly increases with t.  \nFurthermore, t he 90° AFM domain wall  could be available in real materials such as \nCuMnAs using the current polarity  and may play an essential role in future applications .33  \nHere,  the dynamics of the 90° AFM wall is also investigated  by introducing the cubic  \nanisotropy  K(S2 \nxS2 \ny+S2 \nyS2 \nz+S2 \nzS2 \nx) and additional  hard axis anisotropy  KyS2 \ny in the model .34 In this \ncase, the wall energy goes EDW = (JKc/2)1/2, exactly a quarter of the energy of the 180° wall \nunder  the same parameters . However, the Rayleigh dissipation function R of the 90°  wall is \nalso one -fourth that of the 180° wall, as explained in the appendix, resulting in the same \nvelocity. As a matter of fact , this behavior has also been confirmed by  the LLG simulations,  \nand the corresponding results are not shown here for brevity.  \n \nB. Lattice dimension  and DM interaction  \nSo far, the anisotropy gradient driven AFM wall motion has been studied based on the \none-dimensional model. However, the results and conclusions also apply to magnetic films \nand bulks, as shown in Fig. 4(a) which presents the simulated wall velocity as a function of \nK for various lattice dimensions fo r K0 = 0.01 J and  = 0.002  with the in -plane exchange \ninteraction J. All the curves well coincide  with each other especially for small ΔK, indicating \nthe independence of the  wall velocity on the lattice dimension. This phenomenon can be \nqualitatively understood from the energy landscape.  Under a small K, the wall structure is \nalmost the same as the static wall. In this case , the neighboring spins in every cross -section \nperpendicular to the z axis still align antiparallel to each other, and the exchange energy is satisfied. Thus, the energy difference between neighboring unit cross sections only arises from \nthe single -ion anisotropy term , which hardly  be affected by  the lattice  dimensions, resulting in \nthe invariant force density exert ing on the wall.  \n \nFig.4.  (color online) (a) The simulated domain  wall velocity as a function of ΔK for various lattice sizes, \nand (b) The simulated (empty points) and calculated (solid lines) velocities as functions of D for various \nΔK. \n \nMoreover , we check the dependence of the domain wall dynamics on DM interaction  \nwhich may be available in some realistic materials.35,36 Here, the DM interaction  energy  \nΣiD(Si  Si+1) with D = D(0, 0, 1) is introduced in the one -dimensional model , and t he LLG \nsimulated results are presented in Fig. 4(b), where the v(D) curves upon various ΔK are \nplotted. For a fixed ΔK, v gradually increases with the increase of D, indicating that the DM \ninteraction  enhances  the wall motion, similar to the earlier report.37,38 When the DM \ninteraction  is considered, t he domain  wall energy decreases to EDW = 2(2 JKc  D2)1/2,9,39 \nresulting in the increase of the wall width and the enhancement of t he wall mobility . In detail, \nthe wall width is increased to40  \n \n 1\n2 2\n012c D JK  ,  (5) \nwith 0 = a(J/2Kc)1/2.  \nIn Fig. 4(b), the calculated velocity curves are also plotted with solid lines, which are \nwell consistent with the simulated results.  \n \nC. Antiferromagnet beat s ferromagnet  \n \nFig.5. (color online) (a) The displacements of the domain walls of the AFM and FM systems as functions of \ntime, and (b) the local my (top half) of FM system and ny (bottom half) of AFM system, and the inserts are \nthe schematic depictions of the precession torques, and (c) the evolution of my at various positions, and  the \ndomain  wall center is located at z = 350 a, and (d) the simulated FM wall velocity as a function of Ky.  \n \nIn this section, we discuss  the physical mechanism  for the ultrafast dynamics of an AFM \nwall under the anisotropy gradient through a detailed comparison with FM domain wall. \nComparing with the AFM wall, the FM wall can hardly be driven by the anisotropy gradient, \nas clearly shown in Fig. 5(a)  where presents the simulated wall displacements of the AFM \nsystem and FM system with the exch ange interaction –J for ΔK/K0 = 1 × 10-3 a-1 and  = \n0.002 . For a FM wall, the precession torque s on SR and SL from the gradient are antiparallel \nwith and compete with each other, resulting in a strong spin fluctuation, as shown in the top \nhalf of Fig. 5(b) where the y component of local  spin my is presented. Moreover, the spin \nfluctuation  cannot be suppressed by the FM interaction and propagates along the nanowire \nfrom the domain wall center, as confirmed  in Fig. 5(c) where gives the evolutions of my at z = \n40a, 160 a, and 280 a, noting that the domain wall center is located at z = 350 a. As a result, the \nexcitation and propagation of  spin fluctuation act  as dominating energy dissipation s since the \nprecession torque is much larger than the damping torque , resulting in  the very low driving \nefficiency of the FM wall. \nOn the other hand, for the AFM system, the high mobility of the AFM wall is obtained , \ndue to the fact that  the spin fluctuation is significantly su ppressed by the AFM interaction , as \nshown  in the bottom half of Fig. 5(b) where presents the y component of the staggered \nmagnetization  ny of the system and the depiction of the precession torque acting on the AFM \nspins.  Thus, this work unveils another origin for ultrafast dynamics of an AFM system. \nHowever , the weak spin fluctuation also slightly speeds down the motion of the AFM domain  \nwall, resulting in the noticeable  discrepancy between the simulations and analytical \ncalculations especially for large ΔK which significantly enhances the spin fluctuation, as \nshown in Fig. 3.    \nIn addition, a  high mobility of the FM wall is expected and also confirmed  in our \nsimulations when the spin fluctuation is suppressed  by introducing additional hard -axis \nanisotropy KyS2 \ny in the model . Fig. 5(d) shows the simulated FM wall velocity as a function of \nKy. With the increase of Ky, the spin fluctuation is gradually suppressed, and the velocity \nincreases to a saturation value approximates  to that of the AFM wall.  \n \nD. Brief discussion  \nWhile we are p ursuing the faster , the smaller, the denser for spintronic devices, the \ndomain wall motion efficiently driven by low energy -consuming can be always the core \nconcern for spintronic study.  Thus, finding new energy -saving methods of controlling the \ndomain  walls in antiferromagnet , regardless of it is metal or insulator, is still of great \nimportance for future applications.  \nIn this work, anisotropy gradient has been  clearly revealed to efficiently drive the AFM \ndomain wall motion. Experimentally, anisotropy gradient could be easil y obtained through \ntuning electric voltage on particular heterostructures , and drive the AFM domain wall motion \nat a considerable speed . Specifically, for ΔK/Kc = 1 × 10-4 a-1 and  = 0.002, the speed  of the \nwall for NiO with the exchange stiffness A ≈ 5 × 10-13 J/m, a ≈ 4.2 Å, s ≈ 1.7 B (B is the \nBohr magneton) is estimated to be ~ 100 m/s. More importantly, the proposed modulating \nmethod is expected to be more energy -saving and with less Joule heat, comparing with those electric current driving m ethods. Thus, our work unveils a promising method of controlling \nAFM wall, and does provide useful information for future spintronic applications.   \n \nIV . C onclusion  \nIn summary, we have studied the AFM wall motion under  the anisotropy gradient  based \non the one -dimensional model . The domain wall velocity depending on the gradient \nmagnitude and intrinsic physical parameters  are simulated based on the LLG equation and \nalso calculated  theoretically  based on the energy dissipation theorem . The domain  wall \nstructure  is rather robust for small gradient  and moves at a constant velocity, while accelerates \nfor large gradient due to the enlargement of the domain  wall width during its motion. The \ndomain  wall mobility  is indep endent of the lattice dimen sion and wall type (180° or 90°), \nwhile  the mobility  is enhanced by the DM  interaction . Moreover,  the physical mechanism of \nthe AFM wall dynamics faster than FM wall dynamics is also qualitatively explained. This \nwork unveils a promising strategy for controlling the AFM domain walls, benefiting to future \nspintronic applications.  \n \n \nAcknowledgment  \nWe sincerely appreciate the insightful discussions with Zhengren Yan and Huaiyang Yuan. \nThe work is supported by the National Key Projects for Basic Research of China (Grant No. \n2015CB921202), and the Natural Science Foundation of China (No. 51971096 ), and the \nScience and Technology Planning Project of Guangzhou in China  (Grant No. 2019040100 19), \nand the Natural Science Foundation of Guangdong Province (Grant No. 2016A030308019).  \n \n  Appendix : The derivation of the AFM wall velocity  \nFollowing the earlier works, w e use domain  wall ansatz tan(/2) = e(z-q)/ for 180° wall \nand tan() = e(z-q)/ for 90° wall with q is the coordination of the wall center . The wall spins \nare restrict ed in the x-z plane  (φ = 0), and the local staggered magnetization is given by : \n \n( )/\n( )/\n( )/sin(2arctan( )) cos( )\nsin(2arctan( )) sin( )\ncos(2arctan( ))zq\nx\nzq\ny\nzq\nzn = e\nn = e\nn = e\n\n\n\n\n\n  (A1)  \nfor 180°  wall, and  \n \n( )/\n( )/\n( )/sin(arctan( )) cos( )\nsin(arctan( )) sin( )\ncos(arctan( ))zq\nx\nzq\ny\nzq\nzn = e\nn = e\nn = e\n\n\n\n\n\n  (A2)  \nfor 90° domain  wall. \nAccording to the classical mechanics, one obtain s dEDW / dt = 2R. The domain  wall \nenergy is calculated by EDW = 2(2 JK)1/2 for 180°  wall and EDW = (JK/2)1/2 for 90°  wall with K \n= K0 + vtΔK. Subsequently , dEDW / dt reads  \n \nDW2 for 180 DW\n2 for 90 DWvKdE dtvK\n . (A3)  \nFurthermore, using the  mathematical transformation  \n \nd d dz d= = = vdt dz dt dzn n nn\n , (A4)  \nR can be described  by  \n \n2\n2for 180 DW=\n4 for 90 DWvR\nv \n     . 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(color online) (a) The position of  the domain  wall as a function of  time for various  K0 \nfor ΔK = 4×10-6 J/a. The simulated (empty points) and calculated (solid lines) velocities as \nfunctions of (b)  ΔK for various K0 and  = 0.002, and (c)  for various ΔK for K0 = 0.01  J. (d) \nThe simulated domain  wall position and velocity as functions of time for large ΔK = 0.0003  \nJ/a.  \n \nFig.4. (color online) (a) The simulated domain  wall velocity as a function of ΔK for various \nlattice sizes, and (b) The simulated (empty points) and calculated (solid lines) velocities as \nfunctions of D for various ΔK. \n \nFig.5. (color online) (a) The displacement s of the domain  walls of the AFM and FM systems \nas functions of time , and (b) the local my (top half) of FM system and ny (bottom half) of AFM \nsystem,  and the inserts are the schematic depiction s of the precession torque s, and (c) the \nevolution of my at various positions , and  the domain  wall center is locate d at z = 350 a, and (d) \nthe simulated FM wall velocit y as a function  of Ky.  "    },    {        "title": "1905.13042v1.Predicting_New_Iron_Garnet_Thin_Films_with_Perpendicular_Magnetic_Anisotropy.pdf",        "content": "1 \n Predicting New Iron Garnet Thin Film s with Perpendicular Magnetic Anisotropy  \nSaeedeh Mokarian Zanjani1, Mehmet C. Onbaşlı1,2,* \n \n1 Graduate School of Materials Science and Engineering, Koç University , Sarıyer, 34450 Istanbul, \nTurkey . \n2 Department of Electrical and Electronics Engineering, Koç University, Sarıyer, 34450 Istanbul, \nTurkey.  \n* Corresponding Author: monbasli@ku.edu.tr  \n \nAbstract:  \nPerpendicular magnetic anisotropy (PMA) is a necessary condition for many spintronic \napplications like spin -orbit  torques  switching , logic and memory devices. An important class of \nmagnetic insulators with low Gilbert damping at room temperature are iron garnets, which only \nhave a few PMA types such as terbium  and samarium iron garnet. More and stable PMA garnet \noptions are necessary for researchers to be able to investigate new spintronic phenomena. In this \nstudy, we predict 20 new substrate/magnetic iron garnet film pairs with stable PMA at room \ntemperature. The effective anisotropy energies of 10 different garnet films that are lattice -matched \nto 5 different commercially available garnet substrates have been calculated using shape, \nmagnetoelastic and magnetocrystalline anisotropy  terms . Strain type, tensile o r compressive \ndepending on substrate choice, as well as the sign and the magnitude of the magnetostriction \nconstants of garnets determine if a garnet film may possess PMA. We show the conditions in which \nSamarium, Gadolinium, Terbium, Holmium, Dysprosium a nd Thulium garnets may possess PMA \non the investigated garnet substrate types . Guidelines for obtaining garnet films with low damping \nare presented. New PMA garnet film s with tunable saturation moment and field may improve spin -\norbit torque memory and compensated magnonic thin film devices.  \n \n \n \n \n 2 \n Introduction  \nWith the development of sputtering and pulsed laser deposition of high -quality iron garnet thin \nfilms with ultralow Gilbert damping, researchers have been able t o investigate a wide variety of \nmagnetization switching and spin wave phenomena1-3. The key enabler in many of these studies \nhas been Yttrium iron garnet (Y 3Fe5O12, YIG)4 which has a very low Gilbert damping allowing \nspin waves to propagate over multiple millimeters across chip. YIG thin films are useful for spin \nwave device applications, but since their easy axes lie along film plane, their utility cannot b e \nextended to different mechanisms such as spin -orbit torques, Rashba -Edelstein effect, logic \ndevices, forward volume magnetostatic spin waves1. At the same time, to have reliable and fast \nresponse using low current densities as in spin -orbit torque switching, magnetization orientation \nneeds to be perpendicular to the surface plane5. The possibility of having Dzyaloshinskii –Moriya \ninteraction (DMI) in TmIG/GGG may enable stabilizing skyrmions and help drive skyrmion \nmotion with pure spin currents6. \n \nThere is a number of studies on tuning anisotropy or obtaining perpendicular magnetic anisotropy \nin insulator thin films7-12. Among the materials studied, insulating magnetic garnet s whose \nmagnetic pr operties can be tuned have been a matter of interest over the past decades13-15 due to \ntheir low damping and high magnetooptical Faraday rotation. In order to obtain perpendicular \nmagnetic anisotropy in magnetic garnets, one needs to engineer the anisotropy terms that give rise \nto out -of-plane easy axis. Angular dependence of total m agnetization energy density is called \nmagnetic anisotropy energy and consists of contributions from shape anisotropy, strain -induced \n(magnetoelastic)  and magnetocrystalline anisotropy. A magnetic material preferentially relaxes its \nmagnetization vector tow ards its easy axis, which is the least energy axis , when there is no external \nfield bias . Such energy minimization process drives magnetic switching rates  as well as the stability \nof total magnetization v ector . Controlling magnetic anisotropy in thin film garnets not only offers \nresearchers different testbeds for experimenting new PMA -based switching phenomena, but also \nallows  the investigation of anisotropy -driven ultrafast dynamic magnetic response in thin film s and \nnanostructures.  \nThe most extensively studied garnet thin film is Yttrium Iron Garnet ( YIG). YIG films display in -\nplane easy axis because of their large shape anisotropy  and negligible magnetocrystalline \nanisotropy3. Although PMA  of ultrathin epitaxial YIG films has been reported16,17, the tolerance 3 \n for fabrication condition variations for PMA YIG is very limited and strain effects were found to \nchange magnetocrystalline anisotropy in YIG. Strain -controlled anisotr opy has been observed in \npolycrystalline ultrathin YIG films17,18. In case of YIG thin film grown on Gadolinium Gallium \nGarnet (GGG), only partial anisotropy control has been possible through significant change in \noxygen stoichiometry19, which increases damping. Since the fabrication of high -quality and highly \nPMA YIG films  is not easy for practical thicknesses on gadolinium gallium garnet substrates \n(GGG), researchers have explored tuning magnetic anisotropy by substituting Yttrium sites with \nother rare earth elements20,21. New garnet thin films that can exhibit PMA with different \ncoercivities, saturation fields, compensation points and tunable Gilbert damping values must be \ndeveloped  in order to evaluate the effect of these p arameters on optimized spintronic insulator \ndevices . \nSince the dominant anisotropy energy term is s hape anisotropy  in thin film YIG , some studies focus \non reducing the shape anisotropy contribution by micro and nanopatterning22-24. Continuous YIG \nfilms were etched  to form rectangular nanostrips with nanometer -scale thickness es, as \nschematic ally shown on Fig. 1(a) . Thus, least magnetic saturation field is needed along the longest \ndimension of YIG nanostrips . By growing ultrathin YIG, magnetoelastic strain contributions lead \nto a negative anisotropy field and thus PMA in YIG film s24. As the length -to-thickness ratio \ndecreases , the effect of shape anisotropy is reduced and in-plane easy axis is converted to PMA17.  \nWhile reducing the effect of shape anisotropy is necessary, one also needs to use m agnetoelastic \nanisotropy  contribut ion to reorient magnetic easy axis towards out of film plane , as schematically \nshown on Fig. 1 (b). Strain-induced perpendicular magnetic anisotropy in rare earth (RE) iron \ngarnets, especially in YIG, has been demonstrated to overcome shape anisotropy16,17,25,26. If \nmagnetoelastic anisotropy term induced by crystal lattice mismatch is large r than shape anisotropy \nand has opposite sign, then magnetoelastic anisotropy overcome s shape . Thus, the easy axis of the \nfilm becomes perpendicular to the film plane and the hyster esis loop becomes square -shaped with \nlow saturation field8. One can also achieve PMA in other RE magnetic iron garnets due to their \nlattice parameter  mismatch with their substrates . PMA has previously been achieved using \nSubstituted Gadolinium Gallium Garnet (SGGG) as substrate and a Samarium Gallium Garnet \n(SmGG) ultrathin film as buffer layer under (and on) YIG16. In case of thicker YIG films (40nm), \nthe magnetic easy axis becomes in -plane again. An important case shown by Kubota et.al19 \nindicates that increasing in-plane strain (ε ||) or anisotropy field (H a) helps achieve perpendicular 4 \n magnetic anisotropy. In ref.8,19, they reported that if magnetostriction coefficient (λ 111) is negative \nand large eno ugh to overcome shape anisotropy, and tensile strain is introduced to the thin film \nsample (ε ||>0), the easy axis becomes perpendicular to the sample plane  as in the case of Thul ium \niron garnet (Tm 3Fe5O12, TmIG ). \nA different form of magnetoelastic anisotropy effect can be induced by using porosity in garnet \nthin films. Mesoporous  Holmium Iron Garnet (Ho 3Fe5O12, HoIG) thin film on Si (001)27 exhibit s \nPMA due to reduced shape anisotropy, increased magnetostrictive and growth -induced anisotropy \neffects. Such combined effects lead to PMA in HoIG. In this porous thin film, the PMA was found \nto be independent of the substrate used, because the mechanical s tress does not result from a lattice \nor thermal expansion mismatch between the substrate and the film. Instead, the pore -solid \narchitecture itself imposes an intrinsic strain on the solution processed garnet film.  This example \nindicates that the film struc ture can be engineered in addition to the substrate choices in order to \novercome shape anisotropy in thin film iron garnets.  \nAnother key method for controlling anisotropy is strain doping through substitutional elements and \nusing their growth -induced aniso tropy effects , as schematically shown on Fig. 1(c) . Bi-doped \nyttrium iron garnet (Bi:YIG and Bi:GdIG) has been reported to possess perpendicular magnetic \nanisotropy due to the chemical composition change as the result of increased annealing \ntemperature21,28. Another reason for PMA in these thin films is strain from GGG substrate29. \nDoping of oxides by Helium implantation was shown to reversibly and locally tune magnetic \nanisotropy30. For TbxY3-xFe5O12 (x=2.5, 2.0, 1.0, 0.37 ) samples  grown by spontaneous nucleation \ntechnique31, magnetic easy axis was found to change from [111] to [100] direction as Tb \nconcentration was decreased. The first -order anisotropy constant K 1 undergoes a change of sign \nnear 190K . Another temperature -dependent lattice distortion effect that caused anisotropy chang e \nwas also reported for YIG films32. These results indicate  that temperature also plays an important \nrole in both magnetic compensation, lattice distortion and change in anisotropy.   \nIn this study, we systematically calculate the anisotropy  energ ies of  10 different types of lattice -\nmatched iron garnet compounds epitaxially -grown as thin films (X 3Fe5O12, X = Y, Tm, Dy, Ho, \nEr, Yb, Tb, Gd, Sm, Eu) on commercial ly available (111) -oriented garnet substrates (Gd 3Ga5O12-\nGGG , Y 3Al5O12-YAG , Gd 3Sc2Ga3O12 -SGGG, Tb 3Ga5O12 –TGG, Nd 3Ga5O12 -NGG).  Out of the \n50 different film/substrate pairs, we found that 20 cases are candidates for room temperature PMA. \nOut of these 20 cases, 7 film/substrate pairs were experimentally tested and shown to exhibit 5 \n characteristics originating from PMA. The remaining 13 pairs, to the best of our knowledge, have \nnot been tested for PMA experimentally.  We indicate through systematic anisotropy calculations \nthat large strain -induced magnetic anisotropy terms may overcome shape when the films are highly \nstrained . We use only the room temperature values of λ 11133 and only report predictions for room \ntemperature (300K) . Throughout the rest of this study, the films are labelled as XIG (X = Y, Tm, \nDy, Ho , Er, Yb, Tb, Gd, Sm, Eu), i.e. TbIG (Terbium iron garnet) or SmIG (Samarium iron garnet) \netc. to distinguish them based on the rare earth element.  Our model could accurately predict the \nmagnetic easy axis in almost all experimentally tested garnet film/substrate cases provided that the \nactual film properties are entered in the model and that the experimental film properties satisfy \ncubic lattice mat ching condition to the substrate.  6 \n  \nFigure 1.  Methods to achieve perpendicular magnetic anisotropy in iron garnet thin films. \n(a) Micro/nano -patterning reduces shape anisotropy and magnetoelastic anisotropies overcome \nshape.  (b) Large strain -induced anisotropy must over come  shape anisotropy to yield out -of-plane \neasy axis.  (c) Substitutional doping in garnets overcome shape anisotropy by enhancing \nmagnetocrystalline, growth -induced or magnetoelastic terms.   \n \nAnisotropy energy den sity calculation s \nTotal anisotropy energy density contains three main contributions; according the Equation 1, shape \nanisotropy ( Kshape ), first order cubic magnetocrystalline anisotropy ( K1), and strain -induced  \n7 \n (magnetoelastic)  anisotropy ( Kindu) parameters determine the total effective anisotropy energy \ndensity16. \nKeff=Kindu +Kshape +K1   (1) \nIn case of garnet film magnetized along [111] direction  (i.e. on a 111 substrate) , the magneto -elastic \nanisotropy energy density, resulting from magneto -elastic coupling is calculated by Equation 2: \nKindu =−3\n2λ111σ||      (2) \nwhere λ111 is magnetostriction constant along [111] direction  and it is usually  negative  at room \ntemperature34. In Eqn. 2,  σ|| is the in -plane  stress induced in the material  as a result of lattice \nmismatch between film and the substrate , and the in -plane stress is calculated from Equation 335: \nσ||=Y\n1−νε||      (3) \nwhere Y is elastic mo dulus, and ν is P oisson’s ratio36. \nFor calculation of in -plane strain, lattice parameter values obtaine d from the XRD characterization \nof the thin films are used . Equation 4  shows the strain relation as the lattice constant difference \nbetween the bulk form of the film and that of the substrate divided by the lattice constant  for the \nbulk form of the film16. \nε||=afilm −abulk\nafilm       (4) \nAssuming the lattice parameter of the thin film matches with that of the substrate, the lattice \nconstant of substrate can be used as the lattice constant of thin film for  calculation of strain in \nEquation 537: \nε||=asub−afilm\nafilm       (5) \nThe lattice constants used for the films and substrates examined for this study are presented on \nTable 1.  \nShape anisotropy energy density depends  on the geometry  and the intrinsic saturation magnetic \nmoment of the iron garnet material. Shape anisotropy has a demagnetizing effect on the total 8 \n anisotropy energy density. These significant anisotropy effects can be observed in magnetic \nhysteresis loop s and FMR measurements24.  \nThe most common anisotropies in magnetic materials are shape anisotropy and magneto -crystalline \nanisotropy38,39. Considering that the film is continuous, the shape anisotropy is calculated as16 \nKshape =2πMs2      (6) \nBy obtaining the values of M s for rare earth iron garnets as a function of temperature40,41, the value \nfor shape anisotropy energy density have been calculated using Equation 6.   \nIntrinsic magnetic anisotropy42, so called magnetocrystalline anisotropy, has the weakest \ncontribution to anisotropy energy densit y compared  to shape, and strain -induced \nanisotropies9,11,16,19. The values for the first order magnetocrystalline anisotropy is calculated and \nreported previously for rare earth iron garnets at different temperatures43. A key consideration in \nmagnetic thin films is saturation field. In anisotropic magnetic thin films, the anisotropy fiel ds have \nalso been calculated using equation 7 as a measure of how much field the films need for magnetic \nsaturation along the easy axis:  \nHA=2Keff\nMs       (7) \nTable 1. List of magnetic iron garnet thin films and garnet substrates available off -the-shelf used \nfor this study. The fourth column shows the lattice constants used for calculating the magnetoelastic \nanisotropy values of epitaxial garnets on the given substrates.  \nGarnet \nmaterial  Chemical \nformula  Purpose  Bulk \nlattice \nconstant  \n(Å) \nGGG  Gd3Ga5O12 Substrate  12.383  \nYAG  Y3Al5O12 Substrate  12.005  \nSGGG  Gd3Sc2Ga3O12 Substrate  12.480  \nTGG  Tb3Ga5O12 Substrate  12.355  \nNGG  Nd3Ga5O12 Substrate  12.520  \nYIG Y3Fe5O12 Film  12.376  \nTmIG  Tm 3Fe5O12 Film  12.324  \nDyIG  Dy3Fe5O12 Film  12.440  \nHoIG  Ho3Fe5O12 Film  12.400  \nErIG  Er3Fe5O12 Film  12.350  \nYbIG  Yb3Fe5O12 Film  12.300  9 \n TbIG  Tb3Fe5O12 Film  12.460  \nGdIG  Gd3Fe5O12 Film  12.480  \nSmIG  Sm 3Fe5O12 Film  12.530  \nEuIG  Eu3Fe5O12 Film  12.500  \n \nResults and Discussion  \nTable s 1 and 2 list in detail the parameters used and the calculated anisotropy energy density terms \nfor magnetic rare earth iron garnets  at 300K . These tables show only the cases predicted to be PMA  \nout of a total of 50 film/substrate pairs investigated . The extended version of Tables  1 and 2 for all \ncalculated anisotropy energy density terms for all combinations of the 50 film/subst rate pairs are \nprovided in the s upplementary tables. The tabulated values for saturation magnetization40,41 and \nlattice parameters44 have been  used for the calculations . In this study , we assumed the value of \nYoung’s modulus and Poisson ratio as 2.00×1012 dyne ·cm-2 and 0.29 for all garnet types , \nrespectively, based on ref.36. We also assume that the saturation magnetization, used for calculation \nof shape anisotropy, does not change with the film thickness. The saturation magnetization (Ms) \nvalues and  shape anisotropy  for iron garnet  films  are presented in the third  and fourth  columns , \nrespectively. The stress values for  fully lattice -matched film s σ calculated using equation 3 and \nmagnetostriction constants  of the films , λ111, are presented  on column s 6 and 7. Magnetoelastic \nanisotropy K indu, magnetocrystalline  anisotropy energy density K 1, and the total magnetic \nanisotropy energy density K eff are calculated  and listed on columns 8 , 9 and 10, respectively . H A \non column 11  is the anisotropy field ( the fields required to saturate the film s). \n \nTable 2 . Anisotropy energy density parameters calculation results. Rare earth iron garnets on GGG \n(as=12.3 83Å), YAG ( Y3Al5O12, as=12.005Å) , SGGG (a s=12.48Å) and TGG (Tb 3Ga5O12, \nas=12.355Å), and NGG ( Nd3Ga5O12, as=12.509Å)  substrates, with K eff < 0, are presented.   \nFilm  Substr ate Ms \n(emu·cm-\n3) Kshape \n(erg·cm-\n3) (× 103) ε \n(× \n10-3) σ \n(dyn·cm-\n2) \n(×1010) λ111 \n(×10-\n6) Kindu \n(erg·cm-\n3) (×104) K1 \n(300K) \n(erg·cm-\n3) \n(× 103) Keff \n(erg·cm-\n3) \n(× 103) HA \n(Oe) \n(× \n103) \nDyIG  GGG  31.85  6.37 -4.58 -1.29 -5.9 -11.4 -5.00 -113 -7.09 \nHoIG  GGG  55.73  19.5 -1.37 -0.386  -4 -2.3 -5.00 -8.66 -0.311  \nGdIG  GGG  7.962  0.398  -7.77 -2.19 -3.1 -10.2 -4.10 -106 -26.5 \nSmIG  GGG  140 123 -11.7 -3.30 -8.6 -42.6 -17.4 -321 -4.58 \nYIG YAG  141.7  126 -30.0 -8.44 -2.4 -30.4 -6.10 -184 -2.60 10 \n TmIG  YAG  110.9  77.2 -25.9 -7.29 -5.2 -56.9 -5.80 -497 -8.97 \nDyIG  YAG  31.85  6.37 -35.0 -9.85 -5.9 -87.2 -5.00 -870 -54.7 \nHoIG  YAG  55.73  19.5 -31.9 -8.97 -4 -53.8 -5.00 -524 -18.8 \nErIG  YAG  79.62  39.8 -27.9 -7.87 -4.9 -57.8 -6.00 -545 -13.7 \nYbIG  YAG  127.3  102 -24.0 -6.76 -4.5 -45.6 -6.10 -360 -5.66 \nGdIG  YAG  7.962  0.398  -38.1 -10.7 -3.1 -49.9 -4.10 -502 -126 \nSmIG  YAG  140 123 -41.9 -11.8 -8.6 -152.3  -17.4 -1420  -20.2 \nTbIG  SGGG  15.92  1.59 1.61 0.452  12 -8.14 -8.20 -88.0 -11.1 \nGdIG  SGGG  7.962  0.398  0.00 0.00 -3.1 0.00 -4.10 -3.70 -0.930  \nSmIG  SGGG  140 123 -3.99 -1.12 -8.6 -14.5 -17.4 -39.3 -0.562  \nDyIG  TGG  31.85  6.37 -6.83 -1.92 -5.9 -17.0 -5.00 -169 -10.6 \nHoIG  TGG  55.73  19.5 -3.63 -1.02 -4 -6.13 -5.00 -46.8 -1.68 \nGdIG  TGG  7.962  0.398  -10.0 -2.82 -3.1 -13.1 -4.10 -135 -33.9 \nSmIG  TGG  140 123 -14.0 -3.93 -8.6 -50.8 -17.4 -402 -5.74 \nTbIG  NGG  15.92  1.59 3.93 1.11 12 -19.9 -8.20 -206 -25.9 \n \nIn this study, we take the same sign convention as in ref. 16 and the film s exhibit PMA when Keff \n< 0. So for obtaining PMA, negative and large values for anisotropy energy density are desired. As \nall the garnets (except TbIG ) possess negative magnetostriction constant s at room temperature, the \nsign of the strain  (tensile or compressive)  determines whether the induced anisotropy is negative \nor positive . In the literature16,20,45,46 however , we observe that PMA was defined for either positive \nor ne gative effective anisotropy energy density (K eff). This inconsistency may cause confusion \namong researchers . Thermodynamically, a higher energy means an unstable state with respect to \nlower energy cases. Easy axis, by definition, is the axis along which the magnetic material can be \nsaturated with lowest external field or lowest total energy. A magnetic material would thus \nspontaneously minimize its energy and reorient it s magnetic moment along the easy axis. As a \nresult, we use here Keff < 0 for out-of-plane easy axis . Due to the thermodynamic arguments \nmentioned above, we suggest researchers to use K eff < 0 definition for PMA.  \n \nEffect of Substrate on Anisotropy Energy Density  \nChanging the substrate alters the strain in the film, which  also changes strain -induced anisotropy \nin the film. Figure 2 show s the calculated anisotropy energy density of rare earth iron garnet  thin \nfilms  grown on five commercially available differ ent substrates : Gadolinium Gallium Garnet \n(Gd3Ga5O12, GGG), Yttrium Aluminum Garnet ( Y3Al5O12, YAG), Substituted Gadolinium 11 \n Gallium Garnet ( Gd3Sc2Ga3O12, SGGG), Terbium Gallium Garnet (Tb 3Ga5O12, TGG ), and \nNeodymium Gallium Garnet (Nd 3Ga5O12, NGG) . As shown on Fig. 2(a), when gr own on GGG \nsubstrate ; Dysprosium Iron Garnet (DyIG), Holmium Iron Garnet (HoIG), Gadolinium Iron Garnet \n(GdIG), and Samarium Iron Garnet (SmIG) possess compressive strain  (afilm>asubstrate ). Considering \nthe large negative magnetostriction constant (λ 111) for each case, the strain -induced anisotropy \nenergy densit ies are estimated to cause negative total effective anisotropy energy density . As a \nresult, DyIG, HoIG, GdIG, and SmIG on GGG are predicted to be PMA cases.  \nBased o n the shape, magnetoelastic and magnetocrystalline anisotropy terms (room temperature \nK1), Thulium iron garnet (TmIG) on GGG  (111)  is estimated to be in -plane easy axis although \nunambiguous experimental evidence indicates that TmIG grows with PMA on GGG (111)  [1,19] . \nThe fact that only considering shape, magnetocrystalline and magnetoelastic anisotropy terms does \nnot verify this experimental result suggests that the PMA in TmIG/GGG (111) may originate from \na different anisotropy term such as surface anisotropy , growth -induced  or stoichiometry -driven \nanisotropy. Since the films used in the experiments are less than 10 nm or 5 -8 unit cells thick, \nsurface effects may become more significant and may require density functional theory -based \npredictions to account for surface anisotropy effects .  12 \n  \nFigure 2.  Calculated effective  anisotropy values for each rare earth iron garnet thin film when they \nare epitaxially grown on ( a) GGG, ( b) YAG, ( c) SGGG, ( d) TGG, (e) NGG  substrates . Note that \nthe scales of the axes are different in each part.  \n13 \n Yttrium Aluminum Garnet (YAG) is a substrate with smaller lattice parameter than all the rare \nearth iron garnet films  considered . With a substrate lattice parameter of as=12.005Å, YAG causes \nsignificant and varying amounts of strain on  YIG (af=12.376Å) , TmIG (af=12.324Å) , DyIG \n(af=12.440Å) , HoIG (af=12.400Å) , ErIG  (af=12.350Å) , YbIG (af=12.300Å) , TbIG (af=12.460Å) , \nGdIG (af=12.480Å) , SmIG (af=12.530Å)  and EuIG (a f=12.500Å) . Strain from YAG substrate \nyields negative strain -induced anisotropy energy density for these films . The strain -induced \nanisotropy term overcomes the shape anisotropy in these garnets when they are grown on YAG. \nConsequently , effective anisotropy energy densit ies become negative and these garnet  films are \nestimated to  possess perpendicular magnetic anisotropy . In the e xceptional case s of Terbium Iron \nGarnet (TbIG)  and Europium Iron Garnet (EuIG) , compressive strain is not enough to induce \nnegative strain anisotropy  because  the magnetostriction coefficient s of TbIG and EuIG are positive. \nSo the strain -induced anisotropy term s are also positive for both TbIG (Kindu(TbIG)  = 1.85×106 \nerg·cm-3) and EuIG (K indu(EuIG)  = 3.01×105 erg·cm-3) and do not yield PMA.  \nOther potential PMA garnets as a film on SGGG substrate are GdIG , TbIG , and SmIG. TbIG and \nGdIG cases are particularly interesting as growth conditions of these materials can be further \noptimized to achieve room temperature compensation and zero saturation magnetization. This \nproperty enables PMA garnet -based room temperature terahertz magnonics. The lattice parameters \nof GdIG (af=12.480Å) and SGGG (as=12.48Å) match exactly, so the in -plane strain value is zero \nand the effect of strain -induced anisotropy is eliminated completely. Consequently, due to small \nvalue for saturation magnetizat ion of GdIG, shape anisotropy (3.98 ×102 erg·cm-3) cannot compete \nwith magnetocrystalline anisotropy ( -4.1×103 erg·cm-3). In other words, in this case, the influence \nof magnetocrystalline anisotropy is not negligible compared to the other anisotropy terms.  \nConsequently , the anisotropy energy density is negative for GdIG when grown on SGGG due to \nthe influence of magnetocrystalline anisotropy energy density.  \nOne other candidate for a PMA rare earth iron garnet on SGGG substrate is SmIG. Since the film \nlattice parameter is greater than that of the substrate, compressive strain ( -3.99 ×10-3) is induced in \nthe film such that the resulting anisotropy energy density possesses a negative value of an order of \nmagnitude ( -1.45×105 erg·cm-3) comparable to the sh ape anisotropy energy density (1.23×105 \nerg·cm-3). With its relatively large magnetocrystalline anisotropy energy density ( -1.74×104 \nerg·cm-3), SmIG has a perpendicular magnetic anisotropy due to negative value for effective 14 \n anisotropy energy density (-4.80×105 erg·cm-3). TbIG film on SGGG substrate is a PMA candidate \nwith positive strain  and this film was also recently experimentally demonstrated to have PMA47.  \nSince TbIG’s lattice constant is smaller than that of the substrate, the film becomes subject to tensile \nstrain ( +1.61  ×10-3). Since TbIG also has a positive λ 111 (in contrast to that of SmIG), the film’s \nmagnetoelastic anisotropy term becomes large  and negative  and overcomes the shape anisotropy. \nIn case of TbIG, magnetocrystalline anisotropy alone overcomes shape and renders the film PMA \non SGGG. With the additio nal magnetoelastic anisotropy contribution ( -8.14×104 erg·cm-3), \nsignificant stability of PMA can be achieved.  \nTerbium Gallium  Garnet (TGG) is a substrate with lattice parameter (as=12.355Å ) such that it can \ninduce tensile strain on TmIG, ErIG and YbIG  and it induces compressive strain on the rest of the \nrare earth iron garnets (YIG, DyIG, HoIG, TbIG, GdIG, SmIG, EuIG) . In none of the tensile -\nstrained cases, PMA can be achieved since the sign of the magnetoelastic anisotropy is positive \nand has the same  sign as the shape anisotropy.  Among the compressively  strained cases, YIG, TbIG \nand EuIG are found to have weak magnetoelastic anisotropy terms which cannot overcome shape. \nAs a result, YIG, TbIG and EuIG on TGG substrate are expected to have in -plane easy axis. DyIG, \nHoIG, GdIG and SmIG films on TGG achieve large and negative effective  total anisotropy energy \ndensities  due to their negative λ 111 values . In addition, since the materials have compressive strain, \nthe signs cancel and lead to large magnetoelastic anisotropy energy terms that can overcome shape \nin these materials. So these cases are similar to the conditions explained for GGG substrate,  on \nwhich  only DyIG, HoIG, GdIG, and SmIG films with compressive strain can gain a large negative \nstrain -induced anisotropy energy density which can overcome shape a nisotropy . \nNeodymium gallium garnet  (NGG)  is a substrate 45 used for growing garnet thin films by pulsed \nlaser deposition method. NGG has large lattice constant compared with the rest of the bulk rare \nearth garnets and yield compressive strain in all rare earth garnets investigated except for Samarium \niron garnet (SmIG). For all cases other than SmIG, the sign of the magnetoelastic a nisotropy term \nis determined by the respective λ 111 for each rare earth iron garnet. YIG, TmIG, DyIG, HoIG, ErIG, \nYbIG cases have positive magnetoelastic anisotropy terms, which  lead to easy axes along the ir film \nplane s. For SmIG and EuIG on NGG, magnetoel astic strain and anisotropy terms are not large \nenough to overcome large shape anistropy. For GdIG, the weak tensile strain on NGG substrates \nactually causes in -plane easy axes as magnetoelastic strain offsets the negative magnetocrystalline 15 \n anisotropy. The only rare earth iron garnet that can achieve PMA on NGG is Terbium Iron Garnet \n(TbIG) due to its large negative strain -induced anisotropy energy density ( -2.44×105 erg·cm-3). Its \nlarge and negative magnetoelastic anisotropy can offset shape (1.59×103 erg·cm-3) and first order \nmagnetocrystalline anisotropy term ( -8.20×103 erg·cm-3), leading to a large negative effective \nmagnetic anisotropy energy density ( -2.51×105 erg·cm-3). Consequently, we predict that  growing \nTbIG on NGG substrate may yield PMA . \n \nFigure 3 show s the substrates on which one may expect PMA rare earth iron garnet films (or \nnegative Keff) due to strain only. Figures  3(a)-(d) compare the calculated effective energy densities \nas a function of strain type and sign for YIG, TmIG, YbIG, TbIG. For comparing the calculation \nresults presented here with the experimentally reported values for the anisotropy energy density of \nTmIG, we added the K eff directly from the experimental data in20 to Fig. 3(b). As shown on Fig. \n3(b), the experimental TmIG thin film shows positive K eff as the result of tensile in -plane strain \nand large negative magnetostriction constant.  \nFigure 4(a) -(f) shows the calculated effective energy densities as a function of strain type and sign \nfor GdIG, SmIG, EuIG, HoIG, DyIG, ErIG, respectively. K eff may get a positive or a negative value \nin both compressive and tensile strain cases due to vary ing signs of λ 111 constants of rare earth iron \ngarnets. In almost all cases that yield PMA on the given substrates, PMA iron garnets form under \ncompressive lattice strain. The only exception s in which tensile strain can yield PMA in garnet thin \nfilms is Tb IG on SGGG  and TbIG on NGG . In both of those cases, a small tensile strain enhances \nPMA but the magnetocrystalline anisotropy could already overcome shape and yield PMA without \nlattice strain. Therefore, experimental studies should target compressive latti ce strain.  \n 16 \n  \nFigure 3.  Effect of substrate strain on the effective anisotropy energy densities of ( a) YIG, ( b) \nTmIG, the data inserted on the graph, with red square symbol , GGG (Exp.) , is the experimental \nvalue of effective anisotropy energy of TmIG  on GGG based on ref.20 (c) YbIG, ( d) TbIG. Note \nthat the axes scales are different in each part.  \n17 \n  \nFigure 4. (a) GdIG, ( b) SmIG, ( c) EuIG, ( d) HoIG, ( e) DyIG, ( f) ErIG films on GGG, YAG, \nSGGG, TGG, and NGG substrates. Note that the axes scales are different in each part.  \n \n18 \n Based on Fig. 2 -4, the calculations  in this paper  numerically match with the reported values in the \nexperimental demonstrations in literature  both in sign and order of magnitude . However, since \nthere are inconsistent sign conventions for predictin g the magnetic anisotropy state  of the iron \ngarnet samples in the literature so far, some of the previous studies draw dif ferent conclusions on \nthe anisotropy despite the similar K eff. \nIn case of TmIG, as shown in Fig. 3(b), our model p redicts that there is a tensile strain -induced \nanisotropy resulting from the difference in film and substrate lattice parameters  and the film’ s \nnegative magnetostriction constant. The experimental results of magnetic anisotropy in \nTmIG/GGG8,20,48 are consistent with our model predictions . Previous studies identify PMA , if the \nfilm Keff is positive. A shortcoming of this approach is that such a definition would also identify \nYIG/GGG as PMA although its in-plane easy axis behavior has been experimentally de monstrated \nin numerous studies3,32,49. Keff < 0 for PMA definition  would be thermodynamically more \nappropriate and would also accurately explain almost  all cases including YIG/GGG.  Further \nexplanation about TmIG exceptional case is included in the Supplementary Information.   \nSensitivity of Anisotropy to Variations in Saturation Magnetic Moment and Film Relaxation  \nThe films with predicted effective anisotropy energy and field  may come out  differently when \nfabricated due to unintentional variability  in fabrication process conditions , film stoichiometry \n(rare earth ion to iron ratio and oxygen deficiency) as well as process -induced non -equilibrium \nphases in the garnet films.  These changes may partially or completely relax the films or increase \nstrain further due to secondary crystallin e phases. Practical minor changes in strain may \ndramatically alter both the sign and the magnitude of magnetoelastic ani sotropy energy and may \ncause a film predicted as PMA to come out with in -plane easy axis.  On the other hand, o ff-\nstoichiometry may cause reduction in saturation magnetic moment.  Reduction in saturation \nmagnetic moment decreases shape anisotropy term  quadratically (Kshape = 2πM s2), which implies \nthat a 1 0% reduction in Ms leads to a 19% decrease in K shape and anisotropy field may increase  (HA \n= 2K eff/Ms). Therefore, sample fabrication issues and the consequent changes in anisotropy terms \nmay weaken or completely eliminate the PMA of a film/substrate pair  and alter anisotropy field . \nWhile these effects may arise unintentionally, one can also use these effects deliberately for \nengineering garnet films for devices.  Therefore , the sensitivity of anisotropy properties of garnet 19 \n thin films such as anisotropy field and effective anisotropy energy density need s to be evaluated \nwith respect to changes in film strain and saturation magnetic moment.  \nFigure 5 shows the sensitivity of  the effective magnetic anisotropy energy density to deviation of \nboth strain and saturation magnetization, M s for five PMA film/substrate  combinations:  (a) \nHoIG/GGG, (b) YIG/YAG, (c) SmIG/ SGGG, (d) HoIG/TGG , and (e) SmIG/NGG ). The negative \nsign of effective magnetic anisotropy energy indicates PMA . The change of anisotropy energy from \nnegative to positive indicates a transition from PMA to in -plane easy axis . In these plots, calculated \nanisotropy energies are presented for saturation moments and strains scanned from 60% to 140% \nof tabulated bulk garnet Ms and of the strain s of fully lattice -matched films on the substrate s. The \ncolor scale indicates the anisotropy energy density in erg·cm-3. Although magnetocrystalline \nanisotropy energies are negative for all of the thin film rare earth garnets considered here, these \nterms are negligible with respect to shape anisotropy (K 1(300 K) ~ -5% of K shape). Therefore, \nmagnetoelastic anisotropy term must be large enough to overcome shape anisotropy. A derivation \nof anisotropy energy  as a function of Ms and strain in equations ( 8)-(10) shows that the negative \nλ111 values and negative strain states (compressive strain) for garnet films in Fig. 5(a)-(e) (HoIG, \nYIG, SmIG) enable these films to have PMA . To retain PMA state; λ 111 must be negative and large  \nassuming elastic moduli and the Poisson’s ratio are constant . The necessary condition for \nmaintai ning PMA is shown in equation 11 . \nKeff=Kindu +Kshape +K1    (8) \nKeff=−3\n2λ111Y\n1−vε||+2πMs2+K1   (9) \nKeff=3\n2λ111Y\n1−v|ε|||+2πMs2+K1   (10) \nKeff<0    if   |3\n2λ111Y\n1−v|ε|||+K1|>2πMs2  (11) \nRelaxing each fully strained and lattice -matched thin film towards unstrained state (ε → 0  or \nmoving from left  to right on each plot in Fig. 5  causes  the magnetoelastic anisotropy energy term \nto decrease in magnitude and gradually vanish . The total anisotropy energy decreases  in intensity \nfor decreasing strain and constant M s. When M s increases, shape anisotropy term also increases \nand overcomes magnetoelastic anisotropy term. As a result, higher M s for relaxed films (i.e. \nrelatively thick and iron -rich garnets) may lose PMA. Therefore, one needs to optimize the film 20 \n stoichiometry and deposition process conditions, especially growth temp erature, oxygen partial \npressure  and film thickness, to ensure that the films are strained and stoichiometric. Since strains \nare less than 1% in Fig. 5(a), 5(c)-(e), these samples are predicted to be exp erimentally more \nreproducible. For YIG on YAG, as shown in Fig. 5 (b), the strains are around 3%, which may be \nchallenging to reproduce.  The cases presented in Fig. 5 (a)-(e) are the only cases among 50 \nfilm/substrate pairs where reasonable changes in M s and strain may lead to complete loss of PMA. \nThe rest of the cases have not been found as sensitive to strain and M s variability and those \npredicted to be PMA are estimated to have stable anisotropy.  Effective a nisotropy energy plots \nsimilar to Fig. 5(a) -(e) are presented in the supplementary figures for all 50 film/substrate pairs.   \nWhile PMA is a useful metric for garnet films, the effective anisotropy energy of the films should \nalso not be too high ( < a few 105 erg·cm-3) otherwise the saturation fields for these films would \nreach or exceed 0.5 Tesla (5000 Oe). Supplementary figures present the calculated  effective \nanisotropy energy and anisotropy field values for all 50 film/substrate pairs for changing strain and \nMs values. These figures indicate that one can span anisotropy fields of about 300  Oersteds up to \n12.6 Tesla in PMA garnets. For practical integrated magnonic  devices, the effective anisotropy \nenergy should be large enough to have robust PMA although it shoul d not be too high such that \neffective anisotropy fields (i.e. saturation fields) would still be small and feasible.  Engineered strain \nand M s through controlled oxygen stoichiometry may help keep anisotropy field low while \nretaining PMA.  In addition, according to the recently published paper  on magnetic anisotropy of \nHoIG50, the lattice matching in case of the thick samples becomes  challenging to sustain, and the \nstrain relaxes inside the film. Thus, the decrease in the anisotropy field is one consequences of the \nlower strain state, which is an advantage for magnonics or spin -orbit torque devices. Below a \ncritical thickness, HoIG gr own on GGG has PMA. However, as the film reaches this critical \nthickness, the 40% or more strain relaxation is expected and the easy axis becomes in -plane. So \nthinner films are preferred to be grown in integrated device applications.  21 \n  \nFigure 5. Effect of partial film relaxation and saturation magnetic moment variability on the \neffective anisotropy  energy density of the films . Variation of effective magnetic anisotropy energy \ndensities for (a) HoIG on GGG, ( b) YIG on YAG, ( c) SmIG on SGGG, ( d) HoIG on TGG and ( e) \nSmIG on NGG are presented when strain relaxation and magnetic saturation moments change \nindependently. Film strain may vary from a completely lattice -matched state to the substrate to a \nrelaxed state or a highly strained state due to microparticle nucleation . Strain variability alter s \nmagnetoelastic  anisotropy and cause a PMA film become in -plane easy axis.  On the other hand, \n22 \n magnetic saturation moments may deviate from the tabulated values because of process -induced \noff-stoich iometry in the films (i.e. rare earth ion to iron ratio or iron deficiency or excess , oxygen \ndeficiency) . Relaxing the films reduces the magnetoelastic anisotropy term and diminishes PMA. \nIncreasing M s strengthens shape anisotropy and eliminates PMA for lo w enough strains for all five \ncases presented.  \nMinimizing Gilbert damping coefficient in garnet thin films is also an important goal for spintronic \ndevice applications. First principles predictions of physical origins of Gilbert damping  51 indicate \nthat magnetic materials with lower M s tend to have lower damping. Based on this prediction, DyIG, \nHoIG and GdIG films are predicted to have lower Gilbert damping with respect to the others. Since \nthe compen sation temperatures of these films could be engineered near room temperature, one may \noptimize their damping for wide bandwidths all the way up to terahertz (THz)52 spin waves  or \nmagnons . The first principles predictions  also indicate that higher magnetic susceptibility (χm) in \nthe films helps reduce damping (i.e. lower saturation field). Therefore, the PMA garnet films with \nlower anisotropy fields are estimated to have lower Gilbert dampi ng parameters with respect to \nPMA garnets with higher anisotropy fields . \n \nConclusion  \nShape, magnetoelastic and magnetocrystalline m agnetic anisotropy energy terms have been \ncalculated for ten different garnet thin films  epitaxially grown on five different garnet substrates. \nNegative K eff (effective magnetic anisotropy energy) corresponds to perpendicular magnetic \nanisotropy  in the convention used here .  By choosing a substrate with a lattice parameter smaller \nthan that of the film, one can  induce compressive strain in the films  to the extent that one can \nalways overcome shape anisotropy  and achieve PMA  for large and negative λ 111. Among the PMA \nfilms predicted, SmIG possesses  a high anisotropy energy density and this film is estimated to be \na robus t PMA when grown on all five different substrates.  \nIn order to obtain PMA, magnetoelastic anisotropy term must be large enough to overcome shape \nanisotropy. Magnetoelastic anisotropy overcomes shape anisotropy when the strain type \n(compressive or tensile) and magnetoelastic anisotropy constants λ 111 of the garnet film  have the \ncorrect signs (not necessarily opposite or same) and the magnetoelastic anisotropy term has a \nmagnitude larger than shape anisotropy. Both compressive and tensile -strained films can , in \nprinciple, become  PMA as long as shape anisotropy can be overcome with large magnetoelastic 23 \n strain effects.  Here, in almost all cases that yield PMA on the given substrates, PMA iron garnets \nform under compressive lattice strain, except TbIG on SGGG and  TbIG on NGG . These two cases \nhave tensile strain and relatively large magnetocrystalline anisotropy, which could already \novercome shape anisotropy without strain. Experiments are therefore suggested to target mainly \ncompressive lattice strain.  \n20 different garnet film/substrate pairs have been predicted to exhibit PMA and their properties are \nlisted on Table 2 . For 7 of these 20 potential PMA cases, we could find unambiguous experimental \ndemonstration of PMA. Among the  20 PMA cases, HoIG/GGG, YIG/Y AG, SmIG/SGGG, \nHoIG/ TGG and SmIG/ NGG  cases have been found to be sensitive to fabrication process or \nstoiochiometry -induced variations in M s and strain.  In order to control  effective anisotropy in rare \nearth iron garnets (RIGs), shape anisotropy could be tuned  by doping garnet film s with Ce 53, Tb \n31 and Bi 54 or by micro/nano -patterning . Saturation magnetization could also be increased \nsignificantly by doping, which results in increasing the shape anisotropy in the ma gnetic thin films.  \nAmong the cases predicted to possess PMA , anisotropy fields ranging from 310 Oe (0.31 T) to \n12.6 T have b een calculated . Such a wide anisotropy field range could be spanned and engineered \nthrough strain state, stoichiometry as we ll as substrate choice. For integrated magnonic devices and \ncircuits, garnets with low M s and lower anisotropy field s (HA < 0.5 T)  would require less energy \nfor switching and would be more appropriate due to their lower estimated Gilbert damping.  \nMethods  \nCalculation of anisotropy energy density. We used  Keff = K indu + K shape + K 1 equation to calculate \nthe total anisotropy energy density for each thin film rare earth iron garnet/substrate pair. Each \nanisotropy term consist of the following parameters:  Keff=−3\n2λ111Y\n1−vε||+2πMs2+K1. The \nenergy density is calculated based on the parameters reported in previous  references16,34,36,40,41,43. \nFirst order magnetocrystalline anisotropy, K 1, is an intrinsic, temperature -dependent constant \nreported for each REIG material. Young’s modulus  (Y), poison ratio  (ν) and magnetostriction \nconstant (λ111) parameters evolving in the magnetoelastic anisotropy energy density term  (first \nterm)  are considered to be constant according to the values previous ly reported . For shape \nanisotropy energy calculations (second term), bulk s aturation magnetization (Ms) for each film was \nused. Since each film may exhibit variability in M s with respect to bulk, the model presented here \nyields the most accurate predictions when the actual film Ms, λ 111, Y, ν and K 1, and in -plane strain 24 \n are enter ed for each term. 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Optical Materials \nExpress  7, 2248 -2259 (2017).  \n Acknowledgments  \nM.C.O. acknowledges BAGEP 2017 Award and TUBITAK Grant No. 117F416.  \nCompeting interests  \nThere is no financial and non -financial competing interest among the authors.  \nAuthor contributions  \nM.C.O. designed the study. S.M.Z. performed the calculations and evaluated and analyzed the \nresults with M.C.O. Both authors discussed the results and wrote the manuscript together.  \n \n "    },    {        "title": "1906.03782v1.Giant_anisotropic_magnetoresistance_and_nonvolatile_memory_in_canted_antiferromagnet_Sr2IrO4.pdf",        "content": " \nGiant anisotropic magnetoresis tance and nonvolatile memory in canted antiferromagnet \nSr2IrO 4  \n \nHaowen Wang1, Chengliang Lu1*, Jun Chen2, Yong Liu3, S. L. Yuan1, Sang-Wook Cheong4, Shuai \nDong2†, and Jun-Ming Liu5,6 \n \n1 School of Physics & Wuhan National High Magne tic Field Center, Huazhong University of \nScience and Technology, Wuhan 430074, China \n2 School of Physics, Southeast Un iversity, Nanjing 211189, China \n3 School of Physics and Technology, Wuhan University, Wuhan 430072, China \n4 Rutgers Center for Emergent Materials and Department of Physics and Astronomy, Rutgers \nUniversity, Piscataway, New Jersey, 08854, USA.  \n5 Laboratory of Solid State Microstructures and Innov ation Center of Advanced Microstructures, \nNanjing University, Nanjing 210093, China \n6 Institute for Advanced Materials, Hube i Normal University, Huangshi 435001, China \n \n  \n\n* cllu@hust.edu.cn  \n† sdong@seu.edu.cn   Abstract \n Antiferromagnets have been generating intense interest in the spintronics community, owing \nto their intrinsic appealing properties like zero  stray field and ultrafast spin dynamics. While \nthe control of antiferromagnetic (AFM) orders ha s been realized by vario us means, applicably \nappreciated functionalities on the readout side of AFM-based devices are urgently desired. \nHere, we report the remarkably enhanced anis otropic magnetoresistance (AMR) as giant as ~ \n160% in a simple resistor structure made of AFM Sr\n2IrO 4 without auxiliary reference layer. \nThe underlying mechanism for the giant AMR is an  indispensable combination of atomic scale \ngiant-MR-like effect an d magnetocrystalline anisotropy energy, which was not accessed earlier. \nFurthermore, we demonstrate th e bistable nonvolatile memory states that can be switched \nin-situ without the inconvenient heat-assisted pr ocedure, and robustly preserved even at zero \nmagnetic field, due to the modified in terlayer coupling by 1% Ga-doping in Sr 2IrO 4. These \nfindings represent a straightforward step toward the AFM spintronic devices.  \n \nKeywords:  antiferromagnetic spintroni cs, anisotropic magnetoresistance, memory effect \n  Introduction \nAntiferromagnets represent the overwhelming majo rity of magnetic materials in nature, and \nexhibit fascinating physical properties, such as diverse spin textures a nd topologically protected \nstates of matter 1, 2, 3, 4. Although the antiferromagnetic (AFM) orde r serves as the second basic type \nof magnetic order, and has been discovered for nearly a century si nce the 1930s, it ha s so far been \nused as passive elements, such as pinning of th e ferromagnetic (FM) layer in modern spintronics \ntechnology 5. The main obstacle that keeps the AFM ma terials away from even more extensive \napplications is the great challenge in detection and manipulation of AFM orders because of their \nimperviousness to external magnetic field. Recently, this perception has been largely modified, and \nthe emerging concept of AFM-spintronics has been garnering considerable interest 6, 7, 8, 9.  \nThe AFM-spintronics, where magne to-transport is governed by an  antiferromagnet instead of a \nferromagnet, opens perspectives for both fundame ntal research and device technology, since the \nAFM orders offer some unique and irreplaceable a dvantages compared to ferromagnets. One of the \nmajor appealing features is the promise of ultr afast writing speed (on picosecond time scale), \nenabled by the THz scale AFM resonance which is  three orders of magn itude higher than the \nferromagnets with GHz resonance frequency 10, 11, 12. The control of AFM orders, which was \nthought to be precisely difficult, has been expe rimentally realized by va rious means, including \nelectrical switching 1, 13, 14, 15, 16, optical excitation 11, and heat-assisted magnetic recording 17, 18. On \nthe detection side, anisotropic magnetoresist ance (AMR), which is the magneto-transport \ncounterpart of magnetic anisotropy energy, has been  generally and successfully  utilized, in analogue \nto the traditional FM-based spintronics 7, 18, 19, 20, 21, 22, 23, 24, 25. However, the readout  signal, i.e. the \nAFM-based AMR (AFM-AMR), is commonly found to be  small at a level of ~1% (or even smaller) \n7, 22. This is incompatible with the scalability of the readout time a nd cell size in m odern spintronics \ndevices, and conceptual breakthroug h on the readout side of AFM spin tronics is required to obtain \nsignificantly larger signals.   \nMagnetocrystalline anisotropy energy (MAE) de termined by spin-orbit coupling (SOC) has \nbeen utilized as a generic princi ple to realize the AFM-AMR in prev ious studies. For instance, when \nthe AFM spin axis is aligned along different dir ections with respect to the crystal axes, various \nresistive states are expected because of th e induced anisotropy of electronic structure 19, 23, 24. \nAlthough calculations predicted large density of st ates (DOS) anisotropy in antiferromagnets with strong SOC, experimentally observed AMR ratio is unfortunate ly small overall 19, 20, 26. The \nsubstantial gap between theories and experiments based on MAE sole ly indicates that additional \ningredients should be involved to  enhance the AFM-AMR. Regard ing this, AFM junction may be \nan alternative, and indeed giant AM R exceeding 100% at low temperature ( T) was reported in an \nAFM IrMn-based tunnel junction 25. However, spintronics effects in AFM heterostructures are \nstrongly sensitive to disorder and perfect epitaxy, which hindered their experimental realization  22, 27, \n28.  \nRecently, a number of exotic phenomena , such as spin polarized current 29 and anomalous Hall \neffect 30, 31 which were assumed to only happen in ferromagnets, were revealed in bulk AFM \nmaterials. This provides an exce llent opportunity to combine the typi cal spintronics phenomena, i.e. \ngiant or tunneling magnetoresistance (GMR or  TMR) with AFM-AMR without expense of \nstructural complexity. Note that AMR is also  a bulk effect. Such a scheme may harvest two \nadvantages. First, significantly enhanced AFM-AMR is expected because the typical spintronics effects are usually remarkable. Second, the synergy of different types of magneto-transports may \nlead to unconventional AF M-based functionalities.  \nIn the present work, we demonstrate the giant AMR reaching ~160% at low- T in a simple \nresistor plate of spin-orbit coupled AFM Sr\n2IrO 4 single crystal. Furtherm ore, by tiny Ga-substitution \nat Ir-site, nonvolatile resistive me mory can be achieved, and the bist able states can be manipulated \nin-situ simply by varying the magnetic field ( H) direction. Collaborative ac tion of an atomic scale \nGMR-like effect and MAE is responsible for these super AFM-spintronics functionalities in this \nJeff=1/2 antiferromagnet. In the first part of the ar ticle, we illustrate the physical descriptions of \nanisotropic magneto-transport in Sr 2IrO 4. In the second part, we present remarkably enhanced AMR \nand its reversal behavior. In the third part, we  demonstrate the AFM nonvolatile memory effect in \nthe tiny Ga-doped Sr 2IrO 4.  \n \nResults \nFundamental aspects of the Jeff=1/2 antiferromagnet Sr 2IrO 4. We first briefly recall some \nfundamentals of Sr 2IrO 4. The Jeff=1/2 magnetic moments, entangling both spin and orbital momenta \ndue to the strong SOC (~ 0.5 eV) at Ir site, deve lop a canted AFM order with uniform in-plane \ncanting angle ϕ ~ 12o below TN ~ 240 K 32, 33, 34. Neutron scattering demonstrated that the AFM state is in a twinned manner with equal domain populations 32, 33. Because of the canting, net magnetic \nmoments appear alternatively in each IrO 2 planes, and are coupled an tiferromagnetically along the \nc-axis. Therefore, Sr 2IrO 4 is fully compensated at the ground state without macroscopic \nmagnetization ( M). A magnetic field H larger than the flop critical field Hflop drives the Jeff=1/2 \nmoments flop transition into a weak FM phase in which the net moments of the IrO 2 layers are \nferromagnetically aligned, accompanied concurre ntly with a sharp drop in resistance ( R) at Hflop 23, \n35, 36, 37, 38. As revealed by microscopic investigations in Sr 2IrO 4, the basal-plane first \nnearest-neighboring exchange constant is about 60 meV  39, and the exchange interaction along the \nc-axis is only about 1 μeV 40. Such large anisotropic exchange interaction in Sr 2IrO 4 makes the flop \ntransition possible. The AFM to weak-FM transition in Sr 2IrO 4 resembles the GMR effect in \ntraditional FM-based spin valves, and we call it the GMR-like effect. More details of the GMR-like effect can be found in the Supplementary informa tion (see Supplementary Fig. 1). A similar effect \nwas first revealed in AFM La\n2CuO 4 41, and then identified in  double layered manganites  42. This is \nan atomic scale effect and appears in a Jeff = 1/2 AFM state, which is unusual. While the previous \nstudies of magnetotransport in Sr 2IrO 4 have illustrated some interesting features  19, 23, 24, 36, 37, 38, the \nAMR phenomenon related to the Jeff=1/2 AFM lattice remains elusive. One possible reason is \nrelated to the crystal quality, as revealed by Kim et al.  43. For instance, evident in-plane magnetic \nanisotropy was identified ve ry recently when the Sr 2IrO 4 crystal quality was improved  35, 43, 44.  \n \nAbnormal anisotropic magneto-transport.  To check the physics highli ghted above, we measured \nthe MR = [ R(H) / R(0) - 1] data over a broad H (up to 5 T) and temperature T (35 K < T < 240 K) \nrange, with the measuring geometry shown in  the inset of Fig. 1. Here, electric current I is always \napplied along the [001] direction (the c-axis), and H is rotated within the ba sal plane with an angle \nΦ relative to the [100] direction. The magnetic easy axis is [100] with Φ = 0o and the in-plane hard \naxis is [110] with Φ = 45o. Crystal quality is an important  ingredient for observing evident \nanisotropy in Sr 2IrO 4 43. Therefore, X-ray diffraction (XRD) and energy dispersive spectroscopy \n(EDS) measurements were performed, which suggest  that the samples are pure phase and high \nquality (see Spplementary Figs 2 and 3). It is seen that below TN, the MR( H) curves at different Φ \ninter-cross at several H, as shown in Fig. 1a for Φ = 0o and 45o at three temperatures (For more MR \ndata see Supplementary Fig. 4). A summarized phase  diagram is shown in Fig. 1b which can be divided into three regions I, II, and III, respectively.  \nThis phase diagram reflects the abnormal anisotropic magneto-transport behavior in Sr 2IrO 4. In \nfact, both Hc1 and Hc2 are intimately related to Hflop where the MR drop emerges. This sharp MR \ndrop is due to the atomic scale GM R-like effect associated with the Jeff=1/2 moments flop-transition. \nIt is seen that the deri ved critical fields at Φ = 45o are approximately √2 times larger than those at Φ \n= 0o (see Supplementary Fig. 5), following a geometri c relationship simply. This can be directly \nunderstood by considering the MAE. For instance, when H is applied along the magnetic easy axis \n(Φ = 0o), relatively low Zeeman energy gain is require d to overcome the MAE, and thus to trigger \nthe Jeff=1/2 moments flop transition and thereby the atomic-scale GMR-like effect. This is \nconfirmed by the M(H) data at various T, as shown in Fig. 2. The Jeff=1/2 moments flop transition is \naccompanied by a drastic enhancement in M in the low field range for both Φ = 0o and Φ = 45o, \nmarking the conversion from the fully compensated st ate to the weak FM phase . In particular, it is \ntrue to see that the transition deve lops with a smaller critical field Hflop at Φ = 0o (easy axis) than at \nΦ = 45o (hard axis), in consistent with the MR  data. An advantage is a very small Hflop ~ 0.2 T, \nfavorable for practical AFM devices with Sr 2IrO 4.  \nFurthermore, it is seen that M at Φ=0o is overall larger than that at Φ=45o up to H=5 T (the \nmaximum field used for M(H) measurements in the present wo rk). This can be seen up to TN, \ndemonstrating the unchanged magnetic easy axis after the flop transition 43, 45. By plotting ΔM \n(=M(Φ=0o) - M(Φ=45o)) as a function of H (the insets of Fig. 2), the  M-discrepancy can be seen \nmore clearly, and the maximal ΔM is found at H~Hflop. However, at H>Hc2 shown in Fig. 1a, the \nrelatively more conductive be havior is identified at Φ=45o, i.e. H // [110] which is the in-plane hard \naxis. Similar phenomenon was also identified in Sr 2IrO 4 thin films previously, and bandgap \nengineering due to rotation of the Ir moments was revealed to be re sponsible for this through first \nprinciples calculations 24. For instance, the bandgap of Sr 2IrO 4 would be reduced when rotating Ir \nmoments from the [100] direction (easy axis) to the [110] direction (hard axis) 19, 24. This is \nphysically anticipated considering the essential ro le of the strong SOC in build up the electronic \nstructure in Sr 2IrO 4. In Fig. 1a, a clear slop-change can be seen at Hc1<H<Hc2 in the MR curves at \nΦ=45o, after which the [110] direction is getting to be relatively more conductive than the [100] \ndirection. This is indicative of the onset of  the reduced bandgap dominating the transport. \nComplementary magneto-transport data measured under pulsed high magnetic field up to 50 T are shown in the Supplementary Fig. 6.  It is seen that the unusual an isotropic magneto-transport due to \nbandgap engineering can persist up to very high field.   \nRegarding the MR bumps at H<Hflop, domain wall scattering may be qualitatively relevant \nbased on three considerations. First, in region-I with H<Hflop, H is insufficient to trigger the flop \ntransition, but may tilt the magnetic moment s evidenced as gradual increase in M. However, the \npositive value of the finite MR in region-I excludes a possible origin arising from H-suppressed \nmagnetic scattering effect, but indicates an enhancem ent of scattering effect in the system. Second, \nintensive domain wall motion has been demonstrated as H<Hflop in SIO  43. For instance, with \nincreasing H//[100], the domain with Jeff=1/2 moments along [100] gr ows, but the domain with \nJeff=1/2 moments along [010] shrinks. The domain repopulation involves complex domain wall \nmotions reflected as deviations from linear behavior and hysteresis in the magnetization \nmeasurements, as shown in Suppl ementary Fig. 7. Meanwhile, evid ent MR-bumps with hysteresis \nis seen at H<Hflop, coinciding with the hysteresis effect in M(H). This close relationship among the \nMR( H), M(H), and domain wall motion leads us to pr opose a domain wall scattering scenario \nresponsible for the observed  MR-bumps. However, with H//[110], the domains remain populated as \nH<Hflop, since the field has no pr eference to the domains  43. As shown in Fig. 1 and Supplementary \nFig. 4, with H//[110], the MR-bumps are largely supp ressed indeed, and instead continuous \ndecrease can be seen in the MR curves. Simultaneously, the hysteresis in M(H) curves are also \nsuppressed. These phenomena are consistent with  the proposed domain wall  scattering scenario. \nThe remained weak MR-bumps may due to slight misalignment of H from the [110] direction, since \nany tilt of H leads to an imbalance in  the domain population. Third, significant domain wall \nresistance has been theoretically revealed in SIO very recently by Lee et al. 37, further supporting \nthe proposed domain wall scattering mechanism re sponsible for the MR-bum ps in region-I. With \nincreasing T, the MR bumps and its hysteresis are evidently suppressed at T>50 K, indicating that \nH-suppressed magnetic scattering gets to dominate the magnetotransport. Surely, further \ncomprehensive investigations are desired to clarify the low-field MR in region-I.      \n \nGiant anisotropic magnetores istance and its reversal. Having established the physical \ndescriptions of the abnormal an isotropic magneto-transport in Sr 2IrO 4, we now are in a position to \ntrack the AMR effects in our simple AFM resistor. In Fig. 3, we plot R as a function of Φ measured at T=35 K and T=90 K. Complementary AMR data can be seen in Supplementary Figs 8 and 9.  Two \nfeatures can be seen clearly, such as the H-driven AMR reversal and large R-variation at \nintermediate H region.  \nFig. 3b and d show estimated AMR ratio (AMR= R(Φ)/Rmin-1) as a function of H obtained at \nT=35 K and 90 K, respectively. Here, Rmin means the minimal R in the AMR traces. The AMR( Φ) \ncurves exhibit pronounced peak around H~0.2 T. This is even remarkable at T=35 K, and the \nmaximum AMR ratio is found to be as large as 160%  which is the largest value compared to those \nreported so far in AFM material s. AMR phenomenon was studied usi ng different experimental setup \nin Sr 2IrO 4 previously, and a detailed comparison with the present work can be seen in the \nSupplementary information (see Supplementary Fig. 10). As compared to the phase diagram shown \nin Fig. 1b, we can see that the striking AMR enhancement develops  right within the region II \n(shadowed with olive). Therefore, the observed gian t AMR effect can be ascribed to a combination \nof the atomic scale GMR-like effect and the MA E energy which dominate the magneto-transport \ntogether in this intermediate field region ( Hc1<H<Hc2). In addition, inhomoge neities (i.e. domains \nand defects) within the sample resulting in regions with slightly different tr ansition fields might also \ncontribute a little to the AMR.    \nHere we note that the AMR symmetry in  region II is more like twofold at both T=35 K and 90 \nK, although a fourfold rotation symmetry is in princi ple anticipated because of the square lattice of \nSr2IrO 4. As revealed by several recent theoreti cal works, the in-plane anisotropy in Sr 2IrO 4 can be \naccurately described by a magnetic-lattice coupling model 43, 45. Importantly, it was found that the \nmagnetic-lattice coupling effectivel y acts like a uniaxial anisotropy , once the magnetic moments are \naligned along a certain direction, such as the Jeff=1/2 moments flopping. Therefore, it is physically \nreasonable to expect a tw ofold AMR symmetry at H~Hflop, such as the present case, according to \nthe proposed magnetic-lattice coupling model. In addition, weak R-tips can still be seen in the basin \nof the AMR trace, indicating existenc e of retained biax ial anisotropy.  \nHaving understood the drastic AMR enhancement in  region II which is a key finding in the \npresent work, we now turn to look at the AMR effect  happening in Region I a nd III. Within region I, \nthe magneto-transport is dominated by domain wa ll scattering as discussed above. For instance, \nclear hysteresis can be seen in the AMR traces within region-I, wh ich may also duo to the domain \nwall motion akin to the MR-bumps. As revealed by Porras et al. , only the cases with H//[110] are not expected to cause domain repopulation 43. Therefore, the AMR hystere sis appears at both easy- \nand hard-axes.  In Fig. 3a, the induced AMR symmetry at H=0.1 T is twofold with peaks at Φ~45o \nand ~225o. First, the application of H=0.1 T is insufficient to tr igger the flop transition at T=35 K. \nSecond, the domain wall motion becomes more impetuous when applying H along the magnetic \neasy axes (i.e. at Φ=0o) than along the hard axes (i.e. at Φ~45o). Therefore, rotating H =0.1 T from \nhard- to easy-axis would signifi cantly promote the domain wall moti on, giving rise to the increase \nin R. If taking a closer look at the AMR trace, step-l ike anomalies with hysteresis can be identified \n(indicated by arrows), marking the booming of  domain wall motion. Such domain wall motion \nprocess will be conti nued as further rotating H from Φ=0o to 45o, since only the cases of H along \nthe hard axes do not change the domain populati on. This gives rise to the ongoing increase in R till \nΦ=45o. From Φ=45o to 135o, the  H direction is partly reversed, and thus the domain walls are \ngradually removed. As a consequence, R decreases with Φ from 45o to 135o. This process will be \nrepeated when subsequently rotating H=0.1 T from Φ=135o to 315o, leading to the twofold AMR \nsymmetry. In addition, such tunneling AMR is not observable at T=90 K (Fig. 3c), since the thermal \nfluctuation is sufficiently high to overcome the en ergy barrier at domain walls for electron transport. \nThis is consistent with the phase diagram shown in Fig. 1b.  \nRegarding the fourfo ld AMR symmetry at H=3 T, it can be ascribed to the bandgap engineering \nakin to the analogous phenomenon in Sr 2IrO 4 thin films 19, 24. Because H=3 T is much larger than H \nflop, the weak FM phase is well stabilized, and its net magnetic moment and the corresponding \ncanted AFM orders can (mostly) follow H rotating within the basal plane of Sr 2IrO 4. For instance, \nthe Ir moments can factually reach both the [100] direction and the [110] direction when rotating H \nwithin the ab-plane. In this sense, the anisotropi c magneto-transport dominated by bandgap \nengineering emerges, for example it is relativel y more conductive as Ir moments pointing along the \n[110] direction (with smaller ba ndgap) compared with the [100] direction (with larger bandgap) 19, \n24. As a consequence, a fourfold AMR symmetry with  minima appearing right at the in-plane hard \naxes is obtained. More details of giant AM R phenomenon and its reve rsal can be found in \nSupplementary Fig. 11.  \n  \nNonvolatile memory in tiny Ga-doped Sr 2IrO 4. The above data have shown a highly efficient \nscheme, which is the combination of various types of magneto-transports dominated by either spin polarized current or MAE energy, to enhance the AFM-AMR effect in Jeff=1/2 Sr 2IrO 4. Along this \nline, a natural proposal is if the various magneto-transport  behaviors can be preserved after \nremoving H. By this, we should be able to observe  the interesting AFM-based spintronics \nfunctionalities at H=0 T, i.e. the AFM memory effect. A key step is to retain the flopped magnetic \nstate from recovery after the H-writing. It is found that the non-doped Sr 2IrO 4 is not proper to \nrealize this purpose, since the pristine state can be recovered after different H sweeping (Fig. 1a). \nWe then dope Sr 2IrO 4 with 1% Ga at Ir-site, which was found to be efficient in triggering the flop \ntransition without breaking the Jeff=1/2 state akin to the effect of applying external H 46. As shown \nin Fig. 4a, it is true to see evident remnant magnetization Mr in the M(H) curves taken at T=10 K. \nImportantly, Mr at Φ=45o is found to be much larger than that at Φ=0o. The reason includes that first \nthe larger Mr means the stronger suppres sion of magnetic scatteri ng. Second, more Ir moments \nremaining at the [110] direction would lead to relatively smalle r bandgap in the resistor. Both \neffects are all pointing to a higher conduction along the [110] direction at H=0 T, as compared with \nthe [100] direction in the tiny Ga-doped Sr 2IrO 4.  \nThis is confirmed by further tran sport characterizations. We first zero field cooled the resistor \nto T=10 K, and then measured th e MR by performing successive H-cycles either at Φ=0o or Φ=45o. \nAs shown in Fig. 4b, after the first H-cycle, two reversible resis tive states can be obtained by \nfurther H writing procedures, demonstra ting an AFM memory effect. Th e two memory states can be \nswitched by in-situ H-writing without heat-assisted procedure,  and the resistive ratio of the two \nstates is as large as ~4.5%. This is a major point  deserved for high appreciation and it is different \nfrom the operation utilized in AFM MnTe 17 and FeRh 18. The drift effect seen in the resistance over \ntime (Fig. 4c) may due to inhomogeneities (such as domains) within the sample. Full R(H) data can \nbe seen in the Supplementary Fig. 12. The memory effect is nonvol atile and fully reproducible in \nthe successive write-read cycles, as shown in Fig. 4c. The same protocol was repeated several times \nwith H applied initially either at Φ=45o or Φ=0o, and the same results were  obtained. In our devices, \nthe memory states are stable against a field up to  ~1 kOe. Although this is smaller than that in \ncompensated AFM materials showing very high rigidity, it is robust against electromagnetic \ndisturbances, and the relatively small switching fi eld would facilitate th e operation in devices.       \nHaving demonstrated the AF M-based nonvolatile memory in the tiny Ga-doped Sr 2IrO 4, we \nthen go to some details of the physical properties of the resist or. First, both the high and low resistances ( RH and RL) are smaller than the pristine state. Th is can be understood  directly by the \ndifferent Mr for these cases. For instance, the initial state after zero fiel d cooling has zero net \nmagnetization 46. Second, M at Φ=45o is larger than at Φ=0o, different from the case in the \nnon-doped Sr 2IrO 4. This should be ascr ibed to the modification of in terlayer couplin g by Ga-doping. \nAs revealed by recent calculations, the interlayer  coupling can effectively change the anisotropic \nmagnetization in Sr 2IrO 4 43, 45, 47. If considering bare effect of interlayer coupling in Sr 2IrO 4, the \n[100] direction is easier than th e [110] direction as AFM interaction dominates. However, this \nsituation is reversed, i.e. the [ 100] direction behaves even harder than the [110] di rection, when the \ninterlayer coupling becomes FM-like (which can be realized by the flop tran sition). The significant \nrole of interlayer coupling in de termining anisotropic magnetism l ooks to be genera l in iridates 48. \nAs shown in Fig. 4a, the clear M(H) hysteresis evidences the app earance of FM coupling of the \nIrO 2 layers, in analogous to the Rh-doping effect 49. Therefore, it is physical ly reasonable to expect \ndifferent anisotropic magnetization in the tiny Ga-doped Sr 2IrO 4 compared to the non-doped one.  \n \nDiscussion \nWe have demonstrated giant AMR and nonvolatile memory in simple AFM resistors made of \nJeff=1/2 antiferromagnets without aux iliary reference layer. Note th at coupling to FM layer would \nlose the unique merits of antife rromagnets in AFM spintronics devi ces. The AMR ratio  is found to \nreach ~160% at T=35 K in the present work, which is much larger than previously available values \n17, 18, 19, 20, 23, 37. It is noted that very larg e tunneling AMR exceeding 100% at T=4.2 K was identified \nin AFM IrMn-based tunnel junc tions, while additional FM NiFe  layer was used for inducing \nexchange spring effect 25, different from the present case. Regarding the AFM-based nonvolatile \nmemory, it has been rarely reported before. In the previous works, to obtained different memory \nstates, the sample has to be heated above TN, and then cooled below TN with magnetic field 17, 18. \nHere we show an in-situ manipula tion of the bistable states, which may have advantages in practical \ndevices.   Our work has shown an efficient mean to achieve significantly large AMR and nonvolatile \nmemory in the J\neff=1/2 antiferromagnets. The crucial part of our scheme is to combine various types \nof magneto-transport behaviors, i. e. the atomic scale GMR-like effect  due to spin polarized current, \nand crystal AMR due to the MAE energy, in a single system. Impor tantly, the scheme proposed in the present work is appli cable to a broad class of AFM material s. As a fundamental  piece of the two \nmajor findings, the spin polarized current was theo retically predicted to generally exist in most \nantiferromagnets except those with simple  collinear spin orders very recently 29. In addition, the \nscheme could also be extended to some recently  discovered AFM alloys where exotic anomalous \nHall effect was identified, and distinctive AFM spintronic functionalities would be expected 30, 31, 50.  \n In summary, significantly enhanced AMR reaching ~160% at T=35 K is revealed in a simple \nresistor made of AFM Sr 2IrO 4 hosting Jeff=1/2 moment due to strong spin-orbit coupling, owing to a \ncombination of an atomic scale GMR-like eff ect and the magnetocrystalline anisotropy. The \nenhanced AMR can be observed at higher- T, although its magnitude decreases gradually with \nincreasing T. Further experiments demonstrated that th e plural magneto-trans port behaviors can be \npreserved by only 1% Ga-doping in Sr 2IrO 4 without the assist of external magnetic field. \nImportantly, in the tiny doped resistor, nonvolatile  AFM memory effect is identified, and the \nbistable memory states can be operated in-s itu. The modification of interlayer coupling by \nGa-doping is essential for realizing the nonvolatile  memory. Our work has illustrated a very \nefficient scheme to significantly improve the sp intronics functionalities in antiferromagnets.  \n \nMethods \nSample preparation.  The Sr 2IrO 4 single crystals (2 ×1×0.5 mm3) were synthesized from \noff-stoichiometric quantities of SrCl 2, Sr 2CO 3, and IrO 2 using the self-flux techniques. The \nthoroughly mixed powders were  placed in a platinum crucible covered with a lid, and then melted at \n1250 °C in a programmable furnace in air. After this, the crucible was cooled to 800 °C at a rate of \n6 °C per hour and then furnace -cooled to room temperatur e. The molar ratio of SrCl 2:IrO 2 was set at \n~8:1, which was found to be critical for obtai ning high quality crystals. For a comparison, 1% \nGa-doped Sr 2IrO 4 single crystals were also synt hesized through the same method. The \nstoichiometry of the resulting crystals was c onfirmed using energy di spersive spectroscopy \nmeasurements. The pure phase of the crystals was examined by performing room-temperature \npowder X-ray diffraction measurements on thoroughl y crashed crystals. The crystals were also \nchecked using Rigaku XtaLAB miniTM diffractometer at room temperature.  \nMagnetization and electric transport measurements. Electric transport measurements for the \ncrystals were carried out using a four-probe method in a Quantum Design (QD) physical property measurement system equipped with a rotator module. Silver paste was used to make the electrodes. \nIn order to have homogeneous electric current  I flowing through the sample, the electrodes of \ncurrent source were made as large as possible (>1 mm2). With regard to the magneto-transport \nmeasurements, exciting current I was applied along the [001] direction, and the magnetic field H \nwas always applied within the basal plane of the crystals. Therefore, the AMR refers to the ( I, H) \nbehaviors. Magnetiza tion as function of T and H were measured using a QD superconducting \nquantum interference device. All the M(H) curves were measured after the zero-field cooled (ZFC) \nsequence. For the high field transport measuremen ts, the same device stru cture was used, and the \npulsed magnetic field up to ~50 T with a dura tion time of ~50 ms was generated by using a \nnondestructive long-pulse magnet energized by a 1 MJ capacitor bank.  \n  Data availability.  \nThe data supporting these findi ngs are available from the corresponding author on request.  \n \nAdditional Information \nSupplementary Information  accompanies this paper at doi:    \n \nCompeting interests: The authors declare no competing interests.  \n \nAcknowledgements:  We thank Prof. J. Liu for fruitful discu ssions and Prof. W. Li for single crystal \ncharacterizations. This work is supported by th e National Nature Science Foundation of China \n(Grant Nos. 11774106, 11674055, 51431006, and 51571152), th e National Key Research Projects \nof China (Grant No. 2016YFA0300101). S.W.C. is funded in part by the Gordon and Betty Moore \nFoundation’s EPiQS Initiative th rough Grant GBMF4413 to the Rutgers Center for Emergent \nMaterials, and also by the visiti ng distinguished professorship of Nanjing University sponsored by \nthe State Administration of Fore ign Experts Affairs of China.   \n \nAuthor contributions:  C.L.L. conceived the project. H.W.W.  prepared the samples and performed \nthe physical property measurements. Y .L. participat ed the measurements. J. C., S.L.Y ., and S.W.C. \ncontributed to the analysis of the results. C. L.L., J.M.L., and S.D. wrote the manuscript.  \n \n  References  \n1. Dong, S., Liu, J-M., Cheong, S-W. & Ren, Z. Multiferroic materials and magnetoelectric \nphysics: symmetry, entanglemen t, excitation, and topology. Adv. Phys.  64, 519-626 (2015). \n2. Wang, K., Graf, D., Lei, H., Tozer, S.W. & Petrovic, C. Quantum transport of \ntwo-dimensional Dirac fermions in SrMnBi 2. Phys. Rev. B  84, 220401 (2011). \n3. Kim,  W. 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Magnetotransport in Sr 2IrO 4 single crystal.  (a) Measured magnetoresistance as a \nfunction of H applied at Φ=0o and Φ=45o at T=35 K, 50 K, and 90 K. The intercross points \nindicated by blue ( Hc1) and orange ( Hc2) dashed lines are used for building up the phase diagram. \nThe H-sweeping directions are indica ted by black arrows. For a better version, the curves have been \nshifted vertically. (b) Phas e diagram of the anisotropic magneto-transport in Sr 2IrO 4 single crystal. \nThe inset: the measurement geometry and two ma jor crystalline directions of the crystal.  \n \nFigure 2.  Magnetic properties of Sr 2IrO 4. Magnetization as a function of H applied at Φ=0o and \nΦ=45o at (a) T=10 K, (b) 90 K, and (c) T=220 K. The inset shows the difference of magnetization \nmeasure at two directions.   \nFigure 3.  Giant antiferromagnetic AMR and its reversal.  Anisotropic magnetoresistance \nmeasured with various H=0.1 T, 0.25 T, and 3 T at (a) T=35 K, and (c) T=90 K. The measurements \nwere performed by sweeping H from -45\no to 315o (red curves), and then back to -45o (black curves). \nDerived AMR ratio as a function of H obtained at (b) T=35 K and (d) T=90 K. The orange lines are \nused to highlight the remarkable AMR enhancement.  Regions I, II, and III are labeled according to \nthe phase diagram shown in figure 1.   \nFigure 4.  Nonvolatile memory and the in-situ control.  (a) Measured M(H) curves at Φ=0\no and \nΦ=45o at T=10 K for 1%-Ga doped Sr 2IrO 4, in which different remnant magnetization can be seen. \n(b) R(H) curves measured by successive H cycling along different directions at T=10 K. For \ninstance, Φ=0o_1 means the first H-cycle at Φ=0o. The field sweeping direction is indicated by \narrows. The blue and red dots represent the high ( RH) and low ( RL) resistance state which can be \nswitched by H-cycling. (c) Stable and reproducible memory states in-situ controlled by H at T= 10 \nK. The corresponding net moment arrangements are schematically show n in the inset. In \ncomparison with the case at Φ=45o, smaller net magnetization exists at Φ=0o as H→0.  \n   \n \nFigure 1 \n  \n \n \n \n \nFigure 2 \n  \n   \n \n \nFigure 3 \n \n  \n \n \n \nFigure 4 \n "    },    {        "title": "1906.07566v1.Increased_perpendicular_magnetocrystalline_anisotropy_governed_by_magnetic_boundary_in_an_asymmetrically_terminated_FeRh_001__thin_film.pdf",        "content": "Increased perpendicular magnetocrystalline anisotropy governed by magnetic  \nboundary in an asymmetrically terminated FeRh(001) thin film  \n \nEunsung Jekal ∗ \nDepartment of Materials, ETH Zurich, 8093 Zurich, Switzerland  \n \nRh-terminated FeRh( 001) film is known to be stable in a ferromagnetic (FM) state \ndifferent from a G -type antiferromagnetic (G -AFM) bulk ground state, while an Fe -\nterminated FeRh(001) film has the same ground state as the bulk. In this paper, we \ninvestigate the magnetic prope rties of asymmetrically terminated FeRh(001) films: one \nsurface is Fe -terminated and the other is Rh -terminated. Rh surface only  ] (RhSO ) FM \nstate in asymmetrically terminated FeRh(001) film is identified  to exhibit 40% \nincreased perpendicular MCA energy as compare to that of the whole -layer  (WL) FM  \nstate. This increased MCA energy is governed by Rh atom which is placed at the  \nmagnetic boundary. Since FM and G -AFM states are mixed up at the magnetic  \nboundary, <x2 −y2↑|Lz| xy ↑>, <yz,zx ↑|Lx|x2 −y2↓> and <yz,zx ↑|Lx|z2 ↓> couplings \nwhich  give positive contribution to perpendicular MCA are revealed.  \n \nI.  INTRODUCTION  \nAntiferromangetic  (AFM) materials are insensitive to disturbing magnetic fields due \nto their zero net magnetic moment. This is crucial property to store the information  densely and keep it safely in the memory device such as spintronics [1–3].  In addition, \nthe intrinsic h igh frequencies of AFM dynamics is another benefit of AFM materials \nover ferromagnetic (FM) ones  [4,5].  However, a difficulty in controlling their spin \ndirection is still remained as a major obstacle of AFM material. To overcome this \nproblem, a relativist ic spin -orbit coupling -induced torque  has been suggested.  Another  \nsuggestion  to manipulating the antiferromagnetic spin has been demonstrated in FeRh, \nwhereby heating and cooling the material can control spin orderings. Moreover, since \nFeRh has AFM  spin st ate and an antiferromagnetic anisotropic magnetoresistance \n(AMR) simultaneously, it could be used as  a valuable memory device [6–8]. \nB2 structured FeRh exhibits AFM at room temperature and undergoes  the magnetic \ntransition to the FM  with volume expansion i n just above room temperature  350 K \n[7]. Since this magnetic phase change with respect  to the temperature was  \ndiscovered over 70 years ago,  much theor etical and experimental works have been \ndone  in FeRh. Recently, first principle study clarified that bulk  and the Fe -terminated \nFeRh(001) films are ordered as G -type AFM (G -AFM), however, the Rh -terminated \nfilms are energetically more stable in FM up to 17 monolayers (ML) [7].  Due to the \ndifferent magnetic states which  strongly depend on surface termination of the film, an  \nasymmetrically  terminated film that one surface is Fe  and the other surface is Rh \nterminated has a complicated  spin configuration. In films thicker than 8 -ML, layers \nare magnetically distinguished, whereby 3 -ML from Rh surface a nd other rest layers \nare ordered as FM and G -AFM,  respectively. This mixed magnetic state has been \ndefined  as Rh surface only (RhSO) FM state in previous work.  \nIn this paper, MCA energy of asymmetrically terminated FeRh(001) film is \ninvestigated. In particular, w e found that magnetic boundary placed in the asymmetrically terminated film gives increased perpendicular MC A energy as \ncompare to the identical film wit hout magnetic  boundary. It can satisfy high bit density \nand thermal stability, simultaneously, in magnetic data storage.  \n \nI I.  COMPUTATIONAL METHOD  \nVienna ab initio Simulation Package (VASP)  [9] is used  to perform first -principles \ncalculations.  Generalize d gradient approximation  [10] is employed  for the exchange  \ncorrelation potential. To optimize the structure of the  film, internal coordination is fully \nrelaxed while the two  dimensional  lattice constant was fixed as 2.998  A which  is \ncorresponding to G -AFM bulk value.  MCA energy is  determined by  total energy \ndifference between in -plane  and perpendicular magnetizations, E MCA≡E(→)-E(↑). \nTo obtain  reliable MCA energy, 24x24x1 is used for  k-mesh in MCA energy \ncalculations while reduced value  of 17x17x1 is taken in self -consistent calculations.   \nFor wave function expansion, a plane -wave basis set with a  cutoff energy of 450 eV \nwas used. To describe the G -AFM spin ordering in the film, we adop t an expanded  unit \ncell of c(2x2) along the  xy plane. Film thickness is  considered from 4 - to 14 -ML. Unlike \nthe film calculation,  2x2x2 supercell is applied to obtain MCA energy in the  bulk. Since \nMCA value is very sensitive to  k-point density, completely symmetric unit cell has been \nnecessary  for calculation accuracy.  \n \nIII.  RESULTS AND DISCUSSION  \nThe spin configuration of an  asymmetrically terminated FeRh(001) thin film is depicted \nin Fig.1(a).   \n \nFig. 1. (a) spin configuration of 8 -ML asymmetrically terminated FeRh(001) film.  \nPurple (gray) circle corresponds to Rh (Fe) atom. Red (blue) arrow represents up \n(down) spin. (b) MCA energy as function of film thickness. Red solid and black dotted \nlines e xpress the values of RhSO FM and WL FM states, respectively  \n \n RhSO FM  WL FM   WK G -AFM  \nRh(S)  1.143  1.103  0.000  \nFe(S Rh-1) 3,139(3.142)  3.162  ±3.150  \nRh(S Rh-2) 0.870  1.048.  0.000  \nFe(S Rh-3) 3.119( -3.132)  3.127  ±3.076  \nRh(S Fe-3) 0.074  0.923  0.000  \nFe(S Fe-2) 3.064( -3.075)  3.164  ±3.113  \nRh(S Fe-1) 0.001  0,885  0.000  \nFe(S)  3.139( -3.143)  3.162  ±3.113  \n \nPurple  and gray circles correspond to Rh and Fe atoms, respectively. To  identify the \nmost stable magnetic structure  with respect to the film thickness, different numbe r of \nFM layers from Rh surface is taken into account while  other rest layers are fixed as G -\nAFM [7].  \nIt was clarified that 4 - and 6 -ML films are stable when  whole layers (WL) are ordered \nas FM, however, the RhSO  FM phase as shown in Fig. 1(a) turns out to be the  most \nenergetically favorable in thicker than 8 -ML films.  One crucial point is number of FM \nlayers independents  on film thickness except 4 - and 6 -ML films. The reason  of certain \nFM layers  (3-ML from Rh surface) of RhSO  FM st ate has been reasonably explained \nby magnitude  of induced magnetic moment of Rh atoms . Magnetic moments of 8 -ML \nfilm in RhSO FM, WL  FM and WL G -AFM states are listed in Table1, where  surface \nand n layers below Rh (Fe) surface are denoted  by (S) and (S Rh(Fe )-n), respectively.  \nSince there are two distinct Fe atoms due to complicated spin interaction at the \nmagnetic boundary in RhSO  FM, we denote two different Fe moments. Rh(S) in \nRhSOFM  state has larger magnetic moment by 0.04μ B than that of WL FM state while \nmoment of Rh(S Rh-2) is much  reduced in RhSO FM with respect to the WL FM. This  \ntendency that surface atom of RhSO FM state has large  moment but decrease more \nsharply as compare to W L FM case when n becomes 2 also appears in G -AFM ordered \nFe-terminated part. Fe(S) has 0.03μ B larger moment in RhSO FM state than WL G -\nAFM, but in case  of Fe(S Fe-2), RhSO FM shows smaller value by 0.10μ B than WL G -\nAFM.  Interestingly, in all cases, Fe(S Fe-1) shows comparable magnetic moments to Fe(S).  \nMCA energies of WL FM and RhSO FM states are  plotted in Fig.1(b) as function of \nthe film thickness. Dotted and solid lines represent WL FM and RhSO FM, respectively. \nFrom  the definition of MCA energy in this  paper, positive (negative) sign corresponds \nto perpendicular (in -plane) MCA. Even though perpendicular MCA  \nenergy is obtained in both magnetic states, RhSO FM has  averagely 40% strong MCA \nenergy than WL FM ones.  \nWhile the saturation behavior is clearly show n as the  film is getting thicker, the  \nsaturated values of WL FM  and RhSO FM are 0.70 and 1.25 meV/cell, respectively.  \nTo elucidate the origin of the increased perpendicular  MCA of RhSO FM state with \nrespect to the FM state,  DOS of WL FM and RhSO FM are co mpared in Fig. 2   \n \nThe FM and the G -AFM regions are separated by using black and gray blankets.  To \nemphasize and simplify  crucial parts, only Rh(S), Rh(S Rh-2) and Fe(S Rh-3) are  \npresented while other layers are expressed as dots(···) . Because it has been already \nclarified that most positive  contribution to perpendicular MCA is originated by FM  \nordered Rh(S), and contributions from centered layer  are negligible.  \nFor the MCA analysis through DOS, second perturbation theory is adapted [11].  MCA \nenergy is described as  \n \nE𝑀𝐶𝐴↑↑(↓↓)≈ξ2∑|<𝑜↑(↓)|𝐿𝑧|𝑢↑(↓)>|2−|<𝑜↑(↓)|𝐿𝑥|𝑢↑(↓)>|\n𝜀𝑜↑(↓)−𝜀𝑢↑(↓)𝑜↑(↓),𝑢↑(↓)  \n \nE𝑀𝐶𝐴↑↓≈ξ2∑|<𝑜↑(↓)|𝐿𝑥|𝑢↑(↓)>|2−|<𝑜↑(↓)|𝐿𝑧|𝑢↑(↓)>|2\n𝜀𝑜↑(↓)−𝜀𝑢↑(↓)𝑜↑(↓),𝑢↑(↓) , \nwhere ξ is the spin -orbit coupling constant and ↑(↓) represents majority  (minority) spin.  \no(u) and  𝜀𝑜(𝑢) represent  eigenstate  and eigenvalue of occupied(unoccupied)  \nstate, respectively.  \nDOSs of Rh(S) in two magnetic phases look so similar and there is strong  \n<yz,zx ↑|Lx|z2↓> coupling which gives positive contribution to perpendicular  MCA. On \nthe other hands, the DOS shape of magnetic  boundary  Rh is very different  between \nFM and RhSO  FM, in particular near Fermi level. In case of the magnetic boundary \nRh with RhSO FM spin configuration,  positive contributio ns from  \n<x2−y2↑|Lz|xy↑>, <yz,zx ↑|Lx|x2−y2↓> and < yz,zx ↑|Lx|z2↓> couplings are much \ndominant even though negative contribution from  < yz,zx ↑|Lx|xy↑> coupling somewhat  \nreduces perpendicular MCA. Therefore, a net contribution is positive. However, in case \nof Rh(S Rh-2) in WL  FM state which is corresponding to magnetic boundary  Rh of RhSO \nFM state, there is strong negative contribution,  \n<xy↑|Lz|x2−y2↓>. This coupling is invisible  in DOS of magnetic boundary Rh of RhSO \nFM state.  \nAs a result of DOS analysis, we can argue that increased  perpendicular MCA of \nRhSO FM state is originated by  magnetic boundary Rh atom while the strong perpendicular MCA in both magneti c states are mainly determined  by FM ordered \nRh(S) [15].  \nFor a double check, MCA energies are investigated in  bulk with different spin \nconfigurations. To design a magnetic boundary in the bulk, one layer and another \nalternative layer are fixed as G -AFM and  FM orderings,  respectively. We named this \nartificial magnetic state as  FM+G -AFM.  \n \nAs shown in Fig.  3(a), FM and G -AFM states have  zero MCA energy due to a cubic \nsymmetry.  Even though  A-AFM shows perpendicular MCA energy about 0.06  \nmeV/atom, FM+G -AFM has much stronger MCA energy by 0.10 meV/atom than that \nof A-AFM. DOS of  Rh atom of FM+G -AFM state is presented in Fig. 3(b).  \nSimilar with magnetic boundary Rh in the film (see Fig.  2), strong  <x2−y2↑|Lz|xy↑> is \nfound even in the  bulk.  Although a peak of  yz,zx ↑ orbital which makes  \n<yz,zx ↑|Lx|x2−y2↓> and <yz,zx ↑|Lx|z2↓> couplings near Fermi level is much reduced in \nthe bulk  comparison with film case, it still contributes to the perpendicular MCA energy.  \nTherefore, we can confirm the  increased perpendicular MCA energy of RhSO FM state  \nis originated by somehow mixed magnetic interactions at  the magnetic boundary.  \n \nIV.  CONCLUSION  \nIn summary, RhSO FM state in asymmetrically terminated FeRh(001) film is identified \nto exhibit 40% increased perpendicular MCA energy as compare to that  of the WL FM \nstate. This increased MCA energy is  governed by Rh atom which is placed at the \nmagnetic  boundary. Since  FM and  G-AFM states are mixed up  at the magnetic \nboundary,  <x2−y2↑|Lz| xy↑>, <yz,zx ↑|Lx|x2−y2↓> and <yz,zx ↑|Lx|z2↓> couplings which \ngive positive contribution to perpendicular MCA are revealed.  \n \nV.  REFERENCE  \n[1] Rocha, A. R., Garcia -Suarez, V. M., Bailey, S. W., Lambert, C. J., Ferrer, J., & \nSanvito, S. (2005). Towards molecular spintronics. Nature materials, 4(4), 335.  \n[2] Sanvito, S. (2011). Molecular spintronics. Chemical Society Reviews, 40(6), \n3336 -3355.  \n[3] Bogani, L., & Wernsdorfer, W. (2010). Molecular spintronics using single -\nmolecule magnets. In Nanoscience And Technology: A Collection of Reviews from \nNature Journals (pp. 194 -201).  [4] Jungwirth, T., Marti, X., Wadley, P., & Wunderlich, J. (2016). Antifer romagnetic \nspintronics. Nature nanotechnology, 11(3), 231.  \n[5] Baltz, V., Manchon, A., Tsoi, M., Moriyama, T., Ono, T., & Tserkovnyak, Y. (2018). \nAntiferromagnetic spintronics. Reviews of Modern Physics, 90(1), 015005.  \n[6] MacDonald, A. H., & Tsoi, M. (201 1). Antiferromagnetic metal spintronics. \nPhilosophical Transactions of the Royal Society A: Mathematical, Physical and \nEngineering Sciences, 369(1948), 3098 -3114.  \n[7] Shick, A. B., Khmelevskyi, S., Mryasov, O. N., Wunderlich, J., & Jungwirth, T. (2010). \nSpin-orbit coupling induced anisotropy effects in bimetallic antiferromagnets: A route \ntowards antiferromagnetic spintronics. Physical Review B, 81(21), 212409.  \n[8] Barthem, V. M. T. S., Colin, C. V., Mayaffre, H., Julien, M. H., & Givord, D. (2013). \nReveali ng the properties of Mn 2 Au for antiferromagnetic spintronics. Nature \ncommunications, 4, 2892.  \n[9] Kresse, G., & Furthmüller, J. (1996). Software VASP, vienna (1999). Phys. Rev. B, \n54(11), 169.  \n[10] Perdew, J. P., Burke, K., & Ernzerhof, M. (1996). Genera lized gradient \napproximation made simple. Physical review letters, 77(18), 3865.  \n[11] Jekal, S., Rhim, S. H., Hong, S. C., Son, W. J., & Shick , A. B. (2015). Surface -\ntermination -dependent magnetism and strong perpendicular magnetocrystalline \nanisotropy of an FeRh (001) thin film. Physical Review B, 92(6), 064410.  "    },    {        "title": "1906.09153v1.Strain_and_thickness_effects_on_magnetocrystalline_anisotropy_of_CoFe_011__films.pdf",        "content": "Strain and thickness effects on magnetocrystalline anisotropy of CoFe(011) \nfilms  \n \nEunsung Jekal* , Phuong Dao  \nDepartment of Materials, ETH Zurich, 8093 Zurich, Switzerland  \nWe investigate MCA of CoFe( 011) thin films as a function of strength of strain and  film \nthickness has been studied. It is elucidated that perpendicular magnetocystalline anisotropy \n(MCA) energy ( EMCA) is getting stronger with compressed xy-plane lattice constant while in -\nplane MCA is become an easy -axis by tensile strain on xy-plane. The reason of the EMCA \nbehaviors can be explained by features of electronic structures:  \n \nI.  INTRODUCTION  \nMaterials in a form of thin films show very different properties from bulk because of the \nbroken symmetry. Neel drew the attention of possible appearance of large  \nmagnetocrystalline anisotropy (MCA) of thin films. In addition, the systems with a strong \nperpendicular MCA energy (E MCA) have potential to be used as high density  \nmagnetic recording devices . [1–5] \nFor these applications, the corresponding materials have to be grown on different substrates. \nDue to the mismatch in lattice constants and thermal expansion coefficients between the \nmagnetic material and the substrate, significant amounts of  strain would be occurred, \ndepending on the speci fic growth conditions and substrate materials.  \nIn 1999, extremely large perpendicular magnetic anisotropy (PMA) effect was observed in \nCoFeB/MgO.  \nSince that time, a lot of experimental and theoretical research has been progressed to find \norigin of the large PMA of CoFeB/MgO. Then substrates, interface insertion layer, capping \nlayers and other things are reported that they have an impact on PMA of Co FeB/MgO [6]. \nAccording to one experimental research, 100 nm CoFe/MgO(011) shows perpendicular  EMCA \nabout 0.75  erg/cm2 while  100 nm  CoFe/MgO(001) indicates 0.25  erg/cm2.  Consequently, more than three times stronger  EMCA is obtained by tailoring orientatio n of \nfilms.  \nIn this  study, we investigate magnetic properties of  CoFe(011) thin films via  density \nfunctional theory  calculation. Furthermore, for consider the strain effect by  the substrates, \nvaried  xy-plane lattice constant of [011]  plane was considered . \n \nII.  COMPUTATIONAL METHOD  \n \nFIG. 1. The structure of the CoFe(011)  film. Blue  (yellow)  spheres represent Co  (Fe). Co and  \nFe atoms are placed in  same layer with 1:1 ratio. Axes are  presented in terms of  coordinates of  \nbcc structure . \n \nFirst principles density functional  theory  (DFT)  calculation s [7, 8] were performed using the \nVienna ab  initio simulation package (V ASP)[9]. We used the spin  polarized generalized \ngradient approximation (GGA) [10]  and projector augmented wave (PAW) potentials. 540 eV  \ncut-off energy is used for wave function expansion. For  our 2 -D unit cell which is taken as \nrectangular with 1: √2 ratio to describe the [011] plane of  bcc structure,  k-points  sampling of \na 16x12x1 mesh with Monkhorst -Pack generation schem e is generated. We adapted 20  Å \nspacing  between adjacent layer  to avoid artificial interaction  in film calculations. Fig.  1 shows \nCoFe(011) film and  its xy-plane. Blue and yellow spheres represent Co and Fe  atoms, \nrespectively. In case of CoFe(011) films, Co and  Fe atoms are placed in same layer with 1:1 \nratio. Axes are  explicitly presented in terms of coordinates of bcc structure. We modeled \nCoFe(011) films with thickness of 2 - to7 (2≤n≤7)-mono -layers (MLs). To analyze strain effect, \nwe adjusted  xy-plane l attice constant from 2.68  Å to 3.00 Å \nA while their atomic displacements along z -axis were  fully relaxed until the Hellmann -\nFeynman forces were less  than 0.1 meV/ Å \nTo obtain  EMCA of CoFe(011) films, we considered  four directions  of magnetization.  \nAs shown in Fig.1, [0 1̅1̅], [1̅00] and [ 1̅11̅] represent in -plane while [011]  represents out -of-\nplane.  \nThen we can obtain three  kinds of  EMCA 1: EMCA1=𝐸01̅1̅-𝐸011, EMCA 2=𝐸1̅00-𝐸011, \nEMCA 3=𝐸101̅̅̅̅̅-𝐸011, where  E(011̅̅̅̅), E(100),  E(111)  and E(011)  correspond to the total energies \nwhen  the directions of the magnetization are along the  where (0 11̅̅̅̅), (100), (111) and (011)  \n, respectively.  \nFunctions of  EMCA1-EMCA2=E(011)-E(100) and EMCA2 -EMCA3=E(100)-E(111) are energy to reverse \nmagnetization on  xy-plane. Convergence of the  EMCA was checked  with regard to cut off energy \nand number of  k-point.  \n \nIII.  RESULTS AND DISCUSSION  \nIn Fig.  2, we present  EMCA1, EMCA2 and EMCA3 as a function  of the  xy-plane lattice  constant  \nwith different film thicknesses from 2 - to 7-MLs.  Vertical  dot lines represent 2.84  Å that \nlattice constant of bulk CoFe  [11]. Larger and smaller values than 2.84  Å mean that  \nwe applied tensile and compressive strain on  xy-plane,  respectively.   \n  \n \nFIG.2 EMCA as a function of the lattice constant with  various film thickness.  Square  (circle, \ntriangle) correspond  EMCA1( EMCA2,EMCA3). Vertical dot lines represent lattice  constant of \nbulk CoFe of 2.84  Å. \n  \nAccording to Fig. 2, n≥4 films show linear curve  of EMCA variation that negative strain  leads \nincreased  perpendicular  EMCA but the easy -axis becomes in -plane  with applying tensile strain.  \nBut when n=2 and 3, in -plane EMCA indicates around lattice constant of 2.76 Å  and there is \nperpendicular EMCA while appl ied negative or positive strain since that lattice. One more \nimportant thing is that  EMCA saturates in n≥5. According to Fig. 2(d), 2(e) and 2(f), films with \na 5% decreased lattice constant of 2.68 Å  has a saturated EMCA of about 1.5 erg/cm2 and cases \nof 5% increased lattice constant of 3.00 Å  how saturated  EMCA f about -0.7 erg/cm2 \nFig. 3 presents magnetic moment of each layer from  surface to center.  We compared three \ncases which are 5%  decreased, 5% increased and no strained lattice.  In Fig.  3, magnetic  \nmoments of centered Fe atoms are  reduced compared with surface atoms. In opposite to  this, \nmagnetic moments of Co are increased during Co  atoms place far from surface. Total  \nmagnetic moments are increased (decreased) when apply positive (negative)  strain.  \nFor finding clues of the increased EMCA in strained films, we present density of states (DOS) \nof 2.68 Å  and 3.00 Å  lattice constant cases in n=2 and 7. n=2 shows different EMCA \ntrend in comparison with thicker films,  so we compare n=2 and 7. DOS o f lattice constants \nof 2.68 Å and 3.00  Å are presented in Fig.  3, 4, respectively.  \nEMCA can be explained by perturbation theory [1 2]. \nAccording to perturbation theory, the spin orbit coupling (SOC) interaction between \noccupied and unoccupied states plays i mportant role in determining  EMCA. \nEMCA is expressed by  \n \n \n \nwhere ξ is the SOC strength, ϵo,σ (ϵo,σ’) and oσ(uσ) represent eigenvalues and eigenstates of \noccupied (unoccupied) for each spin state. When σσ’ is ↑↑ or ↓↓, positive (negative)  is \ndetermined b y the SOC interaction between occupied and unoccupied  states with the same \n(different by one) magnetic quantum number  (m) through the  Lz (Lx) operator.  However , in case  \nof σσ=↑↓, positive (negative) contribution come  from the  Lx (Lz) coupling [18].  \nIn case of films with lattice constant of 2.68  Å which  is a 5% reduced value compared to that  \nof bulk, although both n=2 and 7 exhibit perpendicular  EMCA, EMCA of n=2 is about 0.252  \nerg/cm2 while 7 -MLs film  exhibit much larger value of about 1.481 erg/cm2. That  causes are  \ntraceable to their DOS. According to  Fig. 3, <yz↓|Lx|xy↑> couplings of Co of n=2 and 7 lead \nperpendicular  anisotropy. However in Fig. 3(a) Co of n=2, occupied  z2 orbital is found nearby \nFermi level. So <yz↓|Lx|z2↓> coupling abates perpendicular  EMCA in n=2. \n \nFIG. 4. DOS of n=2 and 7 with lattice constant of 2.62  Å. (a) Co and (b) Fe of n=2. (c) Co(S) \nand (d) Fe(S) of n=7. (e) Co(S -1) and (f)  Fe(S -1) of n=7 . (g) Co(C) and (h)  Fe(C) of  n=7.  \nThick solid, thick dash,  dot, thin dash and thin solid  line represent  xy, yz, z2, zx and x2−y2 orbital, \nrespectively.  \n \n \nFIG. 5. DOS of n=2 and 7 with lattice constant of 2.62  Å. (a) Co and (b) Fe of n=2. (c) Co(S) \nand (d) Fe(S) of n=7. (e) Co(S -1) and (f)  Fe(S-1) of n=7. (g) Co(C) and (h)  Fe(C) of  n=7. \n \nIV.  CONCLUSION  \nIn this study, MCA of CoFe(011) thin films as a func tion of strength  of strain  and film  \nthickness  has been studied. It is elucidated that perpendicular  EMCA is getting  stronger  with \ncompressed  xy-plane  lattice constant while in -plane MCA is become an easy -axis by ten sile \nstrain on xy-plane.  The reason of the EMCA behaviors can be explained by features of electronic \nstruc tures:  <yz↓|Lx|xy↑> couplings of Co atom play impor tant role in determining \nperpendicular  EMCA of tensile  strained films, however,  \n<xy↓|Lx|zx↓> coupling which  contributes to in-plane  MCA  appears in compressive  strained \nfilms. Especially, the film with 10% tensile strain  on the  xy-plane exh ibits perpendicular  EMCA \nmore than  1.00 erg/ cm2 for n≥5 \n \nV .  REFERENCES  \n[1] A. Stupakiewicz , K. Szerenos, D. Afanasiev, A. Kirilyuk,  and A. V . Kimel , Nature, 542, 71 \n(2017 ). \n[2] C. V ogler, C.  Abert, F. Bruckner, D. Suess, and D.  Praetorius , Appl . Phys . Let. 108, 102406  \n(2016). \n[3] Y . K. Takahashi, R. Medapalli, S. Kasai, J. Wang, K. Ishioka, S. H. Wee, and E.  E. Fullerton , \nPhys . Rev. Appl . 6, 054004  (2016).  \n[4] S. Jekal, S. H. Rhim,  S. C.  Hong, W. J.  Son, and A. B.  Shick , Phys . Rev B, 92, 064410  \n(2015 ). \n[5] S. Jekal, S. H.  Rhim, and S . C. Hong , IEEE Trans . on Magn . 52, 1 (2016 ). \n[6] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D.  Gan,  M. Endo, and H. Ohno,  Nature \nmaterials , 9, 721 (2010 ). \n[7] R. Car and M. Parrinello,  Phys  rev lett. 55, 2471  (1985 ). \n[8] R. G. Parr, (1980). Density func tional theory of atoms and molecules. In Horizons of \nQuantum Chemistry (pp. 5 -15). Springer, Dordrecht.  \n[9] J. Hafner,  Journal of computational chemistry, 29, 2044  (2008 ). \n[10] H. J. Monkhorst  and J. D  Pack,  Phys  Rev B, 13, 5188  (1076 ). \n[11] E. G.  Kim, S. Y . Jekal, O. Kwon,  S. C. Hong,  J. Kor. Magn . Soc. 24, 35 (2014 ). \n[12] K. Andersson , P. A. Malmqvist,  and R. O. Roos, The Journal of chemical physics, 96 , 1218  \n(1992 ). "    },    {        "title": "1906.11329v1.Concentration_tuned_tetragonal_strain_in_alloys__application_to_magnetic_anisotropy_of_FeNi___1_x__Co__x_.pdf",        "content": "arXiv:1906.11329v1  [cond-mat.mtrl-sci]  26 Jun 2019Concentration tuned tetragonal strain in alloys: applicat ion to magnetic anisotropy\nof FeNi 1−xCox\nAleksander L. Wysocki,1,∗Manh Cuong Nguyen,1Cai-Zhuang Wang,1\nKai-Ming Ho,1Andrey V. Postnikov,2and Vladimir P. Antropov1\n1Ames Laboratory, Ames, IA 50011, USA\n2Universit´ e de Lorraine, Metz, F-57078, France\n(Dated: June 28, 2019)\nWe explore an opportunity to induce and control tetragonal d istortion in materials. The idea in-\nvolves formation of a binary alloy from parent compounds hav ing body-centered and face-centered\nsymmetries. The concept is illustrated in the case of FeNi 1−xCoxmagnetic alloy formed by sub-\nstitutional doping of the L1 0FeNi magnet with Co. Using electronic structure calculatio ns we\ndemonstrate that the tetragonal strain in this system can be controlled by concentration and it\nreaches maximum for x= 0.5. This finding is then applied to create a large magnetocryst alline\nanisotropy (MAE) in FeNi 1−xCoxsystem by considering an interplay of the tetragonal distor tion\nwith electronic concentration and chemical anisotropy. In particular, we identify a new ordered\nFeNi0.5Co0.5system with MAE larger by a factor 4.5 from the L1 0FeNi magnet.\nControlling tetragonal distortion in solids is a com-\nmon approach to improve properties of functional mate-\nrials. The tetragonal strain decreases the symmetry of\nthe system allowing for a number of effects which are,\notherwise, forbidden in cubic structures. From a differ-\nent perspective, such distortionscan be used to tune ma-\nterials properties for desired applications or to induce a\nphase transition in the system creating a new functional\nphase.\nExperimentally,tetragonaldistortionsaretypicallyre-\nalized by coherent growth of the material on a lattice-\nmismatched substrate or buffer. However, such distor-\ntions can exist only for ultrathin layers since for thicker\nfilms the strain is released by formation of dislocations.\nTherefore, this approach is unsuitable when large sam-\nples are required for applications. For bulk systems one\ncan attempt to create tetragonal strain by interstitial\ndoping with small atoms(i.e., C orB) but suchapproach\nis difficult to controland usually only modest strains can\nbe achieved.\nThe problem of controlling tetragonal strains in cu-\nbic systems is of great importance in the field of per-\nmanent magnetism. Due to recent supply shortage of\nrare-earth elements it is, currently, crucial to design\nnew rare-earth-free and high-energy-product permanent\nmagnets.1,2Fromthis perspective, transitionmetalmag-\nnets (Fe, Co, Ni, and their alloys) are especially promis-\ning class of materials since their relatively high magneti-\nzationcouldpotentiallyleadtolargeenergyproducts. In\naddition, these materials typically have large Curie tem-\nperatures that makes them ideal for high temperature\noperations. Unfortunately, transition metal magnets of-\nten crystallize in cubic structures for which the second\norder contribution to the magnetocrystalline anisotropy\nenergy (MAE) is zero by symmetry. This results in a\nlow MAE ( ∼1µeV/atom) which severely limits appli-cations of these materials as permanent magnets. A no-\ntable exception is a marginallystable form ofFeNi called\ntetrataenite3that has a tetragonal L1 0structure and a\nsizable MAE ( ≈40µeV/atom).4This suggests that a\nmuch larger MAE could be realizes in unstable fami-\nlies of transition metal magnets with a built-in tetrag-\nonal distortion. This concept was supported by elec-\ntronic structure calculations.5,6In particular, MAE as\nlarge as 800 µeV/atom has been predicted for a strained\n(c/a≈1.23) FeCo system.6Large MAE was, indeed,\nobserved for ultrathin layer of strained FeCo epitaxially\ngrown on lattice-mismathed substrate.7–9\nHere, we propose a strategy for tuning tetragonality\nin materials by mixing compounds with body- and face-\ncentered symmetries. As a realization of this idea, we\nconsider formation of a FeNi 1−xCoxalloy from B2 FeCo\nand unstable L1 0FeNi parent compounds. Using first\nprinciples electronic structure calculations we demon-\nstrate that the tetragonal strain can be naturally tuned\nby concentration with a maximum around x= 0.5. Fur-\nther, the MAE FeNi 1−xCoxis analyzed. We show that\nfor a random alloy MAE remains low despite the pres-\nence of strong tetragonal distortion. However, a large\nMAE (180 µeV/atom) can be achieved for an ordered\nstructure created by vertical stacking of FeNi and FeCo\nlayers.\nThe key idea follows from the observation that the\nface-centered cubic (fcc) lattice can be viewed as a body\ncentered tetragonal (bct) lattice with the c/aratio equal\nto√\n2 (see Fig. 1). Consequently, the body centered\ncubic (bcc) lattice can be obtained from the fcc lat-\ntice by the appropriate compression so that c/a= 1.\nBoth for c/a= 1 and c/a=√\n2 the system has a cu-\nbic symmetry. However, for intermediate c/avalues the\nsolid has a tetragonal distortion. This transformation is\nknown as continuous Bain transformation and has been2\nobservedexperimentally.10Letusnowconsidertwocubic\n(or nearly cubic) material with similar atomic volumes\nbut different structures. One has a face-centered sym-\nmetry and the other one has a body-centered symmetry.\nAccording to the above discussion, we can expect that\nan alloy formed from these two materials has a crystal\nstructure corresponding to an intermediate point along\nthe Bain transformation path with a tetragonal strain\nthat is controlled by concentration.\nThe above concept can be realized in the FeNi 1−xCox\nalloy formed by substitutional doping of the L1 0FeNi\nmagnet with Co at the Ni site. Within the bct lattice\nL10FeNi has c/a≈1.424 that is slightly larger from the\nideal fcc value ( c/a=√\n2).11If all Ni atoms are replaced\nby Co, we obtain FeCo intermetallic compound with the\nCsCl structure. This system has a cubic body-centered\nsymmetry with c/a= 1. According to the discussion\nin the previous paragraph, we, therefore, expect that\nfor partial Ni-Co substitutions the resulting FeNi 1−xCox\nalloy will develop a sizable tetragonal distortion. Below,\nweinvestigatethissystemusingfirstprincipleselectronic\nstructure calculations.\nIn order to model doping we used 2 ×2×1 and 1×1×N\n(N= 2,3,4) supercells with respect to the primitive bct\ncell of the parent compounds. The 2 ×2×1 supercell was\nFIG. 1. Relation between fcc and bcc lattices. (top) fcc is\nequivalent to the bct lattice with c/a=√\n2. Compression\nof the lattice until c/a= 1 (Bain transformation) results in\ncubic bcc lattice (bottom). For intermediate values of c/a\n(middle) the lattice has tetragonal symmetry.\nFIG. 2. Schematic illustration of supercell models for diso r-\ndered (right) and ordered (left) Ni 1−xCoxalloy in the case of\nx= 0.5. The red, green, and blue spheres denote Fe, Ni and\nCo atoms, respectively.\nused to simulate the random alloy (Co atoms randomly\nsubstitute Ni atoms), see Fig. 2 (right) in the case of\nx= 0.5. In addition, we also considered the case of\nordered Ni 1−xCoxalloy in which some of Ni layers in\nL10FeNi are replaced by Co. As a result, the structure\nconsists of vertical stacking of L1 0FeNi and B2 FeCo\nlayers, see Fig. 2 (left).\nThe calculations were performed using the density\nfunctional theory with PBE exchange-correlation func-\ntional. The Kohn-Sham equations were solved using the\nprojector augmented wave method12as implemented in\nthe VASP code13,14. The cutoff energies for the plane\nwave and augmentation charge were 270 eV and 545\neV, respectively. For the primitive bct cell we used\n12×12×12 24×24×24 Γ-centered k-point mesh for re-\nlaxation and anisotropy calculations, respectively. For\nlargercells the k-point mesh wasscaledaccordingly. The\nlattice parameters and the ionic positions have been re-\nlaxeduntiltheHellmann-Feynmanforceswereconverged\nto less than 0.01 eV/ ˚A. MAE has been calculated using\nthe force theorem. The site-resolved MAE were calcu-\nlated using the approach described in Ref. 15.\nFigure 3 (top) shows the concentration dependence\nofc/aration and the tetragonal strain both for disored-\neredand orderedFeNi 1−xCoxalloy. Here, the tetragonal\nstrain is defined as\nǫT= min/parenleftBig\n|c/a−√\n2|,|c/a−1|/parenrightBig\n×100% (1)\nWe observe that under doping the c/asmoothly de-\ncreasesfromnearlyfccvaluedowntothebccvalue. Con-\nsequently, the tetragonaldistortiondevelopsfor interme-\ndiatedopings. Inparticular, ǫTdependsstronglyoncon-\ncentration and it reaches maximum of 18% at x= 0.5.\nImportantly, we find that both c/aandǫThave a weak\ndependence on doping configuration. In fact, both dis-\nordered and ordered FeNi 1−xCoxalloy have similar a3\nFIG. 3. Calculated c/aratio, tetragonal distortion parameter\n(top), MA energy (middle), and spontaneous magnetization\n(bottom) of FeNi 1−xCoxas a function of the Co content.\nOpen (filled) circles correspond to the random (ordered) al-\nloy case. The dotted horizontal line denotes c/a=√\n2 that\ncorresponds to the face-centered cubic symmetry.\nconcentration dependence of c/aandǫT. These results\nindicate that the tetragonal distortion in FeNi 1−xCox\nalloy can be controlled by doping concentration.\nLet us now consider the magnetic properties of\nFeNi1−xCoxalloy. The calculated spontaneous mag-\nnetization as a function of concentration is shown in\nFig. 3 (bottom). As seen, the magnetization is virtu-\nallyindependentondopingconfigurationanditincreases\nsmoothly with xstarting from 1.6 µB/atom value for\nFeNi to 2.3 µB/atom for FeCo.\nIn the case of MAE the situation is much more com-\nplicated. Concentration dependence of MAE is shown\nin the middle panel of Fig. 3 both for disordered and\nordered alloy. As discussed in the introduction, the de-\nvelopment of a strong tetragonal strain for intermediate\ndoping concentrations may lead to an enhancement of\nMAE. For disordered FeNi 1−xCoxalloy, however, this is\nnot the case. In fact, in this case doping with Co results\nin MAE being even smaller than for L1 0FeNi compound\ndespite the increase of ǫT. This result reflects a fragile\nnature of MAE which, in addition to tetragonal strain,depends also strongly on electronic concentration and\nchemical anisotropy in the system. The complex inter-\nplay between these three factors for FeNi 1−xCoxalloy\ncan be illustrated by plotting site-resolved MAE15,16as\na function of Co concentration, see Fig. 4. As seen, for\npure FeNi, the Fe atom has a large contribution to MAE\nabove 100 µeV. On the other hand, the increase of elec-\ntronicconcentrationfortheNiatomleadstoasignificant\nnegative contribution to MAE resulting in a rather mod-\nerate total MAE for this compound. In the case of the\ndisordered FeNi 1−xCoxalloy, the Co doping results in a\nstrong reduction of the Fe contribution which becomes\neven negative for large x. On the other hand, the Ni\ncontribution to MAE changes somehow with concentra-\ntion but it always remains negative. The Co atoms con-\ntribution to MAE varies strongly with xstarting from\nlarge negative values for low concentrations to moder-\nate positive values for larger x. Note that the electronic\nconcentration for each specie remains approximately the\nsame as xincrease. Therefore, since the tetragonal dis-\ntortion remains positive for all concentrations, we can\nconclude that this is the chemical anisotropy mechanism\nthat is responsible for the reduction of the Fe contribu-\ntion and the strongvariation ofthe Co contribution with\ndoping. Indeed, for the disordered alloy the atomic envi-\nronment of Fe and Co atoms changes with doping. The\nchemical anisotropy mechanism can be, thus, controlled\nby alloy ordering. In the case of the layered ordering\nshown in Fig. 2 (left) the chemical anisotropy of the\nparent compounds is preserved. The corresponding site-\nresolved contributions to MAE are shown in Fig. 4 (bot-\ntom). As seen, the Ni contribution as a function changes\nsomewhat as a function of Co concentration but, simi-\nlarlyasinthecaseofdisorderedalloy,itremainsnegative\nfor all dopings. This indicates that the Ni contribution\nis primarily controlled by the electronic concentration.\nHowever, the concentration dependence of both Fe and\nCo contribution changes completely when the atomic or-\ndering is introduced. Indeed, the Fe contribution has a\nrather weak doping dependence and remains large and\npositive for all x. Therefore, the chemical anisotropy\nmechanism is crucial in this case. The Co contribution\nis also positive for all xbut its magnitude changes sig-\nnificantly with the concentration. and the dependence\nroughly follows ǫT. More specifically, the concentration\ndependence for large and small xthe Co contribution is\nclose to the Fe one but for intermediate doping concen-\ntrations the Co contribution shows a strong increase and\nforx= 0.5 it reaches a gigantic value above 400 µeV.\nThisindicatesthattheCocontributioniscontrolledboth\nby both tetragonal distortion and chemical anisotropy\nmechanisms.\nAs aresult ofthe strongincreaseofthe Fe and Co con-\ntributions, the total MAE of the ordered FeNi 1−xCox\nalloy is significantly larger than in the case of the dis-4\nFIG. 4. Concentration dependence of the site-resolved MAE\nfor disordered (top) and ordered (bottom) FeNi 1−xCoxalloy.\nThe shown values are averaged over different nonequivalent\natoms of the same specie in the considered supercells. The\nerror bars denote the corresponding standard deviation whe n\nthe latter is significant.\nordered alloy (see Fig. 3). More importantly, for the\nordered alloy the MAE dependence on concentration\nroughly follows the ǫTdependence and it becomes sig-\nnificantly enhanced for intermediate dopings. In partic-\nular, for FeNi 0.5Co0.5system MAE is as large as 180\nµeV/atom. The corresponding anisotropy density con-\nstant is equal to 2.4 MJ/m3which is almost half of\nthe room temperature anisotropy density constant of\nNd2Fe14B. Moreover, we found that there is a poten-\ntial for further enhancement of MAE by increasing the\ntetragonal strain (for example by suitable interstitial\ndoping or epitaxial growth). This is illustrated in Fig. 5\nwhere the MAE of the ordered FeNi 0.5Co0.5compound\nis plotted as a function of the c/aratio (the in-plane\nlattice parameter was set to the equilibrium value for\nFeNi0.5Co0.5). Asseen, MAEabove200 µeV/atomcould\nbe realized in this system.\nIt should be also pointed out that the magnetization\nof FeNi 0.5Co0.5is equal to 1.85 T which is significantly\nlarger than that of typical rare-earth-based magnets.\nTherefore, large energy products can be potentially ob-\ntainedin FeNi 0.5Co0.5makingthissystem averypromis-\ning material for permanent magnet applications. Exper-\nimental efforts to realize FeNi 0.5Co0.5material by epi-\ntaxial growth are currently underway.\nIn summary, we investigated a new route to introduce\nand control tetragonality in materials. The idea is basedon the Bain transformation and involves combining ma-\nterials with face- and body-centered symmetries. This\nconcept was illustrated using an example of FeNi 1−xCox\nalloy obtained by doping the L1 0phase of FeNi with\nCo. Using first principles electronic structure calcula-\n/s49/s46/s48/s53 /s49/s46/s49/s48 /s49/s46/s49/s53 /s49/s46/s50/s48 /s49/s46/s50/s53 /s49/s46/s51/s48 /s49/s46/s51/s53 /s49/s46/s52/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48/s49/s54/s48/s49/s56/s48/s50/s48/s48/s69/s32/s40 /s101/s86/s47/s97/s116/s111/s109 /s41\n/s99/s47/s97/s70/s101/s78/s105\n/s48/s46/s53/s67/s111\n/s48/s46/s53\nFIG. 5. MAE of the ordered FeNi 0.5Co0.5alloy as a function\nof thec/aratio. The green circle denotes MAE for equilib-\nriumc/a.\ntions we demonstrated that the the tetragonal strain in-\ncreaseswith doping and it reachesmaximum for x= 0.5.\nThis result was subsequently used to engineer strong\nMAE in the FeNi 1−xCoxalloy. MAE for this system\nwas shown to be a result of complex interplay between\ntetragonaldistortion,electronicconcentrationandchem-\nicalanisotropymechanisms. Wedemonstratedthatlarge\nMAE can be achieved for layered-ordered FeNi 1−xCox\nalloys. In particular, we identified the FeNi 0.5Co0.5com-\npound with large MAE of 180 µeV/atom.\nACKNOWLEDGMENTS\nThisworkwassupportedbytheOfficeofBasicEnergy\nScience, Division of Materials Science and Engineering.\nA. W. acknowledges the support from the Critical Mate-\nrials Institute, an Energy Innovation Hub funded by the\nU.S. Department of Energy (DOE). The research was\nperformed at Ames Laboratory, which is operated for\nthe U.S. DOE by Iowa State University under contract\n# DE-AC02-07CH11358.\n∗alexwysocki2@gmail.com1L. H. Lewis and F. Jim´ enez-Villacorta, Metall. Mater.\nTrans. A 44, 2 (2013).5\n2R. W. McCallum, L. H. Lewis, R. Skomski, M. J. Kramer,\nand I. E. Anderson, Annu. Rev. Mater. Res. 44, 451\n(2014).\n3R. S. Clarke and E. R. D. Scott, Am. Mineral. 65, 624\n(1980).\n4T. Shima, M. Okamura, S. Mitani, and K. Takanashi, J.\nMagn. Magn. Mater. 310, 2213 (2007).\n5T. Burkert, O. Eriksson, P. James, S. I. Simak, B. Johans-\nson, and L. Nordstr¨ om Phys. Rev. B. 69, 104426 (2004).\n6T. Burkert, L. Nordstr¨ om, O. Eriksson, and O. Heinonen\nPhys. Rev. Lett. 93, 027203 (2004).\n7G. Andersson, T. Burkert, P. Warnicke, M. Bj¨ orck, B.\nSanyal, C. Chacon, C. Zlotea, L. Nordstr¨ om, P. Nordblad,\nand O. Eriksson, Phys. Rev. Lett. 96, 037205 (2006).\n8F. Yildiz, F. Luo, C. Tieg, R. M. Abrudan, X. L. Fu, A.\nWinkelmann, M. Przybylski, and J. Kirschner, Phys. Rev.\nLett.100, 037205 (2008).9F. Yildiz, M. Przybylski, X.-D. Ma, and J. Kirschner,\nPhys. Rev. B 80, 064415 (2009).\n10R. Cuenya, M. Doi, S. Lobus, R. Courths and W. Keune,\nSurf. Science 493338 (2001).\n11M. Kotsugi, H. Maruyama, N. Ishimatsu, N. Kawamura,\nM. Suzuki, M. Mizumaki, K. Osaka, T. Matsumoto,\nT. Ohkochi, T. Ohtsuki, T. Kojima, M. Mizuguchi, K.\nTakanashi, and Y. Watanabe, J. Phys.: Condens. Matter\n26, 064206 (2014).\n12P. E. Bl¨ ochl, Phys. Rev. B 50, 17953 (1994).\n13G. Kresse and J. Hafner, Phys. Rev. B 48, 13115 (1993).\n14G. Kresse and J. Furthm¨ uller, Phys. Rev. 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B 88, 024404\n(2013)."    },    {        "title": "1906.12254v2.Zero_field_propagation_of_spin_waves_in_waveguides_prepared_by_focused_ion_beam_direct_writing.pdf",        "content": "1 \n Zero -field propagation of spin waves in waveguides prepared by focused ion beam direct writing  \nLukáš Flajšman1*, Kai Wagner2, Marek Vaňatka1, Jonáš Gloss3, Viola Křižáková4, Michael Schmid3, Helmut Schultheiss2 and \nMichal Urbánek1,4* \n1 CEITEC BUT, Brno University of Technology, Brno, Czech Republic  \n2 Institute of Ion Beam Physics and Materials Research, HZDR, Dresden, Germany  \n3 Institute of Applied Physics, TU Wien, Vienna, Austria  \n4 Institute of Physical Engineering, Brno University of Technology, Brno, Czech Republic  \n \n*E-mail: lukas.flajsman@ceitec.vutbr.cz , michal.urbanek@ceitec.vutbr.cz   \nMetastable Fe78Ni22 thin films are excellent candidates for focused ion beam direct writing  of magnonic structures  \ndue to their favorable  magnetic properties . The focused ion beam transforms  the originally nonmagnetic fcc phase \ninto the ferro magnetic bcc phase  with an additional control over the  direction of uniaxial magnetic  in-plane  \nanisotropy  and the saturation magnetization . The induced anisotropy allows to stabilize transverse direction of \nmagnetization in narrow waveguides . Therefore , it is possible to  propagate  spin waves in these waveguides in the \nfavorable  Demon -Eshbach geometry without the presence of an y external magnetic field.  \n \nNowadays , the vibrant  field of magnonics  stands  on the edge between  development of elementary building blocks of magnonic \ncircuitry and envisioned  all magnon  on-chip devices  [1,2]. The magnonic devices, utilizing physics of spin waves, are \nrecognized to have potential in information processing  in the  frequency range from  gigahertz  to terahertz . High frequencies, \ntogether with low energy of elementary excitations render the magnonic devices  suitable for beyond -CMOS computational \ntechnologies . Many concepts of future device s used for steering and manipulating spin  waves  have been presented  recently  [3-\n6]. To allow further advances in this field, new types of materials  possessing additional means of control over their magnetic \npropertie s together with good spin wave propagation are needed.   \nHere we show, that magnonic waveguides allow ing for fast spin -wave propagation at zero magnetic field  can be \ndirectly written into metastable Fe78Ni22 thin films  by focused ion beam (FIB) .  The local dose and scanning strategy controls  \nboth the saturation magnetization and the magnetocrystalline anisotropy (direction, type and strength) of irradiated  areas.  The \nunique possibilities of this material system allow to overcome the shape anisotropy of long magnonic waveguides and stabilize 2 \n the magnetization along the waveguide’s short side even in the absence of any external magnetic field. This is of great \nimportance for using magnons as information carriers  because the geometry where the magnetization is aligned perpendicularly \nto the propagation direction has the maximum group velocity.  \nThe Fe78Ni22 layers grow on Cu(001) substrate in the metastable nonmagnetic fcc phase and can be transfor med by ion \nirradi ation into the ferromagnetic bcc phase  [7]. Using FIB the fcc->bcc transformation can be precisely controlled which \nresults  in unprecedented  local control over magnetic properties  [8]. This approach remove s the need for complicated multi -step \nlithography processing and allow rapid prototyping of individual structures on the sample with additional possibility to cont rol \nthe magnetic properties of each structure  [8-12]. Classical approaches of nanostructuring of magnetic materials such as optical \nor electron beam lithography combined with lift-off processing  [13], wet [14] or dry  [15] etching or  ion implantation  [16,17] \ndo not offer the same control over local material properties as the fabrication processes rely on binary selection of adding  or \nremoving the magnetic materials .   \nThe FIB-written waveguides  together with  a microwave antenna for spin -wave excitation  are shown  in Fig. 1 a). The \nwaveguides  are 30 µm long with  nominal  widths of 3, 2 and 1.5 µm and are clearly visible  in the SEM  image .  3 \n  \nFig. 1: a) SEM micrograph of FIB prepared waveguides (bcc waveguides appear dark grey) . The crystallographic orientations of the fcc \nmatrix and bcc waveguides are indicated (for the latter, inferred from the magnetic anisotropy, double -headed arrow). The microwave \nantenna is highlighted by ocher color . b) hysteresis loops  of three waveguides for two different orientations of the magnetic field [blue and \nred arrows showing  the respective orientation of the magnetic field are shown in a)]. c) 2D frequency maps of thermal spin -wave spectra \nmeasured by micro -focused BLS scanning over the waveguides in the transverse direction along the red lines in (a)  in zero external magnetic \nfield (top row) and in the applied magnetic field 𝐵𝑒𝑥𝑡−0°=50 mT (bottom row) . The spatially and magnetic field invariant mode  at approx. \n3 GHz is a spurious laser  mode . \nPrior to the FIB processing the metastable  fcc Fe 78Ni22 thin film  of nominal thickness 12 nm was grown  under  UHV \nconditions  on a Cu(00 1) single crystal substrate  using the procedure described in  [7]. After deposition , the sample  was \ntransferred to the FIB-SEM microscope for FIB irradiation . We oriented the long axis of the waveguides  along the fcc[0 10] \ndirection of the Cu substrate  in order to obtain the highest possible magnetocrystalline anisotropy . To imprint  the anisotropy \ndirection perpendicular ly to the long axis of the waveguide, we wrote the structures  with a single pass  of a 30 keV Ga+ ion \n4 \n beam (30 nm spot , beam current of 150 pA and 5 s dwell time ) with the fast  scanning  direction rotated by  80° from the \nwaveguide’ s long axis . The resulting  ion dose 4×1015 ions/cm2 has given reliable growth conditions for all  the waveguides . \nNucleation of the bcc structure was facilitated  by starting the growth in triangular 5 µm wide  region with the double d ion dose \nof 8×1015 ions/cm2 (again in single  FIB scan).   \nAfter the irradiation, the  magnetic wavegu ides were  investigated by Kerr  magnetometry  [18] and Kerr microscopy .  \nWhen the  field 𝐵ext−90° is applied  parallel to the long axis of the waveguide , hard-axis hysteresis loops [blue lines  in Fig. 1 b)] \nwith effective  anisotropy fields [19] in the range of 20−24 mT are observed for all waveguides . When the field  𝐵ext−0° is \napplied perpendicular to the waveguide , easy-axis loops [red lines in Fig. 1 b)] with coercive field s in the range of 4−8 mT \nare observed.   \nThermal spin -wave spectra obtained by micro -focused Brillouin light scattering  microscopy  [20] are shown in Fig. 1 \nc) for zero magnetic field  and for the external field  𝐵ext−0°=50 mT. The signal is proportional to the density of spin -wave \nstates at the detection frequency ( 𝑦-axis) and shows pronounced spin-wave spectra l intensity  localized  solely  in the areas \nirradiated by the FIB.  The bandgap of the spin -wave band  structure  for the middle of the waveguide ( 𝑥=0 µm) can be clearly \nseen as a sudden increase in the density of s pin-wave states for frequencies higher than approx. 6 GHz in zero field and at \napprox.  10 GHz in the external field of  50 mT. \nAll three waveguides show qualitatively the same behavior . The data also reveals the local change in spin -wave spectra \ntowards the sides of the waveguide. Here localized low frequency modes  appear at approx. 4 GHz in zero field and at approx. \n8 GHz in the external field of 50 mT. This is a clear indication of the transverse orientation of the magnetization and its inherent \ndemagnetizing field leading to lower effective fields at the waveguide  edges  and thus directly  result ing in the localized spin-\nwave edge modes  [21,22]. The overall  analysis is  depicted  for three vertically oriented  waveguides only as the behavior of the \nhorizontally oriented waveguides show qualitatively same behavior  (the anisotropy is again imprinted perpendicular to the long \naxis of the waveguides). This also demonstrate unique potential of our approach  when compared to other less versatile \napproaches or materials with global magnetocrystalline anisotropy.  \nIn the following experiments,  we extract the magnetic field -dependenc e of the spin-wave  dispersion relation . By fitting \nthe measured dispersion,  we were able to obtain the full set of magneto -dynamic  parameters  of the material . The µBLS does \nnot directly sense the phase of the detected spin waves and thus it does not allow to determine the wavelength 𝜆 (or equivalently 5 \n the wave propagation vector 𝑘⃗ ) of the spin waves . In order to  extract the wave -vector information  we employed the phase -\nresolved  µBLS t echnique  [20,23]. The method  directly reveals  the spatial profile of the spin -wave phase  by letting the  scattered \nphotons interfere with a reference spatially invariant  signal created by an electro -optic modulator  (EOM) . \nWe record ed the interference signal along the 1.5 µm wide waveguide ( 𝑥=0 µm) with a step size of 120 nm from \nthe edge of the exciting antenna up to 7 µm distance  at the microwave frequency  of 10.2 GHz.  The phase-resolved \nmeasurements are shown in Fig. 2. \n \nFig. 2: Phase -resolved  BLS microscopy  interference intensity map for various external magnetic fields at fixed excitation frequency  of 10.2 \nGHz . Individual linescans have been normalized and space -invariant background was subtracted. The b ottom graph  shows line  profiles \nextracted from the intensity map  for 15 mT (red line)  and for zero external magnetic field  (blue line) .  \nWe measured the BLS interference linescans  in zero and applied external  magnetic field ( perpendicular to the waveguide long \naxis) up to 15 mT (1 mT step) . The BLS interference intensity map  show s gradual transition of the spin-wave wavelength from \nthe lowest value found at zero field up to  the longest wavelengths at 15 mT. This is additional evidence for the presence of the \nDamon -Eshbach geometry even at zero field since otherwise a decrease in wavelength with increasing field would be expected . \nTo extract the  spin-wave  wavelength,  we fitted the measured data with the  simple interference model:  \n6 \n 𝐼(𝑦)=𝐼SW(𝑦)+𝐼EOM+2√𝐼SW(𝑦)𝐼EOMcos(𝜃(𝑦)), (1) \nwhere we assume 𝐼SW=𝐼maxe−𝑦/𝐿att.. The parameters  𝐼EOM and 𝐼max are EOM  and spin-wave  maximum  intensit ies, 𝐿att. is a \nspin-wave attenuation length  and the  total phase difference  is 𝜃=2𝜋𝑦\n𝜆𝑙+𝜃0 with 𝜆𝑙 representing  the longitudinal spin-wave \nwavelength,  𝑦 is the distance from the excitation antenna , and 𝜃0 is an arbitrary phase offset in -between the EOM and the spin -\nwave excitation. We f itted the experimental data  using Eq. (1) with all five parameters unconstrained.  The two important fit \nparameters are the wavelength [shown in Fig. 3 a) as a function of 𝐵ext] and the attenuation  length . The largest attenuation \nlength 𝐿att.=3.1 ±0.4 µm is measured for zero external magnetic field and the frequency of 10.2 GHz.  The spin -wave \ndispersion at zero field obtained from these fit s is shown i n Fig. 3 b).  \nFrom the dispersion, we can determine the magnetic parameters using the model  of Kalinikos and Slavin  [24], while taking \ninto account the finite width of our waveguides by assuming the effective boundary condition s as described by Guslienko et  al. \n[25]. The effective dipolar conditions yield  an effective width 𝑤eff=𝑤𝑑/(𝑑−2) of the waveguide, determined by the \ngeometric width 𝑤 and the  pinning parameter 𝑑=2𝜋/(𝑝+2𝑝ln(1/𝑝)). The parameter 𝑝 is given by 𝑝=𝑡/𝑤, where 𝑡 is the \nthickness of the magnetic material. For the spin wave dispersion model,  we assume d that the external magnetic field  𝐵ext points \nin the direction of the magnetic anisotropy , i.e. perpendicularl y to the long edge of the waveguide  (magnetic anisotropy is \nintroduced  in the form of effective magnetic field 𝐵ani). With all the terms in place the model is following:  \n𝑓2/𝛾2=(|𝐵ext|+|𝐵ani|+𝜇0𝑀s𝑃sin2(∡𝑘)+𝐴ex𝑘2)( |𝐵ext|+|𝐵ani|−𝜇0𝑀s(1−𝑃)+𝐴ex|𝑘⃗ |2), (2) \n𝑤ℎ𝑒𝑟𝑒 𝑓 is the spin-wave frequency and 𝛾 is the gyromagnetic ratio. 𝑀s is the saturation magnetization  and 𝐴ex is an exchange \nstiffness . Since the thickness of the waveguide is small with respect to the other dimensions and the spin -wave wavelength , we \nonly calculate the first branch of spin waves along the thickness, yielding 𝑃=1−(1−exp(−|𝑘⃗ |𝑡)/|𝑘⃗ |𝑡). The total \npropagation vector |𝑘⃗ |=√𝑘∥2+𝑘⊥n2 comprises the longitudinal component  𝑘∥ (parallel to the  long axis of the  waveguide)  and \ntransverse quantized component   𝑘⊥n. The angle of the spin wave  propagation vector  with the long waveguide axis is then  \n∡𝑘=atan (𝑘∥/𝑘⊥n). The amplitude of the transverse  component  𝑘⊥n is calculated from the width quantization condition 𝑘⊥n=\n𝑛𝜋/𝑤eff for 𝑛=1,2,…. In the  micro -focused BLS experiment , as we record only the interference pattern along the long \nwaveguide axis, we see only the longitudinal component of the propagation vector 𝑘∥=2𝜋/𝜆l.  7 \n  \nFig. 3: a) the dependence of the spin-wave wavelength  𝜆𝑙 [extracted from  equation  (1)] on the external magnetic field for three different RF \nfrequencies fitted with a model given by  equation (2). b) experimental and calculated  (blue line) spin-wave  dispersion s together with \ncalculated group velocity ( red line ) at zero external magnetic field.  The error bars have been calculated from the 95% fit confidence bounds.   \nWe performed the fit using Eq . (2) [consider ing 95% confidence intervals obtained by fitting of the Eq. (1)] for various wi dth \nmodes  (and their linear combinations  [26,27]) with one set of unconstrained universal parameters  and we observed the total \nminimal residual s of the fit . The best fit was  found  solely  for single mode of 𝑛=1 (in contrast to experiments in e.g. permalloy \nwavegu ides [26,27 ]), with magnetic parameters of  𝑀s=1.41±0.03 MA/m,  𝛾=29.3±0.1 GHz/T,  𝐵ani=24±1 mT, 𝑡=\n9.5±1.0 nm, 𝐴ex=11±5 pJ/m  and 𝑤=1.62±0.05 µm. The obtained fit parameters lie close to the bulk values of single \ncrystal iron films  [28]. The saturation magnetization  𝑀s=1.41 MA/m is expectedly lower then bulk value of iron  (𝑀sFe=1.7 \nMA/m ). If we consider 22% of nickel  (𝑀sNi=0.51 MA/m)  in our films, we estimate  the expected saturation magnetization to  \n𝑀sFeNi=1.45 MA/m  which is very close to the obtained value 𝑀s=1.41 MA/m . The l arge saturation magnetization together \nwith the Damon -Eshbach geometry result in a high  group velocity  𝑣𝑔=𝜕𝑓/𝜕𝑘∥ of the spin  waves reaching almost 6 km/s [see \nFig. 3 b)]. The anisotropy field resulting from the fit perfectly reproduces the value measured by Kerr microscopy , which  \nfurther support s the validity of the model . We also see that the film is transformed to the magnetic bcc phase throughout the \nwhole thickness . The  fitted thickness is very close to the nominal thickness of 12 nm (the few topmost layers are expected to \noxidize when performing ex -situ experiments ). This is also supported by  the Monte Carlo ion stopping range simulations using \n8 \n the SRIM/ TRIM  package  [29], where more than 90% of the 30 keV Ga+ ions penetrate to the copper substrate.  The width of \nthe waveguide  𝑤=1.62 µm is slightly larger than the nominal value  (1.5 m). First and presumably  the major  contribution \nincreasing the width of the waveguide is the finite size and shape  of the focused ion beam spot. Furthermore,  in our previous \nwork  [30] it was shown that the bcc  crystallites protrude to  the fcc phase  slightly further from the ion impact spot  and thus they \nagain effectively increase the width of the waveguides (the protrusion length is approx. 50 nm). Both effects effectively creat e \na gradient in the magnetization  affect ing the dynamic boundary conditions of our waveguides , which differ s from the \ndiscontinuous boundary conditions found e.g. in structures prepared by  classical lithography techniques  [26,27]. We performed \nmicromagnetic simulations  in mumax3 [31] to study effect s of the magnetization gradient at the waveguide edge s on the spin -\nwave dispersion ( for the approach see supplementary material [ 32]). In the simulations we continuously decreased the \nmagnetization from the bulk value to zero in a defined  region  at the edge of the waveguide . The introduced profile of the \nmagnetization was chosen  as an error function as it resembles the convolution of the nominal shape of the waveguide with a \nFIB spot . As the  gradient of the saturation magnetization is introduced  it is expected that the effective magnetic field , the major \ndriving force affecting the local spin -wave dispersion,  will differ  from discontinuous  case where  the saturation magnetization \nchange s abruptly.  The transverse profiles of the  saturation magnetization together with simulated effective (internal) magnetic \nfield at zero external magnetic field  are shown in Fig. 4 a).  \n \nFig. 4: a) shows the transverse profile s of the saturation magnetization  (grey)  and effective magnetic field  (red)  for discontinuous  (left panel) \nand continuous  (right panel)  transition of saturation magnetization  on the edges of the waveguide  at zero external magnetic field for material \nparameters obtained by the fitting procedure . The continuous transition was  implemented as a quasi -continuous modulation of 𝑀𝑠 expressed  \nby an equation shown above the plots.  Transition width  (𝑤) of the complementary error function is designated in each plo t. b) dispersion \nrelation s of the spin -wave modes  for both  magnetization profiles  extracted from micromagnetic simulations . The density of states (represented \nby color scale) has been normalized to the maximum value . \n9 \n The micromagnetic simulations reveal, that in the case of the infinite  gradient in the magnetization  leads to abrupt increase of \nthe effective magnetic field (𝐵eff) on the edges of the waveguide . In the case of finite magnetization gradient,  the 𝐵eff profile \nis significantly changed . The m aximal value of  the 𝐵eff is lower and the central region is  signif icantly  broadened , as the  lower \ngradient of magnetization leads to  lower and less localized demag netizing  field.  The longitudinal spin -wave dispersion  \nextracted from the micromagnetic simulations is plotted  in Fig. 4 b). There is a very apparent change in the modal structure of \nthe dispersion when we compare the case with 𝑤=0 nm and the case where  𝑤=50 nm. As the gradient is introduced to the \nedge region , the boundary conditions for the dynamic magnetization are altered . The quantization of the modes with higher \ntransverse  mode numbers is  deteriorated . This leads to less effective  excitation of the higher order modes when compared to \ne.g. fundamental mode  (as can be seen from a lower  modal  density of states ). The first waveguide mode does not significantly \nchange even for 𝑤=200 nm. This is likely the  main mechanism  explaining  the absence of  any higher order waveguide modes  \nseen in the linescans.  This statement is  further  supported by careful analysis of the line scans  (see Fig. 2). In additional  analysis \nwe readjusted the equation (1) to allow for more spatial frequencies to be detected since we expect from the modal profiles of \nthe analytical model presented by equation (2) to excite multiple odd modes  [27,33] at certain frequency by the excitation \nantenn a. The analysis confirmed best agreement in the absence of any higher spatial frequencies.  \n In conclusion we studied  the spin-wave propagation in the waveguides prepared by FIB direct writing  into metastable \nfcc Fe 78Ni22 thin films . We have shown  that in these high -aspect ratio waveguides we can propagate spin wave s without the \npresence of an  external magnetic field , and with high group velocities reaching almost  6 km/s . This unique feature comes from \nthe possibility of the local  uniaxial magnetic anisotropy  control . The spin -wave dispersion relation has been  determined  by \nusing phase -resolved BLS microscopy  and the magnetic  properties  of the waveguides  were  extracted . The relatively large \nsaturation magnetization together with high  (controllable)  magnetic anisotropy render the material  suitable  for high frequency \nspin-wave circuits operational  even at zero external magnetic  field. Moreover, the extracted  material properties  of the system  \nwill allow  us to design mo re complex spin -wave devices by utilizing the possibility to spatially control both the saturation \nmagnetization and the direction of the uniaxial magnetic anisotropy in a single magnet ic structure.  Our unique approach paves \nthe way towards many other possibilities to develop and study spin -wave propagation in magnetization landscapes that are \nunattainable in any conventional magnetic system.   10 \n Acknowledgements:  \nThis research has been financially supported by the joint project of Grant Agency of the Czech Republic (Project No. 15 -\n34632L) and Austrian Science Fund (Project I  1937 -N20) and by the CEITEC Nano+ project (ID \nCZ.02.1.01/0.0/0.0/16013/0001728). Part of the work was carried out in CEITEC Nano Research Infrastructure (ID \nLM2015041, MEYS CR, 2016 –2019). L.F. was supported by Brno PhD talent scholarship.  \nReferences:  \n[1] S. Neusser , and D. Grundler, Adv. Mater. 21, 2927 -2932 (2009).  \n[2] A. V. Chumak, V. I. Vasyuchka, A. A. Serga , and B. Hillebrands, Nat. Phys. 11, 453 -461 (2015).  \n[3] E. Albisetti, D. Petti, G. Sala, R. Silvani, S. Tacchi, S. Finizio, S. Wintz, A. Calò, X. Zheng, J. Raabe, E. Riedo , and \nR. Bertacco, Commun. 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Dvo řák, R. Ritter, A. Buchsbaum, D. Stickler, H. P. Oepen, M. Schmid , and Peter Varga, J. Appl. \nPhys. 110, 024309 (2011).  \n[31] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia -Sanchez , and B. Van Waeyenberge, AIP Adv. 4  (10), \n107133 (2014).  12 \n [32] See Supplemental Material  at LINK  for additional information on the micromagnetic simulations.  \n[33] C. Kittel, Phys. Rev. 110, 1295 (1958).  "    },    {        "title": "1907.02318v1.Micromagnetic_modelling_of_magnetic_domain_walls_and_domains_in_cylindrical_nanowires.pdf",        "content": "  Micromagnetic modelling of magnetic domain walls  and domains  in \ncylindrical nanowires  \nJ.A. Fernandez -Rold an1, Yu.P. Ivanov2 and O. Chubykalo -Fesenko1  \n1Instituto de Ciencia de Materiales de Madrid , ICMM -CSIC. 28049 Madrid. Spain  \n2Department of Materials Science & Metallurgy, University of Cambridge, Cambridge \nCB3 0FS, United Kingdom  \n \nAbstract  \nMagnetic cylindrical nanowires are very fascinating objects where the curved geometry allows \nmany novel magnetic effects and a variety of non -trivial magnetic structures. Micromagnetic \nmodelling plays an important role in revealing the magnetizati on distribution in magnetic \nnanowires, often not accessible by imaging methods with sufficient details.  Here we review the \nmagnetic properties of the shape anisotropy -dominated nanowires and the nanowires with \ncompeting shape and magnetocrystalline anisot ropies, as revealed by micromagnetic modelling. \nWe discuss the variety of magnetic walls and magnetic domains reported by micromagnetic \nsimulations in cylindrical nanowires. The most known domain walls types are the transverse and \nvortex (Bloch point) doma in walls and the transition between them is materials and nanowire \ndiameter dependent. Importantly, the field or current -driven domain walls in cylindrical \nnanowires can achieve very high velocities. In recent simulations of nanowires with larger \ndiameter the skyrmion tubes are also reported. In nanowires with large saturation magnetization \nthe core of these tubes may form a helicoidal (“corkscrew”) structure. The topology of the \nskyrmion tubes play an important role in the pinning mechanism, discussed here  on the example \nof FeCo modulated nanowires. Other discussed examples include the influence of antinotches \n(“bamboo” nanowires) on the remanent magnetization configurations for hcp Co and FeCo \nnanowires and Co/Ni multisegmented nanowires.  \n \n1. Introduction  \n \nCylindrical magnetic nanowires constitute the most promising  nanostructures \nfor three dimensional nano -architectures  in future applications for  information \ntechnologies,  nanoelectronics, sensing etc . From  a more fund amental point of view they \ncreate  a fascinating possibility for  study ing nanomagnetism in  curved geometries  which \noffer s a plethora  of novel static and dyna mical properties  [Fernandez -Pacheco2017 , \nStano2019 ].  Nanowires  can be grown by  chemical routes or more frequentl y by \ninexpensive electrodeposition  in porous membranes w ith pore diameters  ranging from \nseveral tens to hundreds of  nanometers  [Vazquez2015 ]. The lengths of nanowires  are \ntypically in the interval between 10 0 nm and 100 m. This geometry presents a high \naspect ratio length/diameter.  \nAmong the properties of cylindrical nanowires  which make them very appealing \nfor future applications,  we ca n summarize the most relevant : • The elongated shape of cylinders i nduce s a natural axial a nisotropy  via the \nmagnetostatic energy effect which favors high stability  of axial magnetic states. \nThe curved geometry (via the exchange interaction) leads to an effect ive \nDzyalo shinkii -Moriya interaction [ Streubel2016 ] with no need to the intrinsic \none.    \n• The overall effect of the intrinsic interactions  and confinement in a curved \ngeometry is to promote  the formation of non -trivial topological structures in \nmaterials  with no need of intrinsic Dzyaloshinskii –Moriya interaction or intrinsic \nanisotropies [Streubel2016 ]. Multiple novel magnetization textures have  been \nreported for this geometry : toroidal, helicoidal,  vortex (Bloch point) domain \nwalls [ Hertel2016a ] and rece ntly, skyrmion t ubes  [Charilaou2018, Fernandez -\nRoldan2018a ].  \n• The dynamical vortex and skyrmion tubes may carry the Bloch point (a \n“hedgehog” skyrmion) [ DaCol2014, Jamet2015, Hertel2016a, Charilaou2018 ]. \nThe Bloch -point is a magnetic singularity with vanishing magnetization \n[Feldtkeller1965 ] which  possesses  some unique properties . The fascinating \nproperty of the Bloch point domain wall is that it is highly mobile  [DaCol2014 ], \nwith velocities higher than the transverse do main wall. Recently it  has been \nshown that  its motion leads to the emergent electromagnetic fields \n[Charilaou2018 ].  \n• Cylindrical nanowires are naturally magneto -chiral materials. F or example, the \nmagnetization mobility and stability  of the vortex domain wa ll depend on its  \nchirality [Hertel2013 , Hertel2016a ].  \n• Magnetic domain walls in cylindrical nanowires  lack the so -called Walker \nbreakdown, which consist s of a sudden drop of the  domain wall  veloci ty under \nthe action of applied magnetic field  or current , observed in  nanostrips \n[Hertel2013, Hertel2016a ,]. However, the domain wall dynamics  in cylindrical \nnanowires is limited by the spin -Cherenkov emission (spontaneous  emission of \nspin waves by the domain wall for velocities higher than the  minimum  spin wave  \ngroup  velocity)  [Hertel2016a ]. This, however, happens at  much higher velocities \n(up to 10 km/s) then the ones achieved in magnetic stripes.  Therefore, ultr afast \ndomain wall velocities can be expected . \n \n \nTherefore, the cylindrical geometry offers multiple possibilities for observing highly non -\ntrivial chiral magnetization structures in simple geometries.  The main problem for \nnanowire applications is the control  of domain wall dynamics:  nucleation, m obility, \npinning , etc. Only recently the development of the  X-ray magnetic circular dichroism \n(XMCD) technique , highly sensitive magnetic force microscopy  and electron holography  \nhas allowed to m ake first steps in observation of some of the se structures such as \ncircular magnetic domains [ Ivanov2016b, Bran2016 , Ruiz-Gomez2018 ] (extended vortex \nstructure with a core parallel to  the nanowire axis) . Micromagnetic simulations play a \ndecisive role in unveiling the observed magnetic contrast since they provide a \ncomplementary 3D information.  The proper dynamical measur ements on domain wall \nin cylindrical magneti c nanowires is a completely open  field of research  doing its steps \n[Ivanov2017 , M. Schöbitz2019 ]. Most of studies report pinned domain walls, such as  the Bloch  point  domain wall [ DaCol2014 ]. The strategies to pin  domain walls involve the use \nof nanowires with different compositions  [Kim2007 , Ivanov2016 , Garcia2018 , Bran2018 ] \nor geometries [ Dolocan2014,  Arzuza2017 , Chandra2014 , Palmero2015 , Mendez2018,  \nFernandez -Roldan2019 ] such as modulated nanowires  or nanowires with notches or  \nantinotches . Here micromagnetic simulations help  to design nanostr uctures with \ndesired properties  and understand the pinning mechanism . \n \n \n2. Hysteresis loops of magnetic nanowires  \n \nThe hysteresis loops of magnetic nanostructures are strongly affected by their geometry  \nthrough the shape anisotropy, and particularly, the aspect ratio  length/diameter. Fig. 1 \n(a) illustrate s the effect of the  shape anisotropy in an ideal individual nanowire of \nPermalloy . Permalloy  (Fe 20Ni80) has a van ishing  magnetocrystalline anisotropy , \ntherefore, the behavior  is dominated by the shape anisotropy effect . A preferential axial \norientation of the magnetization  resulting in a squared loop  and a high remanence  for \nthe field applied along the nanowire axis is observed.  The state before the switching (see \nFig.2a below) is the  two so-called open vortex structure, characterized by the \npredominant magnetization component along the nanowire direction but already \ndeveloping a circular structure for the perpendicul ar components at the nanowire  ends. \nThe magnetization switching  takes pl ace in a unique Barkhausen jump.  On the other \nhand, for a perpendicular magnetic field the hysteresis loop of a single nanowire  exhibits \na completely a nhysteretic (rotational) behavior, i.e. the area enclosed between  both \nbranches of the  hysteresis loop vanishes . Therefore, for a single nanowire  of Permalloy \nthe magnetic properties and the magnetization reversal process are fully  determined by \nthe shape anisotropy induced by the cy lindrical geometr y (magnetostatic  energy).  \nIvanov et al. [ Ivanov2013 ] showed that the magnetization reversal in  other  nanowires \nwith fcc magnetocrystalline anisotropy in either polycrystalline or single crystalline \nnanowires is similar to Permalloy ones because it is  also predominantly determined by \nthe shape anisotropy owing to its large value compared to  the magnetocrystalline \nanisotropy strength [ Ivanov2013 ]. In the array  of nanowire  complex features may arise \nfrom the inter -wire dipolar interactions th at decrease the effect of the shape anisotropy  \nand introduce a “thin film” demagnetizing effect, see Ref.  [Vivas2013, Fernandez -\nRoldan2018b , Stano2018 ]. \n \n \n \nFigure 1. Simulated hysteresis loops for individual  nanowires  of (a) Permalloy  with zero \nmagnetocrystalline anisotropy  (b) Co hcp with perpendicular magnetocrystalline  anisotropy  4.5x105 J/m3. \nBoth nanowires have  40 nm diameter and 2 m length.  Figure adapted from [Ivanov2013].  Materials \nparameters are presented in Table I.  \n \nThe introduction of a  strong  magnetocrystalline anisotropy , non-collinear with \nthe nanowire axis,  changes the hysteresis cycle due to the interplay between the \ninvolved energies . The uniaxial transverse anisotropy reported in Cobalt hcp nanowires \nexhibits a st rong competition with the shape anisotropy  [Ivanov2013 , Ivanov2013b ] \nalmost completely suppressing it  and producing a very soft behavior . For illustration, \nFig. 1 (b ) present s the simulated hysteresis loop  of an individual Co nanowire with \nanisotropy easy axis perpendicular to the nanowire axis. Note also that the saturation \nmagnetization of Cobalt is 75% higher than that of Permalloy, indicating a very high \nshape anisotropy. The inclined hysteresis loops for  the single Co hcp nanowire in both \nparallel and perpendicular applied field s confirm that the magnetocrystalline anisotropy \nis strong enough to compensate the m agnetostatic energy and to promote  a drastic \ndecrease of the coercive field and  a small remanence . The remanent state is \ncharacterized by the formation of a vortex state along the whole length of the nanowire \n[Ivanov2013 , Ivanov2013b ], see Fig.2  (b) below . The perpendicular field hysteresis cycle \nresembles that of the vortex in soft magnetic dots characterized by the nucleation and \nannihilation of the vortices structure.  The magnetostatic interactions  between \nnanowires  also significantly modify the hysteres is cycle due to the “thin film” shape \nanisotropy effect  [Ivanov2013, Fernandez -Roldan2018b ]. \n \n \n \n \nFigure 2.  Magnetic configurations in cylindrical nanowire (a) precursor of the vortex domain wall (also \nknown as open vortex structure) in shape anisotropy dominated nanowire s. The image corresponds to \nthe Permalloy nanowire at fields close to the switching one (b)  the remanent state of hcp Co nanowires \nwith easy axis perpendicular to the nanowire, showing two vortex domains with opposite chiralities.  The \nfield was applied parallel to the nanowire axis. Images taken from Ref.  [Ivanov2013 ]. The same \ngeometrical and m agnetic parameters as in Fig.1 are used.  \n \nExperimental evidence s and micromagnetic simulations confirm that  the properties of \nthe hysteresis loop s of nanowires generally become independent of the  nanowire length \nfor aspect ratio values larger than 8 − 10 [ Bance2014 , Vivas2011] . As an illustration , \nFigure 3 displays the  calculated  hysteresis loops of Permalloy and Co fcc  nanowires with  \ndifferent lengths.  For short nanowires , a lower coercive field is observed and the \nhysteresis  loop s are strongly affected by  small variations of the length. The anisotropy \nof the fcc phase  of Co has a lso a remarkable effect in the inclined  hysteresis loops  for \nshort nanowires . In fact, 3 0 nm length Co nanowire is a pillar  which presents the \ncharacteristic vortex hysteresis cycle.  On the contrary, for nanowires with large aspect \nratios, the coercive  field and the remanence become  practically independent on  the \nlength.  Note that  in this ideal case  in nanowires with lengths smaller than ca . 70 nm \n(estimated for fcc Co ) the open vortices with the same chiralities  are formed at the \nopposite ends due to  the minimization of the  magnetostatic interaction  between them . \nWhe n the length is increasing, the  interaction  of structures at opposite ends  become s \nnegligible  and the vortices  at the opposite ends are formed with the opposite chi ralities  \nas in Fig.2b . The latter happens due to the influence of the  nanowire  internal  \ndemagnetizing field (with an inward and outward component  at opposite  nanowire  \nends), resulting  in opposite magnetic torque s. The latter behavior is common in \nsimulations for all nanowires in the absence of defects and granular structure.  \n \nFigure 3. Simulated hysteresis loops for individual nanowires with several lengths without an d with  \nmagnetocrystalline anisotropy. Figures correspond to individual nanowires of (a) permalloy  and (b) Co fcc \nwith  a 30 nm diameter. Parameters used for simulations  can be found in Table I.   \n \nThe nanowire diameter has a more critical impact on its  hysteresis properties \nthan the  length. As it is presented in Fig.4, t he coercive field  strongly decreases with the  \nincrease of  the NW diameter which reflects the change of the reversal mode from that \nof the transverse domain wall to the Bloch point (vortex) domain wall [ Ivanov2013 ], see \nnext section . At the  same  time, the remanence has a small dependence on the diameter, \ndefined by t he formation of the open vortex structures for shape anisotropy -dominated \nnanowires , see Fig.2a . This dependence becomes more pronounced for nanowires with \nhigh saturation magnetization value such as Fe -based nanowires  (for example Fe bcc or \nFeCo nanowires ) [Ivanov2013 ].  \n \n \n \nFigure 4.  (a) Coercive field and (b) remanence dependence on the nanowire diameters for \nindividual Permalloy  nanowires . \n \n \n3. Magnetic d omain s and domain walls in straight magnetic nanowires  \n \nAs it is indicated in the previous section, shape anisotropy -dominated magnetic \nnanowires  are in the (almost) single domain state in the remanence and  demagnetize  \nvia the domain wall propagation  [Forster2002a,  Hertel2002, Forster2002b , Ivanov2013 ]. \nThe type of domain wall by which the nanowire demagnetizes is determined by the \nnanowire geometry and material [ Ivanov2013 ].   More concretely, only nanowires with \nvery small diameters demagnetize via the transverse do main wall propagation (see \nFig.5a ). Depending on t he relative orientation between domains, the wall is called tail -\nto-tail or head -to-head transverse domain wall [ Jamet2015 ]. The transverse domain wall \ncarries a magnetic charge in volume due to the divergence of magnetization \n[Jamet2015 ]. \n \nLarger -diameter  (or saturation magnetization) nanowires demagnetize  via the Bloch \npoint (vortex) domain wall propagation , see Fig.5b . The vortex domain wall presents an \nazimuthal magnetization component with the flux -closure  structure . In vortex \nstructures, the relative orientation between its azimuthal component and the core \ndirection defines the chirality of the vortex . The direction of the core indicates the \npolarity.  In the case of Co hcp  nanowires with perpendicular diameter , for diameters \nequal and larger than 60 nm  the nucleation  of vortex structures along the whole \nnanowire length  takes place . The vortex domain wall typically  carries a mathematical \nsingularity called Bloch point  [Forster2002a , Hertel2004 , Kim2013  ], which is s  unique \ntopological defect in ferromagnetic materials [ Wartelle2019 ], where the magnetization \nvanishes as displayed in Figure 5(b). Therefore,  it eventu ally received  the name of Bloch \npoint wall to differentiate it from the (surface) vortex wall observed i n ultrathin \nnanostrips [ Jamet2015 ]. This singularity has been already experimentally observed in \nnanowires by XMCD -PEEM technique [ DaCol2014 ]. The transition diameter between  the \ntwo reversal modes decreases as the nanowire saturation magnetization value \nincreases, namely for permalloy this happens for diameters ca. 40 nm, for Co fcc \nnanowires for diameters ca. 20 nm, and the Fe -based nanowires generally demagnetize \nvia the vortex domain wall propagation  [Ivanov2013] . The Bloch point domain wall in \nreality c arries a 3D hedgehog skyrmion, see Fig.5b. Note that the vortex core is not \nnecessary lo cated in the nanowire center, but  may form a spiral  [Fernandez -\nRoldan2018a] . Finally, the proper Neel -type hedgehog 3D skyrmion structure  can be also \nformed  as a stable configuration  in small diameter nanowire s with very large saturation \nmagnetization  such as FeCo , see Fig. 5c.   \n \nThere is another name for a  frequently  metastable domain wall  for nanowire diameters  \nclose to the transition value between the vor tex and the transverse domain walls , \ndepicted in Figure 1.14(e) . This domain wall type presents characteristics of both \nprevious wall types and is called asymmetric transverse domain wall [ Ferguson2015 , \nJamet2015 ], see Fig.5e. In reality it corresponds to the situation when the vortex core \ngoes out  from the nanowire center  to its surface [S. Komineas1996 ]. Together with the \nvortex on the surface of nanowire, there is an anti -vortex structure opposite to it  to \npreserve the topological charge . Recently, the topological transformation from \nasymmetric  transver se domain wall to a vortex domain wall in cylindrical nanowires,  \nmediated by Bloch point injection , has been reported  [Wartelle2019 ]. Other t opological \ntransformation between magnetization textures are  also theoretically possible by Bloch \npoint ejection [ Wartelle2019 ]. \n \n \n \nFigure 5. Domain walls in  nanowires  (Permalloy) . (a) Transverse domain wall (TDW). (b) Vortex -Bloch point \ndomain wall. (c) Hedgehog  Bloch point  separating two magnetic domains . (d) Skyrmion tube from Ref. \n[Charilaou2018 ] formed during the magnetization reversal . (e) Asymmetric transverse (vortex) domain \nwall from Ref. [ Ferguson2015 ]. Cross -sections indicate the inner magnetization of the domain walls at the \nmarked positions .  \n \nFor na nowires with larger diameters ca. 100 nm another reversal mode called recently \na skyrmion tube is possible, especially when the saturation magnetization is high.  \nDifferently to the nanowires with smaller diameter it does not carry a Bloch point but \nconsis ts of a core -shell structure. The outer shell first rotates towards the field direction \nwhile the nanowire core remains magnetized in the opposite direction, see Fig.5d. The \nskyrmion tubes were noticed [ Charilaou2018 ] in Permalloy  during the dynamical \ndemagnetization process. The magnetization in nanowire cross -section indeed has the \nsame topological structure as the classical Bloch skyrmions but here induced by the \ncurvature of the nanowire in the absence of the Dzyaloshinskii -Mor iya interaction \n[Dzyaloshinsky1958 , Moriya1960 ]). During further demagnetization, the inner skyrmion \ntube core breaks and a hedgehog skyrmion (Bloch point) is formed. Its propagation \ncompletes the demagnetization process. We should stress that these struct ures in \nPermalloy are dynamical. In FeCo nanowires (high saturation magnetization) they are \nshown to be stable at negative applied fields [ Fernandez -Roldan2018a ]. \n \nMost of domain walls are dynamical objects, however, they can  separate different \ndomains and  can be pinned at the defects, thus being stable structures.  As many \npromising applications rely on the  nucleation and control of the magnetization \nprocesses and magnetic domain walls, there is a stron g effort in the  nanomagnetism  \ncommunity to understand a nd control their pinning mechanisms [ Kim2007, Ivanov2016,  \nGarcia2018,  Bran2018, Dolocan2014, Arzuza2017, Chandra2014, Palmero2015, \nMendez2018, Fernandez -Roldan2019 ]. \n \nConventionally, a magnetic domain has been considered as a uniformly magnetized  \nregion in  the mesoscale in thermodynamic equilibrium. Then, a magnetic state was  \ndefined as a collection of magnetic domains. However, the 3 D nature of nanowires  and \nthe emerging properties at the nanoscale, make us to redefine the concept of magnetic  \ndomain s as stable configurations of the magnetization which may lack of homogeneity  \ndue to the nanowire curvature.  For example, we define  the part of the nano wire in the \nvortex state as  a vortex domain. Therefore, some materials with large saturation \nmagnetization and  high transverse magnetocrystalline anisotropy (as Co hcp  nanowires ) \nmay show a vortex configuration  along the nanowire length as the ground state \n[Ivanov2013 b, Ivanov2013 ]). In multisegmented nanowires, d epending on the segment \nsize and material, a single vortex, two vortex or multivortex domain states  (see Fig. 6 \nbelow)  have been reported  [Ivanov2013 b, Bran2016, Ruiz -Gomez2018] . For example, \nFig. 2b shows a two -vortex domain, with two opposite vortex chiralities.  Depending on \nthe micromagnetic param eters  and crystalline structure of the nanowire  either  vortex \ndomains, see Fig.7  or transverse domains, see Fig. 7 a or their  combination is possible , \nsee Fig.7 b.  The latter  combination has been observed in Co85Ni15 nanowires by XMCD  \nand confirmed by micromagnetic simulations  [Bran2017] . This shows that the domain \nstructure is not unique and for some parameters set transverse and vortex domains are \nmetastable states.  Another possible domain configuration  discussed in Ref.  \n[Lebecki201 0] is alternating vortex structures with core on the nanowire surfaces  \nformed between the transverse domains . \n \n  \n \nFigure 6. A multivortex  domain state with opposite vortex chiralities along the nanowire. The calculated \nimage corresponds to Cobalt nanowire with perpendicular anisotropy . Figure extracted from Ref. \n[Ivanov2016b] . Paramenters used in simulations are listed in Table I.  \n \n \n \n \nFigu re 7.  Calculated magnetic domain pattern in Co 85Ni15 nanowires (a) transverse domains (b) alternating \ntransverse and vortex domains.  Micromagnetic parameters used in modelling are found in Table I.  \n \n4. Domain wall velocity in cylindrical magnetic nanowires  \n \nOne of the most fascinating characteristics  of cylindrical nanowires are the dynamical \nproperties induced by their cylindrical symmetry and magneto -chiral effects already \nmentioned in previous sections. The propagation of the domain walls driven by applied  \nmagnetic fields and currents in nanostrips is limited in velocities of the order of 100 m/s  \n(in the absence of the Dzyaloshinkii -Moriya interactions) . The steady velocity achieved \nwith the driving force is initially increasing (with  the increase of the  field or current) and \nthen drops  at the so -called Walker b reakdown field. This phenomenon limits the speed \nof fut ure information  technologies. Theoretical considerations demonstrate that in \ncylindrical nanowires the Walker breakdown  [Schryer1974 ] does not  exist due to the \ncylindrical symmetry [ Thiaville2006 , Yan2011 ] or at least it is displaced to higher velocity \nvalues.  Fig. 8  presents the calculated domain wall velocity in a cylindrical Permalloy \nnanowire as a function of the applied fiel d showing veloci ties almost up 10 km/s, \nsaturating at high field values. This saturation occurs due to the so -called spin -\nCherenkov effect [ Yan2011, Hertel2016a ] when the domain wall loses energy with the \nspin-wave emission.  \n \n \n \nFigure 8.  Calculated velocity of a vortex (Bloch point) domain wall in a Co fcc nanowire with diameter of \n40 nm showing saturation due to the spin wave emission.  Micromagnetic parameters used in modelling \nare summarized in Table I.  \n \nAs for the influence of the curr ent, calculations indicate that the direct motion of the \nvortex domain wall with current requires too high current and the assistance with \napplied field will be probably necessary.  The Oersted field being circular in nature can \nchange of the chirality of the vortex domain wall [ M. Schöbitz2019 ]. \n \n \n \n \nFigure 9 . Micromagnetic model for  domain wall propagation for s egmented Co/Ni nanowire  under \napplied magnetic field and current  at two different times . Dashed lines indicate the interph ase between \nCo and Ni segments .  Modelling parameters are summarized in Table I.  \n \nFigure 10 . (a) Domain wall velocity under different current densities J for various external magnetic fields \nvalues with respect to the de -pinning field H dep simulated at 0 K temperature (arrows show the critical \ncurrent density for DW depinning). b Dependence of the critical current density on the external magnetic \nfield values at 0 K (open circles), at 327 K (filled circle) and experimental data at 327 K (fi lled square). The \nline shows linear approximation towards zero field DW motion. Images from Ref. [Ivanov2017] . \n \nIn the Ref. [ Ivanov2017 ] the current -driven VDW dynamics in cylindrical nanowires were \nstudied by  in-situ Lorentz TEM in the presence of the ext ernal magnetic field to promote \nthe DW dynamics for the lower value of the applied current densities. The cylindrical \nnanowire of 80 nm diameter, 20 m length and consisting of segments of Co and Ni of \n700 nm each was used. Additionally, the experiments ha ve been performed only for the \ncase of the constant current which is ruled out the possibility to measure the DW \nvelocity. The interfaces between segments of different magnetic materials was used as \nan effective site for the domain wall  pinning (more detai ls in the section “Domain walls \nin multisegmented Co/Ni nanowires”). Consequently, the value of the external magnetic \nfield needed to nucleate DW in nanowire, the field needed to de -pin the DW from the \ninterface between the segments as well as the  critical  current density for the DW \npropagation  were determined . Those parameters were compared with the ones \nobtained by the micromagnetic simulations  of a 80nm diameter nanowire with 2 \nsegments of 700 nm each, see model in F igure 9  . The direction of the magneto crystalline \nanisotropy was considered in agreement with TEM data (at 850 with respect to the NW \naxis for the Co segment and {220} texture for Ni segments. The spin polarization of 0.42 \nreported for planar structures [ Boulle2011 ] was used.   The simulations  show that these \nnanowires demagnetize via the vortex domain wall propagation.  The simulations has \nbeen done for 0 K and for finite temperatures (extracted from the experiments).  \n \nFig. 10  presents a simulated velocity data for the domain wall propagation f rom Co to Ni \nsegment as a function of  the current density J for several applied field values.  It shows \nthe existence of a critical value of J c above which current -driven DW motion occurs, and \nthis value depends on the applied external magnetic field. The m aximum value for the \ndomain wall speed was found to be as high as 600 m/s in agreement with theoretical \npredictions. Simulations show that the critical density of the spin -polarized current at 0 \nK decreases almost linearly with the increasing of the external magnetic field (Fig. 10 b). \nNote that the simulated value of J c at 0 K is more than one order of magnitude larger \nthan the experimental data. The J c value calculated for 327 K  (the final state  is shown as \na triangle in Fig. 9 b) is  very close to the  one obtained expe rimentally (red cross in Fig. \n10b). Most probably there are additional contributions to the current driven 3D DW \nmotion which were not taken into account  in the micromagnetic model. The complete \nunderstanding  requires more experiments wit h nanoseconds DW dynamics and \nadvanced ultra -fast imaging technique.  \n \n \n \n5. Domain s and domain walls  in nanowires with geometrical modulations  \n \n \nDomain -wall-based technologies as the racetrack memory [ Parkin2015 ] require a \nprecise control over the magnetization and domain wall location. In order to m anage \nthe positioning of domain walls along the nanowire length , multiple strategies have \nbeen proposed to achieve the local pinning of the domain wall. The fundamental idea is \nthe creation of potent ial w ells and barriers where the domain wall may eventually get \npinned. Geometrical strategies consist of creating cons trictions along the nanowire \nlength, artificial notches, antinotches, defects or diameter modulations which act as  \npinning sites for the domain wall  [Dolocan2014, Arzuza2017, Chandra2014, \nPalmero2015,  Mendez2018, Lage2016 ]. In addition, variations of the material along the  \nnanowire length or at specific locations  [Kim2007,  Ivanov2016,  Garcia2018,  Bran2018]  \nalso offer the possibility of promoting domain wall pinning.   Micromagnetic simulations \nplay here a decisive role since they allow to rapidly vary geometrical and material \nparameters with the aim to design  and understand  pinning. Here we discuss two \npossibilities of geometrical constri ctions based on modulations in diameter  (with \nalternating diameters ), see Fig.11 a, and geometrical notches , see Fig.11 b in cobalt -\nbased nanowires. The micromagnetic parameters correspond to Co and FeCo, see Table \nI.  \n \n \nFigure 11. Geometry of  simulated   (a) nanowire with geometrical modulations  and (b) “bamboo” \nnanowires . \n \n \nMaterial  μoMs(T) Aex(pJ/m)  lex(nm)  Crystal structure  Anisotropy \ntype  Easy axis  K1(kJm-3) Ksh(kJm-3) \nPermalloy (Fe 20Ni80) \n[Ivanov2013]  1.0 10.8  5.2 Polycrystalline  - - 0 199 \nCo(111) [Ivanov2013]  1.76  13.0  3.3 FCC  Cubic  [1,1,1] || nanowire axis  -75 616 \nCo(100) [Ivanov2013]  1.76  13.0  3.3 HCP Uniaxial  At 75 º with nanowire axis  450 616 \nCo-hcp [Ivanov2013,  \nMoreno2016 ] 1.76  30.0  4.9 HCP Uniaxial  At 75 º with nanowire axis 450 616 \nFe30Co70 [Bran2013 ] 2.0 10.7  2.6 BCC Polycrystalline  Cubic  A random in -plane  \neasy axis component  \nconsidered in each grain  10 796 \nNi(111) [Ivanov2013]   0.61  3.4 4.8 FCC Cubic  [111] || nanowire axis  -4.8 74 \nCo85Ni15 [Bran2017,  \nKronmuller1997,  \nMoreno2016,  Vega2012, \nMoskaltsova2015 ] 1.60  26.0  5.1 HCP Uniaxial  At 88 º with nanowire axis  350 509 \nCo65Ni35 [Bran2017,  \nKronmuller1997,  \nMoreno2016, Vega2012, \nMoskaltsova2015 ] 1.35  15.0  4.5 HCP Polycrystalline  Uniaxial  At 65 º with nanowire axis  260 362 \nCo35Ni65 [Bran2017, \nKronmuller1997,  \nMoreno2016 ] 1.01  10.0  5.0 FCC Cubic  [111] || nanowire  axis 2 203 \n \nTable I . Materials parameters used in mic romagnetic modelling , microstructure and derived quantities: Saturation magnetization μoMs, exchange stiffness  Aex, exchange \nlength  lex=(2 A ex / μoMs2) 1/2, crystal structure, magnetocrystalline anisotropy type, easy axis direction , first magnetocrystalline constant K 1 and sh ape anisotropy K sh= μoMs2/4. \nSanturation magnetization and exchange stiffness from CoNi alloys have been obtained  by linear interpolation with the Co content of the alloy in Ref [ Bran2017 ]. \nThe modulated nanowires were considered made of FeCo material with high saturation \nmagnetization value. To represent realistic experimental situations, the granular \nstructure was modelled by Voronoi construction and several disorder distributions were \nconsidered. H ysteresis loops  for modulated nanowires  under a magnetic field, applied \nparallel to the symmetry axis of the  nanowire depend  considerably on the particular \npolycrystalline nanostructure  of the nanowire. As  a main characteristic, the hyste resis \nloops become less squared, the remanence  increases and the switching field decreases  \nas the minor dia meter of the nanowire becomes larger , see Fig.12 . The demagnetization \nprocess always starts in the segments of the larger diameter by means of the nucleation \nof the open vortex structures at the constriction. These structures depin from the \nconstriction and propagate first inside the wide diameter segments and later inside the \nsmall diameter segment.  When the difference  between diameters is small the \npropagation in both segments , although consequent and not simultaneous , take s place \nat the same field value  (’weak’ pinning ).  In the opposite case of large diameter \ndifference (“strong pinning ”), the hysteresis loop consists of a propagation stage inside \nthe large di meter segment and depinning (switching) of the magnetization  from the \nconstriction to propagate  inside a small diameter segment occurring  at a different field .  \n \n \n     \nFigure 12. Example of simulated hysteresis  cycles for  individual  modulated FeCo nanowires with three \ndifferent minor diameters for some particular disordered distribution.  Modelling parameters are collected \nin Table I.  \n \n \nImportantly t he open vortex structur es (similar to presented in Fig2a ) at the \nconstrictions between  large and small segments  are formed with arbitrary chirality . \nDifferent realizations of granular structures promote different chirality patterns, \ntherefore in one particular segment v ortices at the opposite ends may be  formed with \nthe same or different ch irality. These vortices propagate towards the center of the large \ndiameter  and span almost the whole segment . In the  case of opposite chirality,  the \npropagation leads to  a topologically  protected structure which  requires a large field to \nannihilate . As the  field progresses , the spins in the outer shell rotate towards its \ndirection fo rming a skyrmion tube, see Fig.13  b. Indeed,  the magnetization structure in \nthe nanowire cross section is formed by a magnetic skyrmion with the core pointing \nagainst magnetic f ield direction and the shell pointing parallel to it.  \n \nIn order to prove that the structure  is indeed a skyrmion,  we evaluate in Fig.15  the \ntopological  charge  as a surface integral evaluated in the nanowire cross -section along \nits length  \n 𝑄=−1\n4𝜋∫𝑚⃗⃗ ∙(𝜕𝑚⃗⃗⃗ \n𝜕𝑥×𝜕𝑚⃗⃗⃗ \n𝜕𝑦)𝑑𝑆𝑆 \n \nFig.14  presents the topological charge along the nanowire for two disorder realization  \nand large diameter difference . In the case of Fig.14 a the vortices at the ends of the large \ndiameter segments are formed with the same chirality, as the field is progressing they \nare transformed into the skyrmion tubes with topological charge almost one in the \ncenter of the  nanowire. In the case of Fig.14  b the vortices in the left segment were \nformed with the opposite chirality , when the field  is increasing in the opposite direction \nthe two tubes meet and the topological charge in the middle of this segment is almost \nzero, reflecting the topological protection. This situation needs additional field in order \nto annihilate and complete the magnetization reversal.  \n \n \nFigure 13. (a) The surface magnetization  configuration of the multisegmented  FeCo nanowire showing \nthe helicoidal (“ cork -screw”) structure (b) The magnetization distribution in the cross sections of the larger \nsegment at indicated positions, showing the skyrmions with displaced core .  \n \n \nFigure 14. Topological charge for magnetization structure in the cork -screw regime for multisegmented \nFeCo nanowires, evaluated along the nanowire. The letter C stands for the clock -wise chirality of the \nvortex /skyrmion  formed at the constriction  or end of the nanowire  while the letter A denotes an anti -\nclockwise chirality.  The colored insert show the surface magnetization distribution with a chiral structure.  \n \nOnce  the demagnetization is completed in the large diameter segments, the skyrmion \nstructure remains pinned at the constrictions. Importantly, the skyrmion center along \nthe segment is not positioned in the center of nanowire s but forms  a spiral, see Fig. 13 a \nand inserts in Figs 14  called in Ref.[ Fern andez -Roldan2018a ] a core -screw. The \noccurrence of spiral for vortex/skyrmion tube center is  a consequence of the \nminimizatio n of magnetostatic charges and is a precur sor of the magnetic domains. \nThese charges are typically created at the constrictions and are re -distributed along the \nnanowire length  to minimize the energy . Ref.[Arrott2016 ] argues that these charges are \nproper to all magnetic nanowires.  But t hey should be especially visible in nanowires with \nlarge saturation magnetization where the minimiza tion of magnetostatic energy may \ncreate structures following this pattern. In our particular case the presence of the \nconstrictions increases the effect since the vortex/tube core is  additionally  displaced \nfrom the center  at the constriction  to minimize th e energy. In simulations the spiral \n(“corkscrew”) may also appear  in straight magnetic nanowires with diameters ca. 100 \nnm and larger, especially for nanowires with large magnetization saturation value. \nSimilar topologically protected structures  and the co re-screw surface structure  have \nbeen recently reported in FeNi nanowires with chemical barriers [Ruiz -Gomez2018]. \nRecently, they were also reported in simulations of magnetization  distribution  and \nobserved by Kerr microscopy  in microwires Ref. [Chizhik2018 ]. However, experimental \nverification of the “core screw” is needed and requires 3D imaging techniques.   \n \n \nFigure 15.  Simulated magnetic configurations of (a) FeCo and (b) Co bamboo nanowires at remanence.  \nThe figures show a half section from the total len gth of the wires, containing an edge and two  modulations \nmarked by green arrows. Below each nanowire, transverse cross sections at the marked  regions along the \nlength  are presented . The black arrows denote the projection of the magnetization in the  corresp onding \nsection while the red -white -blue color gradient of the background corresponds  to the longitudinal \nmagnetization . Figures readapted from Ref . [Bran2016].   Modelling parameters are collected in Table I.  \n \n \nFig.15  present s simulated magnetization config urations in  a “bamboo” ( with \nantinotches) nanowire with FeCo  (a) and hcp Co (b) compositions  in the remanent state . \nIn the case of FeCo nanowire we  observe an overall axial orientation of the magnetic \nmoments (with the exception of the regions close to the  antinotches) that is principally \ndetermined by the shape anisotropy being larger than the magnetocrystalline \nanisotropy of FeCo. The magnetic structure corresponds to a nearly uniform axial \ndomain state with open vortices at each nanowire end. At  the posi tions of an  antinotch \n(pointed by green arrows), we observe a smooth curling of magnetization at the surface \nof the nanowire which occurs in order to minimize the formation of magnetic poles at \nthe modulation . This specific behavior where the magnetization  is deviated from its axial \norientation is in agreement with the periodic contrasts observed in MFM and XMCD \nmeasurements [ Berganza2016 , Bran2016 ] for the FeCo -based “bamboo” nanowires .  \n \nOn the other hand, the magnetic configuration of the individual Co bamboo nanowire  \nat remanence in Fig. 15 (b) displays a more complex structure. Vortex structures with  \nalternating chiralities are observed along the nanowire length as in the case of straight \nCo hcp nanowires , see  Fig.6 . The vortices show an elongated core as a result of the \ndistortion of its shape along the easy axis of the high transverse magnetocrystalline \nanisotropy and displacement i n the direction normal to the easy axis. In addition,  the \nvortex cores are no t at the nanowire center but again form a spi ral structure , similar to \nFig.13 a. Finally, m icromagnetic calculations inform about the lack of correlation \nbetween the multi -vortex structure and the geometric modulations.  These nanowires \nwere studied by MFM and XMCD in Ref.  [Bran2016 ] and micromagnetic simulations here \nprovided complimentary information  helping to interpret the images.   \n \n5. Domain walls  and domains  in multisegmented Co/Ni nanowires.  \n \nAnother approach to create the energy landscape needed to control the DW along the \nlength of the cylindri cal nanowire was reported in the Ref s. [Ivanov2016 , Berganza2017 ] \nwith alternative materials aimed at creating periodically modulated anisotropy . Co and \nNi were chosen due to distinct difference in the magne tization saturation and magneto -\ncrystalline anisotropy [ Ivanov2013 ]. Additionally, it has been shown the nanowires of \nonly Co or Ni are grown perfectly by the electrodeposition in alumina templates  with \nlarge grain size for Ni and being monocrystalline for Co [ Vilanova2015 , Ivanov2013b , \nIvanov2016b ] \n \nShort -segmented nanowires  (with lengths ca. 1 00 nm)  may be used to realize  the \nconcept of the “bar code” see Fig.16 .  \n \n  \nFigure 16 . (a) Magnetic Force microscopy images of a multisegmented 80 nm diameter Co/Ni NW  showing \npinned magnetic structures (b ) Magnified virtual bright field differential phase contrast  transmission \nelectron microsopyc image of the vortex domain wall  pinned at the Co/Ni interface. The arrows show the \ndirection of the magnetic field, which is constructed from the raw data of two orthogonal components of \nthe magnetic field. The dashed lines show the position of the Co/Ni interfa ce (purple) and domain wall  \n(red). (c ) Mi cromagnetic simulation of the pinned domain wall  structure  at the Co/Ni interface  (color \ncones show magnetization direction; black arrows are calculated stray fields).  Modelling parameters are \ncollected in Table I.  \n \nMicromagnetic simulation  (Fig.16 c) of the nanowire consisting of two  segments of Co \nand Ni show  that the Co/Ni interface is able effectively pin the domain wall during \nmagnetization reversal process [ Mohammed2016 ]. The simulation proceeds from a \nsaturated state of the multisegmente d NW. Upon decreasing the field, open vortex \nstates form at the ends of the segments at the remanence. During further increasing of \nthe external field in the opposite direction of the initial magnetization, a subsequent \nreversal of magnetization of the Ni segment was observed. At this point of reversal of \nthe multi segmented NW s, we note that the Ni segment has switched its magnetization \nwhereas the Co segment remains relatively unchanged. This differential  switching of the \nsegments lead to the pinning of a  domain wall at the Co/Ni interface. The MFM and TEM \nexperiments shown on the Figure 16  (a,b)  confirmed the pinning efficiency of the Co/Ni \ninterfaces. The multisegmented Co/Ni nanowires demonstrated periodic pattern of the \nstrong stray fields emanating from interfaces between Co and Ni. Such  field act s as a \npinning center for the DW.  \n \nAnother k ey point is the nature of the domain wall  studied in the multisegmented Co/Ni \nnanowires. The structure of the domain walls  inside 80 nm Co or Ni nanowire should \ncorrespond to Bloch point vortex domain wall [ Ivanov2013 ]. Figure  16(c) shows the  3D \nspin config uration of the domain wall  in Co/Ni nanowire extracted from the \nmicromagnetic simulations together with the calculated stray fields. The very good \ncomparison with the experimentally observed ma gnetic induction map (Figure 16 b) is \nclear.  \n \nA different possibility is offered by longer segments of Co and Ni nanowires.  \nIn Ref.  [Berganza2017 ] CoNi/Ni multisegmented nanowires  of ca. 1\nm segment  lengths  \nwere studies by means of the MFM and interpreted by means of micromagnetic \nmodeling . Importantly, Co 65Ni35 segments were grown in the fcc structure on fcc  Ni. \nHowever, the difference in the saturation magnetization promotes vortex structure in \nCoNi segments wile Ni is always in the axial domain state, see Fig. 17 below.  Importantly, \nthe transition between single vortex and multivortex state in CoNi can be controlled by \nfields perpendicular to nanowire axis.  \n \n \n \nFigure 17.  One of the possible magnetization configurations  simulated in Co 65Ni35 /Ni multisegmented \nnanowires showing axial magnetization configurations for Ni and multivortex state for CoNi segments. \nTwo segments of 1µm long CoNi  separated by a Ni segment of 500 nm were considered. The diameter is \n120nm and we have considered the fcc texture  both in Ni and CoNi , grown in the <110> direction . \n \n \n \nConclusions  \n \nMicromagnetic simulations offer important possibilities to design magnetic cylindrical \nnanowire with the aim to control the type of magnetic domains, domain wall as well as \ntheir pi nning mechanisms which may serve as a basis for future nanotechnological \napplications of these objects. They also offer a complementary tool for interpretation of \nmagnetic imaging techniques and are typically in agreement with them provided the \ncrystallogr aphic structure is known in sufficient details.  Domain walls type in nanowires \nare very rich and probably not all of them are reported and studies yet. Their occurrence \nis magnetic and geometrical parameters dependent. Cylindrical geometry produces a \npleth ora of non -trivial topologies for magnetic structures which will be the subject of \nfuture studies.  The change of magnetic parameters and  geometries  of additional \npossibilities to control domain wall structures and their pinning.  The topological effects \nshould play an important role in the magnetic response of nanowires. Here we have \npresented an example of the magnetization pinning in high -saturation FeCo \nmultisegmented nanowire where skyrmion tubes with a curling c ore structure are \nformed. Other  non-trivial structures will probably be discovered in the future.  \nImportantly, domain walls in cylindrical nanowires are expected to have very high \nvelocities, comparable to those expected from antiferromagnetic spintronics \n[Gomonay2016 ]. 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Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700106, India\n3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n4Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan\n5Diamond Light Source Ltd., Diamond House, Didcot, Oxfordshire, OX11 0DE, UK\n6Institute of Engineering and Management, Sector V, Salt Lake, Kolkata 700091, India\nMaterials suitable for magnonic crystals demand low magnetic damping and long spin wave (SW)\npropagation distance. In this context Co based Heusler compounds are ideal candidates for magnonic\nbased applications. In this work, antidot arrays (with di\u000berent shapes) of epitaxial Co 2Fe0:4Mn0:6Si\n(CFMS) Heusler alloy thin \flms have been prepared using e-beam lithography and sputtering tech-\nnique. Magneto-optic Kerr e\u000bect and ferromagnetic resonance analysis have con\frmed the presence\nof dominant cubic and moderate uniaxial magnetic anisotropies in the thin \flms. Domain imaging\nvia x-ray photoemission electron microscopy on the antidot arrays reveals chain like switching or\ncorrelated bigger domains for di\u000berent shape of the antidots. Time-resolved MOKE microscopy\nhas been performed to study the precessional dynamics and magnonic modes of the antidots with\ndi\u000berent shapes. We show that the optically induced spin-wave spectra in such antidot arrays can\nbe tuned by changing the shape of the holes. The variation in internal \feld pro\fles, pinning energy\nbarrier, and anisotropy modi\fes the spin-wave spectra dramatically within the antidot arrays with\ndi\u000berent shapes. We further show that by combining the magnetocrystalline anisotropy with the\nshape anisotropy, an extra degree of freedom can be achieved to control the magnonic modes in such\nantidot lattices.\nI. INTRODUCTION\nMagnon spintronics has emerged as a future potential\ncandidate for novel computing and data storage tech-\nnology due to several advantages viz. highly e\u000ecient\nwave-based computing, applicability in devices of dimen-\nsions down to\u001810 nm, operation frequency varying from\nsub-GHz to THz range, room temperature transport of\nspin information without generating Joule heating, etc.[1]\nFurther, the spin waves (SW) also \fnd their applications\nin on-chip communication systems due to their wave-\nlength being one order shorter than that of the elec-\ntromagnetic waves, which enables them to minimize the\ndata processing bits. [1] In recent years, magnonic crys-\ntals have been rigorously studied for the propagation and\ncon\fnement of the SWs. There are two kinds of interac-\ntions present in magnonic crystals such as short ranged\nexchange and long ranged dipolar interactions. In large\nwave vector ( k) limit (i.e. short wavelength of SW), the\ninteraction is primarily exchange dominated. Due to such\nstrong exchange interaction (nearest neighbor Heisenberg\nexchange interaction), the atomic spins remain parallel to\neach other in the ground state. However, in case of a fer-\nromagnetic thin \flm, the moments will align themselves\nin the \flm plane under an in-plane bias magnetic \feld.\n\u0003Present address: National Institute for Materials Science,\nTsukuba 305-0047, Japan\nysbedanta@niser.ac.inSuch modes travelling in the \flm plane usually possess\nlong wavelength (hundreds of nm to several \u0016m) which is\nsigni\fcantly larger than the interatomic distance. In such\nlowklimit, the exchange interaction is weak and dipolar\ninteractions dominate. Further, for SWs with relatively\nhigher kvalues both interactions become non-negligible\nwhich is known as the dipole-exchange SW modes. The\ninteractions can be controlled to tune the dispersion re-\nlation in such systems.\nFurther two-dimensional magnonic crystals can be\nbroadly subdivided into two parts: dot and antidot lat-\ntice arrays.[2] Magnetic antidot lattice (MAL) arrays are\narrangement of periodic holes in a continuous thin \flm\nsystem.[3] A major advantage of MAL arrays over dot\narrays is the miniaturization of their dimension not be-\ning restricted by the superparamagnetic limit to the bit\nsize.[4] Additionally, MAL arrays with well-de\fned peri-\nodic holes, provide precise control over the magnetization\nreversal, relaxation, and domain structure in compari-\nson to its thin \flm counterparts.[5{9] Recently several\nother aspects of MALs viz. magnetotransport, ferromag-\nnetic resonance, geometric coercivity scaling, magnetore-\nsistance, domain structure with varying shapes of the\nholes have been reported.[10{15] Further, the MALs are\nsuperior to the dot arrays as magnonic crystals because\nof their larger SW propagation velocity (viz. steeper\ndispersion).[16, 17] The edges of the holes in the anti-\ndot arrays quantize the SW modes and modulate inter-\nnal magnetic \feld periodically due to the demagnetizing\ne\u000bect.[16] Over the last decade several works have been\nreported on the control of SW dynamics in MAL arraysarXiv:1907.02746v1  [cond-mat.mtrl-sci]  5 Jul 20192\nby varying the antidot architecture.[18{24]\nFurther the most desired characteristics of a material\nfor magnonic based applications are low magnetic damp-\ning, high saturation magnetization, high curie tempera-\nture, long spin wave propagation distance, etc.[1] Among\nvarious materials reported till date, Yttrium Iron gar-\nnet (YIG) possesses remarkably low magnetic damping\nof\u00180.0002 and high SW propagation distance of \u001822.5\n\u0016m.[1] However there are certain drawbacks of YIG such\nas its large structure and low saturation magnetization\n(\u00180:14\u0002106A/m).[1] On the other hand Permalloy\n(NiFe) has been used for SW detection due to their low\ndamping (\u00180.008) and relatively high saturation magne-\ntization (\u00180:80\u0002106A/m). However, the SW propaga-\ntion distance is rather short ( \u00183.9\u0016m) in Permalloy thin\n\flms.[1] In this context Co based Heusler compounds are\npromising candidates for magnonic based applications\nbecause of their high saturation magnetization ( \u00181:00\u0002\n106A/m), low magnetic damping ( \u00180.003), and moder-\nately high SW propagation distance ( \u001810.1\u0016m).[1, 25{\n28] It has been shown that due to the lower density of\nstates at the Fermi level in one spin channel, the spin \rip\nscattering gets reduced leading to such remarkable low\ndamping in Co based Heusler compounds.[29] Recently\nit has been observed that low magnetic damping down to\n\u00180.0045 can be achieved from epitaxial Co based Heusler\nalloy thin \flms deposited on Cr bu\u000ber layer.[30] It has\nbeen reported that the Co based Heusler compounds\ncomprise of growth-induced uniaxial magnetic anisotropy\nand cubic anisotropy which can be tuned by varying\nthe thickness as well as the choice of seed layers.[28, 30]\nHence, such antidots of Heusler alloy thin \flms provide\nadditional degrees of freedom (over conventional Permal-\nloy \flms) to control the spin waves and may be useful for\nfuture applications. The combination of magnetocrys-\ntalline and shape anisotropy to tune the SW spectra is\nunexplored. In addition, there are few reports on anti-\ndot arrays with the anisotropic structures like triangular\nand diamond shaped holes which may signi\fcantly mod-\nify the internal \feld distribution in the vicinity of the\nholes leading to further modi\fcations of the SW spectra\nof such systems.[16, 31, 32]\nIn this present work we have chosen Co 2Fe0:4Mn 0:6Si\n(CFMS) thin \flms and their antidot arrays for the in-\nvestigation of SW dynamics. To the best of our knowl-\nedge, there are no reports yet for the study of magne-\ntization dynamics in antidot arrays of similar materials.\nWe show that under the in\ruence of both magnetocrys-\ntalline and shape anisotropy, the magnonic spectra can\nbe tuned with formation of various modes by varying the\nshape of the antidots. We have also explored the possibil-\nity of tuning the domain structure in such antidot arrays\ndepending on the available magnetic area and anisotropy\ndistribution.II. EXPERIMENTAL DETAILS\nEpitaxial thin \flms of Cr (20 nm)/CFMS (25 nm)/Al\n(3 nm) were deposited at room temperature on MgO\n(100) substrates in an ultrahigh vacuum compatible mag-\nnetron sputtering chamber with a base pressure of \u0018\n1:5\u000210\u00009mbar. Prior to deposition, the MgO (100)\nsubstrate was annealed at 600\u000eC for 15 minutes for sur-\nface reconstruction and removal of impurity from the\nsurface. The-20-nm thick Cr seed layer was deposited\non MgO (100) at a rate of \u00180.03 nm/s at 1 :2\u000210\u00003\nmbar. After the deposition of Cr, it was annealed in-\nsitu at 700\u000eC for 1 hour to form a \rat surface. The Cr\nlayer has been placed between MgO and CFMS to re-\nduce the magnetic damping of the system.[30] Further,\nthe CFMS layer was deposited at a rate of \u00180.018 nm/s\nat 1:2\u000210\u00003mbar. Next, the thin \flm heterostruc-\nture was post annealed in-situ at 500\u000eC for 15 minutes\nto promote the B2 (random position of Fe, Mn, and\nSi, with respect to Co) and L21(completely ordered\nstate) ordering of CFMS. Finally, a 3-nm-thick capping\nlayer of Al was deposited at \u00180.039 nm/s at 1 :2\u000210\u00003\nmbar to avoid oxidation of the magnetic layer. The\nCFMS layer shares the following epitaxial relationship\nwith MgO: CFMS(001)[110] kMgO(001)[100].[33] Micro-\nfabrication of the MAL arrays with di\u000berent shapes (cir-\ncular, square, triangular, and diamond) and feature size\nof 200 nm was performed using e-beam lithography and\nAr ion milling. See Supplemental Material at [] for\ndetails of the microfabrication technique of the MAL\narrays.[34] The surface structural quality of the \flms was\ninvestigated in\u0000situ using re\rection high-energy elec-\ntron di\u000braction (RHEED). The x-ray di\u000braction (XRD)\nmeasurement was performed ex\u0000situ to determine the\ncrystalline quality and atomic site ordering in the \flm.\nSaturation magnetization ( MS) of the \flm is extracted\nfrom the room temperature M-H measurement using vi-\nbrating sample magnetometry (VSM). High resolution\ndomain imaging on the nano-dimensional MAL arrays\nwas performed along the easy axis by x-ray photoemis-\nsion electron microscopy at the I06 nanoscience beamline,\nDiamond Light Source, UK. The nature of anisotropy\nhas been extracted from the angle dependent hystere-\nsis loops measured using magneto-optical Kerr e\u000bect\n(MOKE) magnetometer with a micro-size laser spot. To\nquantify the growth induced anisotropy, angle dependent\nferromagnetic resonance (FMR) measurement was per-\nformed using Phase FMR spectrometer manufactured by\nNanoOsc AB, Sweden. The time-resolved precessional\ndynamics was measured using an all-optical time-resoled\nMOKE microscope set-up on two-color collinear pump-\nprobe geometry at applied magnetic \felds signi\fcantly\nhigher than the saturation \felds of the samples.[35] See\nSupplemental Material at [] for details of the time re-\nsolved MOKE measurements.[34] The experimentally ob-\nserved spin-wave spectra have been qualitatively repro-\nduced using micromagnetic simulation (OOMMF).[39]\nThe simulation area is taken as 1600 \u00021600\u000225nm33\nfor an array of 4\u00024 holes (this ensures the feature size of\n200 nm). The cell size for the simulations are considered\nto be 4\u00024\u000225nm3which ensures presence of only one\nshell along the thickness of the sample. Although dis-\ncretization of the sample thickness would have revealed\nadditional information about the mode variation continu-\nously along the thickness of the \flm, the resulting simula-\ntion time would have been computationally challenging.\nHence we have compromised on the small variation along\nthe thickness to focus on the in-plane con\fgurations of\nthe spin-wave modes. We have used the following mate-\nrial parameters: \r= 2:14\u0002105m/As,MS= 9:2\u0002105\nA/m,K2= 3:4\u0002102J=m3,K4= 1:17\u0002103J=m3,\n\u000b= 0:006, and exchange sti\u000bness ( A)= 1:75\u000210\u000011\nJ/m. The values of \r,\u000b,K2, andK4are extracted from\nthe FMR \ftting, whereas A is taken from ref. [30]. Two-\ndimensional periodic boundary condition (2D-PBC) has\nbeen used for approximating the large sample area. It\nshould be noted that although the simulation qualita-\ntively reproduces the experimental observation, however\nthere is quantitative disagreement due to the limitations\nin the simulation viz., edge roughness of the holes, sta-\ntistical di\u000berence in the hole structures, etc.[40] Further,\nthe experiments are performed at ambient temperature\nwhereas the simulations do not consider any e\u000bect aris-\ning from the temperature. The power and phase pro\fles\nof the spin-wave modes were calculated by a home built\ncode Dotmag.[41]\nIII. RESULTS AND DISCUSSION\nThe parent thin \flm is denoted as TF, whereas the\ncircular, square, triangular, and diamond shaped anti-\ndots are de\fned as CA, SA, TA, and DA, respectively.\nSee table ST1 in the Supplemental Material at [] for de-\ntailed sample description.[34] Figure 1(a) - (b) show the\nin-situ RHEED patterns for TF with electron beam inci-\ndence parallel to MgO [110] (CFMS [100]) and MgO [100]\n(CFMS [110]) directions, respectively. The well-de\fned\nre\rected spots in the RHEED patterns ensure the forma-\ntion of epitaxial CFMS \flm on MgO with the relation:\nCFMS [110]kCr [110]kMgO [100]. The formation of\nstreak lines in the RHEED pattern indicates the \rat \flm\nsurface. Superlattice streaks are also observed in TF in-\ndicating the chemical ordering of CFMS into either B2\norL21structure.\nFigure 1(c) shows the XRD pattern obtained for TF in\nthe\u0012\u00002\u0012geometry. MgO (200) corresponds to the\nmost intense peak whereas the peak marked by \u0003appears\nfrom (200) di\u000braction of MgO substrate arising from the\nCu\u0000K\fsource. Both CFMS (200) superlattice and\nCFMS (400) fundamental di\u000bractions have been visible\nwhich is consistent with the results of the RHEED obser-\nvation. The CFMS (200) superlattice peak corresponds\nto the formation of B2 orL21structure. However, it\nis di\u000ecult to quantitatively determine the strength of\ntheB2 orL21ordering individually from the \u0012\u00002\u0012ge-\nFIG. 1. (a) - (b) RHEED images for TF along CFMS [100]\nand CFMS [110] directions. (c) XRD pattern for TF showing\nthe CFMS, Cr, and MgO peaks. The peaks denoted by \u0003\nappearing around 38\u000earise from the (200) di\u000braction of the\nMgO substrate with Cu\u0000K\fsources. (d) M-H curve for TF\nmeasured by VSM at room temperature along the easy i.e.\nCFMS [ \u0016110] direction.\nometry. Figure 1(d) shows the M-H loop measured us-\ning VSM at room temperature for TF while the exter-\nnal magnetic \feld is applied along the easy axis (CFMS\n[\u0016110] direction). See \fgure S1 of Supplemental Mate-\nrial at [] for the orientation of the easy axis and applied\n\feld direction with respect to the crystallographic axes\nof MgO substrate and CFMS thin \flm.[34] Square hys-\nteresis loop with \u0018100% remanence has been observed.\nThe extracted value of saturation magnetization ( MS) is\n920\u0002103A/m. The value of the coercive \feld ( \u00160HC)\nfor TF along CFMS [ \u0016110] is 1.7 mT.\nAngle dependent MOKE measurements reveal that TF\nexhibits presence of cubic anisotropy with the easy axis\nalong 0\u000e, 90\u000e, 180\u000e, and 270\u000e. See \fgure S2(a) of Sup-\nplemental Material at [] for angular dependent MOKE\ndata.[34] It further reveals the presence of an additional\nuniaxial magnetic anisotropy. The uniaxial anisotropy\ncan be introduced in a \flm due to several reasons viz.\noblique angular deposition, anisotropic strain relaxation,\nmiscut in the substrate, interfacial roughness, interfacial\nalloy formation, growth on a stepped substrate, etc. [42{\n45] The strength of uniaxial and cubic anisotropies have\nbeen extracted by \ftting the angular dependent FMR\ndata (See \fgure S2(b) of Supplemental Material at [] for\nangular dependent FMR data.[34]) with two-anisotropy\nKittel equation, which under small angle approximation\ncan be represented as:\nf=\r=2\u0019([H+2K2\nMScos2\u001e\u00004K4\nMScos4\u001e]\u0002\n[H+ 4\u0019MS+2K2\nMScos2\u001e\u0000K4\nMS(3 +cos4\u001e)])1=2\n(1)\nwhere,\ris the gyromagnetic ratio, \u001eis the angle between\nthe easy axis and applied magnetic \feld ( H),K2is the\nuniaxial anisotropy constant, K4is the cubic anisotropy\nconstant. The value of MSwas obtained from the VSM4\nmeasurement and used here to calculate K2andK4. The\n\ftted values of uniaxial and cubic anisotropy constants\nare 0:34\u0002103, and 1:2\u0002103J=m3, respectively. Hence,\nit is concluded that a noticeable contribution of uniaxial\nanisotropy is present in TF, which is \u001828% of the dom-\ninant cubic anisotropy. We have further extracted the\ndamping constant ( \u000b) = 0.0056 from the frequency de-\npendent FMR measurements. The parameters extracted\nfrom the FMR measurements have been used in OOMMF\nsimulation which is discussed in the later part of the\nmanuscript.\nFigure 2 shows the hysteresis loops measured using\nmicro-MOKE along CFMS [ \u0016110] direction for (a) - (d)\nCA, SA, TA, and DA, respectively. The insets of \fgure 2\n(a) - (d) show the SEM images of the corresponding an-\ntidot samples. The values of HCare 7.3, 9.5, 5.5, and 9.4\nmT, for CA, SA, TA, and DA, respectively. The values\nofHSfor CA, SA, TA, and DA are 9.0, 10.1, 6.2, and\n10.3 mT, respectively. The values of HSare signi\fcantly\nhigher for all the antidot samples in comparison to their\nthin \flm (TF) counterpart. This is expected since intro-\nduction of periodic holes pin the magnetic domains and\ndo not allow the DWs to propagate through them lead-\ning to magnetic hardening. The variation in the strength\nofHCandHSbetween the antidot samples can be ex-\nplained by the active area (total magnetic area in the\nantidot samples which is calculated by subtracting the\ntotal area covered by all the holes from the total area) of\nthe samples. The available active area in the triangular\nantidot (TA) is always higher than that in the diamond\nshaped antidot (DA) with the same feature size and lat-\ntice geometry. Hence, the domain walls require higher\nenergy to avoid the holes and propagate in a zigzag path\nfor DA in comparison to that in TA.\nFIG. 2. Hysteresis loops measured using micro-MOKE along\nCFMS [ \u0016110] direction at room temperature for CA, SA, TA,\nand DA, shown in (a) - (d), respectively. The insets show the\ncorresponding SEM images of the antidot samples.\nFigure 3 (a) shows the SEM image of CA. Figure 3\n(b) - (f) show the domain images for CA measured using\nXPEEM at \feld pulses of -40.6, 5.1, 8.1, 9.1, and 40.6mT, respectively. The XMCD measurements have been\nperformed at the L3edge of Fe. The direction of ap-\nplied magnetic \feld as well as the sensitivity direction of\nXPEEM is set along CFMS [ \u0016110] direction. To saturate\nthe sample, initially a high magnetic \feld pulse (-40.6\nmT) has been applied and subsequently set to zero to\nobserve the remanent magnetic state (\fgure 3 (b)). Fur-\nther, gradually \feld pulses in the reverse direction have\nbeen applied followed by removing the \feld and taking\nthe images at remanence states. The values of the re-\nverse \feld pulses have been mentioned above for \fgure 3\n(b) - (f). Presence of the periodic holes in the \flm act as\nFIG. 3. (a) SEM image of CA (circular antidot). (b) - (f)\nshow the domain images in CA measured by XPEEM (XMCD\nat the FeL3edge) at magnetic \feld pulses of -40.56, 5.07,\n8.11, 9.13, and 40.56 mT, respectively. (g) SEM image of\nTA (triangular antidot). (h) - (l) show the domain images in\nTA at \feld pulses of -40.56, 4.26, 4.46, 5.07, and 40.56 mT,\nrespectively. The scale bar for the XPEEM images are same\nfor both the samples which is shown in (b) and (h). The\n\feld has been applied along the EA (CFMS [ \u0016110]) and the\ndirection is shown in (l).\nnucleation centers and the domains remain pinned in be-\ntween two successive holes. The domains propagate in a\nzigzag path avoiding these holes. This leads to formation\nof a chain like domain structure.[36, 37] Further, the DWs\npropagate in the transverse direction with the enhance-\nment of applied \feld pulses to complete the reversal. The\ndomain width in CA near nucleation (\fgure 3 (c)) and\ncoercive \felds (\fgure 3 (d)) are 0.37, and 1.07 \u0016m, re-\nspectively. The circular shape of the antidot itself does\nnot contribute to any shape anisotropy but the lattice\nsymmetry of the two-dimensional magnonic crystal con-\ntributes to a shape anisotropy.[19, 38] This, in conjunc-5\ntion with the magnetocrystalline anisotropy of the CFMS\nHeusler alloy thin \flm, determines its resultant magnetic\nanisotropy. Since during the XPEEM measurements, the\nexternal magnetic \feld is applied along CFMS [ \u0016110], the\ndomains align in a chain like structure (\fgure 3 (d)).\nNevertheless, with enhancement of the Zeeman energy\n(external magnetic \feld), the domain walls are forced to\nmove in the transverse direction before coalescing with\nthe neighboring domains to complete the reversal. Fig-\nure 3 (g) shows the SEM image of TA whereas (h) - (l)\nshow the domain images measured at -40.6, 4.3, 4.5, 5.1,\nand 40.6 mT, respectively. In contrary to the chain like\ndomain formation in CA, the domains are correlated and\nbigger in TA. The domain width in TA near nucleation\n(\fgure 3 (i)) and coercive \felds (\fgure 3 (j)) are 0.44, and\n3.65\u0016m, respectively. The triangular holes in TA are not\nisotropic in nature like the circular ones in CA. Hence,\nthe local \feld distribution as well as the anisotropy is\ndi\u000berent in the vicinity of the holes in the triangular an-\ntidot. This modi\fes the overall anisotropy nature of the\nparent \flm. Hence the domain walls propagate along\nboth the \feld as well as transverse direction leading to\nelevation in size of the domains. The enhancement of\nthe domain size can be further explained by comparing\nthe availability of the active area in CA, and TA. The\navailable active area in CA is 71 :7% of the total sample\narea, whereas the same for TA is 84 :4%. Hence, due to\nthe availability of larger magnetic area in sample TA, the\nDWs have more freedom to expand laterally during the\nreversal. From the above discussion we conclude that\nby varying the shape of the holes in the antidot array,\ndomain engineering can be accomplished.\nFigure 4 (a) - (d) show the spin-wave spectra for CA,\nSA, TA, and DA, respectively, at an applied \feld of 153\nmT along CFMS [ \u0016110]. It should be noted that the ap-\nplied \feld is signi\fcantly higher than the saturation \feld\nso as to employ precession of the magnetization under\nits saturation state. The SW spectra is obtained by tak-\ning fast Fourier transform (FFT) of the background sub-\ntracted time-resolved Kerr rotation data. See \fgure S3 of\nSupplemental Material at [] for the details about obtain-\ning the SW spectra from the Kerr rotation in TF.[34]. It\ncan be observed from \fgures 4 (a) and (b) that there are\ntotal three modes in case of CA and SA. The gap between\nconsecutive modes for these two lattices has slight vari-\nation within the spectral width of 9.90 GHz, and 12.25\nGHz for CA and SA, respectively. In TA, the modes un-\ndergo signi\fcant blue-shift as opposed to those in CA and\nSA, and the frequency gaps between consecutive modes\nalso reduce. The spectral width of TA also reduces to\n5.94 GHz. On the contrary, in DA, four modes appear\nwith similar blue-shift in frequencies like TA and the\ngaps between consecutive modes become narrower with\nrespect to CA and SA. The spectral width of DA is 8.91\nGHz.\nFigures 4 (e) - (h) show the simulated spin-wave spectra\nfor CA, SA, TA, and DA, respectively. The simulated\nspectra agree qualitatively with the experimental results.\nFIG. 4. (a) - (d) Spin wave spectra obtained at \u00160H= 153\nmT from the experimental time-resoled Kerr rotation data\nfor CA, SA, TA, and DA, respectively. (e) - (h) simulated\nspin wave spectra for CA, SA, TA, and DA, respectively. The\nnumbers corresponding to di\u000berent modes are shown in the\nindividual images. The images in the inset of (a) - (d) and\n(e) - (h) show the shape of a single hole in the antidot array\nobtained from SEM images and bitmap images considered in\nthe simulation, respectively.\nIn the simulated spectra, the spectral widths for CA, SA,\nTA, and DA are 9.65, 13.36, 3.96, and 7.43 GHz, respec-\ntively. The spectral width decreases due to the compres-\nsion of the resonant modes in frequency domain for TA\nand DA. There are a few low power modes present in\nthe simulated spectra for TA and DA which are not well\nresolved in the experiment. The remarkable di\u000berence in\nthe mode pro\fle between the di\u000berent antidots arises due\nto the di\u000berence in their internal \feld distribution which\nleads to formation of extended and localized modes.[31]\nAdditionally, asymmetry introduced from the fabrication\nprocess imposes further di\u000berences in the internal \feld\npro\fles in the antidot arrays. Nevertheless, we have tried\nto address this issue by considering roundish edges dur-\ning simulations (see insets of \fgure 4 (f) - (h)) for square,\ntriangular, and diamond shaped antidots (SA, TA, and\nDA). Indeed, the simulation results are closer to the ex-\nperimental \fndings when round corners (see insets of \fg-\nure 4 (f) - (h)) are considered in comparison to the sharp6\nFIG. 5. Simulated power maps for di\u000berent precessional modes as shown in \fgure 4 (e) - (h) for circular (CA), square (SA),\ntriangular (TA), and diamond (DA) antidots, respectively. Phase pro\fles are shown inside the dashed box at the top right\ncorner of the images. The color map for the power distribution is shown at the bottom of the image. The respective mode\nnumbers as per \fgure 4 is shown in the right side of the image. The quantization numbers are calculated from the pro\fles\nwithin the blue colored boxed regions as shown in (g) and (h). An external magnetic \feld of 153 mT is applied along the easy\naxis (CFMS [ \u0016110]) as shown in the \fgure.\ncorners of the holes. Previous reports on such struc-\ntures have considered zero magnetocrystalline anisotropy\nfor the constituent \flms.[31, 32] However, in our study\nboth cubic and uniaxial anisotropies along with the shape\nanisotropy play important roles in determining the total\nanisotropy in the antidot samples. In CA and SA, the\nsymmetric structure of the holes does not lead to any ad-\nditional shape anisotropy along with the anisotropy of the\nparent thin \flm. However, the anisotropy is modi\fed in\ncase of the triangular and diamond shaped antidots (TA\nand DA). This leads to the appearance of high frequency\nmodes. Similar reasons can be corroborated to the ab-\nsence of low frequency mode (mode no. 1) in TA and\nDA.\nTo understand the nature of the precessional modes,\nwe have further simulated the power and phase pro\fles\nof the modes using a code described in ref. [41]. Figure\n5 shows the simulated power maps for the precessional\nmodes for CA, SA, TA, and DA. The phase pro\fles are\nshown inside the dashed boxes at the top right corner\nof each image. See \fgure S4 of Supplemental Material\nat [] for the detailed phase map.[34] Figures 5 (a) and\n(b) show low frequency edge modes (mode no. 1) in CA\nand SA, respectively. From the mode pro\fle, it can be\nobserved that this is a symmetric mode where the max-imum power is stored at the vicinity of the circular and\nsquare holes in CA, and SA, respectively. This low fre-\nquency symmetric edge mode is absent in TA and DA\ndue to the modi\fcation of internal \feld pro\fle. Mode 1\nfor TA, DA and mode 2 for CA, SA are extended modes\nwhich \row in Damon-Eshbach (DE) geometry through\nthe channels in between the holes (\fgure 5 (c), (d)). The\nfrequencies of the extended modes are slightly reduced\nin comparison with TF because of modulation by the an-\ntidots and varying demagnetized regions in the vicinity\nof the holes. These modes can also be characterized as\nquantized modes in backward volume (BV)-like geome-\ntry with quantization number q= 1. Higher frequency\nmodes in these lattices are the standing SW modes ap-\npearing within the potential barrier between two consec-\nutive holes acting like cavity resonator. Mode 2 for TA\nand DA are localized modes in BV-like geometry with\nquantization number q= 4. Mode 3 (\fgure 5 (i) - (l)) is\nalso localized mode with q= 5 for CA, SA, and TA. How-\never, for DA mode 3 and mode 4 are mixed modes where\nthe mode quantization numbers are di\u000ecult to assign.\nThis is attributed to the e\u000bect of the shape of the dia-\nmond antidots in modifying the anisotropy, internal \feld\npro\fles, and pinning energy landscapes in the system.\nTo have a deeper understanding about the magneto-7\nFIG. 6. (a). Magnetostatic \feld distribution of di\u000berent antidot lattices are shown. The line-scans of y-component (CFMS\n[\u0016110]) of the magnetostatic \feld (b) across the antidots (orange dashed line in (a)) and (c) along the channel between the two\ncolumns of antidots (green dotted line in (a)). The magni\fed view of the minima in the line-scan is shown at the inset of (b).\nThe hollow boxes in (c) represent region I and II as highlighted in (a). The color bar is presented at the bottom of (a).\nstatic \feld distributions of the antidot lattices with di\u000ber-\nent shapes, we have performed further simulations using\nLLG micromagnetic simulator [46]. Figure 6(a) shows\nthe contour plots of the \feld distribution and internal\nspin con\fguration for a bias magnetic \feld, \u00160H= 153\nmT along CFMS [ \u0016110]. Due to the shape-dependent\ndemagnetizing e\u000bects, the contours exhibit non-uniform\ndistribution around the antidot edges. For CA and SA,\nthe lines of force at the vicinity of vertical edges of the an-\ntidot are denser than that for DA and TA, which prompts\nthe appearance of edge mode for CA and SA. To quantify\nthe magnetostatic \feld, we have taken line-scan across\nthe antidots and along the channel between two columns\nof antidots. The variation of y-component (along CFMS\n[\u0016110]) of magnetic \feld with distance is plotted in \fgure\n6(b) and (c). The line-scans re\rect the imprint of the\nspatial modi\fcation in the magnetostatic \feld distribu-\ntion due to di\u000berent shape of the antidots. The magni-\ntude of demagnetizing \feld inside the antidots is about\n40 mT which is nearly invariant with antidot shape (\fg-\nure 6(b)). However, the minima in the magnetostatic\n\feld values show asymmetry for TA (shown in the in-\nset of \fgure 6(b)) due to strong interactions between the\nunsaturated spins residing at the vertices of the trian-\ngular antidot. When we analyze the magnetostatic \feld\ninside the channels between the columns of antidot, the\nsituation becomes more interesting. Here, we observe al-\nternate maxima and minima, with the maxima occurring\nat positions between a pair of nearest neighbour antidots\n(region-I), while the minima occurring in the continu-\nous \flm region (region-II) (\fgure 6(c)). This is further\nstrengthened by the observation of increased density of\nmagnetic \feld lines in region-I as opposed to region-II.This e\u000bect is most prominent in DA and that proba-\nbly leads to the appearance of additional high-frequency\nmode in this lattice.\nIV. CONCLUSION\nIn summary we have studied the domain engineering\nas well as spin wave dynamics in antidot arrays of Heusler\nbased CFMS thin \flms. Epitaxial CFMS thin \flms were\ngrown on MgO substrates with highly ordered B2 orL21\nphase. MOKE and FMR measurements con\frmed the\nformation of cubic anisotropy due to epitaxial growth of\nthe \flm along with a moderate uniaxial anisotropy. The\nquanti\fcation of the anisotropy has been performed us-\ning angle dependent FMR measurements which shows\nthat the uniaxial anisotropy is almost \u001828% of the cu-\nbic anisotropy. We have shown that the domain struc-\nture changes from chain like features to wide correlated\ndomains by changing the shape of the holes from circu-\nlar to triangular. This behaviour is explained in terms\nof availability of the active area and the anisotropy dis-\ntribution in the vicinity of the holes in di\u000berent anti-\ndot structures. The continuous thin \flm shows presence\nof a uniform precessional mode in the FFT spectra of\nthe time-resolved Kerr rotation data. However, intro-\nduction of periodic holes leads to formation of extended\nand quantized modes in the antidots. The isotropic cir-\ncular and square antidots show presence of two dominant\nmodes (higher frequency) along with a low frequency\nmode having relatively low power. Appearance of ex-\ntra modes reduces the frequency gaps in the triangular\nand diamond shaped antidots. The extended DE mode8\nalong with other quantized modes are present in all the\narrays. However, the sharp corners of the holes modify\nthe demagnetizing \feld in the vicinity of the antidots in\ntriangular and diamond shaped antidots. Further, the in-\ntrinsic magnetocrystalline and the shape anisotropy are\nnot aligned in case of triangular and diamond antidots.\nDue to its complicated structure and local \feld distribu-\ntion another mixed mode appears at high frequency in\nthe diamond shaped antidot which further reduces the\nfrequency gap which is suitable for spin wave based de-\nvices. This observed tunability of the spin wave spectra\nby varying the shape of the antidots in CFMS thin \flms\nhaving low \u000band highMSmight have signi\fcant impact\nin future applications based on magnonic \flters, splitters,\nother magnonic devices, etc.V. ACKNOWLEDGEMENT\nSM and SB thank Department of Atomic Energy\n(DAE), Department of Science and Technology-Science\nand Engineering Research Board (DST-SERB) (project\nno. SB/S2/CMP-107/2013), Govt. of India, for pro-\nviding the funding to carry out the research. We also\nthank International Collaboration Center of the Institute\nfor Materials Research (ICC-IMR), Tohuoku University,\nJapan, for providing the funding for visit of SM for sam-\nple fabrication, RHEED, XRD, micro-MOKE, and VSM\nmeasurements. We acknowledge Diamond Light Source\nfor time on I06 under proposal SI16582 for XPEEM mea-\nsurements. We further like to thank Visitor, Associates,\nand Students0Programme (VASP), of S. N. Bose Centre\nfor Basic Sciences, Kolkata, India for support to visit the\nS. N. Bose Centre for TR-MOKE measurements and the\nuse of Dotmag code developed by Dr. Dheeraj Kumar.\nSS acknowledges S. N. Bose Centre and SuM acknowl-\nedges DST-INSPIRE scheme for Senior Research Fellow-\nship.\n[1] A. V. Chumak, A. A. Serga, and B. Hillebrands,\nMagnonic crystals for data processing,J. Phys. D: Appl.\nPhys. 50, 244001 (2017).\n[2] S. A. Nikitov, Ph. Tailhades, and C. S. 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Schweitzer, Pessac, F-33608, France\n(Dated: July 10, 2019)\nWe report the structural, magnetic, exchange bias and magnetotransport effect in Sr 4Fe3CoO 11.\nThe material crystallizes in the orthorhombic Cmmmspace group. It shows antiferromagnetic (G-\ntype) transition (T N= 255 K) along with interesting temperature induced magnetization reversal\n(TComp:= 47 K measured at 100 Oe). The magnetic reversal can be elucidated considering the\nincreased magnetocrystalline anisotropy with Co substitution. Magnetoresistance measurements\nshows an interesting crossover from negative to positive side at \u0018100 K. The negative magnetore-\nsistance reaches 80 %at 25 K in 7 T magnetic field. Giant exchange bias effect is observed below\nTNunder field cooling condition. The origin of the negative magnetoresistance and giant exchange\nbias in this sample can be attributed to the magnetic frustration.\nI. INTRODUCTION\nStronglycorrelatedtransitionmetaloxide(TMOs)ma-\nterial systems has attracted widespread attention as a\nplethora of exotic phenomena such as high-temperature\nsuperconductivity (SC, cuprates) [1], colossal magnetore-\nsistance (manganites) [2], exchange bias [3, 4] and multi-\nferroicity (bismuth compounds) [5] arises from correlated\nelectron physics in 3d TMOs. Indeed, the strong corre-\nlation among electrons is a connecting feature of TMOs\nexhibiting extraordinary electronic and magnetic proper-\nties, and that can develop into a promising technological\napplication (such as in electronics and spintronic field).\nAt the same time, purely by changing the valence state of\nthe transition metal ions, unusual physical properties can\nemerge in TMOs This is realized to be one of the critical\nissues in understanding the exotic properties observed in\nhigh-temperature superconductivity[6], multiferroics [7],\nand magnetoresistive compounds [8].\nAmong transition metals, iron is a very attractive\ntransition metal element due to its potential to ex-\nhibit several oxidation states and spin states in differ-\nent environments which often induce interesting proper-\nties. Oxygen deficient strontium ferrites are such sys-\ntems displaying valence changes. In recent years a great\nnumber of studies has been done due to their diverse\nstructural and physicochemical properties relevant for\npotential applications [9–15]. They belong to a class\nof anion-deficient perovskite-like compounds, which are\nknown for the Sr nFenO3n\u00001(or SrFeO 3\u0000\u000e) system. This\nsystem is well recognized to display a rich phase dia-\ngram depending on the oxygen content, ranging from\n3.0 to 2.0, which includes cubic SrFeO 3[14], tetrago-\nnal Sr 8Fe8O23(SrFeO 2:87) [9], orthorhombic Sr 4Fe4O11\n(SrFeO 2:75) [11], Sr 2Fe2O5(SrFeO 2:5) [16] and SrFeO 2\n\u0003rpsingh@iiserb.ac.inphase [15]. The square planer oxygen deficient end mem-\nber exhibits Fe2+cations with an above room temper-\nature magnetic ordering [15]. The other end phase cu-\nbic SrFeO 3possesses an unusually high Fe4+oxidation\nstate [14]. The intermediate member Sr 4Fe4O11( or-\nthorhombic 2p\n2ap\u00022ap\u0002p\n2ap, ap= lattice parame-\nter of the cubic perovskite sub-cell) have an equal num-\nber of tetravalent and trivalent iron. Also, Sr 4Fe4O11\nshows a variety of interesting properties, such as anoma-\nlousthermoelectricpowerandunusualmagneticbehavior\n[11, 13]. Hitherto, all the structural studies have indi-\ncated that Fe ions alternatively occupy square pyrami-\ndal and octahedral sites and undergo a charge ordering\nbelow 675 K with an equal amount of Fe4+and Fe3+\nalternatively arranged in the lattice [11–13, 17–19]. It\nis reported that one Fe site (Fe2, Octahedral) forms a\nlong-range AFM order (T N= 240 K), whereas the other\n(Fe1, square pyramidal) forms a magnetically frustrated\nspin-glass-like configuration below T N, which originate\nunusual magnetic properties in Sr 4Fe4O11.\nIn this paper, we have shown the structure, com-\nplex magnetic and transport properties of the oxygen-\ndeficient perovskite-related compound with composition\nSr4Fe3CoO 11. Giant exchange bias effect and a sizable\nmagnetoresistance are observed below the magnetic tran-\nsition of the material. Detailed structural and complex\nmagnetic properties are presented and discussed here.\nII. EXPERIMENTAL DETAILS\nPolycrystalline samples of composition Sr 4Fe3CoO 11\nwere synthesized by standard solid state reaction method\nusingstoichiometricamountofhighpuritySrCO 3, Fe 2O3\n, Co 3O4at1473K.PhasepuritywasconfirmedbyX-Ray\ndiffraction (XRD) using Cu K \u000b(\u0015= 1.54056 Å) using\nPhilips 1050 diffractometer. Powder neutron diffraction\n(NPD)measurementsweredoneatPSI,SwitzerlandwitharXiv:1907.03986v1  [cond-mat.str-el]  9 Jul 20192\ntemperature using wavelengths of \u0015= 1.494 Å and \u0015\n= 1.8857 Å for room temperature and low temperature\nrespectively. Fullprof suite program used to refine the\ndiffraction patterns. The temperature and magnetic field\ndependent magnetisation and transport properties mea-\nsurements were performed using Squid Quantum Design\nXL-MPMS, MPMS-3 magnetometer and PPMS.\nIII. RESULTS AND DISCUSSION\na. Structural Characterization\nFIG.1. Finalobserved, calculatedanddifferenceprofile(joint\nRietveld refinement pattern, Orthorhombic, Cmmm) of RT\nXRD and RT NPD patterns for the Sr 4Fe3CoO 11sample.\nInsets in the left figure highlight the peaks characteristics of\nCmmmspace group.\nPreliminary Rietveld refinement of the XRD pattern\nindicate that the compound can be isolated as single\nphases, crystallizing in an orthorhombic symmetry, space\ngroup (S. G.) Cmmm(\u00182p\n2ap\u00022ap\u0002p\n2ap). Fig-\nure 1 shows the final plot of joint Rietveld refinement\nof the RT XRD and RT NPD pattern of the material.\nThe results obtained from the joint refinement of the RT\nNPD/RT XRD data, are provided in the supporting in-\nformation file, and they are similar to the reported one\n[19].\nFigure 2 shows the crystal strucrure of Sr 4Fe3CoO 11.\nSimilar to the previous we have found full occupancies\nfor the oxygen positions[19]. Site occupancy refinementindicates a random distribution of Co in both Fe sites.\nSr4Fe3CoO 11shares the same vacancy ordered structure\nof Sr 4Fe4O11.\nFIG. 2. Crystal structure of Sr 4Fe3CoO 11(Orthorhombic,\nCmmm) (oxygen atoms as red spheres).\nb. Magnetic Structure and Magnetic Properties\nTo get the magnetic structure, NPD patterns were\ntaken down to 48 K. Figure 3 shows the tempera-\nture evolution of new reflections of magnetic origin for\nSr4Fe3CoO 11. We have used the Shubnikov space group\nCm’m’m(propagation vector k = 0, 0, 0, \u00007) to refine\nthe low temperature NPD patterns of the present mate-\nrial. It shows a G-type AFM ordering below 250 K. The\ntemperature evolution of the magnetic moments is shown\nas an inset in figure 3.\nFigure 4(a) shows the temperature dependence of the\nFC and ZFC susceptibility ( \u001f) collected at 100 Oe for\nSr4Fe3CoO 11. An AFM ordering (T N= 255 K\u00062 K)\nand then a large irreversibility between the FC and ZFC\nsusceptibility is apparent in the figure. The sudden rise\nof the FC susceptibility curve below T Nhighlights the\nexistence of weak ferromagnetic (WFM) correlations be-\nlow the AFM transition temperature. Below T N, the FC\nsusceptibility curve bifurcate from the ZFC with a broad\nmaximum at\u0018165 K\u00063 K and then on further cool-\ning, it continuously dropping towards a negative value by\ncrossing the \u001fZFCat TCross\u001870 K\u00064 K and reaches\nthe magnetic compensation point ( \u001fFC= 0) at T Comp\n= 47 K.\nBelow this compensation temperature, the FC suscep-\ntibility is negative down to the lowest measured temper-\nature (2 K), i.e., the resultant magnetization is opposite\nto the applied magnetic field. Thus, the FC magneti-\nzation exhibit temperature induced magnetization rever-\nsal phenomenon. The ZFC susceptibility curve shows a\npeak at\u001860 K, indicates a complex magnetic interac-\ntion/magnetic frustration at that temperature. To see\nthe characteristics of negative magnetization observed in3\nFIG. 3. Neutron powder diffraction patterns collected at dif-\nferent temperatures for Sr 4Fe3CoO 11sample. It highlights\n(by solid triangles) the temperature evolution of the mag-\nnetic peaks. Inset shows the temperature dependence of the\nmagnetic moments for the same material.\nthe sample, we have measured the magnetization at dif-\nferent applied magnetic fields. Figure 4(b) shows the\ntemperature dependence of FC susceptibility data col-\nlected under different magnetic field. It shows that the\ncompensation temperature is shifted towards the lower\ntemperaturesidewithincreasingmagneticfieldandwhen\nthe applied magnetic field increases to 500 Oe, the FC\nsusceptibility switches to the positive side. This suggests\nthat the net magnetization, originally opposite to the di-\nrection of the applied magnetic field is oriented along\nthe direction of the applied magnetic field at a higher\nmagnetic field. It is important to recall here that the\nisostructuralcompounds,suchasSr 4Fe4O11andthesam-\nples with lower Co concentration do not show any mag-\nnetization reversal [12, 13, 18]. In this structure, there\nare two different Fe sites, square pyramidal Fe1 sites and\noctahedral Fe2 sites (see figure 2). The Fe2 sub lattice\nform a long range ordered AFM spin structure [11, 17].\nEach Fe1 will have two Fe2 neighbors with up-spin and\ntwo with down spin . As a result, the Fe1 magnetic mo-\nments are topologically frustrated due to a ferromagnetic\nFe1-O-Fe2 double exchange and AFM Fe1-O-Fe1 super-\nexchange interactions. Also, in the present material Co\nand Fe ions are randomly sharing two different crystal-\nlographic (Fe) sites with a net AFM interaction. There-\nfore, the doped cobalt cations also give rise to additional\ncanting of Fe moments; this in fact is observed by the\nsudden rise of \u001fFCbelow TN. In low symmetric anti-\nferromagnetic materials, the existence of WFM is mainly\ndue to the single-ion magnetocrystalline anisotropy or\nDM interactions or both [20, 21]. Also, in certain cases,\nwhere the net moment caused by these mechanisms can\nbe oriented in the opposite field direction [20–22], and\nthey can have different temperature dependencies. Un-\n0 100 200 300-4.5-3.0-1.50.01.53.0\nT (K)FC\n 10 Oe\n 100 Oe\n 500 Oe\n  \n(b)\n0 100 200 300-0.6-0.30.00.30.6( )\nT (K)\n  \n(a)\n100 Oe\n ZFC\n FCFIG. 4. Temperature dependence (a) FC and ZFC magnetic\nsusceptibility curves measured at 100 Oe and (b) FC sus-\nceptibility curves measured under different magnetic field for\nSr4Fe3CoO 11. It highlights the dependence of magnetization\nreversal (negative magnetization) with magnetic field.\nder this circumstances, negative magnetization can arise\nunder low applied magnetic fields. In such a case, the\nobserved magnetization reversal in the present sample\ncan be attributed to the increase of the magnetocrys-\ntalline anisotropy with Co doping. As we reach the 25 %\nsubstitution level (Sr 4Fe3CoO 11), the magnetocrystalline\nanisotropy dominates and the net magnetic moment is\naligned opposite to the applied magnetic field.\nc. Exchange Bias and Training effect\nRecently, the complex magnetic materials have at-\ntracted remarkable interest for exploring the exchange\nbias effect (EB) [4, 23–26]. The EB phenomenon is gen-\nerally demonstrated as a shift [27] in the isothermal mag-\nnetization loop with respect to the field axes. It can\nbe utilized in several technological applications such as\nmagnetic recording read heads [28], random access mem-\nories [29] and other spintronic devices [30, 31]. Figure\n5(a) shows the magnetic hysteresis loop (M-H) between\n\u000690 kOe for the Sr 4Fe3CoO 11) at 10 K, measured in FC\ncondition (cooling field, H FC=\u000640 kOe). For positive\ncooling field (H FC= 40 kOe), the M-H loop is shifted\ntowards the left field axis and it is opposite to the M-\nH loop measured in negative cooling field (H FC= -40\nkOe) at 10 K. This clearly indicates that the observed\nEB effect has not emerged from the unsaturated minor\nloop. The EB field is calculated from the M-H loop shift\n((HEB= -(HC(L)+ HC(R))/2, where H C(L)and HC(R)\nare the left and right intercepts with the field axis). A\nvery high value of H EB\u00181.5 kOe is observed at 10 K.\nThe suitable cooling field was chosen by measuring the\nM-H loops in various cooling fields (H FC) ranging from\n0.5 kOe to 60 kOe at 4 K. To study the nature of ex-\nchange bias properties, and to know its origin, we have\nmeasured the temperature variation of the exchange bias4\neffect. Figure 5(b) shows the temperature dependence of\nthe exchange bias field (H EB). The EB effect is started\nto emerge from the AFM ordering temperature (H EB=\n271 Oe at 250 K). Then, at 25 K the H EBshows a sharp\nrise and shows a very high value of 5.8 kOe at 4 K.\n10\n5\n0\n-5\n-10M (emu/g)\n-80 -40 0 40 80\nH (kOe)Cooling Fields\n \n+4T\n–4T\n (a)\n6\n4\n2\n0HEB (kOe)\n30025020015010050\nT (K)HFC = 4 T(b)\nFIG. 5. (a) Magnetic field dependent magnetization loop (M-\nH) at 10 K measured in field cooling mode and (b) tem-\nperature dependence of the exchange bias field (H EB) for\nSr4Fe3Co O 11. A large value of exchange bias field \u00181.5\nkOe is observed at 10 K.\nSr4Fe3CoO 11shows training effect. In training effect,\nthe uncompensated spin at the interface shows relaxation\ndue to the repetition of the M-H measurement. As a re-\nsult the exchange bias field shows variation on the num-\nber of successive loops studied. Figure 6 highlights the\nvariation of H EBwith number of field cycles (n). In this\ninstance, the exchange bias field shows a decreasing ten-\ndency as the cycle number (n) increases. The decrease in\nHEBcan be described by fitting the data to the following\nempirical power law for n \u00152 [32].\nHEB(n)\u0000HE1=k=pn; (1)\nWhereHEB(n)=exchangefieldinthenthcycle. H E1= exchange field for n = 1, and k is a system dependent\nconstant. The obtained value of H E1is 0.3 kOe, the\nremnantH EBat10K.Also, ourexperimentaldatafollow\nthe recursive formula suggested by Binek [33] i.e.,\nHEB(n+ 1)\u0000HEB(n) =\r(HEB(n)\u0000HE1)3;(2)\nwhere\r= 1/2k2. Notably, the Eq. 1 is applicable for\ndata starting from n = 2, while Eq. 2 is also valid for for\nn = 1. As discussed previously, the Fe1 site is topologi-\n1.6\n1.4\n1.2\n1.0\n0.8 HEB (kOe)\n12 10 8 6 4 2\n n M (emu/g)\n-11.5-11.0-10.5-10.0-9.5\nH (kOe)n\nFIG. 6. The exchange bias field (H EB) vs number of field\ncycle (n) (red open circles) as calculated from the training\neffect magnetic hysteresis loops (M-H) at 10 K. Fitting of\nEqn. 1 (see the text) to the H EBdata is shown as dotted\nline. The data points generated using the recursive formula,\nEqn. 2 (see the text) is shown by green triangles. An enlarged\npart of the consecutive M-H loops (left intercept with the field\naxis) is shown in the inset.\ncally frustrated due to a ferromagnetic Fe1-O-Fe2 double\nexchange and AFM Fe1-O-Fe1 super-exchange interac-\ntions. Also, the random distribution of the Co cation in\ntwo different Fe sites in having a net AFM interaction\noriginates additional frustration in the sample. There-\nfore, the giant EB effect in this material could be due\nto the existence of different magnetic interactions in the\nlayered structure.\nd. Magnetoresistance\nFigure 7(a) shows temperature dependence of the elec-\ntrical resistivity [ \u001a(T)] at different magnetic fields. All\ncurves exhibit a semiconducting-type behavior. We do\nnotobserveanychangeintheresistivitydatameasuredin\nthe presence of a magnetic field down to 250 K. However,\nbelow 250 K \u001a(T)Hstart to deviate from \u001a(T)H=0kOe\nand becomes higher down to 100 K. Then, with further\nlowering the temperature, an interesting crossover from\npositive to negative magnetoresistance is observed. The\ntemperaturevariationofmagnetoresistance(MR)(MR=5\n[\u001a(T)H-\u001a(T)H=0kOe]/\u001a(T)H=0kOe) is highlighted in fig-\nure 7(b). We have observed a huge value of negative\nMR over a wide temperature range (100 - 20 K), and at\n25 K the MR is -80 %. Above 100 K, the magnetoresis-\ntance turns into the positive side ( \u001810%at 150 K). In\norder to confirm the MR behavior and its crossover, re-\nsistivity (\u001a(H)) was measured in the presence of magnetic\nfield at different fixed temperatures. Similar to the \u001a(T)\nmeasurements at different magnetic fields, it confirms\nthe crossover of the magnetoresistance. However, the\nsmall positive magnetoresistance observed above 100 K\nis found to decrease with increasing magnetic field. The\nopening up of the band gap due to the dominant antifer-\nromagnetic interactions at higher temperature could be\nthe origin of the positive magnetoresistance in this sam-\nple. A similar situation is already observed in a few iron\nbased compounds [34, 35]. Then with further lowering\nthe temperature, magnetic frustration increases and this\ndevelops the spin-dependent scattering of carriers which\nin turn originate the conventional (negative) MR in this\nsample. At a lower temperature, the magnetic disorder\ndue to the existence of different magnetic interactions in\nthe layered structure enhances the spin-dependent scat-\ntering of the carriers and therefore the electrical resis-\ntance in zero magnetic field.\nIV. CONCLUSIONS\nWe have investigated the structural, magnetic, ex-\nchange bias and magneto-transport properties for the\noxygen deficient Sr 4Fe3CoO 11materials. The com-\npound shows orthorhombic symmetry, crystallizing in\nthe C mmmspace group, isostructural with Sr 4Fe4O11.\nThe onset of the antiferromagnetic (G-type) transition\nis observed at T N= 255 K. Then, with further lower-\ning the temperature, the material shows a range of excit-\ningphysicalproperties. Temperatureinducedmagnetiza-\ntion reversal is observed at a compensation temperature\n(TComp:) of 47 K. The magnetization reversal is observed\ndue to the increased magnetocrystalline anisotropy with\nCo substitution. Also, two technologically important\nphenomena, magnetoresistance and exchange bias is ob-\nserved near room temperature in this compound. The\nexchange bias is observed below the AFM ordering tem-\nperature when the sample is cooled with a magnetic field.\nThelargeexchangebiaseffectinthissampleisattributed\nto the magnetic frustration in the structure. The mag-\nnetoresistance starts to appear below T N= 255 K and\nreaches a huge value of 80 %at 25 K in 7 T. An interest-\ning crossover in the magnetoresistance value from nega-\ntive to positive side is observed \u0018100 K. The opening up\nof the band gap due to the dominant antiferromagnetic\ninteractions at higher temperature could be the origin\nof the positive magnetoresistance in the sample. On the\nother hand, increasing magnetic frustration in the struc-\nture with lowering the temperature develops the spin-\n01234567-40-30-20-10010 \n MR (%)H\n (T) 300 K \n150 K \n80 K\n1002 003 000.010.1110 \n0 T \n2 T \n7 T \n ρ (Ω−cm)(b)\n1002 003 00-75-50-25025 MR (%)T\n (K)(a)FIG. 7. (a) Temperature dependence of resistivity mea-\nsured at different magnetic field. (b) Temperature depen-\ndence of magnetoresistance at different magnetic field for\nSr4Fe3CoO 11. Magnetic field dependence of magnetoresis-\ntance at different fixed temperature is shown in the inset in\nfigure (b).\ndependent scattering of carriers which originate the con-\nventional (negative) magnetoresistance.\nACKNOWLEDGEMENTS\nPM and SM have contributed equally. SM, CM and\nOT thank CNRS, Université Bordeaux, and SOPRANO\nproject(SeventhFrameworkProgrammeFP7/2007-2013\nunder Grant Agreement No. 214040) for funding this\nwork. R. P. S. acknowledges Science and Engineering\nResearch Board (SERB), Government of India for the\nRamanujan Fellowship through Grant No. SR/S2/RJN-\n83/2012. S. M acknowledges Science and Engineering\nResearch Board, Government of India for the NPDF fel-\nlowship (Ref. No. PDF/2016/000348). This work is6\nbased on experiments performed at the Swiss spallation\nneutron source SINQ, Paul Scherrer Institute, Villigen,\nSwitzerland. 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From this calculation, we have found the interesting effect of epitaxial strain on \nthe magnetocrystalline anisotropy (MCA). Analysis of the energy and k-resolved distribution of the \norbital character of the band structure reveals that MCA largely arises from the spin-orbit coupling \n(SOC) between dx2-y2 andd xz,yz orbitals of Fe atoms at the FeRh/MgO interface. We demonstrate that \nthe strain has significant effects on the MCA: It not only affects the value of the MCA but also induces \na switching of the magnetic easy axis from perpendicular to in-plane direction. The mechanism is the \nstrain-induced shifts of the SOC d-states. Our work demonstrates that strain engineering can open a \nviable pathway towards tailoring magnetic properties for antiferromagetic spintronic applications.\n1. Introduction\nNowadays, the field of antiferromagnetic (AFM) spintronics has been treated as an promising \nmaterial in the science community [1-10]. First priority reason is that they do not produce stray fields \nwhen it is used to device such as random access memory (RAM) [1-7]. Because structures being \nsimilar to their ferromagnetic (FM) counterparts, AFM spintronics complements or replaces \nferromagnets by antiferromagnets in the active components of spintronic devices [8-11]. Second \nreason is that the intrinsic high frequencies of AFM dynamics [12]. It makes them distinct from \nferromagnets. Due to these prominent properties, antiferromagnets have been used as magnetic \nrecording media with good performance. Among various AFM materials the near equiatomic \nchemically ordered bcc-B2 (CsCl-type) bulk FeRh alloy is attracting intense interest due to its unusual first-order phase transition from AFM to FM order at≈350 K. This is accompanied with a volume \nexpansion of ≈1% indicating a coupling between the spin and structural degrees of freedom. Together \nwith the relativistic effects present in FeRh such as magnetocrystalline anisotropy, MCA, thermally \nassisted FeRh-based memory have been successfully fabricated. Most of density functional theory \ncalculations to date have focused on the electronic structure properties solely of the bulk bcc structure \nunder hydrostatic pressure [13]. On the other hand, FeRh thin films are grown expitaxially on MgO, \nBaTiO3, or piezoelectric substrates, where the lattice mismatch inevitably generates strain which may \nmodulate the magnetic properties of the FeRh overlayer [14,15]. Recent ab initio electronic structure \ncalculations have investigated the relative stability of the FM, “bcc-like-AFM”, and “fcc-like-AFM” \nstructures under epitaxial strain [16,17]. Therefore, it is of great importance to understand the effect of \nstrain on the MCA of ultrathin FeRh thin films for further promoting their applications in AFM \nspintronics. In this work, we investigate the effect of epitaxial strain on the MCA of FeRh/MgO by \nperforming ab initio density functional theory (DFT) electronic structure calculations.  We find that the \nvalue of MCA and the direction of magnetic easy axis strongly depend on the strain leading to a spin \nre-orientation from an in- to out-of-plane magnetization orientation in the AFM phase and across the \nmetamagnetic transition. The underlying mechanism is explained by analyzing the band structure and \nthe strain-induced shifts of distribution of orbital characters. Our work demonstrates that strain \nengineering can serve as a viable and efficient approach in tuning the magnetization directions in \nFeRh.\n2. Computational details\nWe use the Vienna ab initio simulation package (VASP) to perform the electronic structure \ncalculations. The projector augmented wave formalism is adopted for describing the electron-ion \ninteractions, and plane waves with a kinetic energy cutoff of 350 eV are used to expand the wave \nfunctions. The generalized gradient approximation (GGA) in the version of Perdew-Burke-Ernzerhof \n(PBE) is adopted for treating the electron exchange and correlation [18-22]. The GGA exchange \ncorrelation functional has been shown to provide a reasonable description of the structural properties \nof FeRh, in contrast to the local density approximation which yields incorrectly that the fcc-like AFM phase is the ground state. With a Monkhorst-Pack k-mesh of 31 x 31 x 31 for Brillouin zone (BZ) \nsampling, the calculated equilibrium lattice constants (a) of bulk G-AFM and FM phase FeRh are \n2.995 Å and 3.012 Å, respectively, in good agreement with available experimental data. The ultrathin \nFeRh (001) film on the rock-salt MgO (001) substrate is modeled by a slab structure, as is shown in \nFig. 1. \n The slab supercell consists of five monolayers (ML) of FeRh with each ML consisting of two Fe or \nRh atoms, which are placed on the top of a MgO slab. A 12 Å thick vacuum region is introduced in the \nsupercell to avoid the artificial interactions between the slab and its images created by the periodic \nboundary conditions. The 110 axis of FeRh is aligned with the 100 axis of MgO and the O atoms at \nthe FeRh/MgO interface are placed atop of the Fe atoms. In this work we consider the Fe-terminated \ninterface and surface in both the G-AFM and FM phases, respectively. For each strain condition, the \nmagnetic and electronic degrees of freedom and the atomic z-positions are fully relaxed until the \nmaximum force acting on each single ion is less than 0.01 eV/Å. A Monkhorst-Pack k-mesh of 15 x 15  \nx 1 is used in the self-consistent DFT calculations followed by a 31 x 31 x 1 k-mesh for the scalar \nrelativistic calculations with the spin-orbit coupling (SOC). The MCA per interfacial area is determined \nby [E[100]-E[001]] where E[100] (E[001]) is the total energy with SOC62included along the [100] and \n([001]) directions, respectively.\n3. Results and discussionsIn Table I, we list the calculated values of the strain dependence of the spin magnetic moments (∆ms), \norbital magnetic moment differences (mo) of the interfacial Fe atom (Fe i) and surface Fe atom66(Fe s), \nrespectively, for both the G-AFM and FM phases, respectively. We also list the MCA values and the \ntotal energy difference (∆E) between the the G-AFM and FM phases under different strain. Note \nbecause the two Fe atoms on each atomic plane in the G-AFM phase have opposite ms.\nTABLE I. Strain dependence of the spin magnetic moment (ms) and91orbital magnetic moment \ndifference (mo = m [001] o m [100] o ) along the [001] and [100] directions, of92the interfacial (Fei ) \nand surface (Fes) Fe atoms, for the AFM and FM phases, respectively. We list also93the strain \ndependence of the MCA values of the G-AFM and FM FeRh/MgO bilayer, as well as the94total \nenergy difference between the two magnetic phases.\nwe only list its magnitude. We find that for the range of strain considered here the G-AFM phase is \nmore stable than the FM phase, where the energy difference ∆E=E FM-EG-AFM= 20 meV/Fe regardless \nof the strain. Previous DFT calculations for free-standing FeRh films reported values of 25 meV/Fe, \nindicating that the MgO substrate slightly decreases the energy difference between the two phases.  \nThe ms values of Fei and Fes in the G-AFM and FM systems range from 3.02 to 3.19 μB, which are \nclose to the bulk value of 3.4 μB, and depend weakly on strain. For the G-AFM phase the variation of \nMCA with strain is large and correlates well with the strain variation of ∆mo of Fei but not with that of \nFes. For the G-AFM phase the compressive strain yields an in-plane magnetic easy axis while tensile \nstrain induces an out-of-plane easy axis, where the spin reorientation occurs around 0. This strain \ninduced MCA energy behavior is similar (opposite) to that in CoFe 2O4 (CoCr 2O4). Furthermore tensile strain induces an in-plane (out-of-plane) magnetic easy axis in Ta/FeCo/MgO (Au/FeCo/MgO) trilayers. \nTo understand the underlying mechanism of the strain effect, we plot in Fig. 2. \nFigure 2 Energy- and k-resolved distribution of orbital character of the minority-spin bands of the \ninterfacial Fe atom (Fei). k points with large negative or positive contributions to the total MCA are \nindicated by ΓΧ-ΧM (based on the second-order perturbation theory of MCA). \nThe energy and k-resolved distribution of the orbital character of the minority-spin bands of the \nspin-up interfacial iron Fei-derived dxz,yz and dx2-y2 states for -0.50% strain. (The analysis of the \nspin-down Fei atom is identical.) Within the second-order perturbation theory, the MCA can be \nexpressed as:\n,\nwhere the coupling matrix elementh <·|·|·> denotes the SOC of the occupied state (o) with eigen \nenergy Eo and the unoccupied state (u) with eigenenergy Eu through angular momentum operator xy \norz (contributes positively) with proper selection rules. Since our analysis for the spin-up Fei is in terms \nof the minority-spin bands, only the first term in Eq. (1) is important, because the majority spin bands \nare well below the Fermi energy. k points with large negative or positive contributions to the total MCA \nare indicated by ΓΧ-ΧM (based on the second-order perturbation theory of MCA). From Fig. 2 one can see that around the ΓΧ and ΧM points the unoccupied dyz,xz states couple to the occupied dx2-y2 \nstate through xy, giving negative contributions to the total MCA. On the other hand, in a wide region \naround k between ΓΧ and ΧM the occupied and unoccupied dyz,xz states couple through z2 yielding \nlarge positive contributions. The sum over these k points gives the negative MCA of 0.47 erg/cm2 for \ns=0.50%. Note that around k points ΓΧ and ΧM cross the Fermi level and are prone to shift under \ndifferent strain. In Fig. 3 we show the minority-spin projected density of states (PDOS) of the spin-up \nFei dyz, dxz, and dx2-y2 orbitals for three strains. As one can see, with increasing there is a redistribution \nof PDOS on dx2-y2 from unoccupied to occupied, indicating a down-shift of this orbital across the Fermi \nlevel around111k4. The PDOS distribution of dxz orbital in the range of [-0.5, 0.5] eV is slightly FIG. 3.\nProjected density of states (PDOS) (minority) on the spin-up Fei dyz, dxz, and dx2-y2 orbitals for \ns=-0.50, 0, and 0.50%, respectively. The PDOS of the dxz and dx2-y2 are shifted by 1.5 and 3 units \nalong the vertical axis, respectively. The peak of unoccupied PDOS shifts from 0.025 to 0.15 eV, \nsuggesting a upward-shift of this orbital around ΓΧ and k between the ΓΧ and ΧM. The occupied PDOS \ndistribution in the range of [0.1, 0.05] eV is largely enhanced due to the down-shift of the band. \nOverall, the band shifts will induce a large variation of MCA. Specifically, the band shifts around ΓΧ \nand ΧM decrease or disable the negative SOC while the band shifts at k3 enable the positive SOC. As \na result, the MCA value changes from negative to positive when the strain changes from -0.5% to \n0.5%.4. Conclusion \nWe have studied the effect of epitaxial strain on the MCA of an ultrathin FeRh/MgO bilayer system by \nperforming ab initio DFT electronic structure calculations. Under a compressive strain of -0.5%, the \nsystem possesses a large MCA value with the magnetic easy axis being in-plane. Detailed analyses \nbased on the perturbation theory show this is due to the <dyz,yz|Lyz,zx|dx2−y2> coupling matrix elements.  \nWhen the strain changes from compressive to tensile, the induced band shifts and the concomitant \nredistributions of PDOS greatly change the contribution from each k point to the MCA, thus causing \nthe reorientation of the magnetic easy axis from in- to out-of-plane. Our work demonstrates that strain \nengineering can be used to tailor the magnetic properties of AFM FeRh spintronic devices.\n5. References \n1. T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nature Nanotech. 11, 231 (2016).\n2. E. V. Gomonay and V. Loktev, Low Temp. Phys. 40, 17 (2014). \n3. F. Keffer and C. Kittel, Phys. Rev. 85, 329 (1952). \n4. S. Loth, S. Baumann, C. P. Lutz, D. M. Eigler, and A. J. Heinrich, Science 335, 196 (2012). \n5. P. Wadley, B. Howells, T. Jungwirth et al., Science 351, 587 (2016). \n6. A. V. Kimel, A. Kirilyuk, A. Tsvetkov, R. V. Pisarev, and T. Rasing, Nature 429, 850 (2004). \n7. J. S. Kouvel and C. C. Hartelius, J. Appl. Phys. 33, 1343 (1962). \n8. G. Shirane, C. W. Chen, P. A. Flinn, and R. Nathans, Phys. Rev. 131, 183 (1963). \n9. X. Marti, I. Fina, C. Frontera, J. Liu, P. Wadley, Q. He, R. J. Paull, J. D. Clarkson, J. Kudrnovský, I. \nTurek, J. Kuneš, D. Yi, J.-H. Chu, C. T. Nelson, L. You, E. Arenholz, S. Salahuddin, J. Fontcuberta, T. \nJungwirth, and R. Ramesh, Nat. Mater. 13, 367 (2014). \n10. M. E. Gruner, E. Hoffmann, and P. Entel, Phys. Rev. B 67, 064415 (2003). \n11. L. M. Sandratskii and P. Mavropoulos, Phys. Rev. B 83, 174408 (2011). \n12. V. L. Moruzzi and P. M. Marcus, Phys. Rev. B 46, 2864 (1992). \n13. C. Bordel, J. Juraszek, D. W. Cooke, C. Baldasseroni, S. Mankovsky, J. Minár, H. Ebert, S. Moyerman, E. E. Fullerton, and F. Hellman, Phys. Rev. Lett. 109, 117201 (2012). \n14. R. O. Cherifi, V. Ivanovskaya, L. C. Phillips, A. Zobelli, I. C. Infante, E. Jacquet, V. Garcia, S. Fusil, P. \nR. Briddon, N. Guiblin, A. Mougin, A. A. Ünal, F. Kronast, S. Valencia, B. Dkhil, A. Barthélémy, and M. \nBibes, Nat. Mater. 13, 345 (2014). \n15. Z. Q. Liu, L. Li, Z. Gai, J. D. Clarkson, S. L. Hsu, A. T. Wong, L. S. Fan, M.-W. Lin, C. M. Rouleau, T. \nZ. Ward, H. N. Lee, A. S. Sefat, H. M. Christen, and R. Ramesh, Phys. Rev. Lett. 116, 097203 (2016). \n16. Y. Lee, Z. Q. Liu, J. T. Heron, J. D. Clarkson, J. Hong, C. Ko, M. D. Biegalski, U. Aschauer, S.-L. Hsu, \nM. E. Nowakowski, J. Wu, H. M. Christen, S. Salahuddin, J. B. Bokor, N. A. Spaldin, D. G. Schlom, \nand R. Ramesh, Nat. Commun. 6, 5959 (2015). \n17. U. Aschauer, R. Braddell, S. A. Brechbühl, P. M. Derlet, and N. A. Spaldin, Phys. Rev. B 94, \n014109 (2016). \n18. P. E. Blöchl, Phys. Rev. B 50, 17953 (1994). \n19. G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). \n20. G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996). \n21. G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). \n22. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).\n "    },    {        "title": "1908.04762v1.Prediction_magnetocrystalline_anisotripy_Fe_Rh_thin_films_via_machine_leaning.pdf",        "content": "Prediction magnetocrystalline anisotripy Fe-Rh thin films via machine leaning\nEun Sung Jekal 1,*, Hyunwoo Park1,\n1 Department of Materials, ETH Zurich, Zurich 8093, Switzerland\n* Correspondence: everjekal@gamil.com, Tel.: +41 76 510 63 96\nAbstract  \n Least absolute shrinkage and selection operator (Lasso) was originally formulated for least \nsquares models and this simple case reveals a substantial amount about the behavior of the \nestimator. It also shows that the coefficient estimates need not be unique if covariates are \ncollinear. Using this Lasso technique, we analyze a magnetocrystalline anisotropy energy which is \na long-standing issue in transition-metal thin films, expectially for Fe-Rh thin film systems on a \nMgO substrate. Our LASSO regression took advantage of the data obtained from first principles \ncalculations for single slabs with seven atomic-layers of binary Fe-Rh films on MgO(001). In the \ncase of Fe-Rh thin films, we have successfully found a linear behavior between the MCA energy \nand the anisotropy of orbital moments.\nKeywords : machine learning, magnetism, magnetocrystalline anisotropy\n1. Introduction\n1.1. Magnetocystalline anisotropy\nAfter Moore’s Law broke few years ago, researchers have started to look for alternatives \nmaterials[1-5]. One of a promising candidate way is magnetic random access memory (MRAM) \ndue its advantages such as the non-volatility, high speed process as well as low power \nconsumption[6-16]. Beside, in another perspective, it is necessary to develop themagnetic tunnel \njunction element which consists of insulating and magnetic metal thin films with strongly \npreferred perpendicular magnetization with respect to the film plane as Fig. 1.\nIn this respect, materials containing strong spin orbit coupling (SOC), for example 4d and 5d \nelements can be expected to have perpendicular magnetocrystalline anisotropy (MCA)[17-19]. This \nhas been confirmed by calculations, for instance, in some Fe-Rh binary systems[20-22]. In such systems, while the magnetic moments are carried out by the Fe atoms or other 3d transition \nmetal (TM) elements, the strong SOC is introduced by the non-magnetic such as Rh which is 5d \nTM elements[20]. However, it was shown theoretically that the MCA of thin films depends strongly \nnot only on the choice of atomic elements, but also on the detail of the atomic layer alignment. \nDesigning novel magnetic metal multilayer films with a strong perpendicular MCA is, therefore, a \nvery demanding task. The difficulty is in part related to the fact that MCA is related to many \nnon-trivial but competing factors such as the SOC strength and crystal field, in a way that the full \nunderstanding of the origin of MCA has been one of the greatest issues in the field of magnetism.\nFigure 1. Calculation model and notation for atomic layer configurations of Fe-Rh/MgO. The\nperpendicular and in-plane directions are [001] and [100], respectively. All atomic layer patterns\n(27 = 128) expressed by binary configurations are considered as illustrated in the right side of the figure\n1.2. Least absolute shrinkage and selection operator (Lasso)\n Lasso was introduced in order to improve the prediction accuracy and interpretability of \nregression models by altering the model fitting process to select only a subset of the provided \ncovariates for use in the final model rather than using all of them[23,24]. It was developed \nindependently in geophysics, based on prior work that used the L1 penalty for both fitting and \npenalization of the coefficients, and by the statistician, Robert Tibshirani based on Breiman’s \nnon-negative garrote. Prior to lasso, themost widely usedmethod for choosing which covariates to \ninclude was stepwise selection, which only improves prediction accuracy in certain cases, such as \nwhen only a few covariates have a strong relationship with the outcome. However, in other cases, \nit can make prediction error worse. Also, at the time, ridge regression was the most popular \ntechnique for improving prediction accuracy. Ridge regression improves prediction error by \nshrinking large regression coefficients in order to reduce overfitting, but it does not perform \ncovariate selection and therefore does not help to make the model more interpretable.\n Lasso is able to achieve both of these goals by forcing the sum of the absolute value of the \nregression coefficients to be less than a fixed value, which forces certain coefficients to be set to \nzero, effectively choosing a simpler model that does not include those coefficients. This idea is \nsimilar to ridge regression, in which the sum of the squares of the coefficients is forced to be less \nthan a fixed value, though in the case of ridge regression, this only shrinks the size of the \ncoefficients, it does not set any of them to zero.\n1.3. Link between MCA and Lasso\n In an effort to understand the MCA of Fe-Rh films on a MgO substrate, using the LASSO \ntechnique, we examine the MCA and the anisotropy of orbital moments obtained from first \nprinciples calculations of Fe and Rh atoms to analyze the Bruno relation. Our analysis indeed \nconfirms that the relation is applicable for systems.\n2. crystal structure and method\n Self-consistent Density Functional Theory calculations were performed based on the generalized \ngradient approximation (GGA) without SOC, by using the Full-potential Linearized Augmented \nPlane-Wave (FLAPW) method and Quantum Espresso. In all calculations, two-dimensional film \ngeometries with additional vacuum regions have been considered. We model single slabs with \nseven-atomic layers of binary Rh-Fe film on top of three layers of MgO (001) substrates. All \natomic-layer configurations, which include as many as 27=128 possibilities for binary system, were \nconsidered in the calculations (see Fig. 1). A planewave cut-off |k + G| of 3.9 a.u. has been chosen, \nand we choose suitable muffin-tin radii of 2.4 bohr for Rh and 2.2 bohr for Fe. The MCA energy, \nEMCA, is defined in our calculations as the energy eigenvalue difference between the \nmagnetizations oriented along the in-plane [100] and perpendicular [001] directions with respect \nto the film plane. To determine the EMCA, the second variational method for treating SOC by using \nthe calculated eigenvectors in the SRA has been employed, and then the MCA energy has been \ncalculated by using the force theorem.The number of special k-points in the two-dimensional Brillouin zone (BZ) was 3600, and it is \nfound that such k-point mesh was large enough to suppress numerical fluctuations in the MCA \nenergy by less than 0.01 meV, which is a sufficient accuracy for the purpose of the present work.    \nIn regression analysis, we consider the linear regression model in which the MCA energy is \nexpressed by using the orbital magnetic moment,\nHere, the EMCA has been expanded by using the descriptors, which for the present work, we \nchoose the anisotropies and averages of orbital moments. The orbital moment anisotropy  is \ndefined as the difference between the orbital moments of the perpendicular and in-plane \nmagnetizations for Fe or Rh species at the ith layer. Likewise, the orbital moment average  is \ndefined as the average of the perpendicular and inplane orbital moments of the respective atomic \nspecies. To eliminate the bias of choosing the descriptor, we consider that is not only the \nanisotropy, but also the average included. A leave-oneout cross validation, where the data is \nseparated to 63 training data and one test data, was applied. Coefficients A, B, C and D in the Eq. \n1 show how the descriptors are correlated to the parameters. For example, a large AAu,1 indicates \nthat the anisotropy of orbital moment of the Au atomic layer placed at the top layer, i.e., layer 1, \nstrongly affects the prediction of EMCA. The A, B, C, and D coefficients are determined by using the \ntraining data, and then used to predict EMCA of the test data.\n3. Results\nThe calculated results of MCA energy and anisotropy of orbital moments as functions of the \natomic composition for each binary system are shown in Fig. 2. \nFigure 1. Results of anisotropy of orbital moments, Manis, (a) and MCA energies, EMCA, (b) as \nfunctions of the composition for Fe-Rh thin film systems. Atomic layer configurations are \nrepresented, e.g., by FRFRRFFR for FeRhFeRhRhFeFeRh/MgO, where F and R indicate Fe and Rh atomic layers, respectively.\n3. Results\n The calculated results of MCA energy and anisotropy of orbital moments as functions of the \natomic composition for each binary system are shown in Fig. 2. Both the computed values diverge \nlargely even within the same atomic composition. For a specific atomic composition, e.g., for a \nFe2Rh6 in the Fe-Rh binary system, the MCA energy varies largely from in-plane anisotropy with \nan EMCA of about 4.0 meV/unit-area, to a perpendicular one with an EMCA of about 1.5 \nmeV/unit-area. Figure 3a shows a scatter plot between EMCA and Manis which suggest a linear \nrelationship, as visualized with the red line. Hence, large values, implying the significance of all \nanisotropy of orbital moments. Figure 3c further shows the comparison between the calculated \nEMCA from first principles and the estimated ones using LASSO. These results show that the EMCA \nvalues predicted using LASSO agree well those obtained exlicitly by using first principles \ncalculations. In the Fe-Rh binary systems, the EMCA varies largely from 4.2 meV/unit-area with a \nperpendicular anisotropy to -3.0 meV/unit-area with an inplane one. Fig. 2b plots between MCA \nenergy, EMCA, and anisotropy of orbital moments, Manis, for Fe-Rh binary system. (b) calculated \ncoefficients for anisotropy and average values of orbital moments in Eq. 1, where Mani and Mave \nindicate, respectively, the anisotropies and averages of the orbital moment. The numbers in the \nhorizontal axis denote the atomic layer positions in the film from the surface to the interface. \nComparison of the MCA energies calculated by the first principle calculations and estimated by \nusing the LASSO coefficients. Red and blue solid circles show the data of training and test. Then I \ncalculated by first principle calculations and those predicted using the LASSO coefficients are not \nsatisfying. Indeed, the induced magnetic moments in Rh due to Fe. Combined with a strong SOC, \nsuch small splitting will lead to important spin-flip term to the SOC energy, resulting 111 in a \nbreakdown of the Bruno relation. Additionally, in the present analysis, we considered only the \norbital moment anisotropy and averages as the data descriptors for the MCA energy. The \ndiscrepancy between the first-principles calculated and the predicted EMCAs reveals that such \ndescriptors might not be sufficient to predict the MCA energy of systems with strong SOC.4. Conclusions\n We found calculation results which are combined first-principles calculations (density functional \ntheory) and LASSO which is machine learning analysis. Finally we show that the EMCA of the Fe-Rh \nthin films is strongly correlated to the anisotropy of orbital moments. Furthermore, the estimated \nEMCA shows a reasonable agreement with the calculated one.\n5. References\n[1] B. Kish, Laszlo, Physics Letters A, 305.3-4  144 (2002).\n[2] Roberts, G. Lawrence,. Computer , 33, 117 (2000): 117.\n[3] N. Theis, Thomas, and H-S. Philip Wong, Computing in Science & Engineering, 19, 41 \n(2017).\n[4] Wu, Jerry, et al, Applied Computational Intelligence and Soft Computing, 2013, 2 (2013).[5] Li, Yingfeng, and B. Kish, Laszlo, Fluctuation and Noise Letters, 6, L127 (2006).\n[6] Pey, Kin Leong, et al, 2019 Electron Devices Technology and Manufacturing Conference (EDTM). \nIEEE, (2019).\n[7] Cubukcu, Murat, et al., IEEE Transactions on Magnetics, 54, 1 (2018).\n[8] Thomas, L., et al., 2017 IEEE International Magnetics Conference (INTERMAG). IEEE, (2017).\n[9] Huai, Yiming, et al., Applied Physics Letters, 112, 092402 (2018).\n[10] S. Jekal, et al., Physical Review B,  92, 064410 (2015).\n[11] S. Jekal, et al., Recent Advances in Novel Materials for Future Spintronics, 25 (2019).\n[12] S. Jekal, et al.,  Applied Sciences, 9.4, 630 (2019).\n[13 S. Jekal, et al., Journal of Applied Physics, 117, 17E105 (2015).\n[14 S. Jekal, et al., Journal of nanoscience and nanotechnology, 15, 2356 (2015).\n[15] Jeong, Tae Sung, et al. J. Kor. Mag. Soc., 27, 41 (2017).\n[16] S. Jekal, J. F. Loeffler, and M. Charilaou, arXiv preprint arXiv:1807.09257  (2018).\n[17] Mokrousov, Yuriy, et al. Physical review letters, 96, 147201 (2016).\n[18] Kikuchi, Nobuaki, et al., Journal of Physics: Condensed Matter, 11, L485 (1999).\n[19] Ležaić, Marjana, and Nicola A. Spaldin. Physical Review B, 83, 024410 (2011).\n[20] Ibarra, M. R., and P. A. Algarabel. \"Giant volume magnetostriction in the FeRh alloy.\" Physical \nReview B 50.6 (1994): 4196.\n[21] Moriyama, Takahiro, et al., Applied Physics Letters , 107, 122403 (2015).\n[22] Loving, M., et al. Journal of Physics D: Applied Physics, 46, 162002 (2013).\n[23] Tibshirani, Rober, Journal of the Royal Statistical Society: Series B (Methodological), 58, 267 \n(1996).\n[24] Bickel, Peter J., Ya’acov Ritov, and Alexandre B. Tsybakov, The Annals of Statistics, 37, 1705 \n(2009)."    },    {        "title": "1908.05836v2.Two_dimensional_magnetic_semiconductors_with_room_Curie_temperatures.pdf",        "content": "Two-Dimensional Magnetic Semiconductors with Room Curie Temperatures\nJing-Yang You,1Zhen Zhang,1Xue-Juan Dong,1Bo Gu,2, 3,\u0003and Gang Su1, 2, 3,y\n1School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China\n2Kavli Institute for Theoretical Sciences, and CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijng 100190, China\n3Physical Science Laboratory, Huairou National Comprehensive Science Center, 101400 Beijing, China\nWe propose two-dimensional (2D) Ising-type ferromagnetic semiconductors TcSiTe 3, TcGeSe 3and\nTcGeTe 3with high Curie temperatures around 200 \u0018500 K. Owing to large spin-orbit couplings,\nthe large magnetocrystalline anisotropy energy (MAE), large anomalous Hall conductivity, and large\nmagneto-optical Kerr e\u000bect were discovered in these intriguing 2D materials. By comparing all\npossible 2D MGeTe 3materials (M = 3d, 4d, 5d transition metals), we found a large orbital moment\naround 0.5 \u0016Bper atom and a large MAE for TcGeTe 3. The large orbital moments are unveiled to\nbe from the comparable crystal \felds and electron correlations in these Tc-based 2D materials. The\nmicroscopic mechanism of the high Curie temperature is also addressed. Our \fndings expose the\nunique magnetic behaviors of 2D Tc-based materials, and present a new family of 2D ferromagnetic\nsemiconductors with large MAE and Kerr rotation angles that would have wide applications in\ndesigning spintronic devices.\nI. INTRODUCTION\nSpin-orbit coupling (SOC) describes the relativistic in-\nteraction between the spin and orbital momentum of elec-\ntrons1. SOC can drive rich phenomena, such as mag-\nnetic anisotropy2, spin relaxation3, magnetic damping4,\nanisotropic magnetoresistance5, and anomalous Hall ef-\nfect6. Recently, a term spin-orbitronics is proposed to\ncover the expanding research \fled, where SOC is a key\nconcept1,7,8. Combining strong SOC and magnetism,\nmany intriguing physical phenomena can be achieved,\nincluding current-driven magnetization reversal9{11, do-\nmain wall propagation12,13, current-driven skyrmion mo-\ntion14{16, etc. Transition metals are usually candidates\nto realize these phenomena and play important roles in\nspin-orbitronics.\nMagnetic anisotropy is one of the fundamental proper-\nties of magnetic materials. It is a key issue in recent\nadvances in two-dimensional (2D) magnetic semicon-\nductors17{20. According to Mermin-Wagner theorem21,\nat \fnite temperatures, the quantum spin-S Heisenberg\nmodel with isotropic and \fnite-range exchange interac-\ntions in 1D or 2D lattices can be neither ferro- nor anti-\nferromagnetism. Thus, for stabilizing long-range ferro-\nmagnetic order in 2D magnetic semiconductors at \fnite\ntemperature, a large magnetic anisotropy, which makes\nthe systems away from the isotropic Heisenberg model,\nis extremely important.\nMagneto-optical Kerr e\u000bect (MOKE), that is closely\nrelated to SOC, is a basic magneto-optic e\u000bect. It de-\nscribes that the plane-polarized light re\rected from a\nmagnetized material becomes elliptically polarized, and\nthe plane of polarization is rotated. MOKE is widely\nused to probe the electronic structure of magnetic ma-\nterials. Many exciting phenomena related to MOKE\nhave been discovered, such as quantum con\fnement ef-\nfects22, oscillations of the Kerr rotation with magnetic\nlayer thickness23and strong correlations between MOKE\nand magnetic anisotropies24. Due to the applicationof MOKE to the readout process in magneto-optical\n(MO) storage devices, many e\u000borts have been devoted to\nsearching for materials with large Kerr rotation angles.\nIn this paper, we propose three stable 2D ferromag-\nnetic semiconductors TcSiTe 3, TcGeSe 3and TcGeTe 3,\nwhich share the same crystal structure as the recently dis-\ncovered 2D magnetic semiconductor CrGeTe 319. These\nTc-based 2D materials have not been observed experi-\nmentally yet. The Monte Carlo simulations give Curie\ntemperatures 538 K, 212 K and 187 K for TcSiTe 3,\nTcGeSe 3and TcGeTe 3monolayers, respectively, which\nare much higher than the Curie temperature in CrGeTe 3.\nThe calculations show that these Tc-based materials have\nspin moment about 2 \u0016Band an extraordinarily large\norbital moment about 0.5 \u0016Bper Tc atom. The large\norbital moment comes from the partially occupied dor-\nbitals, and the partial occupation of dorbitals is due to\nthe comparable crystal \felds and electron correlations in\nthese Tc-based 2D materials. As a result, a large SOC\nis obtained in these materials. Due to the large SOC,\na large magnetocrystalline anisotropy energy (MAE) is\nformed, indicating the Ising behavior of these 2D materi-\nals with out-of-plane magnetization. In addition, a large\nKerr rotation angle about 3.6 degree is achieved in these\nTc-based materials, which is much larger than the value\nof 0.8 degree in metal Fe. Large anomalous Hall conduc-\ntivity of about 7.5 \u0002102(\n\u0001cm)\u00001in p-type TcGeTe 3and\n1.1\u0002103(\n\u0001cm)\u00001in n-type TcGeTe 3is comparable to\nthe anomalous Hall conductivity of 7.5 \u0002102(\n\u0001cm)\u00001in\nbulk Fe25,26and 4.8\u0002102(\n\u0001cm)\u00001in bulk Ni27. The mi-\ncroscopic mechanism of high Curie temperature in these\nTc-based materials is also discussed.\nII. COMPUTATIONAL METHOD\nOur \frst-principles calculations were based on the\ndensity-functional theory (DFT) as implemented in the\nVienna ab initio simulation package (VASP)28, usingarXiv:1908.05836v2  [cond-mat.mtrl-sci]  6 Jan 20202\nthe projector augmented wave method29. The general-\nized gradient approximation (GGA) with Perdew-Burke-\nErnzerhof30realization was adopted for the exchange-\ncorrelation functional. We take the on-site Hubbard in-\nteractionU= 2.3 eV and Hund coupling J= 0.3 eV31\nfor considering electron correlation of 4 delectrons of Tc\natoms, and the e\u000bective Ueff=U\u0000J= 2 eV, be-\ncause the reasonable Ueffis about 2 eV for 4 delec-\ntrons. The plane-wave cuto\u000b energy was set to 550 eV.\nThe Monkhorst-Pack k-point mesh32of size 13\u000213\u00021\nwas used for the BZ sampling. The structure relaxation\nconsidering both the atomic positions and lattice vectors\nwas performed by the conjugate gradient (CG) scheme\nuntil the maximum force on each atom was less than\n0.0001 eV/ \u0017A, and the total energy was converged to 10\u00008\neV with Gaussian smearing method. To avoid unneces-\nsary interactions between the monolayer and its periodic\nimages, the vacuum layer is set to 15 \u0017A. The phonon\nfrequencies were calculated using a \fnite displacement\napproach as implemented in the PHONOPY code33, in\nwhich a 3\u00023\u00021 supercell and a displacement of 0.01\n\u0017A from the equilibrium atomic positions are employed.\nWannier90 code34is used to construct an e\u000bective tight-\nbinding Hamiltonian and to calculate the optic conduc-\ntivity and the anomalous Hall conductivity.\nGe (Si)\nTe (Se) \nTc \na1a2\na3b1b2\nΓMK(a) (b)\nFIG. 1. (a) Top and side views of the crystal structure of\nTcSiTe 3, TcGeSe 3and TcGeTe 3monolayers. The primitive\ncell is noted by a dashed line box. Tc atoms form a honey-\ncomb lattice. (b) The \frst Brillouin zone.\nIII. RESULTS\nThe crystal structure of TcSiTe 3, TcGeSe 3and\nTcGeTe 3monolayers from the prototype CrGeTe 3mono-\nlayer is depicted in Fig. 1(a), where the spcae group\nisP\u001631m(No.191). To determine the ground state of\nTcSiTe 3, TcGeSe 3and TcGeTe 3monolayers, in the ab-\nsence of SOC, we calculated the total energy for ferro-\nmagnetic (FM) and antiferromagnetic (AFM) con\fgura-\ntions as a function of lattice constant, and found that\nthe FM state has an energy lower than AFM state. The\noptimized lattice constants of 2D TcSiTe 3, TcGeSe 3and\nTcGeTe 3are calculated as 6.821 \u0017A, 6.379 \u0017A and 7.029 \u0017A,\nrespectively, which are reasonable according to the radiusof atoms.\nTo con\frm the stability of these three monolayers,\ntheir phonon spectra have been calculated. There is no\nimaginary frequency mode in the whole Brillouin zone\nas shown in Figs. S1(a), (b) and (c) (Supplemental Ma-\nterial), indicating that they are kinetically stable. We\nhave checked the stability of these structures with dif-\nferentUeff(1 eV and 3 eV), and the results show that\nthese structures are always stable. To further examine\nthe thermal stability, we performed ab initio molecular\ndynamics simulations using a 4 \u00024\u00021 supercell contain-\ning 160 atoms. After being heated at 300 K and 500 K\nfor 6 ps with a time step of 3 fs, only little structural\nand energetic changes occur as shown in Figs. S2(d), (e)\nand (f) (Supplemental Material), implying that TcSiTe 3,\nTcGeSe 3and TcGeTe 3monolayers are dynamically sta-\nble.\n/s32\n/s32\n/s32\n/s32/s32\n/s32\n/s32\n/s32-6 0\n3 0PDOS \nE-E F (eV)(a)\nt2g  up\nt2g  down\neg up\neg down6\n-5 0\n-3 3 0PDOS \nE-E F (eV)(b)\nt2g  up\nt2g  down\neg up\neg down5TcSiTe 3\n-3 \nCrGeTe3\nFIG. 2. The partial density of states (PDOS) of (a) TcSiTe 3\nand (b) CrGeTe 3monolayers, calculated by GGA+U method.\nThe structural stabilities of TcSiTe 3, TcGeSe 3and\nTcGeTe 3are also examined by the formation energy,\nwhich is calculated by Ef=E(TcAB 3)\u0000E(Tc)\u0000E(A)\u0000\n3E(B), whereE(Tc),E(A) andE(B) are the total en-3\n/s32/s32\n/g42 /g42 /g42/g42 /g42/g42 M M M K K K0 0 0\n-1 -1 -1 1 1 1E-E F (eV) (a) (b) (c)\nspin upspin down \nFIG. 3. The electronic band structures of TcSiTe 3calculated by (a) GGA+U, (b) GGA+SOC+U, and (c) HSE06 methods.\nTABLE I. The total energy Etotper unit cell for TcSiTe 3, TcGeSe 3and TcGeTe 3monolayers (in meV, relative to Etotof FMz\nground state) for several spin con\fgurations of Tc atoms (see Fig. 4), calculated by GGA+SOC+U method. The spin moment\nhSiand orbital moment hOi(in unit of \u0016B), single ion anisotropy (SIA, in meV) between the out-of-plane and in-plane FM\ncon\fgurations, exchange interaction J(in meV), and Curie temperature TCurie (in K) are calculated. The CrGeTe 3monolayer\nis also calculated with the experimental lattice constant19for comparison.\nFMzFMxFMyNAFMzSAFMzZAFMzPMhSi hOiSIAJ T Curie (K)\nTcSiTe 30.0 52.1 25.9 310.4 165.1 122.0 2000 1.866 0.540 -42.5 7.625 538\nTcGeSe 30.0 53.6 211.9 295.3 95.9 138.0 2065 1.884 0.515 -37.7 2.997 212\nTcGeTe 30.0 112.1 289.7 277.0 84.7 114.2 2300 1.991 0.562 -26.5 2.647 187\nCrGeTe 30.0 3.7 3.8 143.6 84.7 4.3 6000 3.614 0.004 0.032 0.066 19\nergies of the bulk Tc, Ge (Si), and Se (Te) crystals,\nrespectively. The obtained negative values Ef= -0.856\neV, -1.582 eV and -0.694 eV for TcSiTe 3, TcGeSe 3and\nTcGeTe 3, respectively. For CrGeTe 3monolayer, which\nwas discovered in recent experiment19, the formation en-\nergy was calculated as -1.140 eV by the same method.\nThe comparable formation energy of TcSiTe 3, TcGeSe 3\nand TcGeTe 3with CrGeTe 3, suggests that these Tc-\nbased materials may also be feasible in experiment.\nSince TcSiTe 3, TcGeSe 3and TcGeTe 3have similar\nproperties, we will take TcSiTe 3in the following anal-\nysis, and the calculated results of TcGeSe 3and TcGeTe 3\nare shown in supplemental material. The partial density\nof states (PDOS) of TcSiTe 3monolayer was calculated\nby GGA+U method, as illustrated in Fig. 2(a). Because\nof the octahedral crystal \feld for Tc atom, the d orbitals\nof the Tc atoms are split into threefold t2gorbitals and\ntwofoldegorbitals. For Tc2+(4d5) in the TcSiTe 3mono-\nlayer, the spin moment S = 2 \u0016Band orbital moment L\n= 0.6\u0016Bare obtained. The results can be understood\nby the following electron con\fgurations: 3 spin up elec-\ntrons and 0.6 spin down electrons occupy t 2gorbitals,\nand 0.5 spin up electrons and 0.9 spin down electrons\noccupyegorbitals. It re\rects that the crystal \feld and\nCoulomb interaction U are comparable for 4d electrons\nof Tc. In contrast, for Cr2+(3d4) in CrGeTe 3mono-\nlayer (U is taken as 4 eV), 3 spin up electrons are in t2g\norbitals and 1 spin up electron is in egorbitals, which\ngives rise to the spin moment S = 4 \u0016Band orbital mo-\nment L = 0, as shown in Fig. 2(b). It means that the\nCoulomb interaction U is much larger than the crystal\n\feld for 3d electrons of Cr. These results can also beobtained by integrating the total density of states below\nthe Fermi level for spin up and spin down electrons for\nt2gandegorbitals, respectively. The SOC is calculated\nbyHSOC=\u0015S\u0001L, where\u0015is the coe\u000ecient of SOC, S\nandLrepresent the spin and orbital moment operators,\nrespectively. Because of the large \u0015, which is related to\nthe atomic number and large orbital moment L, a much\nlarger SOC is expected in TcSiTe 3monolayer than that\nin CrGeTe 3monolayer.\n/s32\n/s32\n/s32\n/s32\n001\n300 900M (a. u.) \nT (K) (a) (b)TcSiTe 3\nTcGeSe 3TcGeTe 3\nFM NAFM\nSAFM ZAFMCrGeTe3\n600\nFIG. 4. (a) Possible spin con\fgurations of Tc atoms on honey-\ncomb lattice: FM, N\u0013 eel AFM (NAFM), stripe AFM (SAFM),\nand zigzag AFM (ZAFM). (b) Temperature dependence of the\nnormalized magnetic moment of TcSiTe 3, TcGeSe 3, TcGeTe 3\nand CrGeTe 3monolayers by Monte Carlo simulations.\nThe electronic band structure of TcSiTe 3mono-\nlayer was calculated by GGA+U method, as shown in\nFig. 3(a). It is a Weyl half-metal, where only one species\nof electron spin appears at Fermi level. At the high-\nsymmetry lines \u0000 \u0000Kand \u0000\u0000M, there exist Weyl nodes.\nConsidering the inversion and C3vsymmetries, there are4\ntotal 12 Weyl points within the BZ. To demonstrate the\ne\u000bect of SOC, the electronic band structure was calcu-\nlated by GGA+SOC+U method, as plotted in Fig. 3(b).\nAs a result of including SOC, the band gap was opened\nfor TcSiTe 3monolayer. To correctly estimate the band\ngap, since the GGA-type calculations usually underesti-\nmate the band gap, the hybrid functional method HSE06\nwas also employed. The HSE06 calculation shows that\nthe band gap of TcSiTe 3monolayer becomes 0.4 eV, as\nshown in Fig. 3(c). The details of electronic band struc-\ntures were given in Supplemental Material.\nDue to the large SOC, a large magnetic anisotropy\nis highly expected for TcSiTe 3, TcGeSe 3and TcGeTe 3\nmonolayers. To study magnetic anisotropy in these\nmonolayers, we calculated the total energy with possi-\nble spin con\fgurations of Tc atoms on honeycomb lat-\ntice, including paramagnetic (PM), ferromagnetic (FM),\nN\u0013eel antiferromagnetic (NAFM), stripe AFM (SAFM),\nand zigzag AFM (ZAFM) con\fgurations, as shown in\nFig. 4(a). CrGeTe 3monolayer was also calculated in the\nsame way for comparison. The results are summarized\nin Table I. One can observe that the out-of-plane FM\n(FMz) state has the lowest energy among the possible\nspin con\fgurations. The magnetic anisotropy between\nthe in-plane magnetic con\fguration FMxand FMyand\nthe out-of-plane magnetic con\fgurations FMzin Tc-base\nmaterials is extraordinarily lager than that in CrGeTe 3,\nas noted in Table I. We further calculated the energies\nfor FM con\fgurations by rotating the magnetic direction\ndeviated from the z-axis, and found that the FMzstate\nis the most energetically favorable, which shows an Ising\nbehavior of TcSiTe 3, TcGeSe 3and TcGeTe 3monolayer.\nThe calculation reveals that the magnetic anisotropy\noriginates from the single-ion anisotropy (SIA), the lat-\nter can be calculated by four ordered spin states35. The\nresults in Table I show that, for TcSiTe 3, TcGeSe 3and\nTcGeTe 3monolayers, SIA are found to be negative and\nlarge, which determines a strong Ising-type behavior with\nout-of-plane magnetization. For CrGeTe 3monolayer,\nSIA is negligible and approaches to zero, which means the\nHeisenberg-like behavior with weak magnetic anisotropy.\nThus, the magnetism in TcSiTe 3, TcGeSe 3and\nTcGeTe 3monolayers can be described by the Ising-type\nHamiltonian Hspin =\u0000P\nhi;jiJSz\niSz\nj, whereJrepre-\nsents the nearest-neighbor exchange integral, Sz\ni;jis the\nz-component of spin operator, and hi;jidenotes the sum-\nmation over the nearest neighbors. Jcan be determined\nby the di\u000berence of energies between FMzcon\fguration\nand AFM con\fguration, which possesses the lowest en-\nergy among those AFM con\fgurations. In our cases, the\nZAFMzcon\fguration has the lowest energy for TcSiTe 3\nand CrGeTe 3monolayers, and the SAFMzcon\fguration\nowns the lowest energy for TcGeSe 3and TcGeTe 3mono-\nlayers, as shown in Table I. As a result, Jwas estimated\nto be 7.625 meV, 2.997 meV, 2.647 meV and 0.066 meV\nfor TcSiTe 3, TcGeSe 3, TcGeTe 3and CrGeTe 3, respec-\ntively.\nBased on above Ising Hamiltonian and the estimatedexchange parameter J, the Monte Carlo (MC) simulation\nwas carried out to calculate the Curie temperatures of\nthese 2D materials36. The MC simulation was performed\non a 60\u000260 2D honeycomb lattice using 106steps for each\ntemperature. The magnetic moment as a function of tem-\nperature is shown in Fig. 4(b). It can be seen that the\nnormalized magnetic moment decreases rapidly to vanish\nat Curie temperature about 538 K, 212 K, 187 K and 19\nK for TcSiTe 3, TcGeSe 3, TcGeTe 3and CrGeTe 3mono-\nlayers, respectively. The results indicate that TcSiTe 3,\nTcGeSe 3and TcGeTe 3monolayers can be potential can-\ndidates for high temperature 2D ferromagnetic semicon-\nductors.\nDue to the large SOC in these ferromagnetic semicon-\nductors, a large anomalous Hall conductivity (AHC) is\nexpected. We calculated the intrinsic AHC due to the\nBerry curvature of electronic band structure as shown in\n/s32\n/s32\n/s32TcSiTe 3TcGeSe 3TcGeTe 3\n0 0.6 -0.60\n-6 \n-126/g86xy  (10 2 /g58-1 cm -1 )(a)\nE-EF (eV)\n/s32\n/s32\n/s32\n/s32TcSiTe 3TcGeSe 3TcGeTe 3Fe-exp \nFe-bcc \n0 5-1 4\n/g90/g3 (eV)/g84Kerr  (deg) (b)\nFIG. 5. (a) The anomalous Hall conductivity of TcSiTe 3,\nTcGeSe 3and TcGeTe 3monolayers as a function of energy\nnear Fermi level. (b) The Kerr angle \u0012Kerr of TcSiTe 3,\nTcGeSe 3and TcGeTe 3monolayers as a function of photon\nenergy, where the experimental (open squares) and calculated\nvalues (dotted line) of \u0012Kerr for bulk Fe are included for com-\nparison.5\nTABLE II. The dominant hopping matrix elements jVjand energy di\u000berence jEp\u0000Edjbetweenporbitals of Si (Te) and d\norbitals of Tc (Cr) in eV for TcSiTe 3(CrGeTe 3) monolayer.\nmonolayer pz-dz2pz-dxzpz-dyzpz-dx2\u0000y2pz-dxy\nTcSiTe 3jVj 0.444869 0.096455 0.254298 0.179421 0.386326\njEp\u0000Edj0.158048 0.118685 0.445139 0.296614 0.071643\npx-dz2px-dxzpx-dyzpx-dx2\u0000y2px-dxy\nCrGeTe 3jVj 0.447729 0.216707 0.045947 0.720887 0.055202\njEp\u0000Edj0.717856 1.045652 1.118030 0.627825 1.000211\nFig. 5(a). The magnitude of AHC \u001bxyfor the p-type\nTcGeTe 3can reach 7.5\u0002102(\n\u0001cm)\u00001and the n-type\nTcGeTe 3can be up to 1.1 \u0002103(\n\u0001cm)\u00001. The p- or\nn-type TcSiTe 3and TcGeSe 3can be as large as 4 \u0002102\n(\n\u0001cm)\u00001. These values are comparable to the intrin-\nsic\u001bxyin some ferromagnetic metals, such as 7.5 \u0002102\n(\n\u0001cm)\u00001in bcc Fe25,26, and 4.8\u0002102(\n\u0001cm)\u00001in fcc\nNi27due to the Berry curvature of band structures.\nLarge magneto-optical Kerr e\u000bect (MOKE) is also pos-\nsible in 2D ferromagnetic materials with large SOC37.\nWe investigated the MOKE for TcSiTe 3, TcGeSe 3and\nTcGeTe 3monolayers. The Kerr rotation angle is given\nby:\n\u0012Kerr(!) =\u0000Re\u000fxy\n(\u000fxx\u00001)p\u000fxx; (1)\nwhere\u000fxxand\u000fxyare the diagonal and o\u000b-diagonal com-\nponents of the dielectric tensor \u000f, and!is the photon en-\nergy, respectively. The dielectric tensor \u000fcan be obtained\nby the optical conductivity tensor, \u001b(!) =!\n4\u0019i[\u000f(!)\u0000I],\nwhereIis the unit tensor. We performed the calculations\nwith VASP along with the wannier90 tool to obtain the\noptical conductivity tensor \u001band the Kerr angle \u0012Kerr.\nThe calculated \u0012Kerr as a function of photon energy for\nTcSiTe 3, TcGeSe 3and TcGeTe 3monolayers is shown in\nFig. 5(b). Our calculated and previous experimental re-\nsults for Fe metal are also included for comparison38. It\ncan be seen that a large Kerr angle \u0012Kerr is obtained for\nTcSiTe 3, TcGeSe 3and TcGeTe 3monolayers, particularly\nfor photon energies !near 1 eV. The maximal Kerr an-\ngle for TcSiTe 3, TcGeSe 3and TcGeTe 3monolayers is an\norder of magnitude larger than that for CrGeTe 3mono-\nlayer19, and about 5 times larger than that for bulk Fe.\nIV. DISCUSSION\nHow to understand the enhanced Curie temperature of\nTcSiTe 3monolayer compared with CrGeTe 3monolayer?\nAccording to the superexchange interaction39{41, the FM\ncoupling is expected since the Tc-Te-Tc and Cr-Te-Cr\nbond angles are close to 90 degree. The indirect FM\ncoupling between Tc (Cr) atoms is proportional to the\ndirect AFM coupling between neighboring Tc(Cr) and\nTe atoms. The magnitude of this direct AFM coupling\ncan be roughly estimated as J=jVj2=jEp\u0000Edj, where\njVjis the hopping matrix element between porbitals ofTe anddorbitals of Tc (Cr), and jEp\u0000Edjis the en-\nergy di\u000berence between porbitals of Te and dorbitals\nof Tc (Cr). By using maximally-localized Wannier or-\nbital projections, the dominant hopping matrix elements\njVjand their corresponding energy di\u000berences jEp\u0000Edj\ncan be obtained for 2D TcSiTe 3and CrGeTe 3, respec-\ntively, as listed in Table II. The results suggest that the\ndirect AFM coupling for TcSiTe 3is dominated by the pz\norbitals of Te and dz2anddxyorbitals of Tc. Because\nthepzorbitals of Te and dz2anddxyorbitals of Tc for\nTcSiTe 3monolayer are very close to each other in energy,\nand at the same time the su\u000eciently large hoppings exist\nbetween them, a large AFM coupling between Te and Tc\natoms of TcSiTe 3is obtained. Although the hopping pa-\nrameter is quite large between the pxorbitals of Te and\ndz2anddx2\u0000y2orbitals of Cr for CrGeTe 3monolayer,\nbecause of the large energy di\u000berences among them, the\nAFM coupling between Te and Cr atoms is much weaker\nthan that for TcSiTe 3monolayer.\nAre the giant orbital moments in TcSiTe 3, TcGeSe 3\nand TcGeTe 3monolayers unique? To answer this ques-\ntion, we study the 2D MGeTe 3monolayers with M =\n3d, 4d, 5d transition metals. The results of orbital mo-\nment and magnetocrystalline anisotropy energy are listed\nin Fig. 6. By the spin-polarization calculations of these\nmonolayers, only ten are found to be magnetic, and these\nmagnetic materials are colored in red for M metals in\nFig. 6. Among these magnetic materials, the largest or-\nbital moment is 0.53 \u0016Bin TcGeTe 3, two times larger\nthan the 2nd largest orbital moment in CoGeTe 3, and\nabout an order of magnitude larger than the orbital mo-\nment in the rest 2D materials. The same unique behavior\nof Tc is also found in the results of MAE. As listed in\nFig. 6, one may \fnd that among these 2D MGeTe 3ma-\nterials, 2D TcGeTe 3has an extraordinarily large MAE.\nV. CONCLUSION\nBy \frst-principles calculations, we proposed three\nstable 2D Ising-type ferromagnetic semiconductors of\nTcSiTe 3, TcGeSe 3and TcGeTe 3with high Curie temper-\natures of 538 K, 212 K and 187 K, respectively. Due to\nlarge spin-orbit couplings, the large magnetocrystalline\nanisotropy energy, large anomalous Hall conductivity,\nand large magneto-optical Kerr e\u000bect are found in these\nintriguing 2D ferromagnetic semiconductors. Compar-6\nScTi\n0.057\n1.0V\n0.078\n0.4Cr\n0.004\n0.93Mn\n0.004\n1.3Fe\n0.107\n0.3Co\n0.233\n4.0Ni\n0.090\n1.8CuZn\nYZrNbMo\n0.023\n0.2Tc\n0.562\n28.0RuRhPdAgCd\nHfTaW\n0.021\n0.5ReOsIrPtAuHg\nFIG. 6. For MGeTe 3(M=3d, 4d, 5d metals), the orbital moment (number in the \frst line, unit in \u0016B) and magnetocrystalline\nanisotropy energy (MAE, number in the second line, unit in meV). That the compound MGeTe 3is a magnetic material is\nindicated with M in red color. The MAE is calculated by the energy di\u000berence per M atom between the FMzand FMx\ncon\fgurations.\ning all possible 2D MGeTe 3materials (M = 3d, 4d, 5d\ntransition metals), the unique behavior of Tc is high-\nlighted with extraordinarily large orbital moment near\n0.5\u0016B. 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Rev. 115, 2 (1959)."    },    {        "title": "1908.07710v2.Two_dimensional_Ferromagnetic_van_der_Waals_CrX3__X_Cl__Br__I__Monolayers_with_Enhanced_Anisotropy_and_Curie_Temperature.pdf",        "content": " 1 / 12 \n Two-dimensional Ferromagneti c van der Waals CrX 3 (X=Cl, Br , I) \nMonolayers with Enhanced Anisotropy and Curie Temperature  \n \nFeng Xue1, Yusheng Hou2, Zhe Wang1, Ruqian Wu2,* \n \n \n1 State Key Laboratory of Surface Physics and Key Laboratory for Computational \nPhysical  Sciences (MOE) and Department of Physics, Fudan University, Shanghai, \n200433, China  \n2 Department of Physics and Astronomy, University of California, Irvine, CA 92697 -\n4575, USA  \n \n \nAbstract  \n \nAmong the recently wide ly studied van der Waals layered magnets  CrX 3 (X=Cl, Br , I), \nCrCl 3 monolayer (ML) is particularly puzzling  as it is solely shown by experiments to \nhave an in -plane magnetic easy axis and, furthermore, all of previous first -principles \ncalculation results contradict this. Through systematic al first -principles calculations, \nwe unveil that its in -plane shape anisotropy that dominates over its weak perpendicular \nmagnetocrystalline anisotropy is responsible for the in -plane magnetic easy axis of \nCrCl 3 ML. To tune the in -plane ferromagnetism of C rCl 3 ML into the desirable \nperpendicular one, we propose substitut ing Cr with isovalen t tungsten (W). We find that \nCrWCl 6 has a strong perpendicular magnetic anisotropy and a high Curie temperature \nup to 76 K. Our work not only gives insight into understan ding the two -dimensional \nferromagnetism of van der Waals MLs but also sheds new light on engineering their \nperformances for nanodevices.  \n \n \n \nEmail: wur@uci.edu  \n \n \nKEYWORDS:  CrCl 3 monolayer, two -dimensional magnetism, van der Waals Magnet, \nperpendicular ferromagnetism  \n \n \n \n \n \n \n \n \n \n \n \n   2 / 12 \n 1. Introduction  \nEver s ince the discovery of grapheme [1], two -dimensional  (2D) materials have \nattracted tremendous attention due to their unique  properties and immense  potential in \nnano device applications. Over the past decades, a variety of novel 2D materials have \nbeen successfully fabricated and extensively studied, including hexagonal boron nitride  \n[2], silicone [3], transition metal dichalcogenides  [4,5], and black phosphorous  [6], to \nname a few . Recent discoveries of atomically thin ferromagnet ic films such as \nCr2Ge2Te6 [7] and CrI 3 [8] added a new thrust in this realm as they allow integration of \nmagnetic materials in multi functional 2D heterostructures, essential for the design of \nvarious spintronic  and topotronic devices [9-12]. Fundamentally, the long-range \nmagnetic order ing in a 2D system is vulnerable to  thermal fluctuations  even at \nextremely low temperature, a ccording to the Mermin -Wagner theorem  [13] that was \nestablished from the  isotropic Heisenberg model. A sizeable magnetic anisotropy  \nenergy which forces magnetic moments  in a lattice to align in certain crystallographic \ndirection known as easy axi s becomes critical  to stabilize  the 2D magnetic order ing. \nIndeed, it  was shown that the high Curie temperatures of  Cr2Ge2Te6 and CrI 3 \nmonolayers (MLs) stem from  their out -of-plane (perpendicular) magnetic anisotropy , \nthat open s up an spin wave gap to resist the thermal agitation  [7]. \nChromium trihalides CrX 3 (X = Cl, Br, I)  have the v an de r Waals (vdW) layered \nstructure and share similar physical properti es. Their monolayers have been targeted as \nthe most promising 2D magnetic materials for fundamental studies and technological \ninnovations. T heoretical studies suggest that t he magnetic  easy axes  of all CrX 3 \nmonolayers are out -of-plane  [14,15] just like their bulks. While recent experiment s [16-\n21] indeed confirm this conjecture for CrBr 3 and CrI 3 MLs, an in -plane magnetic easy \naxis was observed for the CrCl 3 ML [17,18]. As t he perpendicular easy direction  of \nmagnetic MLs is essential for the sustainability of  their ferromagnetic ( FM) order ing \nand, subsequently, for their performance in heterostructures such as inducing spin \nsplitting on the topological surface states  [22], it is importan t to solve the puzzle  for \nCrCl 3 ML. Furthermore, it is desired to explore effective strateg ies to control  the \nmagnetic anisotropy and to enhance the Curie temperature ( TC) of CrX 3 MLs.  \nIn the present work, we performed systematical  first principles calculations for  \nCrX 3 MLs and their derivations . We found that the inclusion of magnetic shape \nanisotropy (MSA) is crucial for studies of these thin films and is the driving force for \nthe in-plane magnetization of the CrCl 3 ML. By substituting part of Cr atoms with \nisovalent W  atoms, the magnetic easy axis  of Cr 1-xWxCl3 MLs can be tuned to the  \nperpendicular direction with a large magnetic anisotropy  energy  (MAE)  of 1.07 meV \nper magnetic atom . Moreover, the exchange interactions among magnetic atoms are \nalso enhanced by W substitution , resulting in a higher TC up to  76 K. Therefore, our \nfindings not only solve puzzles for the understanding of 2D magnetism in  CrX 3 MLs, \nbut also provide a viable strategy for optimizing their performance as needed for the \ndevelopment of spintr onic materials and devices.  \n \n2. Theoretical approach  \nThe density functional theory  (DFT)  calculations were performed using the projector \naugmented wave (PAW)  [23,24] method as implemented in the Vienna ab initio   3 / 12 \n simulation package (V ASP)  [25,26]. The exchange -correlation effect among electrons \nwas described within the framework of generalized -gradient approximation (GGA), \nusing the functional proposed by Perdew, Burke, and Ernzerhof (PBE)  [27]. Note that \nthe semi -core states of W 5 p and Cr 3 p were treated as valence electrons  as well . For \nthe plane wave basis expansion , we used  an energy cutoff of 550 eV .  We utilized a  \nMonkhorst –Pack k-point mesh of 8× 8× 1 (4× 4× 1) for the CrX 3 (Cr 1-xWxCl3) in the \nstructural optimization  but a finer k-point mesh of 16× 16× 1 (8× 8× 1) in calculat ing the  \nelectronic and magnetic properties. The convergence with respect to k-point sampling \nwas carefully tested. The vacuum s pace between adjacent slabs was set to be larger than \n15 Å, which is enough to eliminate the spurious image interactions. Both  in-plane lattice \nconstant s and atomic coordinates were fully relaxed using the conjugate gradient \nmethod until the force acting o n each atom was less than 0.01 eV/Å.  The electron \ncorrelation  effect for the  localized d orbitals  of Mn and W atoms w as treated by an \neffective on -site Hubbard term U of 3 eV .  \n \n3. Magnetic anisotropies and exchange interactions of CrX 3 ML \nBulk c hromium trihalides  have a rhombohedral structure with the  \nR3  symmetry \nat low temperature, and a monoclinic structure with the C2/m symmetry at high \ntemperature. After the exfoliation, only one type of ML is found (Fig. 1a). CrX 3 MLs \nconsist  of halide -Cr-halide triple layers, and Cr atoms are arranged in the honeycomb \nstructure. The unit cell used in the first-principles calculations is shown by the red \ndashed quadrilateral . The calculate d geometric parameters and magnetic properties are \nsummarized in Table 1 and Table S1. It can be  clearly see n that lattice parameter \nincrease s from CrCl 3, CrBr 3 to CrI 3, which is in accordance with previous reports  \n[15,28,29]. The calculated band structure (Fig. S2a) shows that the single -layer CrCl 3 \nis a FM  insulator with an indirect band gap of 2.59 eV .  \n  \nFigure 1.  (a) Top and side views of CrX 3 (X = Cl, Br, I)  ML. The unit cell is indicated \n 4 / 12 \n by the red dashed quadrilateral. Blue and green balls represent Cr and X atoms, \nrespectively. The first (J 1), second (J 2) and third (J 3) nearest neighboring exchange \ninteraction parameters are indicated by red arrows.  (b) Sketch of the FM \nsuperexchange interaction in CrCl 3 ML via the \n22 / xy x y xyd p d orbitals.  (c) The \nprojected density of states (PDOS) of Cr 3d orbitals in CrCl 3 ML. (d) The PDOS of Cr \nand W 3d orbitals in CrWCl 6 ML. Positive (negative) values refer to up (down) spins. \nFermi level is set at zero energy.  \nNow we investigate the m agnetic anisotropy  energy (\nMAEE ), a critical  parameter \nto stabilize the long -range magnetic ordering against thermal fluctuations  in 2D \nmaterials . In principle, the magnetic anisotropy  is mainly contributed by two terms: (1) \nmagnetocrystalline anisotro py (\nMCAE ) induced by the spin -orbit coupling  (SOC), and  \n(2) magnetic shape anisotropy (\nMSAE ) resulting  from the dipole -dipole interaction, i.e.,\nMAE MCA MSA E E E\n. Here, \nMCAE\n is obtained by calculating the total energy change  \n(with SOC included)  as the magnetization rotates from the in -plane to the perpendicular \ndirections with respect to the CrX 3 layer (\nSOC SOC\nMCA  \n E E E ). On the other hand, \nMSAE\n(\nDipole Dipole\nMSA  \n E E E ) is calculated  according to the energ y of magnetic dipole -\ndipole interactions  as: \nmax\nDipole 0\n1 2 1 2 32\n1113\n24\n\n     \n\nrN\nij ij\nijij ijE M M M r M rrr\n.       (1) \nHere Mi represents the local magnetic moments  and rij are vectors that connects the \nsites i and j. Since the this energy converges very slowly with respect to the cutoff of  \nijr\n (Fig. S2b), we expanded the range to  \nmax 1000r Å to ensure the numerical \nreliability .  \nTable 1. The calculated \nMCAE , \nMSAE , \nMAEE , exchange interaction parameters  and \nCurie temperatures of CrX 3 and CrWCl 6 ML. The units of \nMCAE , \nMSAE  and \nMAEE  \nare µeV per magnetic at om. Experimental Curie temperatures are also listed for \ncomparison.  \n \ndRef. [30] \neRef. [8] \n*For CrCl 3, we assumed an out -of-plane magnetization solely comes from the \nEMCA term to get a finite value of TC. \n 5 / 12 \n The calculated magnetic anisotropy energies  of CrX 3 MLs are summarized in \nTable 1. It is interesting that they all have positive \nMCAE , as found in previous \ntheoretical studies. This indicates that their magnetocrystalline anisotropy parts favor  \nthe perpendicular magnetization , even though the magnitude of \nMCAE decreases \nmonotonically as we move up from I, Br to Cl in the periodic table. Note that the \nmonotonic decrease of \nMCAE  is mainly due to the SOC weakening from I, Br to Cl \n[31]. On the contrary, their shape anisotropy favors an in-plane alignment of \nmagnetization and the magnitude\nMSAE changes in a narrow range as the local magnetic \nmoments are the same in the three MLs.  As a result of the competition, the net magnetic \nanisotropy energy, \nMAEE , is negative (in -plane easy axis) for CrCl 3 ML but is positive \n(perpendicu lar easy axis) for  CrBr 3 or CrI 3 MLs , in perfect agreement with experiment. \nObviously, \nMSAE can be comparable to or even larger than \nMCAE , especially when the \nSOC of chalcogenides is weak. This calls for attention on the consideration of the \nMSAE  \nfor studies of 2D magnetic materials, wh ich has been neglected in most works in this \nrealm  [14,15]. \nAnother fundamental parameter  for the determination of the Curie temperature of  \nmagnetic  systems  is the exchange interaction. To obtain  the exchange param eters, Ji, \nwe mapped the  DFT total energ ies of several configurations onto the following classical  \nHeisenberg Hamiltonian : \n1 2 3            \n i i i\nij ij ijj j j H J S S J S S J S S\n,            (2) \nwhere J1, J2 and J3 represent the nearest, next nearest and third nearest  neighbor  \nexchange interaction s, respectively (Fig. 1a). For CrX 3 MLs, we designed  four different \nmagnetic configurations  as shown in Fig. S 4. As summarized in Table 1, the nearest \nneighbor exchange interaction ( J1) is dominating for the establishment of the FM \nground state.  The next nearest neighbor interaction ( J2) prefers the FM ordering as well , \nbut the weak third nearest neighbor interaction ( J3) tends to be antiferromagnetic ( AFM ). \nNote that t he CrI3 ML has the largest J1 and J2. \nThe FM J1 originates from the competition between a strong FM superexchange \nvia the near -90o Cr-X-Cr bonds and a weak Cr -Cr AFM direct exchange.  Here we take \nCrCl 3 ML as an example. As shown in Fig. 1c, the Cr -d orbitals split into three low -\nlying half -occupied \n2gt  and two high -lying  empty \nge  states. The Cr -Cl-Cr angle \n(95.5 o) is close to 90o in the superexchange path (Fig. 1b) and thus gives rise to  a FM \nsuperexchange interaction  according to the Goodenough−Kanamori−A nderson  [32-34] \nrules (Fig. 1b). As the electronic configuration of Cr3+ ion is \n30\n2ggte , the direct  exchange \nbetween two nearest neighboring Cr3+ ions is AFM . However, the direct AFM  exchange \nis weak because of the large distance (3.53 Å) between adjacent Cr ions . As a result , \nthe FM superexchange overtakes the AFM direct exchange, resulting in a net FM J1. \nAs for  J2, it involve s several possible Cr -Cl-Cl-Cr superexchange paths  which \ncontribute either FM or AFM coupling, depending on the corresponding angles  [15]. \nAs a result of the  competitions  between multiple FM and AFM coupling  mechanisms , \nJ2 is weak and FM .  6 / 12 \n  \n4. Enhanced Anisotropy and Curie Temperature in CrWCl 6 \nAs a strong perpendicular anisotropy  [35] is mostly required for 2D magnetic films, \nit is desirable to find ways to enhance the weak magnetic anisotropy of CrX 3 MLs. We \nperceive that substituting Cr by isovalent W atoms might be a good approach, since W \nhas the same electronic configuration as Cr but much larger SOC. Therefore, we may \nexpec t a larger perpendicular magnetic anisotropy and thereby a higher TC. To \ndemonstrate this concept, we used CrCl 3 ML as our modeling system. At first, we need \nto investigate whether Cr and W atoms can be intermixed. We considered 9 different W \nconcentration s by constructing a \n22  supercell Cr 8-xWxCl24 with \n0 ~ 8x . For each \nW concentration, there are a large number of possible substitution configurations. In \norder to find out the optimal distribution patterns of Cr and W atoms, we considered all \npossible distributions of Cr and W in the \n22  supercell. The detailed structure \nconfigurations for different W concentration ( x) in the alloyed systems are given in \nsupporting information (Fig. S1). For each ground state of Cr 8-xWxCl24 (see supporting \ninformation), the formation energy \n 8 24Cr W Clf x xE  is obtained from:  \n  8 24 8 24 3 3Δ Cr W Cl Cr W Cl 8 CrCl WCl   f x x x xE E x E E\n.      (3) \nThe calculated formation energy as a function of W concentration ( x) is shown in Fig. \n2a. We see that the relative energies of the mixed phases are lower by ∼0.031 eV per \nmagnetic atom  compared to those of the pure phases, namely, CrCl 3 and WCl 3 MLs. \nThis indicates that the mixed phases can stably exist without phase separation.  \n \nFigure 2.  (a) Formation energy \n 8 24Δ Cr W Cl f x xE  as a function of W concentration \n(x) in Cr 8-xWxCl24. Inserts show the lowest -energy configurations for \n2 ~ 4x . (b) The \nphase diagram of the stable region of Cr 8-xWxCl24 (x = 0~8) with different W \nconcentrations. The number of W atoms in Cr 8-xWxCl24 ML is labeled by x. The right \n(bottom) and left (top) boundaries of chromium (tungsten) chemical potential \ncorrespond to the so -called Cr (W) -rich (Cr or W bulk as the reservoir) and Cr (W) -\npoor (Cr or W atom as the reservoir) conditions, respectively.  \nTo further examine the feasibility of making the (Cr, W) mixed phases in \nexperiments, we calculated the formation enthalpy \n 8 24Cr W Clf x xH\n by varying  \nchemical potentials of Cr, W and Cl as : \n 7 / 12 \n \n   8 24 8 24 8 Cr W ClΔ Cr W Cl Cr W Cl ln 8 24        x\nf x x x xH E kT C x x.   (4) \nIn Eq. (4), \n 8 24Cr W Clxx E  is the ground state energy of the mixed phase. The second \nterm is the configuration entropy  (T = 300 K is used here).  \nCr , \nW  and \nCl  \nrepresent  the chemical potentials of Cr, W and Cl relative to the chemical potentials  of \ntheir elemental phases. Here, w e assume that the source of Cl atoms is the same for \ndifferent W concentrations, so the term, \nCl24 , in the Eq. (4) is a con stant.  \nFig. 2b gives the lowest \nΔfH  of \n8 24Cr W Clxx  in the (\nCr ,\nW ) plane within \nranges of \nCr 2.48 eV 0     and \nW 6.54 eV 0    . One may see that the mixed \nstates can survive in a large region for \n4 ~ 6x , which  means that mixing \nintermediate concentration of W into CrCl 3 ML is feasible in the W -rich condition \naround 𝜇𝑊=−4 eV. The stable region in this phase diagram has no obvious change s \nwith decreas ing the value of U in DFT calculations (see  Fig. S 4 in supporting \ninformation), and the evenly mixed Cr 4W4Cl24 with x = 4 (i.e. CrW Cl6) appears to be \nthe most preferential configura tion for all cases.  \nAs the Cr and W atoms tend to rather uniformly, the unit cell can be reduced back \nto CrWCl 6 ML, which makes the discussion and comparison rather straightforward.  \nBecause of the strong SOC of W, E MCA of the CrWCl 6 ML has a large positive value, \n1114 eV per magnetic atom. As the value of \nMSAE is almost unchanged from that of \nthe Cr2Cl6 ML, the magnetic easy axis of CrWCl 6 ML is stably  perpendicular as desired, \nwith a giant MAE of  1071  eV per m agne tic atom  (Table 1). This value is even \ncomparable with MAEs of many FM transition metal thin films such as Fe, Co, and Ni \non different substrates  [36].  \nTo investigate  the physical origin of the giant perpendicular  EMCA of CrWCl 6 ML, \nwe plotted its band structure in Fig. 3a. After W atoms are introduced, the band gap is \n1.07 eV, with the unoccupied spin up states of the Cr2Cl6 ML moving toward the Fermi \nlevel. However, we can’t draw a simple conclusion for the large EMCA solely from the \nreduced band gap. Here, we analyze the tendency of EMCA of the CrWCl 6 ML with \nrespect to the shift of the Fermi level, using the torque meth od and rigid band model   \n[37,38]. Note that the EMCA obtained from the toque method, 1089 μeV per magnetic \natom is very close to that listed in Table 1 (1114 μeV), indicating the reliability of the \ntorque method for these systems. We decompose EMCA into contributions from different  \natoms or  spin channels and show the results in Fig. 3b. It can be seen that W atoms \ncontribute the most to EMCA, i.e., 990 μ eV per magnetic atom . From the spin resolutions, \nwe find that the  dominating part is \n/ E (Fig. S5b), which contribute s 60% of EMCA. \nIf we move the Fermi energy down by 0.5 eV, \n/ E  exhibits a sudden change as \nshown in Fig. S5 b, indicating that electronic states near -0.5 eV pl ay the main role in \nproducing the large positive EMCA of the CrWCl 6 ML. By projecting wave functions of \nstates  around this energy to the W site (Fig. 3 a), we reach a conclusion that the \nperpendicular  EMCA mainly results  from the SOC interactions between the occupied \nspin-up \n/xz yzd  (band s -1, -3 in Fig. 3a) and unoccupied spin -down \n2zd  states ( band \n1 in Fig. 3a) of W through the \nxL  operator.  As W has a large SOC strength, this \ncoupling  produces a large perpendicular  EMCA.  8 / 12 \n  \nFigure 3 . (a) Orbital resolved band structure of CrWCl 6 ML. The |m| = 0, 1, 2 orbitals \nof W are represented by red, green, blue lines, respectively, and the line width scales \nwith their weights. Band indexes are marked by numbers. The up and down spin states \nare indicated by the light blue and red shades, resp ectively. (b) The decomposition of \nthe total E MCA to the contributions of different atoms in CrWCl 6 ML. The black line \nshows the total E MCA, and the red, green and purple lines represent the contributions \nfrom Cl, Cr and W atoms, respectively. (c) Distribu tion of \nMCAE  in the BZ. The blue \nand red colors represent negative and positive contributions, respectively. (d) \nContributions of each pair of spin -up occupied states and spin -down unoccupied states \nfor summation over all the K and M points in the first BZ.  \nWe note that the nearest neighbor exchange coupling parameter, J1, is also large \nfor CrWCl 6 ML, up to 14.8 meV  (Table 1), due to the stronger W -Cl hybridization.  The \nenergy separations between the  occupied W -\n2gt orbitals and empty Cr -\nge orbitals in \nCrWCl 6 are largely reduced compared to their counterparts in CrCl 3 [39] (Fig. 1c and \nFig. 1d). Since the FM superexchange arises from the virtual exchange between \n2ggt p e\n orbitals, this reduction in energy separation enhances the FM \nsuperexchange interaction,  and thus lead s to a larger J1. \nInterestingly, the  perpendicular magnetic anisotropy of CrWCl 6 ML can be further \nenhanced by strain s. For example,  an in-plane tensile strain of 5% may increase EMCA \nof the CrWCl 6 ML to  2375 μeV per magnetic atom, more than twice as large as the \nunstr ained one . By decomposi ng \nstrained unstrained\nMCA MCA MCA  E E E  to the contributions of \ndifferent k-points in the first Brillouin zone  (BZ)  (Fig. 3c), we found that \nMCAE\nmainly occur s around the M point . We further resolve  the \nMCAE  to contributions of \ndifferent  pairs of occupied -unoccupied states according  to the second -order \nperturbation formula  (see Part IV in supporting information ). The results are shown in \nFig. 3d, where the band indices  are given  according to Fig. 3a. Clearly, th e eye-catching \n 9 / 12 \n pairs are (-1, 2), ( -3, 4) and ( -3, 7). Fig. S 5 gives more details about their energy \nseparation and matrix elements.  \n \nFigure 4.  (a) The renormalized magnetization \n\n0MT\nM  as a function of temperature T. \nTC is determined by the temperature at \n0 MT . (b) Spin -wave excitation (magnon) \nspectrums of CrCl 3 ML and CrWCl 6 M. \nFinally, w e determine d the TC of CrWCl 6 using the renormalized spin -wave theory \n(RSWT ) [40,41] to show the benefit of Cr -W intermixing . It is recognized that the linear \nspin-wave theory (LSWT) typically overestimates TC of 2D magnetic films [40,41], and  \nthe inclusion of multi -magnon scattering (i.e., the non-linear corrections in [31]) is \nessential to get reliable estimation of  their TC. Through the studies  of magnetic field \ndependence of TC in few -layer and bulk Cr 2Ge2Te6, it was shown that the RSWT \nmethod can reach quantitative  agreement with experiments  [7]. Here, we outline  the \nessence of the RSWT. In LSWT , through the Holstein -Primakoff transformatio n, the \nspin operator \n ,,x y z\nv v v vSSSl l l lS  on site \nv  in the l-th unit cell  in Eq. (2) is rewritten \nby the deviation creation and annihilation operators \nva\nl  and \nval , i.e., \n2vvS Sall , \n2vvS Sall\n, and \nz\nv v vS S a al l l . In the RSWT, however, \nvlS  is expanded up to  \nsecond  order terms of \nva\nl  and \nval  to take into account the magnon -magnon \ninteraction at finite temperatures  [41]. In this case,  the operator \nz\nvSl  is the same as that \nin LSWT  but spin ladder operators \nvS\nl  and \nvS\nl  are in the form of  \n  24v v v v vS S a a a a Sl l l l l\n,                             (5) \n  24v v v v vS S a a a a S   l l l l l\n.                            (6) \nUsing Eq. (5 -6) and the Fourier transform ation  \n1i\nvva e b\nN  kb\nlb k\n  and \n1i\nvva e b\nNkb\nlb k\n, we transform the spin Hamiltonian Eq. (2) from the real space to \nthe reciprocal space and the resulting Hamiltonian contains the magnon -magnon \n 10 / 12 \n interaction [41]. Following the Bose -Einstein  statistics, th e total magnetization  \n MT  \nas a function of temperature T is expressed as : \n\n01\nex 1 1 p1\nBkMT\nM nN T S\n\n \nk\nk\n             (7)  \nwhere N and n are the numbers of unit cells and magnetic basis atoms in each unit cell, \nrespectively. \n0M  denotes  the fully polarized magnetization at T = 0 K. “” represents \nthe two magnon braches, i.e., the in -phase acoustic mode and the anti -phase optical \nmode. \nk  is the magnon spectrum solved from Eq. (6) . The Curie temperature, TC, \nis determined by the temperature at which the total magnetization becomes zero, i.e. \n0 MT\n. \nThe calculated total magnetizations of CrCl 3, CrBr 3, CrI 3 and Cr WCl 6 MLs by \nRSWT as a function of temperature are shown in Fig. 4a, and their  values of TC are \nlisted in Table 1. It should be pointed out that, in principle , CrCl3 ML cannot sustain a \nlong-range magnetic order since its magnetic  easy axis is in -plane,  as mentioned \npreviously. For a better comparison, we ignore the \nMSAE  term for the CrCl 3 ML and \nassume an out -of-plane magnetization that solely comes from \nMCAE . Nevertheless, this \nhypothetical CrCl 3 ML still has a low  TC, only 17 K. By mixing with W, t he TC of \nCrWCl 6 ML can be as high as 76 K, even higher than that of CrI 3 ML. From the spin \nwave (magnon) spectrum as shown in Fig. 4b, w e see that the optical branch of CrWCl 6 \nML shifts to a much higher energy compared with that of CrCl 3 ML, indicating the \ndifficulty to excite it and hence the TC is significantly improved.  \n \n5. Summary and conclusions  \nIn conclusion, we systematical ly investigate d the 2D ferromagnetism of CrX 3 and Cr 8-\nxWxCl24 (x = 0~8) MLs. We found that the  magnetic shape anisotropy is  responsible for \nthe experimentally observed in-plane magnetic easy axis of CrCl 3 ML, indicating that \nit is vital to consider this contribution for studies of weak magnetic anisotropy thin films . \nTo enhance  the magnetic anisotropy and TC of the CrCl 3 ML, we propose  to substitut e \nsome Cr atoms with the isovalent W atoms. The theoretical feasibility and experimental \nmaneuverability of the intermixed Cr 8-xWxCl24 MLs were thoroughly studied. We find \nthat the intermediate concentration doping with x = 4~6 is energetically  preferred  and \nthereby can be easily synthesized  in experiments . As desired , the CrWCl 6 ML has a \nsignificantly enhanced perpendicular m agnetic anisotropy, a large exchange interaction,  \nand hence a high Curie temperature up to 76 K. Our work gives insights into \nunderstanding the two -dimensional ferromagnetism of van der Waals magnetic MLs  \nand suggests a route toward improving the performance of CrX 3 MLs f or spintronic \napplications.  \n \n  11 / 12 \n Acknowledgement  \nWork was supported by DOE -BES (Grant No. DE -FG02 -05ER46237). Computer \nsimulations were partially performed at the U.S. Department of Energy Supercomputer \nFacility (NERSC). F.X and Z.W. acknowledge support by the Basic Research Program \nof China under Grant No. 2015CB921400.  \n \nReferences  \n \n[1] K. S. Novoselov, A. K. Geim, S. V . Morozov, D. Jiang, Y . Zhang, S. V . Dubonos, I. V . Grigorieva, \nand A. A. Firsov, Science 306, 666 (2004).  \n[2] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. \nKim, K. L. Shepard, and J. Hone, Nature Nanotechnology 5, 722 (2010).  \n[3] S. Cahangirov, M. Topsakal, E. Aktürk, H. Şahin, and S. 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B 54, 61 (1996).  \n[38] J. Hu and R. Wu, Phys Rev Lett 110, 097202 (2013).  \n[39] C. Huang, J. Feng, F. Wu, D. Ahmed, B. Huang, H. Xiang, K. Deng, and E. Kan, Journal of th e \nAmerican Chemical Society 140, 11519 (2018).  \n[40] M. Bloch, Physical Review Letters 9, 286 (1962).  \n[41] Z. Li, T. Cao, and S. G. Louie, Journal of Magnetism and Magnetic Materials 463, 28 (2018).  \n \n \n \n "    },    {        "title": "1908.11761v1.Magnetization_reversal__damping_properties_and_magnetic_anisotropy_of_L10_ordered_FeNi_thin_films.pdf",        "content": "Magnetization reversal, damping properties and magnetic anisotropy of L 10\nordered FeNi thin \flms\nV. Thiruvengadam,1B. B. Singh,1T. Kojima,2K. Takanashi,2M. Mizuguchi,2,a)and S. Bedanta1,b)\n1)Laboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical Sciences,\nNational Institute of Science Education and Research (NISER), HBNI, P.O.- Bhimpur Padanpur, Via Jatni, 752050,\nIndia\n2)Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577,\nJapan\n(Dated: August 2019)\nL10ordered magnetic alloys such as FePt, FePd, CoPt and FeNi are well known for their large magnetocrys-\ntalline anisotropy. Among these, L10-FeNi alloy is economically viable material for magnetic recording media\nbecause it does not contain rare earth and noble elements. In this work, L10-FeNi \flms with three di\u000berent\nstrengths of anisotropy were fabricated by varying the deposition process in molecular beam epitaxy system.\nWe have investigated the magnetization reversal along with domain imaging via magneto optic Kerr e\u000bect\nbased microscope. It is found that in all three samples, the magnetization reversal is happening via domain\nwall motion. Further ferromagnetic resonance (FMR) spectroscopy was performed to evaluate the damping\nconstant and magnetic anisotropy. It was observed that the FeNi sample with moderate strength of anisotropy\nexhibits low value of damping constant \u00184:9\u000210\u00003. In addition to this, it was found that the \flms possess\na mixture of cubic and uniaxial anisotropies.\nIn order to increase the storage density in magnetic\nrecording media it requires reduction in the bit size1. On\nthe other hand, in ferromagnetic materials, the super-\nparamagnetic (SPM) limit is inevitable at a critical ra-\ndius, below which the magnetic moment is thermally un-\nstable and become incapable of storing the information2.\nTherefore, magnetic material with high magnetocrys-\ntalline anisotropy is essential to overcome the SPM\nlimit3. In this context, the L10ordered magnetic alloys\nsuch as FePt, FePd, CoPt and FeNi have potential for\nultra-high density magnetic recording media because of\ntheir large uniaxial magnetocrystalline anisotropy energy\ndensity \u0018107erg cm\u00003.L10ordered alloy is a binary\nalloy system with face centered tetragonal (FCT) crys-\ntal structure where each constituent atomic layers are\nalternatively laminated along the direction of crystallo-\ngraphic c-axis4,5. In these materials the large anisotropy\nenergy is due to their tetragonal symmetry of L10crystal\nstructure6.\nL10-FeNi possess high values of saturation magnetiza-\ntion (1270 emu cm\u00003), coercivity (4000 Oe), and uniaxial\nanisotropy energy density (1.3 \u0002107erg cm\u00003)7{9. In ad-\ndition its Curie temperature is quite high \u0018550\u000eC and it\nexhibits excellent corrosion resistance7. All these above\nproperties make L10-FeNi a promising material for fab-\nricating information storage media and permanent mag-\nnets. Further, L10ordered FeNi alloy is free of noble\nas well as rare-earth elements. Also the constituent ele-\nments (i.e. Fe and Ni) are relatively inexpensive. There-\nfore L10ordered FeNi alloy is economically viable for\ncommercial applications4,10,11. It should be noted that\nalthough, L10- FeNi possess high uniaxial anisotropy,\na)Electronic mail: mizuguchi@imr.tohoku.ac.jp\nb)Electronic mail: sbedanta@niser.ac.inshape anisotropy becomes dominant in thin \flms11. Pre-\nviously the study of the order parameter (S) and Fe-Ni\ncomposition dependence on Kuhas been reported7. Pre-\nviously, magnetic damping constants for L10-FeNi and\ndisordered FeNi have been studied employing three kinds\nof measurement methods12. However, the e\u000bect of di\u000ber-\nent anisotropy values of L10- FeNi thin \flms on evo-\nlution of their magnetic domain structures and damping\nhas not been studied so far. Therefore, focus of this paper\nis to study the domain structures during magnetization\nreversal, anisotropy strength, damping properties of L10\n- FeNi thin \flms with di\u000berent anisotropy values.\nL10-FeNi thin \flms were prepared in a molecular beam\nepitaxy chamber with base pressure of 10\u00008Pa consist-\ning of e-beam evaporators (for Fe and Ni) and Knud-\nsen cells (for Cu and Au). First a seed layer of Fe (1\nnm) and Au (20 nm) were deposited at 80\u000eC on MgO\n(001) substrate followed by Cu (50 nm) layer deposited\nat 500\u000eC. It has been reported that Cu and Au layers\nare prone to form an alloy of Cu 3Au (001)9. In order\nto fabricate highly ordered L10-FeNi \flms, a bu\u000ber layer\nof Au 0:06Cu0:51Ni0:43(50 nm) was deposited on Cu 3Au\nlayer at 100\u000eC by co-deposition13. On top of this bu\u000ber\nlayer, FeNi layer was grown by alternative layer deposi-\ntion (Fe and Ni one after another) or co-deposition (de-\nposition of Fe and Ni simultaneously). Disordered-FeNi\n\flm (sample A) was obtained by co-deposition process\nwhereas L10-ordered FeNi \flms were fabricated by alter-\nnate deposition at 100\u000eC (sample B) and 190\u000eC (sam-\nple C), respectively. Finally, Au capping layer (3 nm)\nwas deposited on top of FeNi layer at 30-40\u000eC. To study\nmagnetization reversal and magnetic domain structures,\nwe have performed hysteresis measurements along with\nsimultaneous domain imaging using magneto optic Kerr\ne\u000bect (MOKE) based microscope manufactured by Evico\nMagnetics Ltd. Germany14. Kerr microscopy was per-arXiv:1908.11761v1  [physics.app-ph]  30 Aug 20192\nFIG. 1. Hysteresis loops measured at room temperature by longitudinal Kerr microscopy at 0\u000e, 30\u000e, 60\u000eand 90\u000efor samples\nA - C shown in (a) - (c), respectively..\nformed at room temperature in longitudinal geometry.\nAngle dependent hysteresis loops were measured by ap-\nplying the \feld for 0\u000e<\b<360\u000eat interval of 10\u000ewhere\n\b is the angle between the easy axis and the direction\nof applied magnetic \feld. Further, magnetization dy-\nnamics of the L10-FeNi thin \flms has been studied by\nferromagnetic resonance (FMR) spectroscopy technique\nin a \rip-chip manner using NanOsc Instrument Phase\nFMR15. Frequency range of the RF signal used in this\nexperiment was between 5 and 17 GHz. To understand\nthe symmetry of anisotropy and quantify the anisotropy\nenergy density, angle dependent FMR measurement has\nbeen performed by applying the magnetic \feld in the\nsample plane with a \fxed frequency of 7 GHz.\nMagnetic hysteresis loops measured using Kerr mi-\ncroscopy for samples A-C are shown in \fgure 1 for various\nvalues of = 0\u000e, 30\u000e, 60\u000eand 90\u000e. Square shaped hystere-\nsis loops have been observed for all three samples for all\nthe angles (0\u000e<\b<360\u000e). This indicates that the mag-\nnetization reversal occurs via nucleation and subsequent\ndomain wall motion16. It is observed that the coercivity\nvaries signi\fcantly among the samples e.g. HCof samplesA, B and C are 2.2, 1.7 and 25 mT, respectively. Kuof\nthese samples are -1.05 \u0002106erg cm\u00003(A), 3.47 \u0002106erg\ncm\u00003(B), and 4.86 \u0002106erg cm\u00003(C).\nDomain images captured near to coercive \feld HCfor\nall three samples at 0\u000e, 30\u000e, 60\u000eand 90\u000eare displayed in\n\fgure 2. The black and gray contrast in the domain im-\nages represent positive and negative magnetized states,\nrespectively. From the domain images it can be con-\ncluded that magnetization reversal occurs via domain\nwall motion. Sample A (\fgure 2 a to d) exhibits large\nwell-de\fned stripe domains which indicates the presence\nof weak magnetic anisotropy. These stripe domains are\nfound to be tilted for e.g. \b = 30\u000eand 60\u000e(\fgure 2\nb and c). For \b = 90\u000e(\fgure 2d) branched domains\nare observed. In addition to 180\u000edomain wall, sample\nB (\fgure 2 e to h) shows 90\u000edomain walls at certain\nangles of measurement (\fgure S1). Sample C (\fgure 2 i\nto l) shows narrow branched domains and are found to\nbe independent of \b , which indicates that the sample\nC is magnetically isotropic in nature. By comparing the\ndomain images at any particular \b among the samples,\nit is observed that the size (i.e. width) of the domain3\nFIG. 2. Domain images captured for samples A, B and C at angles, 0\u000e, 30\u000e, 60\u000eand 90\u000enear to coercivity are shown. Scale\nbars are shown on the domain images for each sample separately which are valid for the images recorded at other angles for\nthose respective samples.\ndecreases with increasing anisotropy strength.\nIn the following we have investigated the magnetiza-\ntion dynamics of the L10FeNi \flms by FMR. Measured\nFMR spectra (open symbol) for samples A and C at se-\nlective frequencies are shown in \fgure 3 (a) and (b), re-\nspectively. Resonance \feld (H res) and line width (\u0001 H)\nwere extracted by \ftting of Lorentzian shape function\n(equation 1) having anti-symmetric (\frst term) and sym-\nmetric components (second term) to the obtained FMR\nderivative signal17;\nS21=K14\u0001H(H\u0000Hres)\n[4(H\u0000Hres)2+ (\u0001H)2]2\n\u0000K2(\u0001H)2\u00004(H\u0000Hres)2\n[4(H\u0000Hres)2+ (\u0001H)2]2\n+ (slopeH ) +Offset(1)\nwhere S21is transmission signal, K1andK2are co-\ne\u000ecient of anti-symmetric and symmetric component,\nrespectively, slope H is drift value in amplitude of the\nsignal. Frequency ( f) dependence of Hresand \u0001 Hare\nplotted in \fgure 3 (a) and (b), respectively. From the\nfvs.Hresplot, parameters such as Lande g-factor, ef-\nfective demagnetization \feld (4 \u0019Meff), anisotropy \feld\nHKwere extracted by \ftting it to Kittel resonance\ncondition18;\nf=\r\n2\u0019q\n(HK+Hres)(HK+Hres+ 4\u0019Meff) (2)\nwhere\r=g\u0016B=~is gyromagnetic ratio, \u0016Bas Bohr\nmagneton, ~as reduced Plancks constant. The values ofdamping constant \u000bfor all three samples were extracted\nby using the equation17{19;\n\u0001H= \u0001H0+4\u0019\u000bf\n\r(3)\nwhere \u0001H0is called as inhomogeneous line width\nbroadening. Values of all the \ftting parameters obtained\nby using equations (1) and (2) are given in table 1. With\nincrease in anisotropy strength of the samples, 4 \u0019Meff\ndecreases whereas \u000bincreases. Sample A exhibits lowest\nvalue of\u000b, 4:9\u000210\u00003which is the same order with nor-\nmal FeNi alloy thin \flm ( \u000b= 1:2\u000210\u00003)20. The value\nof inhomogeneous line width broadening \u0001 H0, which de-\npends on the quality of the thin \flm17, is highest for\nsample C and lowest for sample A.\nApart from frequency dependent FMR, angle depen-\ndent FMR measurements at a \fxed frequency of 7 GHz\nwere performed to analyze the anisotropy energy density\nas well as symmetry of all three samples. Figure 4 shows\nthe angle dependent Hresplot for all three samples. Sam-\nples A and B clearly show the presence of mixed cubic\nand uniaxial anisotropies with minima in Hresapproxi-\nmately at angles 0\u000e, 90\u000e, 180\u000eand 270\u000e. From the an-\ngle dependent Hresplot, the value of uniaxial and cubic\nanisotropy energy constants have been estimated by \ft-\nting those data to solution of LLG equation that includes\nthe cubic and uniaxial anisotropies and it is written as21;4\nFIG. 3. FMR spectra of (a) sample A and (b) sample C measured at frequency 5 17 GHz. The solid lines in (a) and (b) are\n\ftted with equation 1. (c) fvs.Hresand (d) \u0001Hvs.fplots for samples A C extracted from the FMR frequency dependent\nspectra. The lines in (c) and (d) are the best \fts to the equations (2) and (3), respectively.\nTABLE I. The Parameters extracted from the \ftting of FMR experimental data (Figure 3) of all three samples using the\nequations (2) and (3)\nSample H K(Oe) 4\u0019Meff g-factor \u000b \u0001H0(Oe)\nA 49.41 \u00060.82 18279 \u0006365 1.97 \u00060.01 0.0049 \u00060.0001 42.85 \u00060.65\nB -52.39 \u00064.64 8900 \u0006193 1.93 \u00060.01 0.0277 \u00060.0007 66.89 \u00067.13\nC 680.29 \u0006106.88 3509 \u0006106 1.98 \u00060.01 0.0680 \u00060.0019 143.67 \u000616.20\nf=\r\n2\u0019\u0014\nH+2K2\nMscos2\u001e\u00004K4\nMscos4\u001e\u0015\n\u0002\u0014\nH+ 4\u0019Ms+2K2\nMscos2\u001e\u0000K4\nMs(3 +cos4\u001e)\u00151=2\n(4)\nwhere, K2andK4are cubic and uniaxial anisotropy\nenergy density constants, respectively, MSis saturation\nmagnetization, His the applied magnetic \feld, \b is the\nangle between applied \feld direction and easy axis of\nthe sample. The angle dependence of Hresis \ftted to\nequation 4. From the \ft, value of MS,K2andK4have\nbeen extracted and are shown in table 2. It has been\nfound that both cubic and uniaxial anisotropy energy\nconstants of sample A are greater than that of sample B\nby one order of magnitude. Further in sample A the ratioTABLE II. The parameters extracted from the \ftting of angle\ndependent FMR experimental data of two samples using the\nequation (4)\nSamplesMs(emu cm\u00003)K2(erg cm\u00003)K4(erg cm\u00003)\nA 1645 1.66 \u0002104-2.16\u0002104\nB 1026.8 4.8 \u0002103-1.2\u0002103\nC Magnetically isotropic behaviour\nbetween the cubic to uniaxial anisotropy is about 0.75\nwhereas for sample B the ratio is 4. Therefore for sample\nB the cubic anisotropy is four times stronger than the\nuniaxial anisotropy. This is the reason of occurrence of\n90\u000edomain walls for sample B as shown in Fig. S122. In\ncomparison to samples A and B, sample C shows isotropic\nbehavior which is in consistent with the results obtained5\nFIG. 4. Anisotropy symmetry plot and the \fts for (a) sample A, (b) sample B and (c) sample C measured using FMR by\nkeeping the frequency \fxed at 7 GHz. The red solid lines are the best \fts to equation (4).\nfrom Kerr microscopy.\nL10-FeNi thin \flms (samples A, B and C) show dif-\nferent strength of anisotropy, which were fabricated by\ndi\u000berent deposition processes in a MBE system. Do-\nmain images reveals that magnetization reversal for all\nthree samples occur via nucleation and subsequent do-\nmain wall motion. We have observed lowest values of \u000b\n\u00184:9\u000210\u00003for the sample A which shows moderate\nmagnetic anisotropy. Angle dependent FMR measure-\nment shows the mixed cubic and uniaxial anisotropy in\nsamples A and B, while sample C exhibits magnetically\nisotropic behavior. Our work demonstrates that vari-\nable anisotropic L10-FeNi thin \flms can be fabricated in\nwhich the damping constant and magnetization reversal\ncan be tuned. These results may be useful for future\nspintronics based applications.\nAcknowledgements: We acknowledge the \fnancial sup-\nport by department of atomic energy (DAE), Depart-\nment of Science and Technology (DST-SERB) of Govt.\nof India,DST-Nanomission (SR/NM/NS-1018/2016(G))\nand DST, government of India for INSPIRE fellowship.\nREFERENCES\n1Z. Z. Bandic and R. H. Victora, Proceedings of the IEEE 96,\n1749 (2008).\n2S. Bedanta and W. Kleemann, Journal of Physics D: Applied\nPhysics 42, 013001 (2008).\n3D. Weller and A. Moser, IEEE Transactions on magnetics 35,\n4423 (1999).\n4M. Kotsugi, M. Mizuguchi, S. Sekiya, M. Mizumaki, T. Kojima,\nT. Nakamura, H. Osawa, K. Kodama, T. Ohtsuki, T. Ohkochi,\net al. , Journal of Magnetism and Magnetic Materials 326, 235\n(2013).\n5S. Goto, H. Kura, E. Watanabe, Y. Hayashi, H. Yanagihara,Y. Shimada, M. Mizuguchi, K. Takanashi, and E. Kita, Scienti\fc\nreports 7, 13216 (2017).\n6T. Klemmer, C. Liu, N. Shukla, X. Wu, D. Weller, M. Tanase,\nD. Laughlin, and W. So\u000ba, Journal of magnetism and magnetic\nmaterials 266, 79 (2003).\n7T. Kojima, M. Ogiwara, M. Mizuguchi, M. Kotsugi, T. Ko-\nganezawa, T. Ohtsuki, T.-Y. Tashiro, and K. Takanashi, Journal\nof Physics: Condensed Matter 26, 064207 (2014).\n8K. Takanashi, M. Mizuguchi, T. Kojima, and T. Tashiro, Journal\nof Physics D: Applied Physics 50, 483002 (2017).\n9M. Mizuguchi, S. Sekiya, and K. Takanashi, Journal of Applied\nPhysics 107, 09A716 (2010).\n10M. Kotsugi, M. Mizuguchi, S. Sekiya, T. Ohkouchi, T. Kojima,\nK. Takanashi, and Y. Watanabe, in Journal of Physics: Con-\nference Series , Vol. 266 (IOP Publishing, 2011) p. 012095.\n11K. Mibu, T. Kojima, M. Mizuguchi, and K. Takanashi, Journal\nof Physics D: Applied Physics 48, 205002 (2015).\n12M. Ogiwara, S. Iihama, T. Seki, T. Kojima, S. Mizukami,\nM. Mizuguchi, and K. Takanashi, Applied Physics Letters 103,\n242409 (2013).\n13T. Kojima, M. Mizuguchi, T. Koganezawa, K. Osaka, M. Kot-\nsugi, and K. Takanashi, Japanese Journal of Applied Physics\n51, 010204 (2011).\n14\\EVICOmagnetics,\" http://www.evico-magnetics.de/\nmicroscope.html .\n15\\NanOsc FMR Spectrometers,\" https://www.qdusa.com/\nproducts/nanosc-fmr-spectrometers.html .\n16S. Mallick, S. Bedanta, T. Seki, and K. Takanashi, Journal of\nApplied Physics 116, 133904 (2014).\n17B. B. Singh, S. K. Jena, and S. Bedanta, Journal of Physics D:\nApplied Physics 50, 345001 (2017).\n18C. Kittel, Physical Review 73, 155 (1948).\n19B. Heinrich, J. Cochran, and R. Hasegawa, Journal of Applied\nPhysics 57, 3690 (1985).\n20Z. Zhu, H. Feng, X. Cheng, H. Xie, Q. Liu, and J. Wang, Journal\nof Physics D: Applied Physics 51, 045004 (2018).\n21S. Pan, T. Seki, K. Takanashi, and A. Barman, Physical Review\nApplied 7, 064012 (2017).\n22S. Mallik, N. Chowdhury, and S. Bedanta, AIP Advances 4,\n097118 (2014).\nSUPPLEMENTARY MATERIAL6\nFIG. 5. Magnetic domain images for sample B at various \felds\nclose to the reversal \feld. It shows that the magnetization\nreversal is happening through 90\u000edomain wall in sample B."    },    {        "title": "1909.04308v3.Contributions_of_magnetic_structure_and_nitrogen_to_perpendicular_magnetocrystalline_anisotropy_in_antiperovskite__ε__Mn__4_N.pdf",        "content": " \n1 \n Contributions of magnetic structure and nitrogen to perpendicular \nmagnetocrystalline anisotropy in antiperovskite -Mn4N \n \nShinji Isogami*, Keisuke Masuda**, and Yoshio Miura \nNational Institute for Materials Science, Tsukuba 305-0047, Japan \n*E-mail : isogami.shinji@nims.go.jp  \n**E-mail : masuda.keisuke@nime.go.jp  \n \nTo study how nitrogen contributes to perpendicular magnetocrystalline anisotropy \n(PMA) in the ferrimagnetic antiperovskite Mn 4N, we examined both the fabrication of epitaxial \nMn4N films with various nitrogen contents and first-principles density-functional calculations. \nSaturation magnetization ( Ms) peaks of 110 mT and uniaxial PMA energy densities ( Ku) of 0.1 \nMJ/m3 were obtained for a N 2 gas flow ratio ( Q) of ~10% during sputtering deposition, \nsuggesting nearly single-phase crystalline -Mn4N. Segregation of -Mn and nitrogen-deficient \nMn4N grains were observed for Q ≈ 6%, which were responsible for a decrease in the Ms and Ku. \nThe first-principles calculations revealed that the magnetic structure of Mn 4N showing PMA was \n“type-B” having a collinear structure, whose magnetic moments couple parallel within the c-\nplane and alternating along the c-direction. In addition, the Ku calculated using Mn 32Nx \nsupercells showed a strong dependence on nitrogen deficiency, in qualitative agreement with the \nexperimental results. The second-order perturbation analysis of Ku with respect to the spin-orbit  \n2 \n interaction revealed that not only spin-conserving but also spin-flip processes contribute \nsignificantly to the PMA in Mn 4N. We also found that both contributions decreased with \nincreasing nitrogen deficiency, resulting in the reduction of Ku. It was noted that the decrease in \nthe spin-flip contribution occurred at the Mn atoms in face-centered sites. This is one of the \nspecific PMA characteristics we found for antiperovskite-type Mn 4N. \nI.  INTRODUCTION \nPerpendicular magnetic anisotropy (PMA) in magnetoresistive devices such as magnetic \ntunnel junctions1 has attracted significant attention in view of their potential application in spin-transfer \ntorque (STT) magnetic random access memories.2 Owing to their PMA with small saturation \nmagnetization ( Ms), Mn-based alloys with PMA such as  D022-Mn3Ga,3-6 D022-Mn3Ge,7,8 and L10-\nMnAl9-12 are regarded as satisfying the requirement for small critical STT-switching current density, \nJc ∝ MstHk (where , t, and Hk denote the damping constant, ferromagnetic layer thickness, and \nanisotropy field, respectively), which is proportional to the PMA energy density ( Ku). For example, the \nKu and Ms values for sputter-deposited D022-MnGa films are reported to be ~1 MJ/m3 and 310 mT,3 \nwhich are ~15% and ~20% of the values obtained for bulk  L10-FePt,13 respectively. First-principles \ndensity-functional calculations also predict that the spin-flip scattering process due to spin-orbit \ninteraction is the key for PMA in the D022-Mn3Ga.14 \nMn4N with the  phase has also long been known as a Mn-based PMA ferrimagnetic material  \n3 \n with an antiperovskite structure described by the formula ANB 3, where A and B correspond to Mn(I) \nat corner sites and Mn(II) at face-centered sites, respectively (Fig. 1).15 A neutron diffraction study \nidentified a ferrimagnetic spin order with two distinct magnetic moments, 3.5 B for Mn(I) and －0.8 \nB for Mn(II), at 300 K.16 The Mn 4N films recently fabricated by sputtering and molecular-beam \nepitaxy (MBE) exhibited Ms ≈ 110 mT and Ku ≈ 0.1 MJ/m3.17-20 Because its magnetic properties \nresemble those of the other Mn-based alloys mentioned above, recent research has focused on Mn 4N \nin view of its potential for spintronics applications.21-24 According to previous studies, the high Ku of \nMn4N films can be attributed to tetragonal distortion with c/a ≈ 0.99, where a and c correspond to the \nin-plane and out-of-plane lattice constants, respectively.19 Furthermore, the theoretical calculation of \nthe crystal formation energy reveals that the free-standing Mn 4N antiperovskite cell favors a collinear \nmagnetic structure when c/a approaches 0.99.25 \nThe correlation between c/a < 1 and PMA is encountered in other Mn-based PMA alloys ;3 \nhowever, no theoretical study has yet been conducted on the magnetic structures of Mn 4N that lead to \nc/a < 1 and to the resultant PMA. Furthermore, the magnetic moment of Mn atoms shows a strong site \ndependence originating from the interaction between Mn and nitrogen atoms.15,16 However, the \nspecific roles of nitrogen atoms in PMA are still under debate. Thus, the purpose of this study is to \nclarify, firstly, the magnetic structure of Mn 4N showing high Ku, and secondly, the specific contribution \nof nitrogen to PMA.  \n4 \n In the present study, we prepared Mn 4N films and measured the variation of Ku as a function \nof nitrogen content. Detailed structural analysis was performed to investigate the crystal grain growth \nand the chemical ordering of nitrogen. We also carried out first-principles calculations of Ku for two \npossible collinear magnetic structures for Mn 4N. The Ku variations for the Mn 32Nx supercell were \nanalyzed as a function of x in order to extract the contribution of nitrogen to PMA. \nII.  EXPERIMENTAL AND COMPUTATIONAL PROCEDURES \nIn the sample preparation and characterization, 30-nm-thick Mn 4N films were grown on \nsingle-crystal MgO(001) substrates via the reactive nitridation sputtering technique at a substrate \ntemperature of 450 C. To change the amount of nitrogen in the Mn 4N films, the Ar and N 2 gas flows \nwere controlled by mass flowmeters, and the ratio of N 2 (SN2) to Ar (SAr) is defined as Q = ௌಿమ\nௌಲೝାௌಿమ. \nStructural analysis of the (001)-oriented Mn 4N crystal layers was performed by X-ray diffraction \n(XRD) with Cu K radiation. Magnetic properties were measured using a superconducting quantum \ninterference device vibrating sample magnetometer (SQUID-VSM) at room temperature. Magnetic \nanisotropy was measured via torque magnetometry based on the angle-dependent anomalous Hall \neffect (AHE).26 \n In the theoretical calculation of Ku, we carried out first-principles density-functional \ncalculations with the aid of the Vienna ab initio simulation program (VASP),27,28 where lattice \nconstants estimated from the in-plane and out-of-plane XRD profiles were used. Prior to the estimation  \n5 \n of Ku, we first calculated the formation energy of Mn 4N (Etotal) assuming two types of collinear \nferrimagnetic structures. While the non-collinear ferrimagnetic structure of cubic Mn 4N has been \nexamined,29,30 we did not address this in the present study, because the collinear magnetic structure is \nessential for PMA in the case of our (001)-oriented Mn 4N with distortion along [001]. We next \ncalculated the value of Ku for the stable magnetic structure using the force theorem,31,32 Ku = (E[100] – \nE[001])/V, where E[100] (E[001]) is the sum of the eigenenergies of the unit cell with the magnetization \nalong the [100] ([001]) direction and V is the volume of the unit cell. The force theorem approximates \nthe difference in the Etotal between different magnetization directions as that in the sum of eigenenergies, \nwhich is known to be reasonable in most systems with magnetocrystalline anisotropy.31,32 Actually, we \nrecently confirmed the reliability of the force theorem for the interfacial magnetocrystalline anisotropy \nin magnetic tunnel junctions.33 Using the same method, we also estimated the values of Ku in the \nsupercells Mn 32Nx (x = 1 ~ 13), based on which we can discuss the role of nitrogen in Mn 4N \ntheoretically. To understand the results of these calculations, we further carried out second-order \nperturbation calculations of Ku for the supercells with respect to the spin-orbit interaction.34-36 Further \ndetails of our calculations are provided elsewhere.37 \nIII.  RESULTS AND DISCUSSION \nA. Magnetic properties \nFigure 2(a) shows the magnetization curves for Q = 9%, where the magnetic field ( H) was  \n6 \n swept along the [100] and [001] directions. Comparing the two curves, the magnetic easy axis was \nfound to point along the [001] direction ; that is, the studied Mn 4N film showed sizeable PMA. \nAlthough several different magnetic moments generally coexist in the unit cell of ferromagnetic Mn 4N, \nthe PMA is defined as the energy difference of the net magnetic moment pointing in between easy and \nhard axis, which is consistent with the case of ferromagnets. The 0Ms was measured to be 110 mT, \nwhich was comparable to that of -Mn4N films fabricated by sputtering and MBE.19,20 Figure 2(b) \nsummarizes the 0Ms as a function of Q. The peak value appeared at Q = 9%, whereas it decreased at \nboth lower and higher percentages. The results were in good agreement with a previous report ;20 \ntherefore, possible reasons for the degradation of 0Ms against Q can be attributed to the coexistence \nof pure Mn and/or nano-crystals such as Mn 3N2 with stoichiometric Mn 4N as impurities. Figure 2(c) \nshows a representative saturated magnetic torque curve based on measurement of the AHE.26 By fitting \nthe curve with T = –(Ku1 + Ku2)sin2M + (Ku2/2)sin4M, the first- and second-order uniaxial magnetic \nanisotropy constants Ku1 and Ku2 were obtained, where M denotes the magnetization angle with respect \nto the film plane normal, given by M = arccos[ RAHE(H) / RAHE]. The resultant Ku values were found \nto be dominated by Ku1 (the Ku2 contribution can be neglected), and the peak value was comparable to \nthat of Mn 4N fabricated using MBE19 and pulsed laser deposition.38 Note that the self-demagnetization \nenergy density ( 0Ms2/2) was estimated to be 4.8 kJ/m3 using 0Ms = 110 mT, which was two orders \nof magnitude smaller compared with the Ku of 0.1 MJ/m3. These results show that the experimentally  \n7 \n obtained Ku for the Mn 4N film can be dominated by magnetocrystalline anisotropy. As with 0Ms, the \nQ at which the peak Ku appeared was 9%, so we infer that the optimum Q giving the highest PMA is \nnear 9%. \nB. Structural characterization \nTo consider the correlation between Ku and the crystal structures of our Mn 4N films, structural \nanalysis was conducted via XRD. Figure 3(a) shows the out-of-plane XRD profiles for the \nsubstrate/Mn 4N(30 nm) samples. For Q = 10%, the diffraction peaks at 2 /≈ 22.9 and 46.8  \ncorrespond to the (001) superlattice and (002) fundamental peaks of Mn 4N crystals, respectively. These \npeaks were still observed for Q = 6%; however, additional peaks emerged at 2  / ≈ 40.4. \nTransmission electron microscopy revealed that the additional peaks originated from two impurity \nphases, consisting of -Mn crystal grains and a Mn-O crystal layer formed at the top surface of the \nMn4N layer due to natural oxidation. In addition, the chemical ordering of the nitrogen atoms in the \nMn4N grains decreased with decreasing Q (see Fig. A1 in Appendix). Figure 3(b) shows the in-plane \nXRD profiles with the incident angle  = 0.4, where the X-ray scattering vector is pointing along the \nMgO[200] direction. The diffraction peaks from Mn 4N(100) and Mn 4N(200) were observed as in Fig. \n3(a). The broad peak at 2  / ≈ 41 cannot be attributed to the -Mn crystal grains but instead to the \nnaturally oxidized Mn-O top layer, because in-plane XRD measurement is generally sensitive to the \nfilm surface. These results led us to conclude that the decrease in Ku for lower Q can be attributed not  \n8 \n only to the impurity phase -Mn but also to the increase in nitrogen deficiency compared to \nstoichiometric Mn 4N crystals. Although the mechanisms involved are complex, the point we address \nhere is not the influence of the impurity -Mn grains but the difference between stoichiometric and \nnitrogen-deficient Mn 4N. \n \nC. Calculation of magnetic structure \nWe will now present our first-principles calculation results. First, we determined the possible \nmagnetic structure of the equilibrium state of Mn 4N. Neutron diffraction experiments revealed two \ntypes of magnetic structures with ferrimagnetic order,16 type-A: (positive, negative) = (Mn(I), Mn(II)) ; \nand type-B: (positive, negative) = (Mn(II)X/Y, Mn(I) and Mn(II)Z). Therefore, calculation was carried \nout for the two types. Figure 4(a) plots the Etotal of the Mn 4N unit cell with different magnetic structures, \nnamely, type-A and type-B. The insets show schematic illustration of each magnetic structure. Etotal of \nthe type-B structure was clearly smaller than that of type-A for all c/a. The c/a that gave the smallest \nEtotal was 0.998 for type-A and 0.976 for type-B, and the difference reached ~0.2 eV/cell. These results \nwere in good agreement with a previous report.25 Here, we will present the possible explanation for \nthe stable type-B magnetic structure. For simplicity, cubic structure is assumed for both the type-A and \ntype-B. Considering the exchange interaction between Mn atoms, we consider the following \nHeisenberg Hamiltonian,  \n9 \n 𝐻=𝐽ଵ𝑺∙𝑺\n〈,〉+𝐽ଶ𝑺∙𝑺\n≪,≫         ,                  (1)  \nwhere J1 and J2 represent the 1st nearest Mn-Mn and the 2nd nearest Mn-Mn or Mn-N-Mn exchange \ninteraction constants, respectively. Note that the Jn > 0 (Jn < 0) corresponds to the antiferromagneitc \n(ferromagnetc) coupling. The location of the magnetic moment is denoted by i, and <i, j> and << i, j \n>> indicate sums over 1st and 2nd nearest neighbor pairs of spins. The total classical energy of all Si \npresented at atomic coordinates is given by, \n𝐸=𝐽ଵ𝑆ଶcos𝜃\n〈,〉+𝐽ଶ𝑆ଶcos𝜃\n≪,≫        ,            (2)  \nwhere ij represents the relative angle between two spins, Si and Sj. The E of the type-A and the type-\nB were estimated with respect to the Si of Mn at four atomic coordinates: ( a,b,c) = (0,0,0), (1/2,1/2,0), \n(1/2,0,1/2), and (0,1/2,1/2). In the case of type-A, there are four bonds with J1 and J2 for each xy, yz, \nand zx plane with multiplicity of two, therefore, the resultant energy of the Si at (0,0,0) ( E000) is given \nby, \n𝐸=𝑆ଶ൫𝐽ଵ(4cos𝜋+4cos𝜋+4cos𝜋)+𝐽ଶ(4cos0+4cos0+4cos0 )൯×1\n2\n=𝑆ଶ(−6𝐽ଵ+6𝐽ଶ).        (3) \nTaking into account the site equivalency, the other energy can be estimated similarly to the E000 as, \n𝐸ଵଶ ൗଵଶ ൗ=𝐸ଵଶൗ  ଵଶൗ=𝐸ଵଶൗଵଶൗ=𝑆ଶ(2𝐽ଵ+6𝐽ଶ).        (4)  \n10 \n Then, we obtain Etype-A = 24J2S2 in four sites total, suggesting that only the J2 terms exist in the energy \nof type-A, whereas the J1 terms were canceled out because of the parallel (symmetric) spin \nconfiguration for the Mn(II)X/Y/Z in the octahedral coordination structure. In contrast to the type-A, \nthe E000 of type-B is given by, \n𝐸=𝑆ଶ(−2𝐽ଵ+6𝐽ଶ).        (5) \nThe 𝐸ଵଶ ൗଵଶ ൗ ,𝐸ଵଶൗ  ଵଶൗ ,and 𝐸ଵଶൗଵଶൗ are the same as E000, then we obtain Etype-B = –8J1S2 + 24J2S2 \nin four sites total, suggesting that the total energy of type-B includes not only J2 but also J1 terms. This \nis caused by the presence of low symmetric octahedral coordination consisting of the Mn(II)Z and the \nMn(II)X/Y with antiparallel spin configuration. Comparing the Etype-A with Etype-B, we find the \nrelationship: Etype-A > Etype-B (Etype-A < Etype-B) in the case of J1 > 0 (J1 < 0), independent of J2. Namely, \nthe E of type-A and type-B via spin density functional theory shown in Fig. 4(a) can be explained by \nthe presence of preferential antiferromagnetic (AFM) spin order for the 1st nearest Mn-Mn bonds, \nwhich was more remarkable for the type-B than the type-A. Such the AFM preferential spin order \nmight be caused by the direct exchange coupling mechanism, in an analogy with CrN systems \ndiscussed by Filippetti et al.39 They find J1 = 9.5 meV (AFM) by the Cr-Cr direct interactions and J2 \n= –4 meV (FM) by the Cr-N-Cr interactions. Given the similarities of Cr and Mn ions, the Mn-Mn \ndirect interaction with AFM spin order might be a key to give a stable magnetic structure of type-B. \nAlthough the data is not shown, we additionally examined the E of fcc-Mn unit cell via the same  \n11 \n methods for both the type-A and type-B magnetic structures, but removing the nitrogen atom from the \nMn4N unit cell. The type-B showed stable compared with the type-A, which was consistent with the \ncase of Mn 4N.These results show the presence of Mn-Mn AFM direct interaction as the Cr-Cr bonds. \nThe tetragonal distortion along c-axis could provide somewhat influences to the E; however, the effects \nis not so strong for the present c/a regime [Fig. 4(a)] that the energy difference between the type-A and \nB (~0.2 eV/cell) can be explained. We thus infer such the wide remarkable energy difference can be \ndominantly attributed to the preferential spin order with type-B. \nD. Calculation of magnetic moment \nWe calculated the magnetic moment for each Mn atom in the Mn 4N unit cell with type-A and \ntype-B (Table I). The magnetic moment of Mn(I) showed a large negative value, whereas a large \npositive value was exhibited for Mn(II)X/Y, resulting in the relatively small saturation magnetization \n(Ms) of 179.7 mT for the type-B. In contrast, the type-A resulted in 7.86 mT, suggesting a negligibly \nsmall value. Comparing Ms = 110 mT measured for the 30-nm-thick Mn 4N films and 179.7mT obtained \ntheoretically for the type-B structure, possible reasons for the discrepancy are: the imperfect degree of \norder of nitrogen ( S = 0.79); and the formation of the initial growth layer with different magnetic \nstructure from type-B and the top-surface natural oxidation layer consisting of Mn-O. The Ms value \ntends to decrease by the imperfect degree of order of nitrogen, due to the transformation of magnetic \nstructures and/or the nanocrystals of another phase with negligibly small Ms.20 The calculated Ms (179  \n12 \n mT) for S = 1.0 thus decreases down to 141 mT when assuming S = 0.79. The further discrepancy \nbetween 141 mT and the measured 110 mT can be attributed to somewhat dead layer formation such \nas an initial growth layer near the MgO substrate and an Mn-O top surface layer with a thickness of \naround 3 nm judging from the cross-sectional TEM observation (Fig. A1 in Appendix). Based on our \nprevious study,40 it is revealed that many dislocations are generally formed near an interface between \nan MgO substrate with a thickness of around 2 nm and a transition metal nitride such as Fe 4N, of which \ncrystal structure is the same as Mn 4N. The dislocations would give rise to the transformation of \nmagnetic structure from the stable type-B to the other one such as type-A and/or -Mn; therefore, the \nMs of the initial growth layer might be neglected. In terms of the Mn-O layer on top, negligible small \nMs can be expected because of its antiferromagnetism.41 We can speculate that the initial growth layer \nand Mn-O top layer act as dead layers whose thickness is around 5 nm in total in the present samples. \nWhen we tried to consider these dead layers in the estimation of Ms, then resultant ~117 mT (consistent \nwith the measured Ms of 110 mT) was obtained. The estimation led us to conclude that the discrepancy \nbetween the 110 mT and 178 mT is distinct; however, which is exp lained by the imperfect S as well as \nthe ~5-nm-thick top/bottom dead layer formation. \nThe noticeable characteristics realized here are the magnetic moment for Mn(II)Z, which is \nclose to the values of Mn(II)X/Y for the type-A, whereas significantly different for the type-B. This \ncan be attributed to the near cubic structure for the type-A, whereas tetragonal distortion ( c/a ~ 0.976)  \n13 \n for the type-B in the Mn 4N unit cell. The electron number of Mn ion is generally responsible for the \nmagnetic moment in the case of Mn 4N, we thus infer that the number of electron for Mn(II)Z decreases \nby the stronger hybridization with neighboring N atom in the case of type-B, which is resulted from \nshort interatomic distance between the Mn(II)Z and N due to the tetragonal distortion. \n \nTable I Calculated magnetic moment of Mn(I), Mn(II)X, Mn(II)Y, and Mn(II)Z in the Mn 4N unit \ncell with type-A and type-B magnetic structures. \n Mn(I) \n(B) Mn(II)X/Y \n(B) Mn(II)Z \n(B) Ms \n(mT) \nType-A 3.67 –1.38/–1.47 –1.58 7.86 \nType-B –3.44 2.74 –0.98 179.7 \n \nE. Dependence of Ku on c/a \nFigure 4(b) shows the c/a dependence of the Ku values for both the type-A and type-B \nstructures calculated by using the Mn 4N unit cell. Here, we used 31 ×31×31 k points for the \nestimation of Ku after confirming the convergence of Ku against the number of k points [Fig. 4(c)]. The \nc/a values used for this calculation were based on the XRD analysis for the Mn 4N films with Q = 5%, \n10%, and 15% (solid symbols) (Fig. A2 in Appendix). The Etotal of the type-B structure was found to \nbe smaller than that of type-A for the entire c/a range, and the minimum Etotal was obtained at c/a =  \n14 \n 0.998 (type-A) and 0.976 (type-B) [Fig. 4(a)]. We also calculated Ku at c/a = 0.998 for the type-A  \nstructure and at c/a = 0.976 for the type-B structure, where Etotal had the minimum values (open \nsymbols). It is important to note that the Ku of the type-B structure has positive values, indicating PMA, \nwhereas that of the type-A structure has negative values, indicating in-plane magnetic anisotropy. \nTherefore, it can be concluded that the magnetic structure of Mn 4N films fabricated with Q ≈ 10% and \nexhibiting high PMA is “type-B”. In addition, the Ku values were not strongly influenced by the c/a \nratio, so that the crystal distortion of the type-B structure cannot fully explain the dependence of Ku on \nQ observed in Fig. 2(d).  \nF. Dependence of Ku on the nitrogen deficiency  \nFigure 5 shows the dependence of Ku on the number of nitrogen atoms x calculated using the \nMn32Nx supercell, whose size was 2 ×2×2 that of the Mn 4N unit cell. Here, we used 15 ×15×15 k \npoints to estimate Ku, which is large enough because Ku was confirmed to converge for k points larger \nthan 25×25×25 in the case of the Mn 4N unit cell (1 ×1×1) [Fig. 4(c)].  We set the c/a ratio to \n0.990, which is the experimental value that gave maximum Ku [Fig. 2(d) and Fig. A2(b) in Appendix], \nand the type-B magnetic structure was employed since we confirmed lower Etotal for the type-B \nmagnetic structure than for the type-A in the cases of Mn 32N8 and Mn 32N1. As a result, the Ku value \nfor x ≈ 8 was calculated to be ~3.8 MJ/m3, which dropped to ~2.6 MJ/m3 with decreasing x. Note that \nthe dependence of Ku on x was more prominent than that on the c/a ratio (Fig. 4), suggesting that  \n15 \n nitrogen deficiency dominates the mechanism for the degradation of Ku. The calculated Ku value was \nnearly one order of magnitude higher than the experimental results. We will address the possible \nreasons as follows. Taking in to account that the degree of order ( S) was estimated to be up to 0.79 \n[Fig. A2(c) in Appendix], which was close to that of the sputter-deposited Mn 4N films reported by \nanother group,20 and that degradation of Ku due to shape anisotropy was negligible as mentioned in the \ndescription for Fig. 2(d), we speculate the reason for the discrepancy to be the imperfect chemical \nordering of nitrogen involving the change in magnetic structure from type-B and/or the formation of \ndislocations in the initial growth layer.40 The imperfect nitrogen order might partly form possible \nimpurity phases such as nanocrystals of pure Mn and/or Mn 4N with different (type-A-like) magnetic \nstructures from the type-B. These contaminations might decrease the entire PMA of Mn 4N layer, \nbecause the Ku of contaminations are predicted to be negligible small, judging from Fig. 4(b). In terms \nof the initial growth layer of Mn 4N film deposited on the MgO substrate, many dislocations might be \npresent due to large mismatch by ~ 9%. Such the Mn 4N initial layer with dislocations cannot \nqualitatively contribute to the PMA due to weak crystallinity, resulting in the degradation of Ku. Thus, \nthe presence of nitrogen is presumably indispensable for obtaining a sufficiently large PMA in Mn-\nbased antiperovskite nitrides such as Mn 4N. Nevertheless, the variation of Ku with x in the calculation \nagreed with the variation with Q in the experiments [Fig. 2(d)]. Another reason to be considered is \nsomewhat the reliability of employing the force theorem to estimate Ku values of ferrimagnetic systems.  \n16 \n As we mentioned at the computational procedure, the force theorem has been widely known as a \nreasonable method in most systems with magnetocrystalline anisotropy,31,32 even in that with an \ninterfacial magnetocrystalline anisotropy.33 However, one must consider that there might be some \nmaterial systems that are beyond the reliability of existing force theorem. Although it is not clear, \nferrimagnets might belong to such the systems owing to their completely different magnetic structures \nfrom usual FMs. Extensive study based on both experiments and calculation might be necessary to \novercome these issues. \nG. Second-order perturbation analysis of PMA \nIn order to determine how each Mn atom contributes to PMA due to the presence of nitrogen, \nwe carried out a second-order perturbation analysis of Ku with respect to the spin-orbit interaction in \nthe cases of x = 1 and 8, namely, Mn 32N1 and Mn 32N8. Figures 6(a-1) and 6(a-2) show the contributions \nto Ku from spin-conserving terms and spin-flip terms, respectively, at the Mn(I), Mn(II)X, Mn(II)Y, \nand Mn(II)Z sites of the Mn 32N8 stoichiometric supercell. Here, the terms that can contribute to PMA \nare shown to be positive values, whereas those that contribute to in-plane magnetic anisotropy are \nnegative; the values were averaged over eight atoms for each site. It was revealed that both the spin-\nconserving and spin-flip terms contribute to PMA: the spin-conserving term (the spin-flip term) has a \nrelatively large value at the Mn(II)X/Y/Z (Mn(I)/Mn(II)X) atoms. To be specific, the spin-flip process \nfrom down- to up-spin state was dominant for the Mn(I) site, whereas that from up- to down-spin state  \n17 \n dominated for the Mn(II)X site. Furthermore, the spin-flip process was suppressed only for Mn(II)Y \nand Mn(II)Z [Fig. 6(a-2)]. Note here that the site equivalence in our results between Mn(II)X and \nMn(II)Y sites is lost, which is resulted from our calculation carried out under M // [100] with the spin-\norbit interaction: the spin moment in the Mn(II)X site points to the nitrogen atom when M // [100] \nwhereas that in the Mn(II)Y site does not. Therefore, the second-order perturbation results showed a \nstrong site-dependent contribution to PMA owing to the presence of nitrogen atoms. This must be a \ncharacteristic of antiperovskite nitrides, such as Mn 4N, in which the nitrogen atom occupies the body-\ncentered site. With nitrogen deficiency, in the case of Mn 32N1, all the spin-conserving and the spin-flip \nterms drastically decreased [Figs. 6(b-1) and 6(b-2)]. These results show that the degradation of Ku \ncorrelates not only with the spin-conserving terms but also with the spin-flip terms. \nH. Interpretation of PMA using atom-resolved DOS  \nTo examine the site-dependent spin-flip terms as well as their degradation due to nitrogen \ndeficiency in more depth, we calculated the atom-resolved DOS for the Mn 32N8 and Mn 32N1 supercells \n[Figs. 7(a) and 7(b)]. Although the data is not shown here, we confirmed that the features of the DOS \nof the Mn 32N8 supercell are almost the same as that of the Mn 4N unit cell. We found that the spin-flip \ncontributions to Ku shown in Fig. 6(a-2) agree with the processes expected from the DOSs as follows. \nIn the case of Mn 32N8 [Fig. 7(a)], the Mn(II)X/Y sites with positive magnetic moment have sharp peaks \nin the DOS around the Fermi level both in the occupied up-spin (–0.8 eV) and unoccupied down-spin  \n18 \n (0.2 eV) states, which are composed of occupied d( xy) and unoccupied d( zx) orbitals. We confirmed \nthat these orbitals provide a matrix element < zx|Lx|xy> of spin-flip scattering from up-spin to down-\nspin channels, which positively contributes to the PMA of Mn 4N. In contrast to the Mn(II)X/Y sites, \nthe Mn(I) site with negative magnetic moment has relatively large DOSs in the unoccupied up-spin \nstates (0.5 eV), leading to a contribution to PMA by the spin-flip scattering from down to up spin \nchannels. The DOS of the Mn(II)Z site is similar to that of the Mn(II)X/Y site, however, corresponding \nspin-flip scattering does not contribute to PMA but to in-plane magnetic anisotropy. This is consistent \nwith the fact that a relatively large matrix element of < xy|Lz|x2-y2> contributing to in-plane magnetic \nanisotropy presents in the Mn(II)Z site. To sum up, coexisting of atoms with opposite magnetic \nmoments in ferrimagnets lead to a different spin-flip scattering, resulting in different contributions to \nPMA depending on each atom. This is a characteristic of PMA mechanism in ferrimagnetic Mn 4N \nrevealed by the present second-order perturbation analysis. Note that the measured PMA by \nexperiments corresponds to an effective one including all of the spin-flip scattering for each atom. \nFigure 7(b) shows the DOS of the Mn 32N1 supercell. Compared with the case of Mn 32N8, the clear \npeaks near the Fermi level were strongly smeared out [Fig. 7(b)], which is associated with the \ndegradation of the spin-flip contribution due to the nitrogen deficiencies. \nIV.  CONCLUSION \nWe have studied the role of nitrogen in the PMA of Mn 4N by both experiments and first- \n19 \n principles density-functional calculations. The results show that: \na) Not only the presence of impurity phases such as -Mn but also the decrease in the chemical \nordering of nitrogen atoms, namely, nitrogen deficiency, is responsible for the degradation of Ku as \nwell as Ms. \nb) Mn4N showing PMA is associated with the “type-B” collinear structure with ferrimagnetic order. \nc) Both the spin-conserving and spin-flip processes contribute to PMA. Furthermore, the contribution \nto PMA from each site was found to be clear owing to the presence of nitrogen: the spin-conserving \nprocess dominates the PMA component for the Mn(II)X, Mn(II)Y and Mn(II)Z sites, while spin-flip \nprocesses from down- to up-spin state for the Mn(I) site and from up- to down-spin state for the \nMn(II)X site contribute to the PMA in Mn 4N. \nd) Both the spin-conserving and the spin-flip processes decreased with decreasing x, which can explain \nthe experimentally-observed degradation of Ku in Mn 4N with N deficiency. \nThese are the characteristics we have identified for antiperovskite nitrides such as -Mn4N. \n \n \n \n \n  \n20 \n Appendix \nA1. Transmission electron microscopy (TEM) observation for Mn 4N films \nIn order to identify the coexistence of nano-crystals with different phases in the Mn 4N films, \nTEM observation was carried out. Figures A1(a1) and A1(a2) show the annular dark field (ADF) \nSTEM image and the corresponding energy dispersive spectroscopy (EDS) mapping image for the \nMn4N films with Q = 10%. The samples were covered with a thick Ni layer to prevent the sample from \ndamage during the focused ion beam fabrication process. The Mn 4N layer shows a smooth surface, \nand EDS image indicates a nearly homogeneous nitrogen distribution within the imaged area. Note \nthat the thin layer of a few nanometers adjacent to the Mn 4N surface could be identified as Mn-O \ncrystals formed by natural oxidation, judging from the EDS mapping as well as the nanobeam \ndiffraction (NBD) pattern [Figs. A1(a2) and A2(a3)]. The NBD of the Mn 4N layer suggested high \nchemical ordering of nitrogen [Fig. A1(a4)], because clear spots corresponding to a (001) superlattice \nappeared. We obtained similar NBD from several different points in the Mn 4N layer, so that -Mn4N \ncrystals grew homogeneously for Q = 10%. In the case of Q = 6%, bright and dark parts were observed \nin the STEM image [Fig. A1(b1)]. The contrast observed in the STEM image corresponds to nitrogen \ncontent. Figure A1(b2) shows EDS mapping, and we found that the bright part involved lower nitrogen \ncontent. To obtain the crystal structure, we observed the NBD as shown in Figs. A1(b3) and A1(b4). \nAs a result, the bright part in the STEM image was identified as -Mn. The dark part of the STEM  \n21 \n revealed Mn 4N grains; however, distinct nitrogen deficiency was present judging from the significantly \nweak or invisible NBD spots originating from the Mn 4N superlattice. These results show that pure -\nMn and Mn 4N with remarkable nitrogen deficiency are formed as Q decreases. \nA2. Lattice constants ( a, c, and c/a) estimated using XRD profiles  \nFigure A2(a) plots the lattice constants, a and c, against Q estimated using X-ray diffraction \n(XRD) profiles. The a and c indicate the lattice constants along in-plane and out-of-plane directions of \nthe unit cell, as shown in Fig. 1 of the main text. In general, light elements such as nitrogen atoms act \nas interstitial impurities in nitrides. Therefore, both a and c monotonically increased with Q as expected, \nwhich was similar  to the results in previous report.1 The resultant c/a values decreased with increasing \nQ, which demonstrates that the Mn 4N unit cell undergoes tetragonal distortion [Fig. A2(b)]. We found \nthat the value c/a ≈ 0.99 was associated with the highest Ku and Ms, as mentioned in the main text. \nFigure A2(c) summarizes the degree of order of nitrogen ( S) estimated from the out-of-plane XRD \nresults [Fig. 3(a) in the main text]. The detailed method of estimation is reported elsewhere1,2. The \npeak value of S ≈ 0.79 appeared at Q ≈ 9%, which corresponds to the nitrogen content giving the \nhighest Ku and Ms. \n \n \n  \n22 \n ACKNOWLEDGEMENTS \nThe authors thank S. Mitani, Y. K. Takahashi, J. Uzuhashi, and T. Ohkubo for fulfilling discussions \nand detailed structural analysis. This work was supported by JSPS KANENHI Grant Nos. 19K04499, \n18H03787, and 16H06332. \nS. Isogami and K. Masuda contributed equally to this work. \n \n \n \n \n \n \n \n \n \n \n \n \n  \n23 \n References \n[1]   S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. \nMatsukura, and H. Ohno, Nat. Mater.  9, 721 (2010). \n[2]   J. C. Slonczewski, J. Magn. Magn. Mater.  159, L1 (1996). \n[3]   B. Balke, G. H. Fecher, J. Winterlik, and C. Felser, Appl. Phys. Lett. 90, 152504 (2007). \n[4]   F. Wu, S. Mizukami, D. Watanabe, H. Naganuma, M. Oogane, Y. Ando,  and T. Miyazaki, Appl. \nPhys. Lett. 94, 122503 (2009). \n[5]   H. Kurt, K. Rode, M. Venkatesan, P. 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The Mn(II) atoms arranged in x, y and z directions are described as Mn(II)X, Mn(II)Y \nand Mn(II)Z. They are distinguished in our second-order perturbation analysis of perpendicular \nmagnetocrystalline anisotropy with respect to the spin-orbit interaction. \n \nFigure 2.  (a) Magnetization curves measured along [001] (red) and [100] (blue) directions for the MgO \nsubstrate/Mn 4N (30 nm) sample, with the N 2 gas flow ratio Q = 9%. (b) Saturation magnetization \n(0Ms) as a function of Q. (c) Saturated magnetic torque curve for the Mn 4N sample with Q = 10%. \n(d) Variations of magnetic anisotropy constant, Ku, Ku1, and Ku2. \n \nFigure 3. (a, b) Out-of-plane (a) and in-plane (b) XRD profiles for MgO sub./Mn 4N (30 nm) samples \nfabricated using the reactive nitride sputtering method with various N 2 gas flow ratios ( Q). \n \nFigure 4. (a) Calculation of total formation energy ( Etotal) of Mn 4N unit cell with different spin \nstructures, type-A and type-B, as a function of c/a. (b) c/a dependence of Ku by first-principles \ncalculations for the Mn 4N unit cell. The insets represent different magnetic structures of type-A and  \n28 \n type-B. The c/a ratios for solid symbols correspond to the values estimated from XRD profiles with Q \n= 3%, 10%, and 15% [Fig. A2(b) in Appendix]. The c/a ratios for open symbols correspond to the \nvalues that gave the minimum energy for the Mn 4N unit cell [Fig. 4(a)]. (c) The number of k points \ndependence of the Ku for the type-B with c/a = 0.99. \n \nFigure 5. Calculated Ku as a function of the number of nitrogen atoms ( x) in the Mn 32Nx supercell with \nthe type-B magnetic structure and c/a = 0.99.  \n \nFigure 6. (a, b) Second-order perturbation terms for Mn 32N8 (a) and Mn 32N1 (b) with the type-B \nmagnetic structure and c/a = 0.99. (a-1, b-1) The spin-conserving contributions from up- to up-spin \nstate and from down- to down-spin state are indicated by green and orange bars, respectively. (a-2, b-\n2) The spin-flip contributions from up- to down-spin state and from down- to up-spin state are indicated \nby blue and red bars, respectively. \n \nFigure 7. (a, b) Atom-resolved density of states (DOS) for Mn 32N8 supercell (a) and Mn 32N1 supercell \n(b) with the stable magnetic structure of type-B and c/a = 0.99. The red, green, and red lines correspond \nto the DOS of Mn(I), Mn(II)X/Y, and Mn(II)Z, respectively. \n  \n29 \n Figure A1. (a1) Cross-sectional annular dark-field scanning transmission microscopy (ADF-STEM) \nimage and (a2) energy dispersive spectroscopy (EDS) mapping image for MgO sub./Mn 4N (30 nm) \nfabricated by reactive sputtering with nitrogen gas flow ratio (Q) = 10%. (a3, a4) Nanobeam diffraction \n(NBD) patterns showing Mn-O (a3) and Mn 4N (a4). (b1–b4) are the same as (a1-a4) but for Q = 6%. \n \nFigure A2. (a,b,c) N 2 flow ratio ( Q) dependence of lattice constants, a and c (a), c/a (b) degree of order \nof nitrogen ( S) (c) for the sputter deposited 30-nm-thick Mn 4N films.  \n \n \n \n \n \n \n \n \n \n \n  \n30 \n  \n \n \n \n \n  \n \nFig. 1 \n \n \n \n \n \n \n31 \n  \n \n \n \n \n \nFig. 2 \n \n \n 24681012050100\n00.20.40.6\nQ (%)0M (mT)\n0M (B/cell)(a)\nQ= 9%[001](b)\n(c) (d)\nQ= 10%\n2468101200.51\nQ (%)Ku, Ku1, Ku2 (×105 J/m3)\nKu\nKu1\nKu2[100]\n04590135180-0.100.1\nM (deg.)T (×105 J/m3) Experiment\n Calclated-5 0 5-1000100\n-0.500.5\n0H (T)0M (mT)\n0M (B/cell) \n32 \n  \n \n \n \n \n \n \nFig. 3 \n \n \n \n 2030405060Log intensity (a.u.)\n2 (deg.)2030405060Log intensity (a.u.)\n2 (deg.)◎○\n▽＊\nMn4N (002)MgO sub.\n-Mn\nMnO\nMn4N (001)○\n◎Mn4N (200)\nMn4N (100) ▽-Mn\nMnO＊MgO sub. (a) (b)\n10 %Q = 6 %q|| MgO[200]\n10 %Q = 6 % \n33 \n  \n \n \n \n   \n \nFig. 4 \n \n (c)\n3 3 3\ntype A\ntype B(a)\ntype B(b)\ntype A\n0.97 0.98 0.99 1-4-2024\nc / aKu (MJ/m3)\n0 10 20 303.83.94\nNumber of k pointsKu (MJ/m3)\n0.96 0.98 11.02-45-44.9-44.8\nc / aEtotal (eV/cell) \n34 \n  \n \n \n \n \n \n \nFig. 5 \n \n \n 0 5 10234\nx in Mn 32NxKu (MJ/m3) \n35 \n  \n \n \n \nFig. 6 \n (a-1) (a-2)\n(b-1) (b-2)X           Y           Z\nMn(II) Mn(I)X           Y           Z\nMn(II) Mn(I)\nX           Y           Z\nMn(II) Mn(I)X           Y           Z\nMn(II) Mn(I)-0.100.10.2↑⇒↑\n↓⇒↓Perturbation terms for Mn 32N8 (meV/atom)↑⇒↓\n↓⇒↑\n-0.100.10.2 ↑⇒↑\n↓⇒↓Perturbation terms for Mn 32N1 (meV/atom)↑⇒↓\n↓⇒↑ \n36 \n  \n \n  \n \nFig. 7 \n (a)\n(b)up-spin\ndown-spin\nup-spin\ndown-spinMn(I)Mn(II)X/YMn(II)Z\nMn(I)Mn(II)X/Y\nMn(II)Z-4 -2 0 2-4-2024\nE－EF (eV)DOS for Mn 32N8 (states/(eV atom))\n-4 -2 0 2-4-2024\nE－EF (eV)DOS for Mn 32N1 (states/(eV atom)) \n37 \n  \n \n \n \n \n \n \n \nFig. A1 \n \n \n \n \n \n38 \n  \n \n \n \n \n \nFig. A2 5 10 150.980.991.00\nQ (%)c / a\n5 10 150.3840.3870.3900.393\nQ (%)Lattice constant, c, a (nm)(a)\na\nc(b)\n(c)\n5 10 150.40.60.81\nQ (%)Degree of order, S"    },    {        "title": "1909.07450v2.Dissipation_induced_rotation_of_suspended_ferromagnetic_nanoparticles.pdf",        "content": "arXiv:1909.07450v2  [cond-mat.soft]  2 Oct 2019Dissipation-induced rotation of suspended ferromagnetic nanoparticles\nT. V . Lyutyy1,∗S. I. Denisov1,†and P. H¨ anggi2‡\n1Sumy State University, 2 Rimsky-Korsakov Street, UA-40007 Sumy, Ukraine\n2Institut f¨ ur Physik, Universit¨ at Augsburg,\nUniversit¨ atsstraße 1, D-86135 Augsburg, Germany\nAbstract\nWe report the precessional rotation of magnetically isotro pic ferromagnetic nanoparticles in a viscous\nliquid that are subjected to a rotating magnetic field. In con trast to magnetically anisotropic nanoparticles,\nthe rotation of which occurs due to coupling between the magn etic and lattice subsystems through mag-\nnetocrystalline anisotropy, the rotation of isotropic nan oparticles is induced only by magnetic dissipation\nprocesses. We propose a theory of this phenomenon based on a s et of equations describing the deterministic\nmagnetic and rotational dynamics of such particles. Neglec ting inertial effects, we solve these equations\nanalytically, find the magnetization and particle precessi ons in the steady state, determine the components\nof the particle angular velocity and analyze their dependen ce on the model parameters. The possibility of\nexperimental observation of this phenomenon is also discus sed.\n1I. INTRODUCTION\nThe ferromagnetic single-domain nanoparticles are object s of intense research mainly because\nof their high potential for various applications. Among the m, the most promising are biomedical\napplications such as magnetic particle imaging1,2, drug delivery3,4, magnetic fluid hyperthermia5,6\nand cell separation7–9. The development of methods for the production of ferromagn etic nanopar-\nticles with specified properties is one of the key factors for the realization of these and other\napplications. Up to date, a number of such methods have alrea dy been proposed and demonstrated\n(see, e.g., Refs.10–12and references therein). Another key factor is the developm ent of theoreti-\ncal approaches aimed at a more complete description of the ma gnetic properties of ferromagnetic\nparticles in viscous liquids subjected to external magneti c fields.\nThese systems are often studied in the framework of the rigid dipole model, when the particle\nmagnetization is assumed to be directed along the particle e asy axis. This approximation, which\nholds if the anisotropy magnetic field is large enough, was us ed to study, e.g., the e ffects of particle\nrotation, dipolar interaction and thermal fluctuations13–17. The same approximation was also used\nto describe in an analytical way the directed transport of su spended ferromagnetic nanoparticles\ninduced by the Magnus force18–20.\nHowever, if the anisotropy magnetic field does not strongly e xceed the external field, then the\nmodel of suspended particles with “frozen” magnetization ( i.e., the rigid dipole model) fails. In\nthis case, it is necessary to consider a coupled rotation of t he particle magnetization and particle\nbody. One of the most simple and e ffective methods to determine properties of such systems\nis based on the concept of relaxation times (see, e.g., Ref.21). At the same time, the dynamical\napproach based on both deterministic and stochastic equati ons of motion for the particle and its\nmagnetization provides a much more complete description of the system’s properties. Although\nthese equations were derived many years ago22, only recently they have been rederived and applied\nfor studying the coupling between the magnetic and rotation al dynamics of suspended particles and\neffects in magnetic fluid hyperthermia23–28.\nIn this paper, we use the dynamical approach to obtain two mai n results for magnetically\nisotropic (i.e., without magnetocrystalline anisotropy) ferromagnetic nanoparticles in a viscous\nliquid. First, there exists the dynamical coupling between the magnetic and lattice subsystems in\nsuch particles arising from magnetic dissipation. Second, due to this coupling, a rotating mag-\nnetic field induces the precessional rotation of these parti cles. We hope that these rather surprising\n2results will stimulate experimental studies in this area.\nThe paper is structured as follows. In Sec. II, we introduce the basic equations describing the\ncoupled magnetic and rotational dynamics of ferromagnetic nanoparticles in a viscous liquid. The\ncase of magnetically isotropic nanoparticles subjected to a rotating magnetic field is considered in\nSec. III. Here, assuming that inertial e ffects are negligible, we solve these equations in the steady\nstate, analyze stability of obtained solutions and investi gate, both analytically and numerically,\nthe dependence of the magnetization precession and precess ional rotation of nanoparticles on the\nmodel parameters. Our main conclusions are summarized in Se c.IV.\nII. BASIC EQUATIONS FOR COUPLED MAGNETO-MECHANICAL DYNAMI CS\nWe consider single-domain particles of spherical form susp ended in a viscous liquid and char-\nacterized by the total momentum J=J(t) defined as a sum of the mechanical angular momentum,\nIω, and the spin momentum in the quasi-classical approximatio n,−(V/γ)M:\nJ=Iω−V\nγM. (2.1)\nHere, I=ρmVd2/10 is the particle’s moment of inertia, ρm,Vanddare the particle density, volume\nand diameter, respectively, ω=ω(t) is the particle angular velocity, M=M(t) (|M|=M=const)\nis the particle magnetization, and γ(>0) is the gyromagnetic ratio. In the absence of dissipation\nwe have23,24dJ/dt=VM×H, where H=H(t) is an external magnetic field and the sign ×denotes\nthe vector product. Therefore, by di fferentiating the particle angular momentum Iωwith respect to\ntime and introducing the frictional torque −6ηVω(ηis the dynamic viscosity of the liquid) acting\non the particle, one obtains the equation\nId\ndtω=V\nγd\ndtM+VM×H−6ηVω (2.2)\ndescribing the rotation of ferromagnetic particles in a vis cous liquid. Note that for vacuum (when\nη=0) this equation was introduced in Ref.23, and for a viscous liquid it was introduced in Ref.24.\nBecause the magnetization value Mis assumed to be time-independent, the dynamics of the\nmagnetization vector Mcan be described, e.g., by the Landau-Lifshitz (LL) equatio n29. An im-\nportant feature of this equation describing the magnetizat ion dynamics in rotating nanoparticles is\nthat its dissipation term should be properly modified23:\nd\ndtM=−γM×Heff−γα\nMM×/parenleftigg\nM×/parenleftigg\nHeff−ω\nγ/parenrightigg/parenrightigg\n, (2.3)\n3whereα(>0) is the LL damping parameter, −ω/γis the so-called Barnett field originating from\nthe particle rotation (see, e.g., Ref.26), and Heffis the effective magnetic field acting on M. In\nparticular, in the case of uniaxial particles the e ffective magnetic field is given by\nHeff=Ha\nM(M·n)n+H. (2.4)\nHere, Hais the uniaxial magnetic anisotropy field, the dot denotes th e scalar product, and the unit\nvector nis directed along the easy axis of magnetization and satisfie s the following equation of\nmotion (kinematic relation):\nd\ndtn=ω×n. (2.5)\nEquations (2.2) ,(2.3) and(2.5) supplemented by the e ffective magnetic field (2.4) completely\ndescribe the coupled dynamics of magnetization and rotatio nal dynamics of uniaxial nanoparticles\nin a viscous liquid. Introducing the dimensionless variabl es and parameters\nm=M\nM,ν=ω\nγM,heff=Heff\nM,h=H\nM,\nτ=γMt, κ=Iγ2\nV, β=6γη\nM,ha=Ha\nM, (2.6)\nthese equations can be rewritten in the dimensionless form\nκ˙ν=˙m+m×h−βν, (2.7a)\n˙m=−m×heff−αm×(m×(heff−ν)), (2.7b)\n˙n=ν×n, (2.7c)\nwhere the overdot denotes the derivative with respect to the dimensionless time τand, according\nto(2.4) ,\nheff=ha(m·n)n+h. (2.8)\nIf the magnetization dynamics is assumed to be governed by th e Landau-Lifshitz-Gilbert (LLG)\nequation30, then, to take into account the influence of particle rotatio n, this equation should be\nmodified as follows24:\nd\ndtM=−γM×Heff+α′\nMM×/parenleftiggd\ndtM−ω×M/parenrightigg\n(2.9)\n(α′is the LLG damping parameter). Using the relation\nM×d\ndtM=−γM×(M×Heff)−α′Md\ndtM−α′M(M×ω)\n4that follows directly from (2.9) and notations (2.6) , Eq. (2.9) can be reduced to the dimensionless\nLLG equation\n(1+α′2)˙m=−m×(heff+α′2ν)−α′m×(m×(heff−ν)), (2.10)\nwhich in this case should be used instead of Eq. (2.7b) .\nBy comparing Eqs. (2.7b) and(2.10) , we can make sure that at ν=0(when nanoparticles do\nnot move) these equations are, in fact, equivalent29. Strictly speaking, at ν/nequal0these equations\nare different. However, in the most common case, when α′≪1, this difference can be neglected.\nTherefore, in further analysis we will use the set of Eqs. (2.7) .\nIt is important to emphasize that Eqs. (2.7) are written in the deterministic approximation. In\nprinciple, thermal fluctuations can also be accounted for by introducing in these equations the\nGaussian white noises (see, e.g., Refs.22,24,25). However, if these noises are not too strong, they do\nnot destroy the deterministic e ffects. This means that Eqs. (2.7) can be used as a starting point for\nstudying the coupled magnetic and rotational dynamics of su spended ferromagnetic particles. It\nis also possible to formulate conditions under which therma l fluctuations can safely be neglected.\nIn particular, the magnetization fluctuations in magnetica lly isotropic nanoparticles (when ha=0)\nmay be considered as small if the magnetic energy MHV exceeds the thermal energy kBT(kBis\nthe Boltzmann constant and Tis the absolute temperature). This situation occurs when th e particle\ndiameter satisfies the condition d>d1, where d1=(6kBT/πMH)1/3(see also Sec. III D ).\nIII. PRECESSIONAL ROTATION OF MAGNETICALLY ISOTROPIC NANO PARTICLES\nOur next aim is to study the coupled magnetic and rotational d ynamics of isotropic ferromag-\nnetic nanoparticles. Since in this case ha=0 and, according to (2.8) ,heff=h, this dynamics is\ndescribed by a set of only two equations, (2.7a) and(2.7b) . As to Eq. (2.7c) , for these nanoparticles\nit can be excluded from further consideration.\nAssuming that the left-hand side of Eq. (2.7a) is negligibly small (for more detail, see Sec.\nIII D ), we can rewrite the set of Eqs. (2.7a) and(2.7b) in the form\n˙m=−m×h+βν, (3.1a)\n˙m=−m×h−αm×(m×(h−ν)). (3.1b)\nSubstituting ˙mfrom Eq. (3.1b) into Eq. (3.1a) and taking into account that, according to (3.1a) ,\n5ν·m=0, one obtains\nν=−α\nα+βm×(m×h). (3.2)\nThen, substituting this expression for the dimensionless a ngular velocity into Eq. (3.1b) , it is not\ndifficult to derive a closed LL equation for the unit magnetizatio n vector of a magnetically isotropic\nparticle in a viscous liquid\n˙m=−m×h−qm×(m×h), (3.3)\nwhere q=αβ/(α+β). Note that the limit β→∞ corresponds to immobile particles. In this limit,\nν→0,q→α, and Eq. (3.3) reduces to the standard LL equation.\nNext, we use Eq. (3.3) and expression (3.2) to study the magnetic and rotational steady-state\ndynamics of isotropic nanoparticles subjected to the rotat ing magnetic field\nh=h(cosυτex+ρsinυτey). (3.4)\nHere, h=|h|=const is the dimensionless amplitude of the rotating magnet ic field,υ=Ω/γM,\nΩis the rotating field frequency, ρ=±1 is the parameter that determines the direction of the\nmagnetic field rotation, and ex,eyandezare the unit vectors of the Cartesian coordinate system.\nA. Magnetization precession\nLet us represent the unit magnetization vector min the form\nm=sinθcosϕex+sinθsinϕey+cosθez, (3.5)\nwhereθ=θ(τ) andϕ=ϕ(τ) are the polar and azimuthal angles of m, respectively. Then, intro-\nducing the lag angle\nψ=ρυτ−ϕ, (3.6)\nwe can reduce the vector LL equation (3.3) to a set of differential equations for θandψ\n˙θ=hsinψ+qhcosθcosψ,\n(˙ψ−ρυ) sinθ=hcosθcosψ−qhsinψ. (3.7)\nAssuming that in the steady state (when τ→∞ ) the angles θandψdo not depend on time,\nθ=θρ=const, ψ=ψρ=const (3.8)\n6(0≤θρ≤π,−π<ψρ≤π), from Eqs. (3.7) one gets a set of equations for θρandψρ\nsinψρ+qcosθρcosψρ=0,\nρχsinθρ+cosθρcosψρ−qsinψρ=0, (3.9)\nwhereχ=υ/h. If these angles are represented in the form\nθρ=π\n2(1+ρ)−ρθ0, ψρ=ρψ0, (3.10)\nthen new variables θ0(0≤θ0≤π) andψ0(−π<ψ 0≤π) do not depend on the parameter ρ. Indeed,\ntaking into account that sin ψρ=ρsinψ0, cosψρ=cosψ0, sinθρ=sinθ0and cosθρ=−ρcosθ0,\nthe set of Eqs. (3.9) readily yields\nsinψ0−qcosθ0cosψ0=0,\nχqsinθ0−(1+q2) sinψ0=0. (3.11)\nAccording to the last equation in (3.11) , the angleψ0(likeθ0) must belong to the interval [0 ,π],\ni.e., only non-negative values of sin ψ0are permitted. Introducing parameters c=q//radicalbig\n1+q2and\nk=χ//radicalbig\n1+q2, it can be easily shown from Eqs. (3.11) that sinψ0satisfies the biquadratic equation\nsin4ψ0−(1+k2) sin2ψ0+c2k2=0. (3.12)\nSince sinψ0∈[0,1], its unique solution is given by sin ψ0=R, where\nR=1√\n2/radicalig\n1+k2−/radicalbig\n(1+k2)2−4c2k2. (3.13)\nFrom this, using the second equation in (3.11) , one obtains sin θ0=R/ck[note that, according to\n(3.13) , the conditions R≤1 and R/ck≤1 always hold]. In addition, the first equation in (3.11)\nshows that both angles ψ0andθ0must lie either in the interval [0 ,π/2) or in the interval ( π/2,π].\nIn the former case, the solution of Eqs. (3.11) is written as\nθ(1)\n0=arcsinR\nck, ψ(1)\n0=arcsin R, (3.14)\nwhile in the latter case it is written as θ(2)\n0=π−θ(1)\n0,ψ(2)\n0=π−ψ(1)\n0.\nThus, the rotating magnetic field (3.4) could, in principle, induce in magnetically isotropic\nnanoparticles two steady-state precessional states of the magnetization, m(1)andm(2). Using (3.10)\nand(3.14) , we find the angles\nθ(1)\nρ=π\n2(1+ρ)−ρarcsinR\nck, ψ(1)\nρ=ρarcsin R (3.15)\n7form(1), andθ(2)\nρ=θ(1)\n−ρandψ(2)\nρ=ρπ+ψ(1)\n−ρform(2). These expressions, together with the definition\n(3.6) of the lag angle, allow us to determine the components of the v ector m(l)(l=1,2) as follows:\nm(l)\nx\nm(l)\ny\nm(l)\nz=(−1)1+lR\nck×cos/parenleftig\nυτ+(−1)larcsin R/parenrightig\n,\nρsin/parenleftig\nυτ+(−1)larcsin R/parenrightig\n,\n−ρ/radicalbig\nc2k2/R2−1.(3.16)\nAccording to them, the steady-state magnetization precess ions, if they are stable, occur about the\nzaxis with the magnetic field frequency and their direction co incides with the direction of the\nmagnetic field rotation. The time-averaged magnetization i n these precessional states, defined as\n/angbracketleftm(l)/angbracketright=(υ/2π)/integraltext2π/υ\n0m(l)dτ, is given by/angbracketleftm(l)/angbracketright=(−1)lρ/radicalbig\n1−R2/c2k2ez, i.e., the magnetic field\nrotating in the xyplane magnetizes the isotropic nanoparticles in the zdirection. We note in this\ncontext that a similar e ffect, the magnetization of nanoparticle systems by a rotatin g magnetic field,\nwas earlier predicted and analyzed for anisotropic (uniaxi al) and immobile nanoparticles31–33. But\nits nature is quite di fferent: the magnetization of those systems occurs due to the p resence of the\nanisotropy magnetic field, which in our case is absent.\nB. Stability analysis of the magnetization precession\nNow we analyze the linear stability of the precessional stat esm(l). Substituting θ=θ(l)\nρ+θ1\n(θ1=θ1(τ),|θ1|≪1) andψ=ψ(l)\nρ+ψ1(ψ1=ψ1(τ),|ψ1|≪1) into Eqs. (3.7) , we obtain a set of\nordinary differential equations for θ1andψ1\n˙θ1=h/parenleftbigcosψ(l)\nρ−qcosθ(l)\nρsinψ(l)\nρ/parenrightbigψ1−qhsinθ(l)\nρcosψ(l)\nρθ1,\n˙ψ1sinθ(l)\nρ=−h/parenleftbigcosθ(l)\nρsinψ(l)\nρ+qcosψ(l)\nρ/parenrightbigψ1+h/parenleftbigρχcosθ(l)\nρ\n−sinθ(l)\nρcosψ(l)\nρ/parenrightbigθ1. (3.17)\nAssuming that\nθ1=˜θ1eλlhτ, ψ 1=˜ψ1eλlhτ, (3.18)\nwhere the parameters ˜θ1,˜ψ1andλldo not depend on τ, Eqs. (3.17) are reduced to a homogeneous\nsystem of linear equations with respect to ˜θ1and˜ψ1, which can be written in the matrix form as\nλl+(−1)1+la11 (−1)la12\n(−1)1+la21λl+(−1)la22˜θ1\n˜ψ1=0\n0. (3.19)\n8Taking into account that, according to (3.10) and(3.15) ,\nsinθ(l)\nρ=R\nck,cosθ(l)\nρ=(−1)lρ1\nck√\nc2k2−R2,\nsinψ(l)\nρ=ρR,cosψ(l)\nρ=(−1)1+l√\n1−R2, (3.20)\nthe coefficients anmin Eqs. (3.19) are expressed through the parameters candk(recall that c<1\nandk<∞) as follows:\na11=R\nk/radicalbigg\n1−R2\n1−c2,\na12=√\n1−R2+R\nk/radicalbigg\nc2k2−R2\n1−c2,\na21=√\n1−R2+k\nR/radicalbigg\nc2k2−R2\n1−c2,\na22=√\nc2k2−R2−c2k\nR/radicalbigg\n1−R2\n1−c2. (3.21)\nIt is well known that non-zero solutions of the system of Eqs. (3.19) exist only if the determinant\nof the coefficient matrix vanishes, i.e., if\nλ2\nl+(−1)1+l(a11−a22)λl+a12a21−a11a22=0. (3.22)\nThis occurs at λl=λ+\nlandλl=λ−\nl, where\nλ±\nl=(−1)l1\n2(a11−a22)±i1\n2/radicalbig\n4a12a21−(a11+a22)2 (3.23)\n(iis the imaginary unit) are solutions of Eq. (3.22) . It can be verified (analytically or numerically)\nthata11−a22>0 and 4 a12a21−(a11+a22)2>0 for all values of candk. Therefore, using (3.23) ,\none can conclude that the steady-state precessional state o f the magnetization with l=1 (i.e.,\nm(1)) is stable (because Re λ±\n1<0), while the precessional state with l=2 (i.e., m(2)) is unstable\n(because Re λ±\n2>0). Note also that, according to (3.23) , the magnetization approaches the stable\nsteady state m(1)in an oscillatory manner.\nC. Particle precession\nSince the steady-state magnetization m(2)is unstable, we only determine the components of the\n(dimensionless) particle angular velocity νthat correspond to the stable steady-state magnetization\n9m(1). To this end, using (3.4) and(3.2) , we first represent the Cartesian components of νin the\nformνx\nνy\nνz=αh\nα+β×cosυτ−m(1)\nx/parenleftig\nm(1)\nxcosυτ+ρm(1)\nysinυτ/parenrightig\n,\nρsinυτ−m(1)\ny/parenleftig\nm(1)\nxcosυτ+ρm(1)\nysinυτ/parenrightig\n,\n−m(1)\nz/parenleftig\nm(1)\nxcosυτ+ρm(1)\nysinυτ/parenrightig\n.(3.24)\nThen, substituting the magnetization components m(1)\nx,m(1)\nyandm(1)\nzfrom (3.16) into(3.24) , one\nstraightforwardly obtains\nνx=αh\nα+β/bracketleftig\ncosυτ−R2\nc2k2√\n1−R2\n×cos(υτ−arcsin R)/bracketrightig\n,\nνy=ραh\nα+β/bracketleftig\nsinυτ−R2\nc2k2√\n1−R2\n×sin(υτ−arcsin R)/bracketrightig\n,\nνz=ραh\nα+βR\nc2k2/radicalbig\n(1−R2)(c2k2−R2). (3.25)\nThese expressions show that a rotating magnetic field causes a dissipation-induced precessional\nmotion of magnetically isotropic nanoparticles. The prece ssion occurs with the magnetic field\nfrequency, the particle and magnetic field are rotated in the same direction, and the (dimensionless)\nmagnitude of the particle angular velocity can be cast as\n|ν|=αh\nα+β1\nck√\nc2k2−R2+R4. (3.26)\nIt should be also pointed out that, according to (3.16) and(3.25) ,ν·m(1)=0, i.e., the magnetization\nprecession is synchronized with the particle rotation.\nIn order to get more insight into the dissipation-induced me chanism of nanoparticle rotation, we\nfirst determine the particle angular velocity in the cases of small fluid dynamic viscosity ( β→0)\nand magnetic damping parameter ( α→0). Using (3.25) and(3.26) , one finds\nνz=ρhχ\n1+χ2,|ν|=hχ/radicalbig\n1+χ2(3.27)\nand\nνz=ρhχα\nβ(1+χ2),|ν|=hχα\nβ/radicalbig\n1+χ2(3.28)\nfor the first and second cases, respectively. According to th ese results, the particle rotation exists\natβ→0, while it vanishes at α→0. We therefore conclude that it is the magnetic dissipation that\nis responsible for the nanoparticle rotation.\n10Because the viscosity parameter βis usually large (see below), it is also reasonable to consid er\nthe limiting case β→∞ . In the main approximation in 1 /β, expressions (3.25) and(3.26) yield\nνz=ρh√\n2βb/radicalig\na2−a√\na2−4b2−2b2 (3.29)\nand\n|ν|=h√\n2β/radicalig\na−√\na2−4b2, (3.30)\nwhere a=1+α2+χ2andb=αχ. They show that in this limit the nanoparticle angular veloc ity\ndecreases to zero inversely proportional to the viscosity p arameter. Note also that\nνz=ρqhχ\nβ(1+q2),|ν|=qhχ\nβ/radicalbig\n1+q2(3.31)\nasχ→0 and\nνz=ρqh\nβχ,|ν|=qh\nβ(3.32)\nasχ→∞ .\nAs it was mentioned in Sec. II, a correct description of the magnetization dynamics in rot ating\nnanoparticles is achieved by introducing in Eq. (2.3) the Barnett field. This emergent magnetic\nfield is responsible for the Barnett e ffect (magnetization by rotation)34and its existence has been\nrecently confirmed experimentally for di fferent spin systems35–37. In this context, it is of interest to\nanalyze the role of the Barnett field in the dissipation-indu ced rotation of suspended ferromagnetic\nnanoparticles. In its absence, when the term −ω/γin Eq. (2.3) is not taken into account, the general\nexpressions for the components and magnitude of the particl e angular velocity, (3.25) and(3.26) ,\nshould be modified by the replacement α+β→β(i.e., q→α,c→α/√\n1+α2, and so on).\nSince this replacement corresponds to the limiting case β→∞ [see (3.29) and(3.30) ], one can\ncheck that the Barnett field does not practically influence th e particle rotation at β≫α. At the\nsame time, for β/lessorsimilarαthe difference between the exact results and those obtained by the ab ove\nreplacement is significant [cf. (3.27) with (3.29) and(3.30) ]. Therefore, we may conclude that the\nrotational properties of isotropic ferromagnetic nanopar ticles suspended in a viscous liquid and\nsubjected to a rotating magnetic field are determined not onl y by the Barnett field, but also by the\nfrictional torque [see Eq. (2.2) ].\n11D. Numerical results\nFor numerical analysis, we use two main assumptions of the mo del to choose its parameters.\nThe first one was that the time derivative of the particle angu lar momentum is assumed to be much\nless than the frictional torque. This assumption, which hol ds whenκυ≪β, i.e.,Ω≪60η/ρ md2,\nallowed us to neglect the left-hand side of Eq. (2.7a) . Since in the single-domain state d<dcr(dcr\nis the critical diameter below which this state is realized) , the last condition is not too restrictive.\nThe second assumption was that the nanoparticle material is assumed to be magnetically\nisotropic. At first sight, according to the definition (2.8) of the effective magnetic field, this as-\nsumption seems to be valid for h≫ha. However, analysis of Eqs. (2.7) shows that it certainly\nholds if h≫ha/α(since, as a rule, α < 1, this condition is more strict than h≫ha), i.e., if\nH≫Ha/α. Because the rotating magnetic field of large amplitude His difficult to generate, mag-\nnetically soft nanoparticles the anisotropy field Haof which is relatively small are most suitable\nfor experimental verification of our predictions.\nAs an illustrative example, we consider permalloy nanopart icles (Ni 80Fe20) characterized by\nthe parameters38M=8×102emu cm−3,Ha=4 Oe,ρm=8.7 g cm−3, andα=0.04. Note that\nfor these particles dcr=36.8 nm and d1=10.7 nm at h=0.1, i.e., the magnetization fluctuations\nare negligible if d1<d<dcr. For particles suspended in water at room temperature T=298 K\nwe haveη=8.9×10−3P,β=1.18×103(we takeγ=1.76×107G−1s−1), and so q=αwith\nexcellent accuracy. Using these parameters, we numericall y solved a set of equations (3.7) for\ndifferent values of the parameters h,υandρcontrolling the rotating field characteristics. It was\nestablished that, according to our predictions, the angles θandψevolve in an oscillatory manner\nto the steady-state values (3.15) that correspond to the stable magnetization state m(1). The time\ndependence of these angles is illustrated in Fig. 1for the particular case of the rotating field.\nDependencies of the z-component and magnitude of the precessional angular veloc ity on the\nmagnetic field amplitude calculated for permalloy nanopart icles from (3.25) and(3.26) , respec-\ntively, are shown in Fig. 2. According to (3.32) , at small h(whenχ=υ/h≫1)νzis a quadratic\nfunction of h,νz=αh2/βυ(we use the relation q=αand assume that ρ=+1), and|ν|grows\nlinearly with h,|ν|=αh/β(see inset in Fig. 2). If his large enough (i.e., χ≪1), then, using\n(3.31) and the condition α2≪1, one can make sure that the functions νzand|ν|approach al-\nmost the same value: νz=|ν|=αυ/β . Specifically, νz=|ν|=1.69×10−6forυ=0.05 and\nνz=|ν|=2.37×10−6forυ=0.07, or, in dimensional form, ωz=|ω|=2.39×104rad s−1and\n12/endash.cap10123\n0 1 2\n10 −2 τψ(τ) θ(τ)θ(τ), ψ(τ)  (rad) \nFIG. 1. Plots of the functions θ=θ(τ) andψ=ψ(τ) obtained via numerical solution of Eqs. (3.7) . The\nparameters and initial conditions are chosen to be q=0.04,h=0.5,υ=0.2,ρ=+1,θ(0)=2 rad\nandψ(0)=1 rad. In the long-time limit, the functions θ(τ) andψ(τ) tend to constant values 1 .95 and\n1.48×10−2rad, respectively. From (3.15) it follows that these limiting values are in complete agreem ent\nwith the analytical ones θ(1)\n+1andψ(1)\n+1.\nωz=|ω|=3.34×104rad s−1, respectively. Note that, since ( |ν|−νz)/(αυ/β )≈α2/2≪1, the\nnanoparticle rotation about the axes xandyis negligibly slow.\n00\n0.3 0.6 12\n0\n0 0.02 h1 12\n12\nh10 6  vz , | v  |\n10 6  vz , | v  |\nFIG. 2. The z-components (dashed curves) and magnitudes |ν|(solid curves) of the precessional angular\nvelocityνof permalloy nanoparticles as functions of the magnetic fiel d amplitude hfor different values of\nthe magnetic field frequency υ. Curves 1 correspond to υ=0.05 and curves 2 correspond to υ=0.07.\nThe behavior of νzand|ν|as functions of the magnetic field frequency υis illustrated in Fig.\n3. Ifυis rather small (i.e., χ≪1), then, according to (3.31) ,νzand|ν|grow approximately\nlinearly with υ:νz=|ν|=αυ/β . In contrast, if υis rather large (i.e., χ≫1), then, according to\n13(3.32) ,νzdecreases with υasνz=αh2/βυ, and|ν|increases up to|ν|=αh/β. Thus, while|ν|is\na monotonically increasing function of υ,νzas a function of υhas a global maximum (recall that\nρ=+1) atυ=υm, whereυmcan be estimated as υm∼h. In other words, the rotation of isotropic\nnanoparticles about the zaxis, which is induced by the magnetic field of a fixed amplitud eh\nrotating in the xyplane, occurs with the maximal angular velocity, if the magn etic field frequency\nυis of the order of h(in dimensional form, if Ω∼γH).\n012\n03\n0.1 0.2 \nυυm10 6  vz , | v  | 10 6  vz , | v  |\n12\nFIG. 3. The z-components (dashed curves) and magnitudes |ν|(solid curves) of the precessional angular\nvelocityνof permalloy nanoparticles as functions of the magnetic fiel d frequency υfor different values of\nthe magnetic field amplitude h. Curves 1 correspond to h=0.05 and curves 2 correspond to h=0.1.\nIt is important to note that, because the magnetization and p article precessions are completely\ncorrelated,ν·m(1)=0, experimental confirmation of the existence of dissipatio n-induced rotation\nof isotropic ferromagnetic nanoparticles could be obtaine d by analyzing some unique magnetic\nproperties of such systems. In particular, according to (3.16) , the zcomponent of the steady-state\nmagnetization m(1)is given by m(1)\nz=−ρ/radicalbig\n1−R2/c2k2. Using (3.13) , it can be shown that m(1)\nzat\nυ≪his a linear function of υ,m(1)\nz=−ρυ/h(1+q2), and m(1)\nzapproaches−ρatυ≫h(see Fig. 4).\nThe experimental observation of these features would confir m the proposed theory of nanoparticle\nrotation.\nIV . CONCLUSIONS\nWe have predicted and analyzed the precessional rotation of magnetically isotropic ferromag-\nnetic nanoparticles in a viscous liquid generated by a rotat ing magnetic field. A remarkable feature\nof this phenomenon is that it occurs when coupling between ma gnetic and lattice subsystems aris-\n140 0.1 0.2 \nυ00.5 1(1) \n  /endash.cap mz  1\n21 /emdash.alt h = 0.05 \n2 /emdash.alt h = 0.1 \nρ = +1 \nFIG. 4. Dependence of the magnetization component −m(1)\nzon the magnetic field frequency υfor different\nvalues of the magnetic field amplitude h.\ning from magnetocrystalline anisotropy is absent. We have s hown explicitly that the reason for\nthis rotation is the dissipation-induced coupling between these subsystems.\nOur approach to the problem is based on a set of the Landau-Lif shitz equation describing the\nmagnetization dynamics of a magnetically isotropic nanopa rticle and the mechanical equation de-\nscribing the particle rotation in a liquid. Assuming that in ertial effects are negligible, we solved\nthese equations analytically and showed that in the steady s tate both the magnetization and the\nnanoparticle are precessed. These precessions are fully sy nchronized and occur about the axis per-\npendicular to the plane of the magnetic field rotation. We hav e determined their characteristics and\nestablished, in particular, that the precessions occur wit h the magnetic field frequency. It should\nbe emphasized that, in contrast to an ordinary spinning top, the frequency of particle rotation is\nmuch less than the frequency of its precession.\nWe have also discussed the possibility of experimental dete ction of the dissipation-induced\nrotation of isotropic ferromagnetic nanoparticles by a rot ating magnetic field. Since direct ex-\nperimental observation of nanoparticle rotation seems to b e problematic, we expect that this phe-\nnomenon can be verified by comparing the predicted and experi mental magnetic properties of\nthese systems. 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Staunton\nDepartment of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom\n(Dated: October 17, 2019)\nComputational design of more e\u000ecient rare earth/transition metal (RE-TM) permanent magnets\nrequires accurately calculating the magnetocrystalline anisotropy (MCA) at \fnite temperature, since\nthis property places an upper bound on the coercivity. Here, we present a \frst-principles methodol-\nogy to calculate the MCA of RE-TM magnets which fully accounts for the e\u000bects of temperature on\nthe underlying electrons. The itinerant electron TM magnetism is described within the disordered\nlocal moment picture, and the localized RE-4 fmagnetism is described within crystal \feld theory.\nWe use our model, which is free of adjustable parameters, to calculate the MCA of the RECo 5(RE =\nY, La{Gd) magnet family for temperatures 0{600 K. We correctly \fnd a huge uniaxial anisotropy for\nSmCo 5(21.3 MJm-3at 300 K) and two \fnite temperature spin reorientation transitions for NdCo 5.\nThe calculations also demonstrate dramatic valency e\u000bects in CeCo 5and PrCo 5. Our calculations\nprovide quantitative, \frst-principles insight into several decades of RE-TM experimental studies.\nThe excellent properties of rare earth/transition metal\n(RE-TM) permanent magnets have facilitated a number\nof technological revolutions in the last 50 years. Now,\nthe urgent need for a low carbon, low emission economy\nis driving a global research e\u000bort dedicated to improving\nRE-TM performance for more e\u000ecient deployment in the\ndrive motors of hybrid and electric vehicles [1]. RE-TM\nmagnets combine the large volume magnetization and\nhigh Curie temperature of the elemental TM magnets\nFe or Co with the potentially huge magnetocrystalline\nanisotropy (MCA) of the REs [2]. The REs consist of\nSc, Y and the lanthanides La{Lu, but it is Y and the\nlight lanthanides, i.e. those with smaller atomic masses\nthan Gd, which are most attractive for applications due\nto their lower criticality [3]. Nd and Sm stand out thanks\nto the highly successful Nd-Fe-B and Sm-Co magnets [4{\n7], but Ce has the attraction of having a low cost and\nhigh abundance compared to the other REs [8].\nTraditionally, RE-TM magnet research has been driven\nby experiments. First-principles computational mod-\nelling can uncover fundamental physical principles and\nprovide new directions for RE-TM magnet design [9, 10],\nbut faces two challenges. First, RE-TM magnetism orig-\ninates from both itinerant electrons and more localized\nlanthanide 4 felectrons [11]. Although the local spin-\ndensity approximation to density-functional theory [12]\n(DFT) satisfactorily describes the itinerant electrons, the\n4felectrons require specialist techniques like dynamical\nmean \feld theory [13, 14], the local self-interaction cor-\nrection (LSIC) [15], the open-core approximation [16{18],\nor Hubbard + Umodels [19, 20]. Second, DFT calcula-\ntions are most easily performed at zero temperature, but\nunder actual operating conditions the RE-TM magnetic\nmoments are subject to a considerable level of thermal\ndisorder [21]. The disordered local moment (DLM) pic-\nture accounts for this disorder within DFT [21] and, com-\nbined with the LSIC has been used successfully to calcu-\nlate magnetizations, Curie temperatures and phase dia-\nattractiverepulsive\nNd\nSmFIG. 1. Preferred orientation of RE-4 fcharge clouds (with\nmagnetic moments indicated by arrows) for Nd and Sm in the\nRECo 5crystal \feld, shown on the contour plot. The Co and\nRE atoms are shaded in grey and purple, respectively. Repul-\nsive (attractive) corresponds to high (low) potential energy.\ngrams of itinerant electron and RE-based magnets [22{\n24]. DFT-DLM studies of the MCA have also been per-\nformed on itinerant electron and Gd-based magnets [25{\n27], but it is important to realise that these materials are\nspecial cases, where there is no contribution to the MCA\nfrom 4felectrons interacting with their local environ-\nment (the crystal \feld). A \frst-principles, \fnite temper-\nature theory which accounts for these crystal \feld e\u000bects\n(and is therefore applicable to general RE-TM magnets\nlike Nd-Fe-B or Sm-Co) has, up to now, proven elusive.\nIn this Rapid Communication, we rectify this situation\nand demonstrate a \frst-principles theory of the MCA of\nRE-TM magnets including the crystal \feld interaction.\nFundamentally, the theory takes a model originally devel-\noped by experimentalists, and obtains all of the quanti-\nties required by the model from DFT-based calculations.\nWe demonstrate the theory on the RECo 5family of mag-\nnets, (RE = Y, La, Ce, Pr, Nd, Sm and Gd). The RECo 5\nphase, shown in Fig. 1, is important due to its presence\nin SmCo 5and in the cell-boundary phase of commercial\nSm2Co17[7, 28, 29]. We calculate anisotropy constants\nand spin reorientation transition temperatures to analyse\nexperimental data obtained 40 years ago [30, 31].arXiv:1910.07436v1  [cond-mat.mtrl-sci]  16 Oct 20192\nThe model partitions the RE-TM magnet into an itin-\nerant electron subsystem (originating from the TM, and\nRE valence electrons), and a subsystem of strongly-\nlocalized RE-4 felectrons [32]. Critically, the RE ions\ntend to adopt a 3+ state with a common s2dvalence\nstructure [33]. As a consequence, for each RE-TM mag-\nnet class the itinerant electron subsystem is essentially\nindependent of the speci\fc RE [32], so its properties can\nbe obtained for the most computationally convenient pro-\ntotype (e.g. RE = Y or Gd) The itinerant electrons drive\nthe overall magnetic order, primarily determining the\nCurie temperature TC; for example, TCdi\u000bers by only\n20 K in Y 2Fe14B and Nd 2Fe14B [34]. The itinerant elec-\ntrons also drive the RE-4 fmagnetic ordering through an\nantiferromagnetic exchange interaction [35], with an ex-\nchange \feld of a few hundred Tesla at cryogenic temper-\natures [36]. RE-RE interactions are relatively weak [37].\nThe RE-4fsubsystem contributes to the magnetic mo-\nment and can have a small e\u000bect on TC[24], but its most\nimportant contribution is to the MCA, which in turn\nprovides an intrinsic mechanism for coercivity [38]. The\norigin of the potentially huge MCA of RE-TM magnets\nis illustrated in Fig. 1. The itinerant electrons and sur-\nrounding ions set up a (primarily) electrostatic poten-\ntial with the symmetry of the RE site [32], known as\nthe crystal \feld (CF). The CF calculated for RECo 5is\nshown as a contour plot in Fig. 1, with electrons attracted\nto the blue regions and repelled by the red. Meanwhile\nthe un\flled RE-4 fshells form non-spherically-symmetric\ncharge clouds coupled to the magnetic moment direction\nthrough a strong spin-orbit interaction [32]. The charge\nclouds are elongated either parallel or perpendicular to\nthe moment direction depending on Hund's rules, with\nthe opposing examples of Sm and Nd shown in Fig. 1.\nPlaced in the RECo 5CF, the charge clouds will prefer-\nentially orientate with their elongated part lying in the\nattractive region, generating the MCA [39]. A secondary\ncontribution to the MCA is provided by the itinerant\nelectrons, with YCo 5(which has no RE-4 felectrons)\nhaving a MCA energy of 5 MJm\u00003at room tempera-\nture [30], ten times larger than hexagonal Co [40].\nThe above ideas are formulated mathematically by in-\ntroducing a Hamiltonian for the RE-4 felectrons ^H[41]:\n^H=\u0015^L\u0001^S+\u00160\u0016B(^L+2^S)\u0001H+2\u0016B^S\u0001Bexch+X\niV(ri):\n(1)\nHere, ^Land^Sare the orbital and spin angular momen-\ntum operators, where for each RE3+ionLandSare\n\fxed by Hund's rules. \u0015quanti\fes the spin-orbit inter-\naction, and \u0016Bis the Bohr magneton. His an external\nmagnetic \feld, and Bexchis the exchange \feld originat-\ning from the itinerant electrons. V(ri) is the crystal \feld\npotential, where ilabels each 4 felectron. The Hamilto-\nnian in Eq. 1 acts upon the RE-4 fwavefunction which\ncan be expressed schematically as a radial part multi-plied by the angular part jJ;L;S;M Ji, where the quan-\ntities within the ket are standard quantum numbers [32].\nEquation 1 is diagonalized within the manifold of states\njJ;L;S;M J=J;J\u00001;J\u00002;:::;\u0000jJji. We consider the\ngroundJ=jL\u0000Sjmultiplet, along with the \frst excited\nmultipletJ=jL\u0000Sj+1 for Pr and Nd, and also the next\nexcited multiplet J=jL\u0000Sj+ 2 for Sm. Angular parts\nof the matrix elements of the CF term in Eq. 1 are cal-\nculated by decomposing the states into jL;S;M L;MSi\nform, and then using the operator form of the poten-\ntial [42], which introduces Clebsch-Gordan coe\u000ecients\nandl-dependent (Stevens) prefactors [32, 43]. The radial\nparts are incorporated into the CF coe\u000ecients Blm[33],\ndescribed in more detail below. For a given temperature\nT, we use the eigenvalue spectrum of ^Hto construct the\npartition function ZREand the RE-4 ffree energy con-\ntributionFRE(T;Bexch;H) =\u0000kBTlnZRE.\nThe quantities forming the itinerant contribution to\nthe free energy Fitinare determined from DFT-DLM cal-\nculations.Fitindepends on the Co magnetization MCo:\nFitin(T;^MCo;H) =K1sin2\u0012+K2sin4\u0012\n\u0000\u00160M0\nCo(1\u0000psin2\u0012)^MCo\u0001H(2)\nwhereK1andK2quantify the itinerant electron contri-\nbution to the anisotropy, and cos \u0012=^MCo\u0001^ z, with ^ z\npointing along the caxis.pquanti\fes the magnetization\nanisotropy, which in the RECo 5compounds can cause\nthe Co magnetization to reduce by a few percent from\nits maximum value M0\nCo[44].Fitindepends on tempera-\nture through the quantities K1,K2,M0\nCoandp.\nThe two subsystems are coupled together by noting\nthat ^MCo=\u0000^Bexch, i.e. the exchange \feld felt by the\nRE-4felectrons points antiparallel to the Co magneti-\nzation [16]. The equilibrium direction of MCo(equiva-\nlently, of Bexch) minimizes the sum of FitinandFRE. The\nRE magnetization is obtained as MRE=\u0000\u0016Bh^L+2^SiT,\nwithhiTdenoting the thermal average over the eigenvalue\nspectrum of Eq. 1 at the equilibrium value of Bexch, and\nthe total magnetization is MTot=MRE+MCo. The\nmagnetization measured along the \feld direction Mis\nMTot\u0001^H, whilst the easy direction of magnetization \u000bis\nobtained as cos\u00001(^MTot\u0001^ z) in zero external \feld.\nBexch,K1,K2,M0\nCoandpare obtained using the FP-\nMVB procedure developed to calculate First-Principles\nMagnetization VsB-\feld curves for GdCo 5[27]. FP-\nMVB \fts DFT-DLM torque calculations [45] for di\u000ber-\nent magnetic con\fgurations to extract the desired quan-\ntities [27, 46]. For all RE3+Co5compounds considered\nhere, we exploit the isovalence of the RE3+ions and\nsubstitute the RE with Gd in the FPMVB calculations.\nThis step ensures no erroneous double counting of the\nCF contribution which is already accounted for by FRE,\nbut still captures the coupling between TM-3 dand RE-\n5dvalence states. We do however use the (experimen-\ntal) lattice parameters appropriate for each RE [47, 48].3\nK1,K2,M0\nCoandpwere \ftted to calculations where \u0012\nwas varied between 0{90\u000ein 10\u000eintervals. Bexchwas\nobtained by \ftting the torque induced by introducing a\n1\u000ecanting between the Gd and Co sublattices to a free\nenergy of the form \u0000Bexch\u0001MGd;swhere MGd;sis the\nspin moment of Gd including thermal disorder, i.e. the\nlocal spin moment weighted by the Gd order parame-\nter [46]. For itinerant CeCo 5and PrCo 5(see below),\nthe quantities in Eq. 2 were obtained directly from DFT-\nDLM calculations on these compounds. The calculations\nused the atomic sphere approximation, angular momen-\ntum expansions with maximum l= 3, and an adaptive\nreciprocal space sampling to ensure high precision [49].\nExchange and correlation were modelled within the local\nspin-density approximation (LSDA) [50], with an orbital\npolarization correction applied to the Co- delectrons [51]\nand the LSIC applied to Gd. The calculated quantities\nare given as Supplemental Material [52].\nWe calculate the RECo 5CF coe\u000ecients using an\nyttrium-analogue model [33]. The basic premise here is\nthat due to the isovalence of RE3+ions, the RE3+Co5\nCF potential (which originates from the valence elec-\ntronic structure) can be substituted with that of Y3+Co5.\nThis step ensures no double-counting of RE-4 felec-\ntrons, and allows the use of projector-augmented wave-\nbased DFT calculations to calculate the CF potential\nto high accuracy without needing special methods to\ntreat the 4 felectrons [33]. The CF potential is com-\nbined with the radial RE-4 fwavefunctions obtained in\nLSIC calculations. At the RE site (symmetry D6h)\nthere are four independent components of the CF poten-\ntial which a\u000bect the 4 fanisotropy, with ( l;m) = (2,0),\n(4,0), (6,0) and (6,6)[=(6-6)]. The calculated CF co-\ne\u000ecients are given as Supplemental Material [52]. We\nnote that this method implicitly neglects any tempera-\nture dependence of the CF coe\u000ecients themselves, and\nfuture work must evaluate the e\u000bects of \fnite temper-\nature, e.g. due to charge \ructuations or lattice expan-\nsion [47]. The calculations were performed using the\nGPAW code [53] within the LSDA. A plane wave basis\nset with a kinetic energy cuto\u000b of 1200 eV was used,\nand reciprocal space sampling performed on a 20 \u000220\u000220\ngrid. The spin-orbit parameter \u0015was calculated using\nthe RE-centred spherical potential V0(r) from the LSIC\ncalculation as \u0015=R\ndrr2n0\n4f(r)\u0010(r)=(2S) [54], where\nthe normalized spherically-symmetric 4 fdensityn0\n4f(r)\nwas also obtained from the LSIC calculation [33] and\n\u0010(r) =\u0016h2\n2m2c21\nr@V0\n@r. The calculated values of \u0015are 1205,\n703, 540 and 417 K for Ce, Pr, Nd and Sm respectively.\nThese\u0015values yield anisotropy constants indistinguish-\nable from those calculated using experimental \u0015values\nextracted from spectroscopic measurements [41, 54].\nWe now demonstrate the theory by calculating\nexperimentally-measurable quantities. Figure 2 (left\npanel) reproduces anisotropy constants measured by Er-\n100 200 300 400 500 600\nT(K)\nY\nLa\nNd\nSm\nGd\nK20 100 200 300 400 500 600-30-20-10010203040\nT(K)K(MJ m )-3\nK2\nExperimentCalculatedFIG. 2. Experimental anisotropy constants \u00141(and\u00142for\nNdCo 5) [30] of RECo 5, compared to the current calculations.\nmolenko in 1976 [30] for YCo 5, LaCo 5, NdCo 5, SmCo 5\nand GdCo 5. They were extracted using the Sucksmith-\nThompson (ST) method [40], which is based on the ex-\npression for the dependence of the free energy of a uni-\naxial ferromagnet on the magnetization direction \u0002:\nFFM(\u0002) =\u00141sin2\u0002 +\u00142sin4\u0002 +O(sin6\u0002) (3)\nAs explained in detail in the Supplemental Material [52],\nmeasuring the magnetization along the hard direction\nand plotting the data as an Arrott plot ( H=M vs.\nM2) [55] allows \u00141and\u00142to be extracted from the gradi-\nent and intercept. Equation 3 and the ST method strictly\napply to ferromagnets, but the same technical procedure\ncan be applied to RE-TM ferrimagnets too [27]. How-\never, the fact that the external \feld can induce a canting\nbetween the RE and TM moments means that the ex-\ntracted anisotropy constants for the ferrimagnet are ef-\nfective ones, which measure both the anisotropy of the\nindividual sublattices and the strength of the exchange\ninteraction keeping the spin moments antialigned [27, 46].\nThe experimental data in Fig. 2 demonstrates the di-\nversity in\u0014among RECo 5. The behavior of YCo 5and\nLaCo 5, where the RE is nonmagnetic, is rather similar.\nBoth compounds display uniaxial anisotropy associated\nwith the itinerant electron subsystem. GdCo 5is still uni-\naxial, but is softer than YCo 5and LaCo 5. Since the\n\flled Gd-4fsubshell makes no CF contribution to the\nanisotropy, this reduction in \u00141is due to the \feld-induced\ncanting of the Gd and Co magnetic moments [27]. SmCo 5\nstands out for having the largest uniaxial anisotropy over\nthe entire temperature range, with a room temperature\nvalue of 17.9 MJm\u00003[30]. NdCo 5has a negative \u00141at\nlow temperatures which switches to positive at approxi-\nmately 280 K, and also has a non-negligible \u00142, at vari-4\nExperiment\nCalculated\n200 250 3000306090\nT(K)α(degrees)\nFIG. 3. Evolution of the easy magnetization direction \u000bin\nNdCo 5, as reported experimentally [31] and calculated here.\nance with the other compounds. As discussed below, this\nvariation results in NdCo 5undergoing spin reorientation\ntransitions from planar !cone!uniaxial anisotropy [31].\nThe right panel of Fig. 2 is the main result of this work,\nshowing the anisotropy constants obtained entirely from\n\frst principles. We calculated hard axis magnetization\ncurves, and then performed the ST analysis on the data\nto extract\u00141and\u00142, in exact correspondence with the\nexperimental procedure [52]. The calculations reproduce\nall of the experimentally-observed behavior. SmCo 5and\nNdCo 5have strong uniaxial and planar anisotropy at zero\ntemperature, respectively. NdCo 5has a non-negligible \u00142\nand a value of \u00141which changes sign between 280{290 K.\nYCo 5and LaCo 5have uniaxial anisotropy, with LaCo 5\nslightly stronger. GdCo 5also has uniaxial anisotropy but\nis softer, and has a complicated temperature dependence.\nComparing in more detail, we \fnd agreement be-\ntween experimental and calculated \u0014values to within a\nfew MJm\u00003for all but the lowest temperatures, where\nthe classical statistical mechanics of DFT-DLM calcu-\nlations leads to inaccuracies [37], and high tempera-\ntures, where experimentally the compounds might un-\ndergo decomposition [27]. We calculate \u00141at room\ntemperature for SmCo 5to be 21.3 MJm\u00003(experiment\n17.9 MJm\u00003). Even at zero temperature the value of\n36.3 MJm\u00003shows improved agreement with experiment\n(29.5 MJm\u00003) compared to open core (19.7 MJm\u00003) [17]\nor Hubbard+ Ucalculations (40.5 MJm\u00003) [20].\nThe calculations also reproduce more subtle features,\nfor instance the slightly enhanced anisotropy (by less\nthan 1 MJm\u00003) of LaCo 5over YCo 5. The least good\nagreement is for GdCo 5, especially at higher tempera-\ntures; however, more recent measurements of \u00141found\ndi\u000berent behavior at elevated temperatures [27, 30]. At\nlower temperatures, we note that the present calculations\ndo not include the magnetostatic dipole-dipole contribu-\ntion to the MCA, or the Gd-5 dcontribution to the itiner-\nant electron anisotropy, which we previously calculated to\nbe 24% the size of the Co contribution [27]. We conclude\nthat omitting the magnetostatic and RE- dcontribution\nto the anisotropy is reasonable for nonmagnetic REs or\nthose with un\flled 4 fshells (whose RE-4 fanisotropy is\nmuch larger), but is less suitable for Gd-based magnets.Unlike the other materials in Fig. 2, the easy direction\nof magnetization of NdCo 5lies in the abplane at low\ntemperature, with polar angle \u000b= 90\u000e. The anisotropy\nwithin theabplane is determined by the B6\u00066CF coef-\n\fcients. Both our calculations and experiments \fnd the\neasy direction to be the aaxis, which points from the RE\nto between its nearest neighbour Co atoms [56]. Exper-\nimentally, as Tis increased from zero to past approxi-\nmately 240 K, a spin reorientation transition (SRT) oc-\ncurs and the magnetization begins to rotate towards the\nc-axis, i.e. planar!cone anisotropy. This rotation con-\ntinues (decreasing \u000b) until approximately 280 K, when\na second SRT (cone !uniaxial) occurs. Further increas-\ningTsees\u000bremaining at 0\u000eup toTC. The presence of\nthe SRTs close to room temperature led to the proposal\nthat NdCo 5may be a candidate material for magnetic\nrefrigeration [57]. The evolution of \u000bas measured exper-\nimentally in Ref. 31 is shown in Fig. 3.\nThe calculated variation of \u000bwith temperature is also\nshown in Fig. 3. We see that the agreement with experi-\nment is remarkably good, with calculated SRT tempera-\ntures ofTSRT1 = 214 K (plane!cone) and and TSRT2 =\n285 K (cone!uniaxial). There is also good agreement be-\ntween calculated and experimental \u0014values, especially in\nthe SRT region. Indeed, the SRTs are intimately linked\nto the temperature dependence of \u00141and\u00142, with the\nplane-cone SRT occurring when \u00141=\u00002\u00142and the cone-\naxis SRT occurring when \u00141crosses zero [58, 59].\nThe calculations provide the underlying physical ex-\nplanation of the SRTs, which result from a competition\nbetween the uniaxial anisotropy of the itinerant electrons\nand a preference for the oblate Nd3+charge cloud to have\nits axis lying in the abplane (Fig. 1). As the tempera-\nture increases, the Nd moments disorder more quickly\nthan the Co, due to the relative weakness of the RE-\nTM exchange \feld Bexchcompared to the exchange in-\nteraction between itinerant moments [24]. As a result,\nthe negative contribution to \u00141from Nd reduces quickly\nwith temperature, leaving the positive uniaxial contri-\nbution from the itinerant electrons. Obtaining realistic\nSRT temperatures therefore requires accounting for the\nitinerant electron anisotropy, the crystal \feld potential\nand the exchange \feld at a comparable level of accuracy.\nWe \fnally consider CeCo 5and PrCo 5. Ce has a well-\nknown tendency to undergo transitions between trivalent\nand tetravalent valence states, as seen for instance in the\n\u000b-\rtransition [60, 61]. The LSIC describes strongly-\ncorrelated RE-4 felectrons, with them forming a narrow\nband several eV below the Fermi level [24]. Without the\nLSIC, the 4 felectrons are less correlated and more itin-\nerant, appearing as wider bands close to the Fermi level.\nThe LSIC \fnds a lower-energy ground state if the en-\nhanced correlation o\u000bsets the energy penalty associated\nwith the stronger localization [15, 24]. Of the RECo 5\ncompounds, the LSIC predicts a higher energy ground\nstate only for CeCo 5, implying that the Ce-4 felectron5\n0 100 200 300 400 500 600-40-30-20-1001020\nT(K)K(MJ m )-3\nCe\nPrExperimentCalculated\n100 200 300 400 500 600\nT(K)\nCe\nPritinerant\n3+\nFIG. 4. Comparison of experimental and calculated\nanisotropy constants for CeCo 5and PrCo 5. Values are shown\nfor both itinerant and localized (RE in 3+ state) 4 felec-\ntrons, with the ground state predicted by the LSIC calcula-\ntions shown in the darker color.\nis not strongly localized.\nPractically, we model compounds with more itinerant\n(weakly correlated) RE-4 felectrons by performing non-\nLSIC DFT-DLM calculations on RECo 5, with only Fitin\ncontributing to the free energy. The values of \u00141cal-\nculated in this way for CeCo 5and PrCo 5are labelled\n\\itinerant\" in Fig. 4. We also show \u00141values labelled\n\\3+\", calculated for strongly localized RE-4 felectrons\nusing the same method as in Fig. 2. The Ce and Pr\nmoments are held collinear to the Co moments in the\nitinerant calculations [52].\nFigure 4 shows a dramatic di\u000berence in the anisotropy\nconstants calculated for the di\u000berent RE-4 fvalences.\nPr3+Co5has anabplane anisotropy at low temperature,\nwhich is stronger than NdCo 5. This behavior is in fact ex-\npected; the leading crystal \feld contribution to the MCA\nscales asJ(J\u00001\n2)\u000bJ(\u000bJbeing the Stevens factor), and\nthis quantity is larger for Pr than Nd [32]. The calculated\nplane!cone and cone!uniaxial SRTs occur at 235 K and\n297 K respectively, which are higher temperatures than\nthose calculated for NdCo 5. Ce3+Co5meanwhile is calcu-\nlated to have cone anisotropy at zero temperature, with\n\u000b= 80\u000e. A cone!axis SRT occurs at 100 K, after which\nthe compound has uniaxial anisotropy. The presence\nof only one SRT shows that Ce3+has a weaker planar\nanisotropy than Pr3+or Nd3+. This weaker anisotropy is\ndue to CeCo 5having a reduced B20CF coe\u000ecient, which\ncorrelates with a contracted alattice parameter [52].\nIf instead the RE-4 felectrons are treated as itinerant,\nboth CeCo 5and PrCo 5are found to have strong uniaxial\nanisotropy. At low temperatures CeCo 5has the higher\nvalue of\u00141, (23.5 MJm\u00003at 0 K), while above 200 K, \u00141\nof PrCo 5is larger. The \u00141values exceed those calculated\nfor YCo 5and LaCo 5, showing that delocalizing the RE-\n4felectrons boosts the uniaxial anisotropy.\nThe experimental anisotropy constants from Ref. 30\nare also shown in Fig. 4. The experiments support\nthe picture obtained from the total energy calculations,that the Ce-4 fand Pr-4felectrons are more itiner-\nant/localized (weakly/strongly correlated) respectively.\nCeCo 5has uniaxial anisotropy across the entire tem-\nperature range. For PrCo 5, although \u00141is negative\nat low temperature, its magnitude is smaller than that\nmeasured for NdCo 5(-6.5 MJm\u00003vs. -33.5 MJm\u00003at\n4 K). As a result, at low temperature the easy magne-\ntization direction of PrCo 5does not lie in the abplane,\nbut rather in a cone with \u000b= 23\u000e[62]. Increasing the\ntemperature decreases \u000b, and a cone-axis SRT occurs at\n105 K [62]. Therefore, although the Pr ions do favour ab-\nplane anisotropy, their contribution is weaker than from\nthe Nd ions in NdCo 5, at variance with the CF picture.\nOverall, in CeCo 5(PrCo 5) experiments \fnd a smaller\nuniaxial (planar) contribution from Ce (Pr3+). As a re-\nsult, the calculated uniaxial anisotropy of CeCo 5is larger\nthan found experimentally, while the plane !cone SRT of\nPrCo 5at 235 K is missing from experiments.\nThe anomalous behavior of PrCo 5in the context of\nCF theory was identi\fed in Ref. [63], where it was pro-\nposed that in PrCo 5, Pr assumes a mixed valence state,\ne.g. Pr3:5+, whose properties lie between Pr and Ce. The\ncalculations shown in Fig. 4 support this view, if we make\nthe reasonable assumption that the anisotropic proper-\nties of the mixed valence state are bounded by those of\nthe strongly localized and more itinerant (strongly and\nweakly correlated) Pr-4 felectrons. In a similar way,\nthe experimentally-observed reduction in CeCo 5uniax-\nial anisotropy compared to the calculations could be ex-\nplained if the Ce-4 felectron was more localized (cor-\nrelated) than predicted by the \\itinerant\" calculations.\nFrom Fig. 4, such an electron would be expected to have\na reduced contribution to the uniaxial anisotropy. Within\nthis picture, encouraging the itineracy of the Ce-4 felec-\ntron through e.g. chemical pressure, would boost the uni-\naxial anisotropy of CeCo 5.\nApart from highlighting the 4 f-electron physics of Ce\nand Pr, our calculations serve as a reminder of the re-\nmarkable properties of SmCo 5. As well as its huge zero\ntemperature uniaxial anisotropy, the large spin moment\nof Sm strengthens its coupling to the exchange \feld, so\nthat the Sm moments stay ordered up to higher temper-\natures. Furthermore, mixing of the higher- Jmultiplets\nalso boosts the anisotropy [64]. As a result, as shown in\nFig. 2, the\u00141value of SmCo 5remains larger than that of\nYCo 5and LaCo 5(where the RE is nonmagnetic), even\nat 600 K. Previously we have shown that the electronic\nstructure of SmCo 5close to the Fermi level also gives it\nthe highest TCof the RECo 5compounds [24].\nIn summary, we have demonstrated a framework to\ncalculate the \fnite-temperature MCA of RE-TM mag-\nnets. Combined with the previously established DFT-\nDLM method which provides \fnite-temperature magne-\ntization and TC[24], we have a full framework to cal-\nculate the intrinsic properties of RE-TM magnets which\nrequires no experimental input beyond the crystal struc-6\nture. The validation of our method on the RECo 5mag-\nnet class opens the door to tackling other RE-TM mag-\nnets, like Nd-Fe-B, REFe 12and Sm 2Co17. The good per-\nformance of the calculations for SmCo 5will allow us to\npropose strategies to improve this magnet, e.g. through\nmodi\fcation of the CF potential and/or exchange \feld\nthrough TM doping or application of pressure. More gen-\nerally, our work realizes the proposal made two decades\nago in Ref. 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G.\nPastushenkov, and T. I. Ivanova, Giant rotating magne-\ntocaloric e\u000bect in the region of spin-reorientation transi-\ntion in the NdCo 5single crystal, Phys. Rev. Lett. 105,\n137205 (2010).\n[58] M. Ohkoshi, H. Kobayshi, T. Katayama, M. Hirano,\nT. Katayama, M. Hirano, and T. Tsushima, Spin reori-\nentation in NdCo 5single crystals, AIP Conf. Proc, 29,\n616 (1976).\n[59] S. Kumar et al. , Study of the spin reorientation transi-\ntion and magnetocrystalline anisotropy in NdCo 5using\ntorque magnetometry, in preparation (2019).\n[60] H. C. Herper, T. Ahmed, J. M. Wills, I. Di Marco,\nT. Bj orkman, D. Iu\u0018 san, A. V. Balatsky, and O. Eriksson,\nCombining electronic structure and many-body theory\nwith large databases: A method for predicting the na-\nture of 4fstates in Ce compounds, Phys. Rev. Materials\n1, 033802 (2017).\n[61] M. J. Lipp, D. Jackson, H. Cynn, C. Aracne, W. J. Evans,\nand A. K. McMahan, Thermal signatures of the Kondo\nvolume collapse in cerium, Phys. Rev. Lett. 101, 165703\n(2008).\n[62] E. Tatsumoto, T. Okamoto, H. Fujii, and C. Inoue, Sat-\nuration magnetic moment and crystalline anisotropy of\nsingle crystals of light rare earth cobalt compounds RCo 5,\nJ. Phys. Colloques 32, C1 (1971).8\n[63] R. Radwa\u0013 nski, The rare earth contribution to the magne-\ntocrystalline anisotropy in RCo 5intermetallics, J. Magn.\nMagn. Mater. 62, 120 (1986).\n[64] K. Buschow, A. van Diepen, and H. de Wijn, Crystal-\feldanisotropy of Sm3+in SmCo 5, Solid State Commun. 15,\n903 (1974).\n[65] M. Richter, Band structure theory of magnetism in 3d-4f\ncompounds, J. Phys. D: Appl. Phys. 31, 1017 (1998)."    },    {        "title": "1910.10531v1.Computational_screening_of_Fe_Ta_hard_magnetic_phases.pdf",        "content": "Computational screening of Fe-Ta hard magnetic phases\nS. Arapan1,\u0003P. Nieves1;2, H. C. Herper3, and D. Legut1\n1IT4Innovations, V ˇSB - Technical University of Ostrava,\n17. listopadu 15, 70833 Ostrava-Poruba, Czech Republic\n2ICCRAM, International Research Center in Critical Raw Materials and Advanced\nIndustrial Technologies, Universidad de Burgos, 09001 Burgos, Spain and\n3Department of Physics and Astronomy, Uppsala University, Box 516, 75121 Uppsala, Sweden\n(Dated: October 24, 2019)\nIn this work we perform a systematic calculation of the Fe-Ta phase diagram to discover novel hard magnetic\nphases. By using structure prediction methods based on evolutionary algorithms, we identify two new ener-\ngetically stable magnetic structures: a tetragonal Fe 3Ta (space group 122) and cubic Fe 5Ta (space group 216)\nbinary phases. The tetragonal structure is estimated to have both high saturation magnetization ( µ0Ms=1.14 T)\nand magnetocrystalline anisotropy (K 1=2.17 MJ/m3) suitable for permanent magnet applications. The high-\nthroughput screening of magneto-crystalline anisotropy also reveals two low energy metastable hard magnetic\nphases: Fe 5Ta2(space group 156) and Fe 6Ta (space group 194), that may exhibit intrinsic magnetic properties\ncomparable to SmCo 5and Nd 2Fe14B, respectively.\nPACS numbers: 31.15.A-,75.50.Ww, 75.30.Gw, 07.05.Tp\nI. INTRODUCTION\nMany technological applications used for information stor-\nage and green energy generation, like motors for hybrid and\nelectric cars and direct-drive wind turbines, rely on high qual-\nity permanent magnets (PMs)1. The increasing importance of\nPMs in modern society has resulted in a renewed interest in\nthe design of new magnet materials that are cheaper and con-\ntain less critical components like Rare-Earth (RE). One possi-\nble alternative to RE-PM could be RE-free Fe(Co)-rich inter-\nmetallic compounds. To be viewed as a good PM a ferromag-\nnetic compound must have a high Curie temperature ( TC>\n400 K), a high saturation magnetization ( µ0MS>1T) and a\nuniaxial anisotropy energy larger than 1 MJ/m3, since large\nanisotropy is a key factor for the large coercivity needed for\nhigh-performance PMs2. The main contribution to the mag-\nnetic anisotropy is usually magnetocrystalline (K), that is, a\ncombined effect of crystal-field splitting (or band formation)\nand spin-orbit coupling. This mechanism is also responsible\nfor the surface, interface and magnetostrictive anisotropies.\nSince none of the 4d and 5d dopants is ferromagnetic at room\ntemperature, the 3d sublattice must spin-polarize the partially\nfilled 4d/5d shells. This ensures a net spin-orbit effect, as re-\nquired for the creation of the anisotropy. Therefore, the addi-\ntion of heavy elements for a large spin-orbit coupling, as Hf,\nTa, Bi, Sn or Zr, to the Fe-Co alloys could form hard magnetic\nphases suitable for PM applications with a low raw materials\ncost. However, a large magnetocrystalline anisotropy in Fe-\nbased alloys can only be found in non-cubic uniaxial struc-\ntures, like FePt where a large K=7 MJ/m3is observed in the\ntetragonal L1 0structure.3Therefore, the theoretical research\nof such compounds should be focused on non-cubic uniaxial\nstructures as tetragonal, hexagonal or rhombohedral.\nIn Fe-Co alloys, Ta can induce some interesting features.\nFor instance, recent gradient-composition sputtering exper-\niments made by Phuoc et al.4,5showed that Co-Fe-Ta ex-\nhibits a peculiar increased magnetic anisotropy with temper-\nature much larger than Fe-Co-M where M =Hf, Zr, Lu.An improved coercivity has been reported in magnets such\nas (Fe,Co) 2B and Ce(Co,Fe,Cu) 5after the incorporation of\nsmall amount of Ta6,7. While the Co-Ta phase diagram shows\nup to seven stable binary phases8,9, only two structures have\nbeen found in the phase diagram of Fe-Ta: Laves phase (C14)\nFe2Ta space group (SG) 194, P6 3/mmc and µ-phase FeTa SG\n166, R-3m. The Fe 2Ta structure is a paramagnet in which ei-\nther Fe or Ta excess can induce a ferromagnetic ordering at\nlow temperatures ( <\u0018150 K)10. Theoretical calculations show\nan easy cone magnetocrystalline anisotropy in the ferromag-\nnetic state11. Recently, Gabay et al. reported melt-spun al-\nloys made of Fe-rich Fe 2Ta and Fe-bcc without sufficiently\nhigh coercivities (around to 0.5 kOe at room-temperature) for\nPM applications12. The other known stable crystalline phase,\nµ-phase FeTa, is an antiferromagnet with N ´eel temperature\naround 336 K13. Additionally, small amount of Ta can also be\ninserted into the Fe-bcc structure8. In the case of structurally\namorphous systems, Fe 9Ta thin films have been found to ex-\nhibit characteristics of a soft ferromagnetic material with very\nlow coercivities (1\u000010 mT), saturation magnetization around\n2 T and Extraordinary Hall Effect14.\nIn this work we performed a systematic computational\nexploration of Fe-rich Fe-Ta compounds to find new mag-\nnetic structures with intrinsic properties suitable for high-\nperformance PMs. Theoretical search for new magnetic\nphases is done in few stages. First, we searched for sta-\nble and low-energy metastable structures for different Fe-rich\nFe1\u0000xTaxbinaries. Details of calculations and results of the\ncrystal phase exploration are presented in the Section II. Sec-\nond, a set of structures with intrinsic magnetic properties ful-\nfilling the criteria of a performance PM, i. e., exhibiting nega-\ntive enthalpy of formation, DH<0, high saturation magneti-\nzation and uniaxial lattice, are selected from the collection of\npredicted structures as well as from the AFLOW database15.\nWe calculate the magnetocrystalline anisotropy (MCA) of all\nof these phases and identify a smaller subset of structures,\nwhich exhibit intrinsic properties of hard magnets. These few\nphases are analyzed in more detail to understand the possi-arXiv:1910.10531v1  [cond-mat.mtrl-sci]  23 Oct 20192\nble mechanisms of a high MCA in RE-free intermetallic com-\npounds. We present the corresponding results and calculation\ndetails in Section III and Appendix A. At the third stage, we\ninclude the effects of finite temperature on the phase stability\nand performed calculations of the exchange integrals of the\nfew selected structures to estimate the Curie temperature (T C),\nthus, having screened all considered Fe 1\u0000xTaxbinaries ac-\ncording to all three criteria to select a structure as a promising\nPM. Calculation detail and results of this stage are presented\nin Section IV. We finalized our work by performing a study\nof a possible stabilization of metastable phases as thin films\nby epitaxial growth on suitable substrates. We performed cal-\nculations of the elastic properties of selected structures and\nresults are given in Section V. The paper is completed by a\nConclusions section.\nII. CRYSTAL PHASE SPACE EXPLORATION\nWe began our study by exploring the phase space of ordered\nFe-rich Fe 1\u0000xTaxbinaries for several compositions (Fe 2Ta,\nFe3Ta, Fe 4Ta, Fe 5Ta, Fe 5Ta2, Fe 6Ta, Fe 7Ta, Fe 7Ta2, Fe 17Ta3,\nFe8Ta) using crystal predicting methods. We searched for\npossible structures by using the USPEX software16,17, an im-\nplementation of the evolutionary algorithm, and the Vienna\nAb Initio Simulation Package (V ASP)18–20. As in our recent\nwork21, we used an optimized USPEX-V ASP interface for\nthe efficient computational search of magnetic structures. We\nhave run the USPEX with the evolutionary algorithm method\nfor 3-D structures and choosing the enthalpy as a fitness cri-\nterion. The population size was set to be twice the number\nof atoms in the system and the maximum number of gen-\nerations to be calculated was set to 40. We used 15 gener-\nations for convergence and best 65% of the population size\nwas used for new generation. Out of the new structures 45%\nof were obtained by heredity, 5% by soft-mutations, 5% by\nlattice mutations, and 45% were randomly generated. For\nthe random generation, we used all space groups except tri-\nclinic and monoclinic lattice systems. All of the V ASP calcu-\nlations were done with the projector augmented wave (PAW)\nmethod22and the generalized gradient approximation (GGA)\nof Perdew, Burke, and Ernzerhof (PBE)23to the exchange cor-\nrelation part of the energy functional (PAW PBE potentials\nversion 5.4). We performed calculations with the psemi-core\nelectrons treated as valence ones. Best generated structures\nwere fully relaxed until maximum force component became\nless than 5\u000210\u00003eV/˚A. We used an automatic k-points gen-\nerating scheme with the length l=40 and an energy cut-off\nup to 1.4 of the default V ASP energy cut-off. Additionally,\nwe also ran calculations for a set of low-energy Fe-Ta phases\navailable in the AFLOW database15,24, which we use for the\nreference. We present relevant results of this study in Fig. 1.\nThis figure shows the convex hull diagram of the Fe-Ta binary\nsystem. The hull, shown by solid black lines is formed by\nusing calculated energies of Fe, Ta, FeTa and Fe 2Ta phases.\nSymbols correspond to the values of enthalpy of formation\nDHof various phases with respect to single elements Fe andTa as:\nDH(FenTam) =E(FenTam)\u0000n\u0001E(Fe)\u0000m\u0001E(Ta);(1)\nwhere E(:)is the energy at equilibrium conditions ( PV=0),\nnandmare the number of atoms of Fe and Ta in the for-\nmula unit of the Fe nTamcompound, respectively. E(Fe)and\nE(Ta)correspond to the energy of computationally optimized\nbcc phases of Fe and Ta. The relative position of these sym-\nbols with respect to the hull provides the information of the\nenergy stability of structures at T=0K. Our search reveals a\nset of new metastable magnetic phases (shown by filled disks)\nwith two structures, a tetragonal Fe 3Ta (SG 122) and a cu-\nbic Fe 5Ta (SG 216), in the proximity of the convex hull. In\naddition, we show in Fig. 1 the magnitude of saturation mag-\nnetization of calculated phases, represented by the intensity\nof the gray scale. We can observe that all ferromagnetic Fe-\nTa compounds with Fe content above 75 at.% have µ0MS>\u00181.\nIn this respect, the new tetragonal Fe 3Ta also exhibits a sat-\nuration magnetization, which qualifies it as a promising PM\nstructure. While the new energetically stable Fe 5Ta phase is\nnot suitable as a PM structure because of its cubic symmetry,\nit is interesting to analyze this phase in view of possible exper-\nimental synthesis as a validation of our theoretical predictions.\nMore details of the Fe-Ta phases calculated with USPEX can\nbe found in the Novamag database25,26.\nFIG. 1. Convex hull diagram of Fe-Ta showing phases predicted\nby using the USPEX (disks) and the reference structures from the\nAFLOW database (hollow red triangles). The magnitude of satura-\ntion magnetization is represented by the intensity of the gray scale.\nIII. SCREENING OF MAGNETO-CRYSTALLINE\nANISOTROPY\nAt the next stage of our study, we have performed calcu-\nlations of magnetocrystalline anisotropy energy (MAE) on a\nset of selected Fe 1\u0000xTaxbinary structures. We screened a set\nof structures consisting of phases predicted by the USPEX\nsearch and phases available in the AFLOW database to select\na subset of structures with DH<0,µ0MS>\u00181 and uniaxial\nlattice system (tetragonal, rhombohedral and hexagonal). We3\nTABLE I. Space group, enthalpy of formation, saturation magnetization, MCA constants, lattice parameters, Curie temperature and magnetic\nhardness parameter of main Fe-Ta phases discussed in this work. Data of SmCo 5and Nd 2Fe14B are also shown for comparison. The chemical\nformula of known stable structures are written in bold and superscripts stand for references.\nCompoundSpace\ngroupDH\n(eV/atom)µ0Ms\n(T)K1\n(MJ/m3)K2\n(MJ/m3)a\n(˚A)b\n(˚A)c\n(˚A)Temp\n(K)TC\n(K)k\nFe2Ta 194 -0.2053 0.63 -0.73 1.48 4.776 4.776 7.824 0 - -\nFe2Ta11194 -0.2350 0.66 -0.27 1.52 4.811 4.811 7.874 0 - -\nFe3Ta 122 -0.1588 1.14 2.17 -0.84 6.733 6.733 13.455 0 364 1.45\nFe5Ta 216 -0.1083 1.53 - - 6.678 6.678 6.678 0 - -\nFe5Ta2 156 -0.1143 1.00 10.43 6.22 4.713 4.713 4.744 0 724 3.62\nFe6Ta 194 -0.0370 1.62 5.77 0.60 4.627 4.627 9.353 0 886 1.66\nSmCo 527191 - 1.08 17.20 - 4.990 4.990 3.980 300 1020 4.30\nNd2Fe14B27136 - 1.61 4.90 - 8.790 8.790 12.180 300 588 1.54\nTABLE II. Crystallographic data, spin magnetic moment ( µspin), orbital magnetic moment ( µorb) and SOC energy (E so) of Fe 5Ta2(SG 156).\nCompound AtomWyckoff\npositionx y zµspin[001]\n(µB)µorb[001]\n(µB)µspin[100]\n(µB)µorb[100]\n(µB)Eso[100]\n(meV)Eso[001]\n(meV)DEso\n(meV)\nFe5Ta2 Fe1 1a 0 0 0.514 1.456 0.189 1.442 0.077 -17.403 -20.441 3.038\nFe2 1b 1/3 2/3 0.746 2.353 0.206 2.358 0.095 -17.395 -19.373 1.978\nFe3 3d 0.498 0.502 0.258 1.707 0.111 1.702 0.067 -16.559 -17.926 1.367\nTa1 1c 2/3 1/3 0.741 -0.536 0.072 -0.537 0.028 -288.187 -289.123 0.936\nTa2 1a 0 0 0 -0.524 0.052 -0.516 0.027 -297.208 -300.756 3.548\ncalculated the MAE by performing V ASP non-colinear spin-\npolarized calculations in a high-throughput manner26. Calcu-\nlations of MAE require, usually, a higher accuracy, and, espe-\ncially, a denser k-mesh to sample the reciprocal space. This,\ninevitably, increases the amounts of computational time and\nmemory. To decrease the computational demand, we used the\nPAW PBE potential with minimum number of valence elec-\ntrons. Extended test calculations showed that the addition of\nthepsemi-core electrons do not change considerably calcu-\nlated values (some detailed examples of the MAE calculations\nare provided in Appendix A). For all of the MAE calcula-\ntions we used an energy cut-off 1.50 times larger than the\ndefault one (ENCUT =401:823 eV) and the energy of a sys-\ntem was calculated with a tolerance EDDIF=10\u00009eV . We also\nfound that an automatic k-point mesh with the length l=60\nis enough to provide reliable MAE. The MAE was calculated\nas energy difference between the configurations with different\ncolinear spin arrangements:\nDE=Eq=0\u0000Eq; (2)\nwhere qis the angle between the direction of spins and the z\u0000\naxis. The energy of a given qconfiguration, Eq, was calcu-\nlated in a non-self-consistent way by using the charge density\nand wavefunctions of a colinear spin-polarized calculation.\nWe estimated the anisotropy constants K1andK2for uniax-\nial systems by fitting the MAE to the following equation:\nDE(q) =K1sin2(q)+K2sin4(q): (3)\nIn Fig. 3 we show the convex hull diagram of uniaxial Fe-Ta\nphases obtained by performing the MAE calculations, where\nFIG. 2. Unit cells of four interesting Fe-Ta structures studied in this\nwork: Fe 3Ta (SG 122), Fe 5Ta (SG 216), Fe 6Ta (SG 194) and Fe 5Ta2\n(SG 156).\nthe magnitude of the first MCA constant K1is represented by\nthe intensity of the gray scale and its sign by a given sym-\nbol. This figure reveals two hard magnetic phases: Fe 5Ta2\n(SG 156) with µ0MS=1 T, K1+K2=16:65 MJ/m3and4\nmagnetic hardness parameter k=q\nK1=(µ0M2\nS) =3:62, and\nFe6Ta (SG 194) with µ0MS=1:62 T, K1=5:77 MJ/m3and\nk=1:66. We can see that these values are comparable to the\nstate of the art RE-PM SmCo 5(µ0MS=1:08 T, K1=17:2\nMJ/m3) and Nd 2Fe14B (µ0MS=1:08 T, K1=4:9 MJ/m3) at\nroom temperature, respectively27. Here, we also highlight\nFe3Ta (SG 122) which combines both high phase stability\n(lays on the enthalpy convex hull) and good magnetic proper-\nties ( µ0MS=1:14 T and K1=2:17 MJ/m3). In Fig. 2 we show\nthe unit cells of the above mentioned structures: two new en-\nergetically stable phases Fe 3Ta (SG 122) and Fe 5Ta (SG 216),\nand two low energy metastable hard magnetic phases Fe 5Ta2\n(SG 156) and Fe 6Ta (SG 194). In Tables I, II and Appendix\nB we also provide various structural and magnetic properties\nof these phases. In the following sections we analyze these\nfour structures in more detail. To some extent, these results\nmay also be applied to Fe-Nb or Fe-Ta-Nb systems due to\nthe similarity between Ta and Nb. For instance, Fe 5Nb2(SG\n156) shows a large easy axis MAE with K1=7:7 MJ/m3and\nK2=2:2 MJ/m3.\nFIG. 3. Convex hull diagram of set of uniaxial Fe-Ta phases selected\nfrom the USPEX predicted phases and the AFLOW database. Gray\nscale represents the magnitude of the first MCA constant K1. Various\nsymbols correspond to the sign of K1.\nIn order to identify the source of the large MAE found in\nFe5Ta2we follow a similar analysis as in Refs. 28 and 29.\nNamely, we analyzed the spin-orbit coupling energy of each\natom with all spin orientation along z and x axis, Eso[001]and\nEso[100], see Table II. We observe that main contribution to\ntotal MAE comes from Fe and Ta atoms at the Wyckoff (1a)\nsite with DEso=Eso[100]\u0000Eso[001] =3:04 meV and 3 :55\nmeV , respectively. Next, we try to find the electronic states\nresponsible for the this large MAE. To do so, we analyzed the\npartial density of states (DOS) projected on d states of Fe (1a)\nwithout spin-orbit coupling (SOC) interaction, see Fig. 4(a).\nIt shows that there is a large DOS in the minority spin channel\nof d xyand dx2\u0000y2states (d orbitals that lay on the hard plane)\nright at the Fermi level. This peak is decomposed into two\nsmaller peaks below and above the Fermi level when the SOC\nis included and the magnetization is aligned along the easy\naxis [001], Fig. 4(b), decreasing the total energy and inducing\na large MAE. The fact that the coupling between the minority\nFIG. 4. a) Partial DOS projected on d states of Fe atom at Wyckoff\nposition (1a) of Fe 5Ta2without SOC interaction. b) Total DOS pro-\njected on d xyand dx2\u0000y2states of Fe (1a) in Fe 5Ta2when the mag-\nnetization is aligned along the easy axis (blue line) and hard plane\n(red line) including SOC interaction. c) Change of the SOC matrix\nelements of Fe (1a) in Fe 5Ta2, when magnetization goes from hard\nplane [100] to easy axis [001].\nspin channel gives the largest contribution to the SOC is also\nsupported by the maximum values of the orbital magnetic mo-\nments in the easy direction of magnetization (see Table II)11.\nIn this process, where the magnetization goes from hard plane\nto easy axis, the SOC matrix element hdxyjˆHsojdx2\u0000y2iexhibits\na much greater change than the other ones, Fig. 4(c). It is\ninteresting to note that Fe 5Ta2is capable to transfer very ef-5\nficiently its spin-orbit coupling energy to total MAE with a\nratioDEso=DE=1:41, where DE=E[100]\u0000E[001] =9:55\nmeV is the total energy difference between the states with all\nspins orientated along [100] and [001] direction. For instance,\nthis spin-orbit reduction factor is smaller than in the L1 0FePt\n(1.84) and CoPt (1.67)28. Typically, when the second order\nperturbation theory holds, this factor is close to 2. This analy-\nsis may also explain the dependence of calculated MAE with\nenergy smearing methods shown in Fig. 9 (Appendix A), since\nthe energy is an integral of the density of states weighted by a\nsmearing function.\nIV . FINITE-TEMPERATURE EFFECTS: PHASE\nSTABILITY AND THE CURIE TEMPERATURE\nA. Phase stability at finite temperatures\nFor the four selected phases in the Fe 3Ta, Fe 5Ta, Fe 5Ta2\nand Fe 6Ta binaries, we further studied the effect of tempera-\nture and pressure on their structural stability. We have calcu-\nlated the free energy by considering the lattice contribution in\nthe quasi-harmonic approximation (QHA) and the entropy of\nelectron system:\nF(T) =E0+Fph(T)\u0000T Sel(T); (4)\nwhere E0is the Density Functional Theory (DFT) energy at\nT=0K,Fphis the phonon free energy, and Selis the entropy\nof electrons, estimated by the Sommerfeld formula30:\nSel(T) =\u0000N(EF)\n6p2k2\nBT; (5)\nwhere N(EF)is the density of states (DOS) at the Fermi level.\nCalculations were performed in the following manner. For\neach structure we performed DFT calculations over a set of\nvolumes within an interval (V0\u0000DV;V0+DV)about equilib-\nrium volume V0, with DV=0:1V0. Structures were relaxed\nat each volume and accurate total energy E0and DOS calcu-\nlated. To calculate the phonon free energy Fphwe used PHON\ncode31, an implementation of the small displacements method\nto calculate phonon dispersion of a harmonic crystal. At each\nvolume supercells were generated from relaxed structures and\nforces were calculated with the V ASP program. For these cal-\nculations we used the PAW potentials containing the semi-\ncore pelectrons, and performed accurate calculations with a\ncut-off energy of ENCUT =513:167 eV (1 :75 of the default\ncut-off energy) and the tolerance of the electronic convergence\nset to EDIFF =10\u00007eV . We used a fully automatic k-points\ngeneration mesh with length l=40 to sample the reciprocal\nspace. We have performed free energy calculations for the\nfour above mentioned structures, as well as for the two stable\nreference phases: bccFe and Fe 2Ta. All phases were dynam-\nically stable within the whole considered pressure range (no\nimaginary frequencies). To calculate the phase stability at fi-\nnite temperatures and different pressures we have estimated\nparameters of the equation of state (EOS) F=F(V;T)at a\nset of temperature values within the (0K;1000 K)interval by\nFIG. 5. Gibbs free energy difference DG(T;P)of selected structures:\na)Fe3Ta (blue filled symbols) and Fe 5Ta (red open symbols); b)\nFe5Ta2(blue filled symbols) and Fe 6Ta (red open symbols); relative\nto stable bccFe and Fe 2Ta phases according to Eq. 7.\nfitting at each temperature Tthe setfF(Vi)gof calculated free\nenergy values to the Vinet EOS32:\nF(V) =F0+4B0V0\n(B0\u00001)2\u00002B0V0\n(B0\u00001)2\u0002(6)\n\u0000\n5+3B0(h(V)\u00001)\u00003h(V)\u0001\nexp\u0014\n\u00003\n2\u0000\nB0\u00001\u0001\n(h(V)\u00001)\u0015\n;\nwhere h(V) = ( V=V0)1=3andF0=F0(T),V0=V0(T),B0=\nB0(T), and B0=B0(T)are the equilibrium free energy, vol-\nume, bulk modulus and its derivative at a given T. Within the\nVinet approximation to the Helmholtz free energy we calcu-\nlated the Gibbs free energy G(P;T) =F(V;T)+PVand esti-\nmated the formation energy with respect to stable bccFe and\nFe2Ta as:\nDG=G(FenTam)\u0000XnG(Fe)\u0000YmG(Fe2Ta); (7)\nwhere XnandYmare appropriately chosen for each consid-\nered Fe nTamphase. The temperature dependence of the for-\nmation energy of Fe 3Ta and Fe 5Ta, which were close to the\nenthalpy hull at T=0 (Fig. 1) is shown in Fig. 5 afor dif-\nferent pressures. Calculations show that the predicted cubic\nFe5Ta phase should be stable at ambient pressure (data shown\nby open disks) over the whole considered temperature range.\nThe promising new magnetic phase Fe 3Ta (shown by filled\nsymbols) also remains very close to the stability line for the\nambient pressure. At finite positive pressures (compression)\nboth phases get energetically less stable. The effect is oppo-6\nsite for the negative pressure values (dilatation). This pres-\nsure effect suggests that the new phase may get energetically\nmore stable by adding a third element, which would slightly\nincrease the inter-atomic distances. In Fig. 5 bwe show the\ntemperature and pressure trends for the formation energies of\nmetastable Fe 5Ta2(filled symbols) and Fe 6Ta (open symbols)\nphases with high magnetic anisotropy. Both phases remain\nunstable with temperature and formation energies show sim-\nilar trends with pressure as in the case of Fe 3Ta and Fe 5Ta\nphases.\nB. Exchange integrals and Curie temperature\nUsing the optimized geometries of the new phases as the\nstarting point, the exchange coupling constants were calcu-\nlated. Using Liechtenstein’s approach33the magnetic inter-\nactions for arbitrary arrangements of magnetic moments can\nbe calculated. The effective exchange interaction parameters\nwere obtained using Lichtenstein et al. method33,34, as imple-\nmented in SPR-KKR35. In this technique the energy of the\nsystem is mapped onto a classical Heisenberg model with the\nfollowing Hamiltonian:\nEex=\u0000å\ni;jJi jsi\u0001sj; (8)\nwhere Ji jare the exchange parameters and siis the unit vector\nalong the magnetic moment of atom i-th. The fast SPR-KKR\ncore within atomic sphere approximation was used to obtain\nthe exchange coupling constants. After a self-consistent run\nneeded to create the potential for the systems the Ji jwere cal-\nculated using a dense k-point mesh 34x34x29 for Fe 5Ta2(SG\n156). The cut-off for the Ji jcouplings was chosen to be 3\nlattice constants. The Ji jvalues obtained from the SPR-KKR\ncode depending on the neighbour distance are shown in the\nFig. 6.\nDue to the complexity of the structure the Fe-Fe couplings\nof the Fe 5Ta2phase are very diverse. The contribution from\nFe1-Fe1(1a site, along c axis) is quite small being 5 meV for\nthe nearest neighbors. The main contributions stem from Fe 2\nand Fe 3couplings. For these ions the next nearest neighbor\ncouplings are between 20 and 30 meV . However, the coupling\nstrength rapidly decreases with the distance and is almost zero\nafter 5 neighbor shells. It should be pointed out that the ma-\njority of the coupling constants is positive which means fer-\nromagnetic coupling. Only very tiny small antiferromagnetic\ncontributions were observed for Fe 5Ta2(Fe3). However, even\nthough Ta has only an induced moment there are significant\ncouplings between Ta and the Fe ions especially Fe 1. Un-\nfortunately, the coupling is antiferromagnetic and therewith\ncounteracts to a high T C.\nFinally, we estimated the T Cof these novel phases by means\nof atomistic spin dynamics (ASD) simulations36using as in-\nputs the calculated lattice parameters, exchange parameters\nand magnetic moments. For each structure, we consider a\nsystem of 15x15x15 unit cells with periodic boundary con-\nditions, which is thermally relaxed integrating the stochas-\ntic Landau-Lifshitz-Gilbert equation. We performed this task\nFIG. 6. Calculated exchange coupling constants for: (a)-(b) Fe 3Ta,\n(c)-(d) Fe 5Ta2and (e)-(f) Fe 6Ta, depending on the distance between\nthe ions. Left: The couplings between the Fe ions in the system are\nshown. Right: The couplings between the magnetic Fe ions and the\nTa ions (which have an induced moment antiparallel to the Fe atoms)\nare plotted. Note the different scale between the two graphs.\nwith the software UppASD37,38. The obtained values for\nFe3Ta, Fe 5Ta2and Fe 6Ta are T C= 364 K, 724 K and 886 K, re-\nspectively, that make them suitable for PM applications. How-\never, the performance of Fe 3Ta may be highly deteriorated at\nroom temperature since T Cis not so large. It should be noted\nthat despite the fact that the Fe-Ta coupling parameters for\nFe5Ta2an Fe 6Ta are larger in size i.e. larger AF coupling ex-\nists, the Curie temperature for these systems are higher than\nfor Fe 3Ta. This is partially related to the multiplicity of the in-\ndividual couplings but also to the fact that the Fe-Fe couplings\nin Fe 3Ta are smaller.\nV . SCREENING OF SUBSTRATES FOR EPITAXIAL\nSTABILIZATION\nAs we mentioned in the Introduction, only two stable\nphases have been experimentally identified for the Fe-Ta bina-7\nries so far. Our theoretical predictions seem to contradict this\nfact, but there may exist a possibility that observing these new\nphases is difficult when synthesizing the Fe-Ta alloys with\ntraditional metallurgical methods39. Of course, these meth-\nods are preferable to produce a bulk PM suitable for applica-\ntions. Right now, however, we would be more interested in\nthe possibility to synthesize predicted phases and probe their\nmagnetic properties to assess the predictive accuracy of our\ncomputational methods. One may try to stabilize some of\nthese phases as thin films by epitaxial growth. Thus, as a final\nstage of our work, we performed a screening of suitable sub-\nstrates that may help to stabilize these new phases. Ding et\nal.proposed two types of filters based on the unit cell topol-\nogy (geometric unit cell area matching between the substrate\nand the target film) and strain energy density of the film in or-\nder to identify ideal substrates for epitaxial stabilization40. We\nhave applied this screening approach to 66 widely used single-\ncrystalline substrates40available in Materials Project41,42for\nthe films Fe 5Ta2(SG 156), Fe 3Ta (SG 122), Fe 6Ta (SG 194)\nand Fe 5Ta (SG 216). We performed this task using pymatgen\nlibrary43. The elastic constants needed for the calculation of\nthe film elastic energy were obtained through AELAS code44\ncombined with V ASP using PAW method and GGA-PBE with\ndefault settings. The elastic stiffness matrix of these phases is\ndefinite positive, so that they can be considered mechanically\nstable crystalline structures.\nFIG. 7. Computational screening of substrates for the epitaxial sta-\nbilization of Fe 5Ta2.\nIn Fig. 7 we show the high-throughput calculation of the\nminimal coincident interfacial area (MCIA) and elastic en-\nergy of Fe 5Ta2for all considered substrates and orientations.\nOptimal substrates are those with both low MCIA and elastic\nenergy. In this case, the best substrates are (001)-MgF 2(001)-\nTiO 2and (001)-Al 2O3with Fe 5Ta2film orientation (100),\n(100) and (001), respectively. The obtained values of MCIA\nand elastic energy for the substrates with the lowest value of\nMCIA are given in Table IV (Appendix C). Although bulk\nFe5Ta is not a good PM (cubic phase), its phase stability is\ngreater than the above mentioned ones (see Fig.5). Our calcu-\nlations show that potential good substrates for Fe 5Ta could be(001)-MgF 2, (100)-InSb and (100)-CdTe with film orientation\n(100).\nVI. CONCLUSIONS\nDespite of the fact that the two experimentally observed\nordered Fe-Ta alloy phases do not exhibit magnetic proper-\nties suitable for PMs, our computational study suggests that\nnew phases, with intrinsic magnetic properties appropriate for\nPMs, might exist within this binary system. The structure pre-\ndiction study based on evolutionary algorithm revealed two\nnew energetically stable structures for the Fe 5Ta and Fe 3Ta bi-\nnaries, respectively. Fe 3Ta is a uniaxial phase (tetragonal sym-\nmetry) with calculated saturation magnetization µ0Ms=1.14 T,\nmagnetocrystalline anisotropy K 1=2.17 MJ/m3and the Curie\ntemperature TC=364 K, which makes it a potentially promis-\ning phase for PMs. Calculations of MAE of various low en-\nergy metastable phases also showed the existence of structures\nwith extraordinary high MAE in the Fe-Ta system. We identi-\nfied two phases in the Fe 5Ta2and Fe 6Ta binaries with intrinsic\nmagnetic properties comparable to SmCo 5and Nd 2Fe14B, re-\nspectively. For the Fe 5Ta2structure, for example, our calcula-\ntions predict µ0Ms=1.00 T, K 1+K2=16.65 MJ/m3andTC=724\nK. In this phase, we found that there is a large DOS in the\nminority spin channel of d xyand dx2\u0000y2states (d orbitals that\nlay on the hard plane) right at the Fermi level, especially at\nWyckoff (1a) site for both Fe and Ta atoms, which may be\nresponsible for the observed high MAE. Is is necessary, of\ncourse, to analyze more hard intermetallic Fe-based phases in\norder to identify possible general mechanism of high MAE.\nThe analysis of the Gibbs free energy shows that these hard\nmagnetic phases are energetically unstable at finite tempera-\ntures too, so this may prevent the synthesis of these phases\nin bulk. It might be possible, however, to synthesize these\nphases as thin films and we provide some possible substrates\nfor the epitaxial growth. These new findings might consti-\ntute a step forward towards the discovery and design of novel\nhigh-performance RE-free PMs.\nACKNOWLEDGEMENT\nThis work was supported by the European Horizon 2020\nFramework Programme for Research and Innovation (2014-\n2020) under Grant Agreement No. 686056, NOV AMAG. Au-\nthors acknowledge the European Regional Development Fund\nin the IT4Innovations national supercomputing center - path\nto exascale project, project number CZ 02.1.01/0.0/0.0/16-\n013/0001791 within the Operational Programme Research,\nDevelopment and Education.8\nAPPENDIX\nAppendix A: Some details of calculations of MAE\nIn this appendix, we show some tests of our MAE calcula-\ntions for Laves phase Fe 2Ta, and theoretical predicted Fe 3Ta\n(SG 122) and Fe 5Ta2(SG 156). The Fe 2Ta phase was the-\noretically investigated previously11,45with contradictory re-\nsults for the MAE calculations using different DFT codes. In\nFig. 8 we show calculated MAE with V ASP code for differ-\nent k-point meshes as well as different PAW PBE potentials\nand cut-off energies. We can see that calculated values of the\nMAE are quite robust against different PAW PBE potentials\nand are converged for a number of k-points in the reciprocal\nspace larger than 1500, which corresponds to the length l=60\nin the V ASP automatic k-points generation scheme. These re-\nsults are in agreement with those reported by Edstr ¨om11, who\nalso obtained an easy cone for the MCA with K1=\u00000:27\nMJ/m3andK2=1:52 MJ/ m3. The value of the MAE of 1 :25\nMJ/m3is larger than one we had obtained (Fig. 8) due to the\nlarger cell volume used in Ref. 11, see Table I.\nFIG. 8. Magneto-crystalline anisotropy energy of the ferromagnetic\nLaves Fe 2Ta phase calculated for different number of k-points in the\nreciprocal space: a)calculations performed with PAW PBE poten-\ntials with the minimum number of valence electrons and cut-off en-\nergy of 401 :823 eV (1 :50 of the default cut-off energy); b)calcula-\ntions performed with PAW PBE potentials with the psemi-core elec-\ntrons added to the valence electrons and cut-off energy of 513 :167 eV\n(1:75 of the default cut-off energy)\nThe Fe 3Ta and Fe 5Ta2phases present a particular interest.\nThe first one is energetically very close to the convex hull ofthe Fe-Ta binary diagram and, thus, could be experimentally\nobtained under appropriate conditions. The relatively high\nsaturation magnetization and magnetocrystalline anisotropy\nmake this phase a promising candidate for RE-free PM ma-\nterial. The second one, despite being metastable, provides an\nexample of huge MCA for a RE-free intermetallic compound.\nWe have performed a series of MAE calculations for different\nparameters which affects the energy calculations, like the type\nof PAW PBE potentials, energy cut-off and energy smearing.\nThese additional calculations are shown in Fig. 9. We see that\nresults of the MAE calculations are robust against the varia-\ntions of these parameters.\nFIG. 9. Magneto-crystalline anisotropy energy of predicted a)Fe3Ta,\nandb)Fe5Ta2phases calculated for different settings for the smear-\ning parameter ISMEAR = 0,1,-5 (Gaussian, 1st order Methfessel-\nPaxton and tetrahedron with Bl ¨ochl corrections methods, respec-\ntively). For all of the calculations, except those marked with an as-\nterisk, we used the PAW PBE potentials with minimum number of\nvalence electrons, a k\u0000mesh corresponding to length l=60 and en-\nergy cut-off of 401 :823 eV (1 :50 of the default cut-off energy). For\nother two calculations marked with an asterisk, we used the PAW\nPBE potentials with the psemi-core electrons added to the valence\nelectrons and cut-off energy ofT 513 :167 eV (1 :75 of the default cut-\noff energy).9\nAppendix B: Crystallographic and magnetic data\nTABLE III. Crystallographic data and spin magnetic moment of\nFe3Ta (SG 122), Fe 6Ta (SG 194) and Fe 5Ta (SG 216).\nCompound AtomWyckoff\npositionx y zµspin[001]\n(µB)\nFe3Ta Fe 1 16e 0.876 0.374 0.436 1.727\nFe2 16e 0.626 0.874 0.061 1.646\nFe3 4b 0 0 1/2 2.564\nTa1 8d 0.731 1/4 0.125 -0.382\nTa2 4a 0 0 0 -0.354\nFe6Ta Fe 1 6g 1/2 0 0 1.921\nFe2 4e 0 0 0.628 2.436\nFe3 2d 1/3 2/3 3/4 2.305\nTa1 2c 1/3 2/3 1/4 -0.890\nFe5Ta Fe 1 16e 0.625 0.625 0.625 1.939\nFe2 4c 1/4 1/4 1/4 2.746\nTa1 4a 0 0 0 -0.438Appendix C: Optimal substrates information for epitaxial\ngrowth\nTABLE IV . Optimal substrates for epitaxial growth of phases Fe 5Ta2\n(SG 156), Fe 3Ta (SG 122), Fe 6Ta (SG 194) and Fe 5Ta (SG 216).\nFilm Film\norientationSubstrate Substrate\norientationMCIA\n(˚A2)Elastic energy\n(meV)\nFe5Ta2<001> Al2O3<001> 19.24 1.645\n<100> TiO 2<001> 22.36 1.058\n<100> MgF 2<001> 22.36 0.239\nFe3Ta<001> MgF 2<001> 45.34 0.793\n<001> InSb <100> 45.34 0.880\n<001> CdTe <100> 45.34 0.954\nFe6Ta<100> MgF 2<100> 43.29 0.213\n<100> MgF 2<001> 43.29 0.332\n<100> TiO 2<001> 43.29 0.041\nFe5Ta<100> MgF 2<001> 44.59 0.124\n<100> InSb <100> 44.59 0.158\n<100> CdTe <100> 44.59 0.189\n\u0003Corresponding author: sergiu.arapan@gmail.com\n1G. Scheunert, O. Heinonen, R. Hardeman, A. Lapicki, M. Gub-\nbins, and R. M. Bowman, Applied Physics Reviews 3, 011301\n(2016), https://doi.org/10.1063/1.4941311, URL https://doi.\norg/10.1063/1.4941311 .\n2K. Skokov and O. 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Here, we report an approach to manipulate the damping factor of \nFeGa/MgO(001) films by oblique deposition. Owing to  the defects at the surface or \ninterface in thin films , two -magnon scatterin g (TMS)  acts as a non -Gilbert damping \nmechanism in magnetization relaxation.  In this work, the contribution of TMS was \ncharacterized  by in-plane angul ar dependent  ferromagnetic resonance (FMR) . It is \ndemonstrated that the intrinsic Gilbert damping is isotropic and invariant , while the \nextrinsic mechanism  related to TMS is anisotropic  and can be tuned by oblique \ndeposition. Furthermore, the two and fourfold TMS related to  the uniaxial magnetic \nanisotropy  (UMA)  and m agnetocrystalline anisotropy were discussed. Our result s open \nan avenue to manipulate  magnetization relaxation in spintronic devices.  \n1 \n Keywords : Gilbert damping , two -magnon scattering, FMR, oblique deposition, \nmagnetic anisotropy  \n2 \n 1. Introduction  \nIn the past decades, controlling magnetization dynamics in magnetic \nnanostructures has been extensively studied due to its great importance for spintronic \nand spin- torque applications  [1,2] . The magnetic relaxation is described within the \nframework of the Landa u-Lifshitz  Gilbert (LLG)  phenomenology  using the Gilbert \ndamping factor  α [3]. The intrinsic  Gilbert damping depends primarily on the spin- orbit \ncoupling (SOC) [4,5] . It has been demonstrated that alloying or doping with non-\nmagnetic transition metals  provides an opportunity to tune the intrinsic damping  [6,7] . \nUnfortunately, in this way the soft magnetic properties will reduce . In addition to the \nintrinsic damping, the two-magnon scatterin g (TMS) process se rves as a n important  \nextrinsic  mechanism i n magnetization relaxation  in ultrathin films  due to the defects at \nsurface or interface [8,9] . This process  describes the scattering between the uniform \nmagnons and degener ate final -state spin wave modes  [10]. The existence of TMS has \nbeen demonstrated in many systems of ferrites  [11-13]. Since the anisotropic scattering \ncenters , the angular dependence of the extrinsic TMS process  exhibi ts a strong in -plane \nanisotropy  [14], which allows  us to adjust the overall magnetic relaxation , including \nboth the int ensity of relaxation rate and  the anisotropic behavior.  \nHere, we report an approach to engineer  the damping factor of Fe81Ga19 (FeGa ) \nfilms by oblique deposition. The FeGa alloy exhibits large  magnetostriction and narrow \nmicrowave resonance linewidth  [15] , which  could assure  it as a  promising material for \nspintronic devices. For the geometry of off -normal deposition, it has been demonstrated \nto provoke shadow effects and create a periodic stripe defect matrix. This can introduce \na strong uniaxial magnetization anisotropy (UMA) pe rpendicular to the projection of \nthe atom flux  [16-19]. Even though some reports have shown oblique deposition \nprovokes a  twofold TMS channel  [20-22], the oblique angle dependence of the intrinsic \n3 \n Gilbert damping and the TMS  still remain in doubt. For our case, on the basis of the \nfirst-principles calculation and the in -plane angular -dependent FMR measurements, we \nfound that the intrinsic Gilbert damping is isotropic and invariant with varying oblique \ndeposition angles, while the extrinsic mechanism related to the two -magnon -scattering \n(TMS ) is anisotropic and can be tuned by oblique deposition. In addition, importantly \nwe firstly observe a phenomenon that the cubic magnetocrystalline anisotropy \ndetermines the area including degenerate magnon modes, as well as the intensity of fourfold TMS. In general , the strong connection between the extrinsic TMS and the \nmagnetic anisotropy , as well their direct impact on the damping constants , are \nsystem ically investigated, which offer us a useful approach to tailor the damping factor.\n \n2. Experimental details  \nFeGa thin films with a thickness of 20 nm were grown on MgO(001) substrates in \na magnetron sputtering system with a base pressure below 3 × 10−7 Torr. Prior to \ndeposition, t he substrates were annealed at 700 °C for 1 h in a vacuum chamber to \nremove surface contaminations and then held at 250 °C during deposition. The incident \nFeGa beam was at different obl ique angles of ψ =0°, 15°, 30°, and 45°, with respect to \nthe surface normal , and named S1, S2, S3, and S4 in this paper , respectively.  The \nprojection of FeGa beam  on the plane of the substrates  was set perpendicular to the \nMgO[110] direction, which induces a UMA perpendicular to the projection of FeGa \nbeam , i.e., parallel to the MgO[110] direction, due to the we ll-known  self-shadowing \neffect.  Finally , all the samples  were covered with a 5 nm  Ta capping layer to avoid \nsurface oxidation [see figure  1(a)]. The epitaxial relation  of \nFeGa(001)[ 110]||MgO (001)[ 100] was characterized by using t he X -ray in -plane Φ-\nscans , as described elsewhere [23]. Magnetic hy steresis loops were measured  at various \nin-plane magnetic field orientations φ H with respect to the FeGa [100] axis using \n4 \n magneto -optical Kerr effect (MOKE) technique at room temperature . The d ynamic \nmagnetic properties were investigated by broadband FMR measurements  based on a \nbroadband vector network analyzer  (VNA) with a  transmission geometry coplanar \nwaveguide (VNA- FMR ) [24]. This setup allows both frequency and field- sweeps \nmeasurements with external field applied parallel to the sample plane.  During \nmeasurements,  the sampl es were placed face down on the coplanar wavegu ide and the \ntransmission coefficient S 21 was recorded.  \n3. Results and discussion \nFigure  1(b) displays the Kerr  hysteresis loops  of sample  S1 and S4 recorded along \nwith the main crystallographic directions of FeGa [100], [110],  and [010] . The sample \nS1 exhibit s rectangular hysteresis curves with sm all coercivities for the magnetic  field \nalong  [100] and [010]  easy axes. In contrast, the S4 displays a  hysteresis curve with \ntwo step s for the magnetic field along the [010]  axis, which indicates  a UMA along the \nFeGa[100 ] axis superimposed on the four fold magnetocrystalline anisotropy . As a \nresult,  with increasing the oblique angle, the angular dependence of normalized remnant \nmagnetization ( Mr/Ms) gradually reveals a four fold symmetry combined with a uniaxial \nsymmetry, as shown i n the inset of f igure 1(b).  \nSubsequently,  the magnetic anisotropic properties can be further precisely \ncharacterized  by the in -plane angular -dependent FMR measurements.  Figure 1(c) and \n1(d) show  typical  FMR spectra for the real and imaginary part s of coefficient S 21 for \nthe sample S2 . Recorded  FMR spectra contain a symmetric and an antisymmetric \nLorentzian peak , from which the resonant field H r with linewidth ∆𝐻𝐻 can be obtained  \n[24,25] . \nFigure 2(a) shows the in -plane angular dependence of H r measured at 13 .0 GHz  \nand can be fitted  by the following expression [26,27] : \n5 \n 𝑓𝑓=𝛾𝛾𝜇𝜇0\n2𝜋𝜋�𝐻𝐻𝑎𝑎𝐻𝐻𝑏𝑏                                                     (1 ) \nHere,𝐻𝐻𝑎𝑎=𝐻𝐻4(3+𝑐𝑐𝑐𝑐𝑐𝑐4𝜑𝜑M)/4+𝐻𝐻u𝑐𝑐𝑐𝑐𝑐𝑐2𝜑𝜑M+𝐻𝐻r𝑐𝑐𝑐𝑐𝑐𝑐(𝜑𝜑M−𝜑𝜑H)+𝑀𝑀eff and 𝐻𝐻𝑏𝑏=\n𝐻𝐻4𝑐𝑐𝑐𝑐𝑐𝑐4𝜑𝜑M+𝐻𝐻u𝑐𝑐𝑐𝑐𝑐𝑐2𝜑𝜑M+𝐻𝐻r𝑐𝑐𝑐𝑐𝑐𝑐(𝜑𝜑M−𝜑𝜑H), H4 and Hu represent the fourfold \nanisotropy field and the UMA field caused by the self -shadowing effect , respectively. \n𝜑𝜑H(𝜑𝜑M) is the  azimuthal angles of the applied field ( the tipped magnetization ) with \nrespect to the [100] direction , as depicted  in figure 1(a). 𝜇𝜇0𝑀𝑀eff=𝜇𝜇0𝑀𝑀𝑠𝑠−2𝐾𝐾out\n𝑀𝑀𝑠𝑠, Ms is \nthe saturation magnetization  and Kout is the out -of-plane uniaxial anisotropy constant . \n𝑓𝑓 is the resonance frequency , 𝛾𝛾 is the gyromagnetic ratio and here used as the accepted \nvalue for Fe films, 𝛾𝛾=185 rad GHz/T  [28] . \nThe angular dependent  Hr reveals  only a  fourfold symmetry for the none -\nobliquely deposited sample , which indicates  the cubic lattice texture of FeGa on MgO . \nWith increasing the oblique angle , a uniaxial symmetry is found  to be superimposed \non the four fold symmetry, clearly confirming a UMA is produced by the oblique \ngrowth , which agrees with the MOKE’ results.  The fitted  parameter  𝜇𝜇0𝑀𝑀eff=1.90±\n 0.05T is found to be independent on the oblique deposition and close to 𝜇𝜇0𝑀𝑀𝑠𝑠=\n1.89 ± 0.02 T  estimated using VSM, which is almost same as the value  of the  \nliterature  [29] . This indicates negligible out- of-plane ma gnetic anisotropy in the thick \nFeGa films . As shown in figure 2(b), it is observed  that the UMA (Ku=HuMs/2) exhibits \na general increasing  trend with oblique angle , which coincides with the fact  the \nshadowing effect is stronger  at larger  angles of incidence [16-19]. Interestingly the \noblique deposition also affects the cubic anisotropy  K4 (K4=H4Ms/2). Different from  \nthe K4 increases slightly with deposition angle  in Co/Cu system  [16], here the value of \nK4 is the lowest at a n oblique angle of 15°. It is well known th at film stress significantly \ninfluences  the crystallization tendency  [30,31] . FeGa alloy is highly stress ed sensitive \n6 \n due to its larger magnetostriction . Thus, t he change in  K4 of FeGa films  may be \nattributed to the anisotropy dispersion created due to the stress variations  during grain \ngrowth. It should be mentioned that the best way to determine magnetic parameters is \nto measure the out -of-plane  FMR . But the effective saturation magnetization 𝜇𝜇0𝑀𝑀eff=\n1.90T of FeGa alloy leads to the perpendicular applied field beyond our instrument \nlimit. Meanwhile, t he results obtained above are also in accord with those extracted \nby fitting field dependence of the resonance frequency with H//FeGa[100]  shown in \nfigure 2(c).  \nThe effective Gilbert damping 𝛼𝛼eff is extracted by linearly fitting the dependence  \nof linewidth  on frequency : 𝜇𝜇0∆H=𝜇𝜇0∆H0+2𝜋𝜋𝜋𝜋𝛼𝛼𝑒𝑒ff\n𝛾𝛾, where ∆𝐻𝐻0 is the inhomogeneous \nbroadening. For the sake of clarity , figure 3(a) only shows the frequency  dependence \nof linewidth for the  samples S1 and S2 along  [110]  and [100]  axes. It is evident that, \nfor the sample  S1, both linear slopes of two direction s are almost  same. While w ith \nregard to the sample S2 , the slope of the ∆H-f curve along  the easy axis is approximately \na factor of 2 greater than that of the hard axis. The obtained values of  𝛼𝛼eff are shown in \nfigure 3(b). Firstly, the results clearly indicate that the effective damping exhibits \nanisotropy , with  higher value along the easy axis . Secondly, f or the easy  axis, the \noblique angle dependence on the damping parameter indicates an extraordinary trend \nand has a peak at  deposition angle 15°. However, the damping shows an increasing \ntrend  with the oblique angle for the field along the hard axis. In the following part, we \nwill explore the effect of oblique deposition on the mechanism of the anisotropic \ndamping and the magnetic relaxation pr ocess.     \nSo far, convincing experimental evidence is still lacking to prove the existence of \nanisotropic damping in bulk magnets. Chen et al. have shown the emergence of \nanisotropic Gilbert damping in ultrathin Fe (1.3nm)/GaAs and its anisotropy disappears \n7 \n rapidly when the Fe thickness  increases  [32]. We perform  the first-principles \ncalculation of the Gilber t damping of Fe Ga alloy considering  the effect induced by the \nlattice distortion. W e artificially make a tetragonal lattice with varying the lattice \nconstant of the c -axis. The electronic structure of Fe -Ga alloy is calculated self -\nconsistently using the coherent potential approx imation implemented with the tight-\nbinding linear muffin- tin orbitals. Then the atomic potentials of Fe and Ga are randomly \ndistributed in a 5× 5 lateral supercell, which is connected to two semi -infinite Pd leads . \nA thermal lattice disorder is included via  displacing atoms randomly from the perfect \nlattice sites following a Gaussian type of distribution [ 33]. The root -mean -square \ndisplacement at room temperature is determined by the Debye model with the Debye \ntemperature 470 K. The length of the supercell is variable and the calculated total \ndamping is scaled linearly with this length. Thus, a linear least- squares fitting can be \nperformed to extract the bulk damping of the Fe -Ga alloy  [34].  The calculated Gilber t \ndamping is plotted in f igure 3(c) as a funct ion of the lattice distortion (𝑐𝑐−𝑎𝑎)𝑎𝑎⁄.  The \nGilbert damping is nearly independent of the lattice distortion and there is no evidence of anisotropy in t he intrinsic bulk damping of Fe Ga alloy.  \nSo the extrinsic contributions  are responsible for  the anisotropic behavior of \ndamping , which can be separated from the in -plane angular dependent linewidth. The \nrecorded FMR  linewidth  have the following different cont ributions  [11] :  \n     𝜇𝜇0∆𝐻𝐻=𝜇𝜇0∆𝐻𝐻inh+2𝜋𝜋𝛼𝛼𝐺𝐺𝑓𝑓\n𝛾𝛾𝛾𝛾+�𝜕𝜕𝐻𝐻r\n𝜕𝜕𝜑𝜑H∆𝜑𝜑H�+�Γ<𝑥𝑥𝑖𝑖>𝑓𝑓�𝜑𝜑H−𝜑𝜑<𝑥𝑥𝑖𝑖>�\n<𝑥𝑥𝑖𝑖>𝑎𝑎𝑎𝑎𝑐𝑐𝑐𝑐𝑎𝑎𝑎𝑎 \n                �(�𝜔𝜔2+(𝜔𝜔0\n2)2−𝜔𝜔0\n2)/(�𝜔𝜔2+(𝜔𝜔0\n2)2+𝜔𝜔0\n2)+Γtwofoldmaxcos4(φM- φtwofold)   (2) \n∆Hinh is both frequency and angle independent  term due to the  sample  \ninhomogeneity . The second term is the intrinsic Gilbert damping  (𝛼𝛼𝐺𝐺) contribution. 𝛾𝛾  \n8 \n is a correction factor owing to the field dragging effect caused by magnetic anisotropy  \n[12], 𝛾𝛾 =cos (φM-φH). The 𝜑𝜑M as a function of φH  for the sample S2 at fixed 13 GHz \nis calculated  and show n in figure 4(a).  Note that the draggi ng effect  vanishes  (𝜑𝜑M=\nφH) when the field is along the hard or  easy axes . The third term  describes the mosaicity \ncontribution originating from the angular dispersion of the crystallographic cubic axes \nand yield s a broader linewidth  [35]. The four th term is the TMS contribution. The \nΓ<𝑥𝑥𝑖𝑖> signifies  the intensity  of the TMS along the principal in -plane crystallographic \ndirection  <𝑥𝑥𝑖𝑖>. The 𝑓𝑓�𝜑𝜑H−𝜑𝜑<𝑥𝑥𝑖𝑖>� term indica tes the TMS contribution depending \non the in- plane direction of the field rel ative to <𝑥𝑥𝑖𝑖> and commonly expressed as \ncos2[2(φM-φ<xi>)] [14].  In addition, 𝜔𝜔 is the angular resonant frequency and  𝜔𝜔0=\n𝛾𝛾𝜇𝜇0𝑀𝑀eff. In our case, besides the fourfold TMS caused by expected lattice geometric \ndefects, the other twofold TMS channel is induced by the dipolar fields emerging from \nperiodic stripelike defects  [20,21] . This term is parameterized by its strength Γtwofoldmax \nand the axis of maximal scattering  rate φtwofold.  \nAs an example, t he angle- dependent linewidth measured  at 13 .0 GHz for  the \nsample S2 is  shown in f igure 4(b).  It clearly exhibits a strong in -plane anisotropy, and  \nthe linewidth along the [100] direction is significantly  larger than that along the [110] \ndirection . Taking only  isotropic Gilbert damping  into account , the dragging effect \nvanishes  with field applied  along the hard and easy axes . Meanwhile, the mosaicity \nterm gives an angular variation of the linewidth proportional to |𝜕𝜕𝐻𝐻𝑟𝑟𝜕𝜕𝜑𝜑𝐻𝐻|⁄ , which is \nalso zero along with the principal <100> and < 110>  directions. This gives direct \nevidence that the rel axation is not exclusively governed only considering the intrinsic \nGilbert mechanism and mosaicity term. Because the probability of defect formation \nalong with <100> directions is higher than that  along the <110> directions  [12], the \n9 \n TMS contribution is stronger along the easy axes , which is in accordance with t he fact \nthat the linewidth s along the [100] and [110]  direction s are non -equivalent. Moreover , \nthe linewidth of [010] direction is slightly  larger  than that along the [100] dire ction, \nsuggesting that  another twofold TMS channel is induce d by oblique deposition. As \nindicated by the red solid line  in figure 4(b), the linewidth can be well fitted. D ifferent \nparts making sense to the  linewidth can therefore be sepa rated  and summarized in Tab le \nI. As we know, the TMS predicts the curved non- linear frequency dependence of \nlinewidth, which not appear in a small frequency range for our case (as shown in f igure \n3(a)).  The linewidth as function of frequency was also well fitted including the TMS -\ndamping using the parameters in Table I  (not shown here) . \nThe larger strength of TMS along the easy axis can clearly explain the anisotropic \nbehavior of da mping , with higher value along the easy axis shown in f igure 3(b). The \nobtained Gilbert damping  factor  of ~ 7×10-3 is isotropic and invariant  with different \noblique angle s. The value of damping is slightly larger than the bulk value of 5.5×10-3 \n[29], which may be attributed to spin pumping of the Ta capping layer.   \n The obtained maxi ma of twofold TMS exhibits  an increasing  trend  with the \noblique angle  [shown in f igure 4(c)]. According  to previous works  on the shadowing \neffect [16-19], the larger  deposition angle  makes  the shadow ing effect  stronger , and the \ndipolar fields within stripe like defects  increase just like the UMA. This  can clearly \nexplain that the intensity of two fold TMS  follows exactly the same trend with the \ndeposition angle as the UMA . The axis of the maximal intensity of two fold TMS is \nparal lel to the projection of the  FeGa atom flux  from the fitting data.   As shown in Table  \nI, amazingly the modified growth conditions also influence  the fourfold TMS, \nespecially the strength of TMS along the <100>  axis. Figure 4(c) also presents the \nchanges of the fourfold TMS intensity as the  deposition angle  and shows a peak at 15° , \n10 \n which follows a similar trend as that of 𝛼𝛼eff along [100] axis as shown in f igure 3(b). \nThis indeed confirm s TMS -damping plays an important role in FeGa thin films.    \nFor the dispersion relation ω(k∥) in thin magnetic films , the propagation angle \n𝜑𝜑𝑘𝑘∥����⃗ defined as the angle between k∥���⃗ and the projection of the saturation magnetization \nMs into the sample plane is less than the critical value : 𝜑𝜑max =\n𝑐𝑐𝑎𝑎𝑎𝑎−1�𝜇𝜇0𝐻𝐻r(𝜇𝜇0𝐻𝐻r+𝜇𝜇0𝑀𝑀eff) ⁄  [9,36,37] . This implies  no degenerate modes are \navailable for the angle 𝜑𝜑𝑘𝑘∥����⃗ larger than φmax. Based on this theory, we propose a \nhypothesis that the crystallographic anisotropy  determine s the area including \ndegenerate magnon modes , as well as  the intensity of the fourfold TMS.  The resonance \nfield along <100>  axis change s due to the various crystallographic anisotropy , which \nhas a great effect on the φmax. The values of φmax of samples are shown in f igure 4(d). \nThe data follow the  same trend with the oblique angle as Γ<100>. During the grain \ngrowth,  the cubic anisotropy is  influenced possibly  since the anisotropy dispersion due \nto the stress. For the lower anisotropy of  sample S2 , a relatively larger  amount of stress \nand defects present in the sample and lead to a larger  four fold TMS.  \n4. Conclusions  \nIn conclusion, the effects of oblique deposition on the dynamic properties of FeGa \nthin films have been investigated  systematically . The pronounced TMS  as non-Gilbert \ndamping  results in an anisotropic magnetic relaxation . As the oblique angle increases, \nthe magnitude of the twofold TMS  increases due to the larger shadowing effect . \nFurthermore, the cubic anisotropy dominates  the area including degenerate magnon \nmodes, as well as the intensity of fourfold TMS. The reported results  confirm  that the \nmodified anisotropy can influence the extrinsic relaxation pr ocess  and open a n avenue \nto tailor magnetic  relaxation in spintronic devices. \n11 \n Acknowledgments  \nThis work is supported by the National Key Research Program of China (Grant Nos. \n2015CB921403, 2016YFA0300701, and 2017YFB0702702), the National Natural \nSciences Foundation of China (Grant Nos. 91622126, 51427801, and 51671212)  and \nthe Key Research Program of Frontier Sciences, CAS (Grant Nos. QYZDJ -SSW -\nJSC023, KJZD -SW-M01 and ZDYZ2012 -2). The work at Beijing Norma l University \nis partly supported by the National Natural Sciences Foundation of China (Grant Nos. 61774017, 61704018, and 11734004), the Recruitment Program of Global Youth \nExperts and the Fundamental Research Funds for the Central Universities (Grant No. \n2018EYT03).  \n12 \n References  \n[1] Slonczewski J C 1996 J. Magn. Magn. Mater. 159  L1 \n[2] Žutić  I, Fabian J and Das Sarma S 2004 Rev. Mod. Phys. 76 323 \n[3] Gilbert T L2004 IEEE Trans. Magn. 40 3443  \n[4] He P, Ma  X, Zhang  J W, Zhao  H B, Lüpke  G, Shi  Z and Zhou S M 2013 Phys. \nRev. Lett.  110 077203 \n[5] Heinrich B, Meredith D J and Cochran J F 1979 J. Appl. Phys. 50 7726 \n[6] Lee A J, Brangham  J T, Cheng  Y, White  S P, Ruane  W T, Esser  B D, \nMcComb  D W, Hammel  P C and Yang  F Y 2017 Nat. Commun. 8 234 \n[7] Scheck C, Cheng L, Barsukov I, Frait Z and Bailey W E 2007 Phys. Rev. Lett.  \n98 117601 \n[8] Azzawi  S, Hindmarch A and Atkinson D 2017 J. Phys. D: Appl. 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B  80 224421  \n \n \n  \n   \n            \n15 \n Figure Captions  \nFigure  1 (color online) (a) Schematic illustration of the film deposition geometry and \ncoordinate system (b) I n-plane hysteresis loops of samples S1 and S4 with the field \nalong [100], [110], and [010]. The inset shows the polar plot of the normalized \nremanence (M r/Ms) as a functi on of the in- plane angle.  FMR spectrum for the sample \nS2 with H along [100] and [110] axes showing the real (c ) and imaginary (d ) part s of \nthe S 21.  \nFigure  2 (color online) (a) H r vs. φH for FeGa films. (b) The  anisotropy  constants  K4 \nand Ku vs. deposition angle. (c) f  vs. Hr plots measured at H //[100], Symbols are \nexperimental data and the solid lines are the fitted results.  \nFigure  3 (color online) (a) ∆H as a function of f  for samples S1 and S2 with  field along \neasy and hard axis. (b) The dependence of the damping parameter on  the oblique angle \nwith field along [100] and [110] directions. (c) The calculated damping of FeGa alloy \nas a function of lattice distortion.  Figure  4 (color online)  (a) φ\nM and (b) ∆H as a function of φH for the  sample S2  \nmeasured at 13.0 GHz. (c) Oblique angle dependences of Γ<100> and Γtwofoldmax. (d) The \nlargest angle including degenerate magnon modes as a function of the oblique angle \nwith the applied field along <100> direction. \nTable Caption  \nTable I. The magnetic relaxation parameters of the FeGa films prepared via oblique \ndeposition (with experimental errors in parentheses).  \n \n  \n16 \n Figure 1  \n \n \n  \n \n  \n \n  \n \n  \n \n17 \n Figure  2 \n \n \n \n \n \n  \n \n  \n \n  \n \n  \n \n18 \n Figur e 3 \n \n \n  \nFigure  4 \n \n \n \n \n \n \n \n19 \n TableⅠ \nSample  𝜇𝜇0ΔHinh\n(mT) 𝛼𝛼G Δ𝜑𝜑H \n(deg.) Γ<100> \n(107Hz) Γ<110> \n(107Hz) Γtwofoldmax\n(107Hz) 𝜑𝜑twofold  \n(deg. ) \nS1 0 0.007  0.62 17(3)  5.8(1.8)  0(2) 90 \nS2 0.7 0.007  1.2 81.4(3.7)  9.3(1.9)  7.4(3) 90 \nS3 0 0.007  1.0 59.2(4.5)  11.1(2) 13(3.7)  90 \nS4 0 0.007  1.1 33.3(6)  14.8(3.7)  26(4)  90 \n \n20 \n "    },    {        "title": "1911.01173v2.DFT_U_Investigation_of_magnetocrystalline_anisotropy_of_Mn_doped_transition_metal_dichalcogenides_monolayers.pdf",        "content": "DFT+U Investigation of magnetocrystalline anisotropy of\nMn-doped transition-metal dichalcogenides monolayers\nAdlen Smiri\nFacult\u0013 e des Sciences de Bizerte, Laboratoire de Physique des Mat\u0013 eriaux: Structure et Propri\u0013 et\u0013 es,\nUniversit\u0013 e de Carthage, 7021 Jarzouna, Tunisia and\nLPCNO, Universit\u0013 e F\u0013 ed\u0013 erale de Toulouse Midi-Pyr\u0013 en\u0013 ees,\nINSA-CNRS-UPS, 135 Avenue de Rangueil, 31077 Toulouse, France\nSihem Jaziri\nFacult\u0013 e des Sciences de Bizerte, Laboratoire de Physique des Mat\u0013 eriaux: Structure et Propri\u0013 et\u0013 es,\nUniversit\u0013 e de Carthage, 7021 Jarzouna, Tunisia and\nFacult\u0013 e des Sciences de Tunis, Laboratoire de Physique de la Mati\u0013 ere Condens\u0013 ee,\nD\u0013 epartement de Physique, Universit\u0013 e Tunis el Manar,\nCampus Universitaire 2092 Tunis, Tunisia\nSamir Lounis\nPeter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich and JARA, 52425 J ulich, Germany\nIann C. Gerber\u0003\nLPCNO, Universit\u0013 e F\u0013 ed\u0013 erale de Toulouse Midi-Pyr\u0013 en\u0013 ees,\nINSA-CNRS-UPS, 135 Avenue de Rangueil, 31077 Toulouse, France\nDoped transition-metal dichalcogenides monolayers exhibit exciting magnetic\nproperties for the bene\ft of two-dimensional spintronic devices. Using density func-\ntional theory (DFT) incorporating Hubbard-type of correction (DFT+ U) to account\nfor the electronic correlation, we study the magnetocrystalline anisotropy energy\n(MAE) characterizing Mn-doped MS 2(M=Mo, W) monolayers. A single isolated\nMn dopant exhibits a large perpendicular magnetic anisotropy of 35 meV (8 meV)\nin the case of Mn-doped WS 2(MoS 2) monolayer. This value originates from the\nMn in-plane orbitals degeneracy lifting due to the spin-orbit coupling. In pairwise\ndoping, the magnetization easy axis changes to the in-plane direction with a weakarXiv:1911.01173v2  [cond-mat.mtrl-sci]  15 Mar 20212\nMAE compared to single Mn doping. Our results suggest that diluted Mn-doped\nMS2monolayers, where the Mn dopants are well separated, could potentially be a\ncandidate for the realization of ultimate nanomagnet units.\n\u0003igerber@insa-toulouse.fr3\nI. Introduction\nInducing magnetism in nonmagnetic semi-conductor atomically thin monolayers (MLs)\nis a current \feld of investigation in material science to reach applications in storage and\nquantum spin processing. Interestingly, only very few experimental works on doping MoS 2\nMLs with other transition metals [1, 2] are available, when most of the reported studies are\ntheoretical ones [3{9]. It has been shown that doping can induce strong ferromagnetism [2{6,\n8{12] and large magnetic anisotropy that corresponds to direction-dependent magnetism [7,\n13{16] in two-dimensional (2D) materials. MoS 2ML appears to be an emblematic case in the\nfamily of transition metal dichalcogenide semiconductors (TMDs) since it possesses peculiar\nphysical properties [17{21]. It is characterized by a robust excitonic binding energies of\nhundreds of meV which suggests several potential applications such as laser or light-emitting\ndiodes \felds [1, 22{24]. Furthermore, MoS 2ML has speci\fc transport properties [25{30]. In\nparticular, this 2D semiconductor has high low-temperature electron mobility (up to 1000\ncm2/Vs) [25{28] and low-power dissipation [27, 29, 30] which makes it a good candidate to\nbuild transistors [27, 29{31]. Thus inducing magnetism in this type of well-featured materials\nis of \frst importance to address spintronic applications [1, 2, 4, 5, 9]. The substitution of\nMo atoms in 2 H-MoS 2ML by Mn ones is one way to realize promising magnetic 2D-TMDs\ncandidates [1, 2, 4, 5, 9]. From a thermodynamical point of view, exchanging Mn at Mo sites\nis found energetically favorable under S-rich regime [9]. Moreover, it has been shown that\nMn dopants clustering within MoS 2ML appears to be a more stable con\fguration than the\nwell-dispersed Mn dopants case [8]. Experimentally, doped Mn-MoS 2ML samples have been\nrealized either by using vapor phase deposition techniques for low doping concentration [17],\nor by a hydrothermal method to reach higher doping concentration [2]. Theoretical studies\nhave shown that doping MoS 2ML with Mn promotes a strong ferromagnetism with high\nCurie temperatures [1, 2, 4, 5, 9].\nIn addition to the desired goal of high Curie temperature in 2D-TMDs, it is highly\ndesirable to have a large magnetic anisotropy energy (MAE), which de\fnes the energy\nbarrier preserving the magnetic moments in a preferential direction against thermal \ructu-\nations at room temperature [32{34]. In order to achieve high density magnetic data storage\ndevices, the magnetic anisotropy induced by single adatoms or single atoms is of great\ninterest [7, 16, 35{37]. For substitutional Mn-doped MoS 2ML, to our knowledge most of4\nthe studies focused on the ferromagnetism aspect without addressing its MAE. The only\nworks dealing with MAE estimates in doped TMDs, have considered (i) magnetic adatoms\n(Mn or Fe) on MoS 2ML containing S di-vacancies [7] which reach MAE values of few meVs\nwith a preferential in-plane magnetization for Mn adatom and a preferential out-of-plane\nmagnetization for a single adsorbed Fe; (ii) the doping of WS 2ML by substitutional Co and\nFe atoms that can achieve large perpendicular magnetic anisotropy of few tenths of meV [16].\nIt is well known that in pristine 2H-MS 2ML (with M=Mo, W), under a trigonal pris-\nmatic coordination, the d-M orbitals are split in three categories [38, 39] in the associated\npoint group D3h(\fgure 1). The M lowest energy orbital corresponds to dz2which re\rects\nits weak coupling with the surrounding atoms. The M intermediate energy orbitals dxyand\ndx2\u0000y2are coupled to each other, when the highest energy orbitals dxzanddyzare strongly\ncoupled with 3 pS atom. As a consequence, those orbitals dominate the conduction and\nvalence band edges characters in MS 2ML [39{41], and are the origin of strong spin-orbit\ncoupling (SOC) in the valence band [42, 43] in K-point of the Brillouin zone, up to 426\nand 150 meV for WS 2and MoS 2MLs, respectively. Any substitution of a M atom by a\ndopant, such as a Mn atom, should lead to a local orbital redistribution with respect to\nthe energy due to crystal \feld e\u000bects [3, 4, 9], with spin splitting. The dopant orbital\nmagnetic moment direction is thus settled by both SOC and the crystal \feld e\u000bect, possibly\ninducing MAE [42, 44]. Understanding and possibly tuning the MAE of Mn-doped MS 2\nML appear crucial to prospect the potential of these materials in information storage devices.\nIn this work, density functional theory (DFT) corrected by a Hubbard term (DFT+ U)\nto account for strong electronic correlation in 3 dorbitals, is considered to study the pos-\nsible magnetic anisotropy in the Mn doped MS 2ML systems. The importance of such\ncorrelations were previously recognized in bulk 3 dNi and Fe atoms, where their account\nled to MAEs with a better agreement with experiments [45, 46], since these e\u000bects can\ntrigger large orbital magnetic moments [45]. Another possibility as in Ref. [38], would be\nto use hybrid functionals, to provide a better localization of the 3 dorbitals, as proposed\nto obtain reliable spin orbit splitting of defect states at the origin of the MAE. However\nthe computational cost associated to this task remains very too large. Hence, taking into\naccount the U-dependence, as done in the present work, is necessary to establish a complete5\nFIG. 1: Schematic diagram explaining the M (M = Mo or W) d-orbitals splitting under\ncrystal \feld of the trigonal prismatic symmetry. The dorbitals of the isolated M atom\n(left) split into three categories (middle) for MS 2MLs. Thedorbitals are expected to\nshow, locally, further splitting when the Mn dopant takes the place of the M atom.\nstudy of the magnetic anisotropy of Mn-MS 2doped MLs.\nUsing DFT+ Umethod, we have estimated and compared the MAE in single and pairwise\nMn-doped MS 2MLs. In particular, we have \frst investigated the MAE induced by a single\nisolated Mn dopant, representative of a moderate dopant's concentration around 4%. Then,\nin view of the preferential clustering of substitutional Mn atoms in MS 2ML, we have studied\nthe MAE's sensitivity to the separation distance between Mn atoms. To this end, various\ncon\fgurations have been constructed by placing two Mn atoms in M sites of a zigzag or/and\narmchair patterns with di\u000berent Mn-Mn separations. In addition, for relevant doping cases,\nthrough the analysis of dopant SOC we have investigated and rationalized the MAE's origin.6\nII. Methods and Computational details\nSpin-polarized DFT as implemented in Vienna ab initio simulation package (VASP)\nwas used in this work [47, 48]. The core potential was approximated by the projected\naugmented wave (PAW) scheme [49, 50]. Perdew-Burke-Ernzerhof (PBE) formulation of\ngeneralized gradient approximation (GGA) was applied to describe the exchange-correlation\ninteraction [51]. In addition, a Hubbard U correction [52] was adopted in order to better\ndescribe the localization of 3 dMn orbitals. The criteria of atom force convergence, used\nfor all structure relaxations, was \fxed to 0.02 eV/ \u0017A, with an energy cuto\u000b of 400 eV. In\norder to model the geometry of Mn-doped MS 2MLs, supercells of size 5 \u00025\u00021 and 7\u00027\u00021\nwere considered, with a 20- \u0017A-thick vacuum region to separate adjacent MLs. The initialized\nlattice parameters are equal to 3.18 \u0017A for WS 2ML and 3.20 \u0017A for MoS 2ML. To calculate the\nMAE, a 4\u00024\u00021, 7\u00027\u00021 and 9\u00029\u00021 \u0000-centered Monkhorst-Pack grid with the tetrahedron\nsmearing method of Bl ochl [53] were tested to obtain accurate results. For density of state\ncalculations, the k-points grid has been increased to 12 \u000212\u00021 and the smearing method\nwas switched to the Gaussian type with 0.02 eV width.\nThe magnetic anisotropy energy of Mn-doped MS 2is determined by performing a fully\nself-consistent SOC and non-collinear calculations with di\u000berent orientations of the magnetic\nmoments of Mn impurities. In particular, we calculate the total energies ExandEzwhen\nthe magnetization direction is parallel to x- andz-axis, respectively. Therefore, the MAE is\ngiven by\nMAE =Ex\u0000Ez; (1)\nwherex;y;z represent the crystalline axes with the z-axis perpendicular to the ML plane.\nPositive MAE stands for preferential out-of-plane magnetization corresponding to perpen-\ndicular magnetic anisotropy (PMA), while negative MAE stands for a preferential in-plane\nmagnetization which is known as in-plane magnetic anisotropy (IMA). Furthermore, in order\nto elucidate the origin of MAE, spin-orbit energy is calculated, using a scalar relativistic\napproximation [54] as implemented in VASP code [55], through the following expression,\nESOC=h~2\n2m2c21\nrdV(r)\ndr^L:^Si; (2)\nHere, ^Land^Sdenote the orbital and spin angular momentum operators, respectively, when7\nV(r) is the spherical part of the e\u000bective potential within the PAW sphere.\nIII. Results and discussion\nA. Magnetic anisotropy of an isolated Mn atom in MoS 2ML\n1.U-dependence of magnetic anisotropy\nTo study the electronic correlation e\u000bects on the magnetic anisotropy of Mn-MS 2doped\nMLs, the calculated MAEs with respect to U-parameter are shown in \fgure (2a). Note here\nthat for each Hubbard term U, the geometrical optimization is repeated. In the absence of\nthe correlation correction, the easy axis of magnetization is oriented along xwhich indicates\nan IMA for the two systems. Here, the MAE value of Mn-WS 2doped ML is in good agree-\nment with that of Ref. [16]. We \fnd that without U, the magnitude of MAEs are almost\nthe same for both investigated systems. However, by introducing the correlation e\u000bects,\na change in the magnetization easy axis occurs at U= 2 eV resulting in an energetically\nfavored PMA states, with MAEs much larger for Mn-WS 2ML than for Mn-MoS 2ML. A\nsimilar change of MAE sign under U-parameter variation was found for Fe and Ni atoms [45].\nWe note that Mn-WS 2ML MAE is larger than Mn-MoS 2ML one.\nIn particular, for U > 1 eV, Mn-MoS 2doped ML exhibits a large MAE of 8 meV com-\npared to the results of Cong et al. , [7] where the largest MAE value was found to be 1.3 meV\nfor Mn adatom implanted in a di-sulfur vacancy in MoS 2ML. Higher MAEs are found in the\ncase of Mn-WS 2doped ML for the same range of U. Indeed, the MAE reaches 30, 35 and 24\nmeV for the Hubbard parameter of 2, 3 and 4 eV, respectively. A thermal stability at room\ntemperature of the magnetic anisotropy can be envisaged in the case of Mn doped-WS 2ML.\n2. In\ruence of crystal structure on the magnetic anisotropy\nOwing to the Hubbard U-parameter, the equilibrium atomic positions can vary and thus\nimpact the electronic structures Hence, a connection between the MAE variation and the8\n(a)\n(b)\nFIG. 2: (a) The magnetic anisotropy energy (MAE) of Mn-MS 2doped MLs as a function\nof the Hubbard parameter U. (b) The length Lof the bonds formed between Mn dopant\nand the \frst nearest neighboring S host atoms as a function of U.\nstructural modi\fcations, both under U-parameter, is possible. To reveal the e\u000bect of cor-\nrelations on the local crystal structure, the S-Mn bond length ( L) is plotted in \fgure (2b)9\nas a function of U-parameter upon geometry optimization. Interestingly when Uis in-\ncreased,Lbecomes larger too. In the meantime, the on-site Coulomb correlation enhances\nthe magnetic anisotropy magnitude. The enhancement of MAE seems to be correlated to\nthe increase in the S-Mn bond lengths suggesting a common electronic origin. In term of\nelectronic structure, the Lincrease results in a decrease of the overlapping between 3 pS\nand 3dMn orbitals. To explore the orbital distribution and hybridization in Mn doped MS 2\nMLs, the projected density of states (pDOS) for DFT and DFT+ Uare shown in \fgure (3).\nFrom DFT, the 3 dMn orbital energy level split into three di\u000berent levels under crystal\n\feld e\u000bect of the trigonal prismatic coordination: a twofold degenerate level containing the\nout-of-plane orbitals ( dxz,dyz) with the highest energies among 3 dMn levels; a twofold\ndegenerate level containing the in-plane orbitals ( dxy,dx2\u0000y2) with an intermediate energies\nand non-degenerate level containing the perpendicular orbital ( dz2) with the lowest energies.\nThedxzanddyzlevel has the highest energy because they are oriented toward the chalcogen\natoms as indicated in the \fgure (1). Hence, dxzanddyzorbitals have a greater overlapping\nwith 3pS orbitals than the rest of Mn orbitals that are lower in energy. Interestingly, once\nthe correlation e\u000bect included, dxzanddyzlevel starts to shift toward lower energies which\nexplains the increase of the S-Mn bond lengths.\nFurthermore, the decrease of the hybridization, under Uvariation, occurs for all Mn 3 d\norbitals and not only dxzanddyzorbitals. The Mn orbitals ( dxy,dx2\u0000y2) anddz2mainly\nhybridize with the dorbitals of the metal M, see \fgure (3). For U2[0;3] for M=W and U2\n[0;4] for M=Mo (\fgure (5b)); the in-plane dxyanddx2\u0000y2levels slightly shift in energy until\nthey cross the Fermi level and exceed the spin-up dz2level. In the meantime, a spin-down\ndz2state appears close to the valence band which maintains the same energetic order of 3 d\norbitals.\nTo further elucidate the MAE connection with local modi\fcations of the geometries, we\nrecalculate the MAE with a \fxed lattice parameters of about 3.2 \u0017A obtained upon a DFT\ngeometry optimization (case (II)) for MoS 2ML. The results are plotted in \fgure (4) and\ncompared to the ones obtained after complete lattice parameter optimization for each U\nas already discussed (case (I)). When comparing the MAEs variations with Uof cases (I)\nand (II), the major di\u000berences appear at U= 2 eV for both systems and at U= 4 eV for\nMn-WS 2ML. In particular, the MAEs of case (I) shows a larger (smaller) values at U= 210\n(a)U= 0 eV\n (b)U= 3 eV\n(c)U= 0 eV\n (d)U= 3 eV\nFIG. 3: The DFT (a and c) and the DFT+ U(b and d) (for U= 3 eV) pDOS of 3 dstates\nof Mn dopant and dand 4pstates of its neighboring M and S atoms, respectively.\n(4) eV with respect to MAEs of case (II) for MoS 2(WS 2) MLs. Therefore, the dependence\nof the crystal structure on Ucan potentially be critical in determining the MAE based on\nthe value of Uitself.11\n(a) Mn-MoS 2doped ML\n (b) Mn-WS 2doped ML\nFIG. 4: The MAE values with respect to the values of the correlation parameter Ufor\nU-dependent (red lines) and -independent (blue lines) crystal structure of (a) Mn-MoS 2\ndoped ML and (b) Mn-WS 2doped ML.12\n(a) Mn-doped MoS 2ML\n (b) Mn-doped MoS 2ML\n(c) Mn-doped WS 2ML\n (d) Mn-doped WS 2ML\nFIG. 5: The total electronic density of states (DOS) of U= 0 to 4 eV for (b) U-dependent\nand (c)U-independent crystal structure of Mn-doped MoS 2ML and Mn-doped WS 2ML.\nWe try now to elucidate the electronic origin of the changes and similarities between\nMAEs obtained in the cases (I) and (II) with or without geometry optimizations. In par-\nticular, for each Uparameter, the DOS is represented for U-dependent and -independent\ncrystal structures in \fgure (5). For U < 2eV, the DOS of case (I) and (II) show comparable\nfeatures which explain the coincidence between the corresponding MAEs. In particular,13\nthe Fermi level lays between the dz2- anddxy/dx2\u0000y2-orbitals. For U= 2 eV, the latter\nelectronic properties are obtained in the case (II), whereas, the Fermi level crosses the\nspin-up bands of the two-fold degenerated dxy/dx2\u0000y2orbitals in the case (I). This explains\nthe sudden increase of MAE at U= 2 eV when the U-dependence of the crystal structure is\nconsidered. For U > 2 eV, a semi-metallic behaviour is observed for both cases. For U= 4\neV, the di\u000berence between the MAEs of the cases (I) and (II) in Mn-WS 2system, originates\nfrom the orbital type present at the Fermi level. Furthermore, by comparing the 3 denergy\npositions in U-dependent and -independent DOS (\fgure (5)), it appears that the on-site\nCoulomb correlation weakens the hybridization between Mn and its environment, Regarding\nMAE variations, the bond-length increase due to Ugeometrical optimization is important\nonly whenU= 2 eV for the two MLs and also for U= 4 eV in the case of Mn-WS 2doped ML.\n3. Connection between magnetic anisotropy and orbital magnetic moments\nU(eV) Mn-MoS 2doped ML Mn-MoS 2doped ML\nmt\nz(\u0016B)mt\nx(\u0016B)mt\nz(\u0016B)mt\nx(\u0016B)\n0 -0.00 -0.01 -0.03 -0.03\n1 -0.00 -0.01 -0.03 -0.02\n2 -0.38 -0.01 -0.46 -0.02\n3 -0.36 -0.02 -0.41 -0.03\n4 -0.34 -0.02 -0.32 0.02\nTABLE I: The orbital magnetic moments and the atomic contributions of Mn dopant to\nand spin-orbit energy for in plane and out of plane magnetization orientation.\nThe relationship between magnetic anisotropy and orbital moment anisotropy (OMA)\nwere treated by di\u000berent models in order to establish a qualitative understanding of MAE\norigin [56, 57]. The OMA is given by the di\u000berence between the in-plane orbital mag-\nnetic moment mxand perpendicular orbital magnetic moment mz, i.e \u0001m=mx\u0000mz.\nTo reveal the relationship between MAE and OMA, the total OMA (\u0001 mt) and the local14\nOMA of Mn dopant (\u0001 mMn) as a function of Uare presented in \fgure (6a). In particular,\nthe MAE can be related to the orbital magnetic moment through Bruno's formula [56],\nMAE=\u0018\n4\u0016B\u0001mMn. Here, the constant \u0018stands for the SOC strength. In other words, the\nmagnetization easy axis coincides with the direction that has the largest orbital moment [56].\nFor both Mn-MoS 2and Mn-WS 2systems, the coincidence between the magnetization easy\naxis and the direction of largest orbital magnetic moment is obtained for each Uvalue.\nIndeed, by comparing MAE to the local OMA in \fgure (6a), for a given Uvalue, the easy\naxis and the largest orbital moment are along the same direction. Regarding the change of\nlocal OMA upon the correlation e\u000bects, the sign reversal is consistent with that of MAE.\nHowever, despite the fact that the increase of Uenhances both the local OMA and MAE,\ntheir variations from U= 2 to 4 eV are not similar. In particular, although the U= 3 eV\nresults in the largest MAE in Mn-MoS 2and Mn-WS 2systems, the local OMA associated\nwith this value is not the largest among the rest of Uvalues.\nFurthermore, the comparison between MAE and the total OMA which includes di\u000berent\ncontributions from the Mn environment, can be also tested. To this end, the total OMA\nversus the Hubbard parameter Uis plotted in \fgure (6a). For the two Mn-MoS 2and Mn-\nWS2systems, the correlation between MAE and the total OMA is shown by their sudden\nincrease at U= 1 eV. Unlike MAE, there is no sign reversal of the total OMA at U=\n1 eV in the case of Mn-WS 2doped ML. This may due to the fact that the Bruno model\nis formulated for single atom but not for structures consisting of multiple atomic species\nwith strong hybridization and large spin-orbit interaction [57]. However, this sign reversal,\nconsistently with MAE, occurs for Mn-MoS 2doped ML.\nOverall, the MAE's enhancement is related to the OMA's increase for both the considered\ndoped MLs. In particular, according to the table (I), the absolute values of the perpendicu-\nlar orbital moment jmt\nzjbecome considerably large when the correlation parameter increases\nfrom 0-1 eV to 2-4 eV. However, the in-plane orbital magnetic jmt\nzjmoments remain more or\nless constant. Therefore, the enhancement in perpendicular magnetic anisotropy is mainly\ndue to the enhancement of the perpendicular orbital magnetic moments imposed by the\non-site electron correlation.\nThe behaviour of both MAE and OMA suggests the perpendicular orbital moment en-15\nhancement and the induced perpendicular magnetic anisotropy in the MLs have a common\nelectronic origin. According to Refs. [46, 58{60], for highly localized 3 d-orbitals including\nspin-orbit interaction and strong intra-atomic electron correlation, the orbital magnetic mo-\nment is anticipated to be large. In particular, this localization of the 3 dorbitals allows atoms\nto retain their large atomic moments [46, 58{60]. However, strong hybridization leads to the\nbroadening of the 3 dbands producing much smaller orbital magnetic moments [46, 58{60].\nAs shown in \fgure (3), pDOS of 3 dstates of Mn dopant and dandpstates of its neighboring\nM and X atoms, respectively, for U= 0 eV and U= 3 eV. In both cases, U= 0 eV and\nU= 3 eV, the high localization of Mn 3 dstates is obvious. The inclusion of Upushes the\n3dMn localized states towards the Fermi level which weakens the overlapping between 3 d\nMn and its neighboring 3 pan 4dorbitals of M and X host atoms. This e\u000bect enlarges the\norbital magnetic moments and hence the MAE.\n4. Origin of magnetic anisotropy\nThe spin-orbit coupling is responsible for both the orbital moment and the magnetic\nanisotropy. The magnetization easy axis can be indicated by the competition between the\nsecond-order SOC energies per Mn dopants for in-plane ( Ex\nSOC) and out-of-plane ( Ez\nSOC)\nmagnetization orientations, \u0001 ESOC=Ex\nSOC\u0000Ez\nSOC[34, 61], where ESOCis given by the\nexpression (2). As we can see in the \fgure (6b), the sign of \u0001 ESOCfollows that of the\nMAE. In particular, apart from the case of U= 0 eV and 1 eV, the out-of-plane spin-orbit\nenergy is larger than the in-plane one which gives rise to the PMA. However, a di\u000berent\nbehavior between the variation of MAE and the variation of \u0001 ESOCunderUis obtained as\ndepicted in \fgure (6b). In fact, these variations of \u0001 ESOCare similar to those of the local\nOMA presented in \fgure (6a). The discrepancy between MAE and \u0001 ESOCvariations under\nU-parameter can be understandable because a large SOC does not necessarily lead to large\nMAE [62].\nThree principal e\u000bects behind the magnetic anisotropy are the spin orbit e\u000bect, the crys-\ntal \feld e\u000bect and the exchange \feld e\u000bect [34, 61]. All these e\u000bects are captured by the\nSOC second-order perturbation method [34] which is largely adopted to investigate the MAE16\norigin. In fact, using the crystal \feld and the exchange splitting of Mn 3 dorbitals together\nwith the angular momentum operators ^Lxand^Lz, the MAE can be expressed as in Ref. [34],\nMAE'\u00182X\nu\u001b;o\u001b0jhu\u001bj^Lzjo\u001b0ij2\u0000jhu\u001bj^Lxjo\u001b0ij2\n\u000eu\u001b;o\u001b0(2\u000e\u001b;\u001b0\u00001) (3)\nHere, the notations \u001b;\u001b0=\";#, indicate the spin directions of the occupied (o) and the unoc-\ncupied (u) states. \u000eu;o=Eu\u0000Eois the energy di\u000berence between (o) and (u) states. The only\nnon-zero matrix elements contributing to MAE are: hxzj^Lzjyzi= 1;hxyj^Lzjx2\u0000y2i= 2;\nhz2j^Lxjxz;yzi=p\n3;hxyj^Lxjxz;yzi= 1 ethx2\u0000y2j^Lxjxz;yzi= 1. It is seen from Eq. 3\nthat the MAE magnitude is governed by the weight of matrix elements and the energy dif-\nferences\u000eu;o. Moreover, the sign of MAE energy depends crucially on heavy matrix element,\nbe it with positive or negative signs.\nTo understand the orbital origin responsible for the easy axis reversal, we consider two\ncasesU=0 eV and U= 3 eV. In particular, we analyse the pDOS of the 3dMn orbitals to\nidentify the nonvanishing matrix elements that contribute negatively or positively to MAE.\nForU= 0 eV, according to the \fgure (3a) of Mn-MoS 2doped ML and the \fgure (3c) of\nMn-WS 2doped ML, the highest occupied states are spin up d\"\nz2orbitals. According to\nEq. (3), negative contribution to MAE comes from matrix elements, hz2\"j^Lxjxz\";yz\"i, while\npositive one corresponds to hz2\"j^Lxjxz#;yz#i. Those matrix elements have similar magni-\ntudes. Owing to the intra-atomic Hund exchange splitting (\u0001 ex), recalling that spin-up and\nspin-down states with the same symmetry have di\u000berent energies, [4] positive contributions\nofhz2\"j^Lxjxz\";yz\"iremain small because of the large energy denominator between the two\nstates. Hence, the MAE is dominated by the negative contribution of hz2\"j^Lxjxz\";yz\"i.\nForU= 3 eV, according to the \fgure (3b) of Mn-MoS 2doped ML and the \fgure (3d)\nof Mn-WS 2doped ML, the band of the up-spin xy=x2\u0000y2\"is half occupied. This behavior\ndominates both the sign and value of the MAE by favoring the PMA. According to the\nexpression (3), the dominant contribution is positive and stems from the non-zero matrix\nelementhxyj^Lzjx2\u0000y2iwhich is consistent with the DFT+ Ucalculated MAE. However, the\ndegeneracy of the in-plane orbitals predict an in\fnite MAE which makes the second order\nperturbation method unreliable in this case. In Ref. [63], it was demonstrated that when the\ntwo orbitals dxyanddx2\u0000y2are degenerate, and being at Fermi level, it favors the out-of-plane17\norientation magnetization. Indeed, the spin-orbit energy splitting reaches its maximum for\nthese degenerate states when the magnetization is parallel to z-axis [63]. Therefore, the\nPMA originates from the spin-orbit splitting of the degenerated Mn ( d\"\nxy;d\"\nx2\u0000y2)-orbitals\nplaced at the Fermi level by means of the correlation e\u000bect.\nFurthermore, using DFT and DFT+ U, the decomposed \u0001 ESOCon Mn 3 d-orbital, are\npresented in \fgure (7) for both Mn-MoS 2and Mn-WS 2doped MLs. The decomposed \u0001 ESOC\nvaries signi\fcantly by including the correlation e\u000bects. For the DFT case, the dominating\ncontributions comes from the spin-orbit matrix elements involving either dx2\u0000y2anddxy.\nHowever, according to the pDOS, these coupled states are not occupied whereas only the\nSOC between the occupied and unoccupied states contributes to the magnetic anisotropy.\nThe remaining large contribution comes from the coupling between dz2anddyzorbitals\nwhich provides a negative contributions to \u0001 ESOC. Therefore, the negative MAE found\nforU= 0 eV originates from the spin-orbit coupling between dz2anddyzorbitals. This\nresult is consistent with the second order perturbation theory method, modeled by Eq. (3),\nwhere the matrix element hz2\"j^Lxjxz\";yz\"iis found dominant. For DFT+ U, an important\nenhancement is observed for the magnitude of spin-orbit matrix elements involving dx2\u0000y2\nanddxystates. These matrix elements are the major contribution to \u0001 ESOC. Together\nwith the semi-metallic character of the in-plane orbitals, it is clear that they are behind the\nenhancement of MAE due to the correlations e\u000bect U.\nThe di\u000berence in MAE between the two MS 2materials is related the behaviour of the\nin plane orbitals under spin-orbit coupling. The DOSs in \fgure (8) show that the spin-\norbit splitting ( \u000eSO) of (dx2\u0000y2,dxy)-orbital energy is more important in Mn WS 2system.\nThis explains why Mn-WS 2has larger MAE than that of Mn-MoS 2ML. This observation\nmeans the the host TM material of Mn plays a critical role in determining the Mn induced\nanisotropy. In fact, for WS 2ML, the spin-orbit coupling is stronger than for MoS 2ML\nbecause W is heavier than Mo [43]. In particular, the upper valence band of MS 2around\nK point, showing strong spin-orbit splitting, is mainly composed by in-plane p-S andd-M\norbitals [43]. These latter hybridise with Mn in-plane orbitals leading to a stronger spin-orbit\nsplitting in the case of M=W.18\nB. E\u000bects of Mn-separation on MAE\nIn this section, we treat the case of Mn pairs implanted in a MoS 2ML using 5\u00025\u00021\nsupercell (see \fgure (9)). Depending on Mn-Mn distance, one can generate four di\u000berent\ndoping con\fgurations: (i) \frst nearest neighbor (nn) con\fguration in which the two Mn\nimpurities are in the nn positions of the zigzag chain (1 nn-z), (ii) the 2-nn con\fguration in\nwhich the two Mn impurities are in the second nn positions of the armchair chain (2 nn-a),\n(iii) the 3-nn con\fguration in which the two Mn impurities are in the third nn positions of\nthe zigzag chain (3 nn-z) and (iv) the 4-nn con\fguration in which the two Mn impurities\nare in the nn positions of the armchair chain (4 nn-a).\n1. Structural stability\nWe study the structural stability of the di\u000berent con\fgurations through the relative\nenergy presented in \fgure (10). The energy as a function of the Mn-Mn distance shows\na behavior that di\u000bers from MoS 2ML to WS 2ML. In particular, for Mn-MoS 2system,\nthe relative energy increases with the increase of the dopant separation distance. Hence,\nthe doped MoS 2structure is more stable when the Mn impurities are close to each other,\ni.e 1 nn-z case. For Mn-WS 2system, the relative energy does not show a monotone de-\npendence on Mn-Mn distance. This may be due to the e\u000bect of the chain type (zigzag or\narmchair) on which the two dopants are placed. Same as Mn-MoS 2system, in Mn-WS 2\ndoped ML, the lower relative energy is found for the doping con\fguration 1 nn-z. Therefore,\nin both systems, Mn-MS 2MLs, Mn impurities prefer to be in the closest substituting M-sites.\n2. Magnetic anisotropy\nThe magnetic anisotropy properties of Mn-MoS 2and Mn-WS 2systems are shown in\n\fgure (11), in which MAEs are given for all doping con\fgurations as a function of Mn-Mn\ndistance. For the case of the host MoS 2, MAE increases and changes its sign with the\nincrease of Mn-Mn distance. Hence, the magnetic anisotropy goes from IMA of -1.2 meV to19\nPMA of about 1 meV when the Mn-Mn distance increases. In particular, it is found that\n1 nn-z and 2 nn-a con\fgurations have IMA, whereas the 3 nn-z and 4 nn-a con\fgurations\ncorrespond to PMA. For WS 2host material, MAE remains negative for all Mn-Mn distances\nwith a minimum of -8 meV for 1 nn-z con\fguration. This latter corresponds to a large in-\nplane magnetic anisotropy. Besides for Mn WS 2doped ML, MAE is not monotonically\ndepending on Mn-Mn distance. Although the Mn-Mn distances are roughly equal in the\ntwo con\fgurations 2 nn-a and 3 nn-z, their corresponding MAEs are di\u000berent. These results\nsuggest that the type of atomic chain has an important role in determining MAE. Thus,\nwe can say that for zigzag or armchair con\fgurations, MAE increases with the increase of\nMn-Mn distance. Overall, by comparing the MAEs obtained here and that of the single Mn\ndoping cases, it is obvious that the diluted doping limit is required to enough the maintain\na large PMA.\n3. Origin of magnetic anisotropy\nIn this section, as the Mn impurities atoms prefer to stay in the nearest positions at\nhigh concentrations, we choose the 1 nn-z MS 2con\fgurations for further study of magnetic\nanisotropy origin. The change of MAE in the case of the Mn-pairwise doping with respect\nto Mn-single doping can be related to the modi\fcation of Mn 3 dstates. Indeed, putting\nanother close dopant in the structure leads to the redistribution of 3 dstates which is cor-\nrelated with structural modi\fcations around dopants. Therefore, we explore the structural\nand electronic feature changes induced by Mn-pairing in the 1 nn-z MS 2con\fgurations. The\nMn-S bond lengths are given in \fgures (9a). In the case of single dopant, the three Mn-S\nbonds are equal ( L) obeying the D3hsymmetry. Contrary to the case of single dopant,\nwhich obeys the D3hsymmetry; the lengths are no longer equal, indicating a symmetry\nreduction. The lengths of Mn 1-S and Mn 2-S bonds that relate the two Mn atoms are the\nlargest lengths. This means that the two Mn dopant repel each other. In the meantime,\nthe distances of Mn 1-S and Mn 2-S bonds, that relate each Mn atom to the nearest M of\nthe zigzag chain, are the lowest lengths. To analyse the electronic structure, the pDOS are\nplotted in \fgures (12a) and (12b). According to the \fgures (12a) and (12b), by placing the\nsecond Mn atom in a metal site close to the \frst one, induces electronic structure modi\fca-\ntions. Considering in-plane orbitals, their degeneracy is lifted as shown in the \fgures (12a)20\nand (12b). In fact, the Mn 1xy-orbital is coupled to Mn 2dx2\u0000y2-orbital as they are pointed\ntoward each other because of the hexagonal geometry. The remaining Mn 1dx2\u0000y2- and Mn 2\ndxy-orbitals are mostly coupled to the dorbital of the surrounding TM host atoms and they\nshowing slightly di\u000berent behaviour compared to each other. A quite similar e\u000bect is shown\nby out-of-plane orbitals ( dxz,dyz) where their associated degeneracy is lifted, as it can be\nseen in \fgures (12a) and (12b). In particular, the Mn 1dyz- and Mn 2dxz-orbital hybridize\nwith the same 3 pof S cation while the Mn 1dxzand Mn 2dyzorbitals are coupled to S atom\nwith Mo or W environment.\nWe now discuss MAE origin of Mn-MoS 2doped ML, upon Mn-pairing. According\nto \fgure (12a), the occupied states correspond to the orbitals dxy,dx2\u0000y2anddz2, while\nthe unoccupied states correspond to dxy,dx2\u0000y2,dxz, anddyz. For Mn 1case, the promi-\nnent coupling elements are hxy\"j^Lzjxz;yz\"iandhz2#j^Lzjxz;yz\"i. In particular, for Mn 2,\nthe prominent coupling elements are hx2\u0000y2\"j^Lzjxz;yz\"iandhz2#j^Lzjxz;yz\"i. Here,\nhxy\"j^Lzjxz;yz\"igives a positive contribution to MAE while hz2#j^Lzjxz;yz\"igives a nega-\ntive contribution. Because of the dominance of hx2\u0000y2\"j^Lzjxz;yz\"iandhxy\"j^Lzjxz;yz\"i,\nthe 1 nn-z con\fguration of Mn-MoS 2ML prefers IMA. In the case of Mn-MoS 2doped\nML, the matrix elements involving minority spin d#\nxy,d#\nx2\u0000y2,d#\nxzandd#\nyzstates yield pos-\nitive contributions to MAE. In particular, for Mn 1, the positive contribution comes from\nhxy#j^Lzjxy;x2\u0000y2#iandhxy#j^Lzjxz;yz#i. Besides, for Mn 1, the positive contribution comes\nfromhx2\u0000y2#j^Lzjxy;x2\u0000y2#iandhx2\u0000y2#j^Lzjxz;yz#i. All these positive contributions\n\fnally lead to IMA for the 1 nn-z con\fguration of Mn-WS 2ML.\nIV. Conclusion\nBy performing DFT+U calculations, we have investigated the magnetic anisotropy in-\nduced by Mn doping in MoS 2and WS 2MLs. In the case of a well-isolated Mn atom\nsubstituting a W center, a large positive MAE of 35 meV is obtained for Mn-WS 2doped\nML. For Mn-MoS 2doped ML a smaller MAE of 8 meV is found. These results mean that\nthe preferential direction of magnetization is perpendicular to the ML plane when the origin\nof large MAE is attributed to the presence of in-plane Mn orbitals located in the vicinity of21\nthe Fermi level. In the case of the Mn pairwise doping, a spin reorientation transition from\nout-of-plane to in-plane magnetization takes place when the Mn-Mn distance decreases. In\ngeneral, it is apparent that the clustering of Mn dopants favors the in-plane magnetization.\nOur \fndings show that important magnetic anisotropy can be found in Mn-doped WS 2and\nMoS 2ML only for considerable Mn-Mn distances.\nAcknowledgments\nS.L. acknowledges funding from the Priority Programme SPP 2244 2D Materials - Physics\nof van der Waals Heterostructures of the Deutsche Forschungsgemeinschaft (DFG) (project\nLO 1659/7-1) and the European Research Council (ERC) under the European Union's Hori-\nzon 2020 research and innovation programme (ERC-consolidator Grant No. 681405 DYNA-\nSORE). I.C.G. thanks the CALMIP initiative for the generous allocation of computational\ntime, through the project p0812, as well as GENCI-CINES and GENCI-IDRIS for grant\nA008096649.\n[1] K. Zhang, S. Feng, J. Wang, A. Azcatl, N. Lu, R. Addou, N. Wang, C. Zhou, J. Lerach,\nV. Bojan, et al., Nano Lett. 15, 6586 (2015).\n[2] J. Wang, F. Sun, S. Yang, Y. Li, C. Zhao, M. Xu, Y. Zhang, and H. 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Phys. 109, 07E143 (2011).\n[63] G. H. O. Daalderop, P. Kelly, and M. F. H. Schuurmans, Physical Review B 50, 9989 (1994).25\n(a)\n(b)\nFIG. 6: (a) The total (left) and local (right) orbital magnetic anisotropy (OMA) of\nMn-MS 2doped MLs as a functions of Hubbard parameter U. (b) The spin-orbit\nanisotropy energy of Mn-MS 2doped MLs as a functions of Hubbard parameter U.26\nFIG. 7: The DFT (a and c) and the DFT+ U(b and d) ( U= 3 eV) di\u000berence between the\nin-plane and out-of-plane d-orbital projected SOC energies of Mn dopant.27\nFIG. 8: The DFT+ UDOS di\u000berence where the spin-orbit coupling is included for\nMn-MoS 2ML (upper panel) and Mn-WS 2ML (down panel).28\n(a)1 nn-z con\fguration\n(b)2 nn-a con\fguration\n (c)3 nn-z con\fguration\n (d)4 nn-a con\fguration\nFIG. 9: Top view of the atomic structures corresponding to 8%-Mn-doped 5 \u00025\u00021 MoS 2\nsupercells. Four doping con\fgurations have considered; (a) 1 NN con\fguration, (b) 2 NN\ncon\fguration, (c) 3 NN con\fguration and (d) 4 NN con\fguration. The yellow atoms are\nthe Sulfur (S), the blue atoms are the Molybdenum (Mo) and the Manganese atom is\ndenoted by purple color.29\nFIG. 10: Mn-Mn distance dependence of relative energy of Mn pair doped (a) MoS 2ML\nand (b) WS 2ML.\nFIG. 11: Mn-Mn distance dependence of (a) magnetic anisotropy energy and (b) the\norbital magnetic anisotropy \u0001 m0values for Mn-doped MoS 2.30\n(a)Mn MoS 2ML\n(b)Mn WS 2ML\nFIG. 12: The projected electron density of states of Mn 1and Mn 2d-orbitals in (a) MoS 2\nML and (b) WS 2ML. The red and blue lines denote spin-up and spin-down channels,\nrespectively. The Fermi level is indicated by the black dashed line."    },    {        "title": "1911.03470v1.Magnetic_i_MXene__a_new_class_of_multifunctional_two_dimensional_materials.pdf",        "content": "arXiv:1911.03470v1  [cond-mat.mes-hall]  8 Nov 2019Magnetic i-MXene: a new class of multifunctional two-dimen sional materials\nQiang Gao and Hongbin Zhang∗\nInstitute of Materials Science, Technische Universit ¨at Darmstadt, 64287 Darmstadt, Germany\n(Dated: November 12, 2019)\nBased on density functional theory calculations, we invest igated the two-dimensional in-plane\nordered MXene (i-MXenes), focusing particularly on the mag netic properties. It is observed that\nrobust two-dimensional magnetism can be achieved by alloyi ng nonmagnetic MXene with magnetic\ntransition metal atoms. Moreover, both the magnetic ground states and the magnetocrystalline\nanisotropy energy of the i-MXenes can be effectively manipul ated by strain, indicating strong piezo-\nmagnetic effect. Further studies on the transport propertie s reveal that i-MXenes provide an inter-\nesting playground to realize large thermoelectric respons e, antiferromagnetic topological insulator,\nandspin-gapless semiconductors. Thus, i-MXenesare anewc lass ofmultifunctional two-dimensional\nmagnetic materials which are promising for future spintron ic applications.\nI. INTRODUCTION\nThe MAX compounds with a general chemical for-\nmula of M n+1AXn(M: early transition metal; A: main\ngroup element; X: C or N; n: integer up to 6)\nhave drawn intensive attention recently, due to poten-\ntial applications as structural, electrical, and tribologi-\ncal materials.1The corresponding two-dimensional (2D)\nnanosheets M n+1Xn, dubbed as MXene, can be ob-\ntained by etching the A elements away, e.g., Ti3C2from\nTi3AlC2.2This defines a new class of 2D materials be-\nyond graphene,3with a number of materials reported\nshowing multi-functionality covering energy, spintronic,\nnano-electronic, and topological applications.4,5For in-\nstance, theoretical calculations predicted that Ti 3C2is\na high performance anode material for lithium ion bat-\nteries,6which was confirmed later by experiments.7Al-\nthough most MXenes are metallic due to the partially\noccupied d-shells of the transition metal atoms, function-\nalization can be applied to open up a finite band gap\nthus further tailor their properties. For instance, passi-\nvated MXene Mo 2MC2(M = Ti, Zr and Hf) with the O 2\ngroup are found to be robust quantum spin Hall insula-\ntors with band gaps from 0.1 eV to 0.2 eV.8Particularly,\nthe magnetic properties of MXenes deserve further inves-\ntigation, driven by the discovery of 2D magnetic mono-\nlayers such as CrI 3and Cr 2Ge2Te6.9–11Gaoet al.12pre-\ndicted that Ti 2C and Ti 2N are nearly half-metals, which\ncan be tuned into spin-gapless semiconductor (SGS) un-\nder biaxial strain. Theoretical calculations also demon-\nstrated that Mn 2N with functional groups O, OH and F\ncan be half-metals with high Curie temperatures.13How-\never, high-throughput density functional theory (DFT)\ncalculationsrevealthat it isdifficult to obtainMAXcom-\npounds with Fe, Co, and Ni,14,15leading to a challenge\nto realize magnetic MXenes.\nRecently, the in-plane ordered MAX (i-MAX) com-\npounds with aformulaof(M 2/3M′\n1/3)2AXhavebeen syn-\nthesized by substituting 1 /3 foreign transition metal or\nrare earth element M′for M in M 2AX compounds.16,17\nBased on theoretical studies for the i-MAX compounds\n(Mo2/3M′\n1/3)2AC (M′= Sc, Y; A = Al, Ga, In, Si,Ge,In), it was found that the stable conditions for an i-\nMAX are: (1) the significant difference between the\natomic radii for the dopant metal M′and parent M; (2)\nsmall A atoms.18For the synthesized Cr-bases i-MAX\ncompounds, DFT calculations suggest (Cr 2/3Zr1/3)2AlC\nis stable in an anti-ferromagnetic spin configuration,19\nwhile (Cr 2/3M′\n1/3)2AlC (M′= Sc and Y) are probably\nstable in a mixed state due to small energy difference be-\ntweentheanti-ferromagneticandferromagneticphases.20\nThis suggests that various magnetic states are possible\nin the i-MAX compounds, and thus interesting for the\ncorresponding i-MXenes in the monolayer limit. On the\nother hand, the i-MXenes can also be obtained from i-\nMAX by etching away the A-element atoms, such as the\nordered i-MXene W 4/3C and Nb 4/3C.21,22\nIn this work, we performed systematic density func-\ntionaltheorycalculationstoinvestigatethe magneticand\nelectronic properties of i-MXenes with a general chemi-\ncal formula (M 2/3M′\n1/3)2X, focusing particularly on the\ncases where M′is magnetic. It is observed that robust\nmagnetism can be induced, and there exists significant\nmagneto-structural coupling, leading to tunable mag-\nnetic ground state and magnetic anisotropy by strain.\nMoreover, our calculations suggest that the i-MXenes\nhost fascinating transport properties, such as large See-\nbeck effect, antiferromagnetic topological insulators, and\nspin-gapless semiconductors.\nII. COMPUTATIONAL METHOD\nTo maintain reasonable computational effort, we con-\nsidered11knownnonmagneticMXeneastheparentcom-\npounds, namely, Sc 2C, V2C, Mo 2C, Nb 2C, Ta2C, Ti2C,\nZr2C, Hf2C, Ti2N, Zr2N, and Hf 2N.23The dopant ele-\nment M′is chosen to be one of the transition metal el-\nements except Tc, resulting in 319 compounds with the\ngeneral chemical formula (M 2/3M′1/3)2X. As sketched in\nFig. 1, the symmetry for the hexagonal MXenes will be\nlowered after substituting the dopants, leading to a rect-\nangular structure. In our calculations, both the hexago-\nnal and rectangular structures are considered in order to2\nunderstand the effect of strain. To determine the mag-\nnetic ground state, we consider non-magnetic (NM), fer-\nromagnetic (FM), interlayer antiferromagnetic (AFM),\nand intralayer AFM configurations for the magnetic M′\nsublattice (Fig. S2 in the Supplementary material).\nOur DFT calculations are performed in an automated\nway using the in-house developed high-throughput en-\nvironment,24–27which is interfaced to Vienna ab initio\nSimulation Package (VASP)28,29and full-potential local-\norbital minimum-basis code (FPLO).30,31The exchange-\ncorrelationfunctional in the generalized gradientapprox-\nimation (GGA) is applied, as parameterized by Perdew,\nBurke, and Ernzerhof (PBE).32To guarantee good con-\nvergence, the plane-wave energy cutoff and k-mesh den-\nsity are set as 500 eV and 60 ˚A−1, respectively. The\ncalculations to obtain optimized structures and vari-\nous magnetic configurations are carried out using VASP,\nwhile the electronic structure and physical properties are\nobtained using the FPLO and WIEN2k33codes, as de-\ntailed in our previous work.24,25,34\nFIG. 1: [a] Top view for i-MXene (M 2/3M′\n1/3)2X in both\nhexagonal lattice (red) and rectangular lattice (green). S ide\nviews for i-MXene in hexagonal [b] and rectangular [c] lat-\ntices. The ”-A” and ”-B” denote the atom above or below\nthe central layer of B in i-MXene. The original pure MXene\ncrystallizes in a hexagonal lattice. After inducing 1/3 for eign\ntransition metal dopant, the hexagonal symmetry is broken\nand tilted to the rectangular lattice.\nIII. RESULTS AND DISCUSSIONS\nAs summarized in Fig. 2, among the 319 i-MXenes\nin the rectangular geometry, 257 compounds are non-\nmagnetic, and the remaining 62 cases are magnetic (with\ntotal magnetic moments greater than 0.2 µB/f.u. in the\nferromagnetic spin configuration), where 36 (26) being\nwith ferromagnetic (antiferromagnetic) ground states.\nFurthermore, for the compounds in the imposed hexag-\nonal geometry, the magnetic M′sublattice forms a tri-\nangular lattice, which is frustrated and thus can lead\nto in-plane noncollinear magnetic structure. We found\nthat there are 13 i-MXenes with AFM intralayer ex-\nchange coupling between the moments on the M′sites(cf. TableS1inSupplementarymaterial), indicatingpos-\nsible 2D noncollinear magnetic states. Such compounds\nwill be saved for detailed investigation in the future and\nwithin the current work we focus only on the collinear\nmagnetic configurations.\nFIG. 2: Classification of the magnetic ground states for\nthe 319 i-MXene, where ”NM to FM” and ”AFM to FM”\nmark the compounds whose ground state changes by impos-\ning hexagonal geometry.\nInterestingly, comparing the magnetic ground states\nfor i-MXenes in the rectangular and hexagonal geome-\ntries, there are 6 compounds whose magnetic configura-\ntions can be changed (Fig. 2), e.g., 3 NM cases become\nFM, and 3 AFM compounds change to FM. For instance,\n(Zr2/3Ti1/3)2N changes from NM (in rectangular lattice)\nto FM (in hexagonal lattice) states with a total magnetic\nmoment of 0.95 µB/f.u.. The same NM →FM transi-\ntion occurs for (Ti 2/3Ru1/3)2C and (Zr 2/3Cu1/3)2C, with\nmagnetic moments of 0.29 and 0.30 µB/f.u., respectively.\nThe underlying mechanism can be understood based on\nthedensityofstates(DOS) ofthe Zr-5dorbitalsasshown\nin Fig. S4 in the in the Supplementary material. Based\non the Stoner model, the criterion for a stable ferromag-\nnet is\nUN(EF)>1 (1)\nwhere U is the exchange integral and N(EF) is the\nDOS at Fermi level in the paramagnetic state. For the\nrectangular lattice, the Fermi level is located around a\npeak of DOS. In the imposed hexagonal structure, the\nDOS at Fermi level is significantly enhanced from 5.2\nstates/eV/u.c. to 6.0 states/eV/u.c. (Fig. S4 in the in\nthe Supplementary material), resulting in the instability\nof becoming ferromagnetic. The same behavior is ob-\nserved also for (Ti 2/3Ru1/3)2C and (Zr 2/3Ru1/3)2C (cf.\nFig. S4 in the Supplementary material).\nFor the compounds with the AFM →FM transition,\nsuch as (Hf 2/3Fe1/3)2C and (Ti 2/3V1/3)2C, the magnetic\nmoments of the Fe (V) atoms change from 1.09 (0.58) µB3\nin the rectangular lattice to as large as 1.79 (0.64) µBin\nthe hexagonal lattice, respectively. For (Hf 2/3Fe1/3)2C,\nthe magnetic moments of Fe atoms areenhanced by 64%,\nwhich can be attributed to the enhanced exchange split-\nting (cf. Fig. S5 in the Supplementary). It also turns out\nto be a gapless semiconductor, which will be discussed\nin detail later. On the other hand, we have shown the\nband structures for (Ti 2/3V1/3)2C in Fig. S6 in the in the\nSupplementary material in both rectangular (Rect.) and\nhexagonal (Hex.) lattices together with the anomalous\nhall conductivity (AHC) for (Ti 2/3V1/3)2C in the hexag-\nonal lattice. In the rectangular lattice, (Ti 2/3V1/3)2C is\nAFM with zero AHC. Obviously, for the FM state in the\nhexagonal lattice there exists a finite AHC of -160 S/cm\nat the Fermi level. In this sense, strain can be applied to\ntune the topological transport properties, leading to the\npiezospintronic effect as discussed in the bulk materials\nas well.35,36\nItisnotedthatthelatticedeformationfromtherectan-\ngular to the hexagonal lattices is of marginal magnitude.\nFor instance, the strains along the a and b directions for\n(Ti2/3Fe1/3)2C are just about 0.14% and 0.18%, where\nthe energy difference between rectangular and hexago-\nnal lattices is as small as 1.2 meV/atom. Thus, it is\nstraighforwardtotailorthe i-MXenesfromlowsymmetry\nrectangular lattice to the hexagonal lattice with higher\nsymmetry. On the other hand, it has been previously\nreported that the piezomagnetic effect can be realized in\npure MXene M 2C (M = Hf, Nb, Sc, Ta, Ti, V, and Zr)\nby applying biaxial strain.37In this regard, we suspect\nthat the piezomagnetic effect is dramaticallyenhanced in\ni-MXene as manifested by the five aforementioned cases.\nAs the hexagonal lattice with high symmetry is more in-\nterestingand easyto obtain, wewill focus onthe physical\nproperties of such systems in the remaining part of this\nwork.\nFor two-dimensional magnets, according to the\nMermin-Wagner theorem, there exists no long range or-\ndering if there is continuous symmetry for the order pa-\nrameters.38In this regard, to stabilize 2D magnets at fi-\nnite temperature, magnetocrystalline anisotropy energy\n(MAE) is essential, which breaks the rotational symme-\ntry of Heisenberg moments.39The MAEs for the mag-\nnetic i-MXenes are evaluated using the force theorem:40\nMAE =/summationdisplay\ni∈occ.(ǫ[100]\ni−ǫ[001]\ni) (2)\nwhereǫ[001]\niandǫ[100]\nidenote the energy eigenvalues of\nthei−thbandforthe magnetizationalong[001]and[100]\ndirections, respectively.\nTable I lists the 5 i-MXenes with a MAE larger than\n0.5meV/f.u. amongallthe magnetici-MXenes(TableS2\nin Supplementary). It is noticed that (Hf 2/3Fe1/3)2C\nhas the largest MAE of 1.39 meV/f.u., favoring out-of-\nplane magnetization direction. In addition, the MAEs\nof (Zr2/3Fe1/3)2C and (Ti 2/3Fe1/3)2C are 0.74 meV/f.u.\nand 0.03 meV/f.u., respectively. Such a trend of increas-\ning MAE as X varying from Ti, Zr and Hf can be at-tributed to the variation of the strength of the atomic\nspin-orbit coupling (SOC) as 11.2 meV (Ti), 42.1 meV\n(Zr)and126.5meV(Hf).41,42Furthermore,theenhanced\nMAE of (Zr 2/3Fe1/3)2C and (Hf 2/3Fe1/3)2C is originated\nfrom the trigonal crystal fields which lead to strongly\nSOC coupled bands around the Fermi energy, as mani-\nfested by the orbital projected band structures (Fig. S9\nin the supplementary material). As shown in the or-\nbital projected band structures (Fig. S9 in the supple-\nmentary material), for (Hf 2/3Fe1/3)2C, the Fe-3d xyand\nFe-3dx2−y2bands are split by 73 meV along the K-Γ\nsection. While for (Zr 2/3Fe1/3)2C, the Fe-3d xyand Fe-\n3dx2−y2bandsaresplitby40meValongtheM-Ksection.\nAs the d xyand dx2−y2orbitals are strongly coupled by\nSOC,suchaspecial electronicstructureleadsto the large\nMAEs. Such a mechanism is in good correspondence to\nthe giant MAE realized in artificial Fe atoms adsorbed\non III-V nitride thin films with local trigonal symmetry\nas well.43\nTo obtain the Curie temperature, we took\n(Hf2/3Fe1/3)2C as an example and evaluated the\nexchange parameters between magnetic Fe atoms by\nmapping the DFT total energies to the Heisenberg\nmodel:\nH=−1\n2/summationdisplay\ni/negationslash=jJijSi·Sj (3)\nwhereJijis the exchange parameter for the local mo-\nments on the iandjsites,Si/jmarksthe on-sitespin op-\nerator. Considering three magnetic configurations, i.e.,\nFM, AFM- α, and AFM- β(cf. Fig. S2 for spin configu-\nrations AFM- βand AFM- α), the energy differences can\nbe formulated in terms of the interlayer ( Jinter) and in-\ntralayer ( Jintra) exchange parameters (cf. Fig. S3 in the\nsupplementary material for JintraandJinter):\nEFM−EAFM−α=−JinterS2, (4)\nEFM−EAFM−β=−8JintraS2. (5)\nHere we use the square of local spin moment length\nto represent the square of the spin operator. The re-\nsulting exchange coupling parameters for (Hf 2/3Fe1/3)2C\nare: J inter= 33.94 meV and J intra= 0.70 meV. Simi-\nlarly, we obtained the exchange coupling parameters for\nthe other ferromagnetic cases such as (Hf 2/3Cr1/3)2N,\n(Ta2/3Fe1/3)2C,(Ti2/3Hf1/3)2N,and(Zr 2/3Fe1/3)2C(Ta-\nble I). Obviously, the value of interlayer exchange cou-\npling is much larger than that of the intralayer coupling\nfor all the listed compounds. Due to the dramatic large\nmagnetic anisotropy, we can use the 2D Ising model to\nestimate the Curie temperature44,45\nTC=2Jinter\nkBln(1+√\n2). (6)\nwith the results listed in Table I. Surprisingly, the Curie\ntemperatures for (Hf 2/3Fe1/3)2C and (Hf 2/3Cr1/3)2N are4\nTABLE I: The basic information of the magnetic i-MXene candi dates in hexagonal lattice with an out-of-plane MAE larger\nthan 0.5 meV/f.u., including the MAE in unit of meV/f.u., the magnetic moment per magnetic atom in unit of µB, magnetic\norder, the exchange coupling parameters and the main magnet ic atom.\nCompound MAE Magnetic Magnetic Magnetic JinterJintraTC\nmeV/f.u. Moment order atom meV meV K\n(Ta2/3Fe1/3)2C 0.86 1.82 AFM Fe -10.05 0.56 -\n(Zr2/3Fe1/3)2C 0.74 1.71 FM Fe 10.16 -2.24 267.54\n(Hf2/3Fe1/3)2C 1.39 1.79 FM Fe 33.94 0.70 893.67\n(Hf2/3Cr1/3)2N 0.76 1.01 FM Cr 13.06 0.20 343.90\n(Ti2/3Hf1/3)2N 0.71 0.30 FM Ti 7.22 0.97 190.11\neven above the room temperature. It is noted that\nfor the recently synthesized 2D magnet CrI 3with a\nCurie temperature of 45 K, its MAE is about 1.71\nmeV/f.u.9,10,39,46Obviously, the out-of-plane MAEs of\n(Hf2/3Fe1/3)2C and (Hf 2/3Cr1/3)2N are almost the same\nas that of CrI 3. Futhermore, for CrI 3the interlayer and\nintralayer exchange parameters are 11.64 meV and 2.37\nmeV.47So, the exchange parameters for (Hf 2/3Fe1/3)2C\nand (Hf 2/3Cr1/3)2N are larger than that of CrI 3. In this\npoint of view, we suspect the i-MXene (Hf 2/3Fe1/3)2C\nand (Hf 2/3Cr1/3)2N being promising 2D magnets with\nhigh Curie temperature.\nIV. ELECTRONIC PROPERTIES\nA. Thermoelectric properties of semiconductors\n[a]\n[b]\nFIG. 3: Band structures and Seebeck coefficients for\n(Sc2/3Cd1/3)2C [a] and (Sc 2/3Hg1/3)2C [b]. The horizontal\ndashed lines denote the Fermi level.\nPreviously, it has been reported that the 2D semicon-ductorshavevery largethermoelectric power.48,49Forin-\nstance, SnSe monolayers are regarded as promising ther-\nmoelectric material, e.g., the Seebeck coefficient is in-\ncreased from 160 µV/K to 300 µV/K when temperature\nis increased from 300 K to 700 K.48Although most i-\nMXenestendtobemetallic, thereexistninenonmagnetic\nsemiconductors i.e., (Sc2/3X1/3)2C (X = Au, Cu, Ir, Ni,\nZn, Cd, Hg), (Hf 2/3Ir1/3)2C and (Ti 2/3Au1/3)2C. Among\nthem, (Sc 2/3Cd1/3)2C and (Sc 2/3Hg1/3)2C are large gap\nsemiconductors (with finite band gap larger than 0.4 eV\nand less than 1.0 eV), which can be good candidate ther-\nmoelectric materials. The band structures of these two\nsemiconductors in the hexagonal lattice are displayed in\nFig. 3 togetherwith the Seebeck coefficients as a function\nof chemical potential. The band gaps of (Sc 2/3Cd1/3)2C\nand (Sc 2/3Hg1/3)2C are as large as 0.41 and 0.79 eV,\nrespectively. Furthermore, the valence band maximum\n(VBM) and conduction band minimum (CBM) are flat\naround the Fermi level. Such a behavior implies that\nboth compounds have the potential to realize large See-\nbeck effect. To confirmthis, the Seebeck coefficientshave\nbeen evaluated for such two compounds, as shown in\nFig. 3. The peak of Seebeck coefficients around Fermi\nlevel for (Sc 2/3Cd1/3)2C ((Sc 2/3Hg1/3)2C) are remark-\nably as high as 2100 (3000) µV/K at 100 K, 820 (1200)\nµV/K at 300 K, and 520 (700) µV/K at 500 K. The en-\nhanced Seebeckcoefficients can be attributed to the large\nderivative of the DOS with respect to the energy and\nhence the flat bands above and below the Fermi energy.\nIt isnotedthat the densityofstatesforthesetwosystems\nare very comparable for the hexagonal and rectangular\nlattices, leadingto negligiblechangesin the Seebeckcoef-\nficients(cf. Fig.S7andS8inSupplementary). Therefore,\nwe suspect that (Sc 2/3Cd1/3)2C and (Sc 2/3Hg1/3)2C are\ngood candidates as 2D thermoelectric materials.\nB. Topological properties\nAs discussed above, most i-MXenes are metallic due\nto the partially filled d-shells of the transition metal\natoms. However, motivated by the reported nontriv-\nial topological state in some functionalized MXenes,8\nwe studied also the topological nature of the i-MXene\n(Ta2/3Fe1/3)2C with a tiny band gap of 1.34 meV. Sur-\nprisingly, (Ta 2/3Fe1/3)2C with a collinear AFM ground5\n[a]\n[b]\nFIG. 4: Band structures [a] and edge states [b] for AFM topo-\nlogical insulator (Ta 2/3Fe1/3)2C. When SOC is turned off, the\nband structures of spin up channel overlap with that of the\nspin down channel. So we only show band structure in one\nspin channel. In both [a] and [b], the dashed line denote the\nFermi level.\nstate hosts a nontrivial state, as shown in the bulk and\nsurface electronic structures in Fig. 4. Without consid-\nering spin-orbit coupling (SOC), the VBM and CBM al-\nmost cross each other at the Fermi energy. According to\nthefatbandanalysis(Fig.S9andS10inSupplementary),\nthese two bands are mainly contributed by the Fe-3d yz\nand Fe-3d zxorbital character, which is strongly coupled\nby SOC. Turning on SOC, a remarkable indirect gap of\n75.0 meV is opened, leading to band inversion and thus\nthe occurrence of the topological nontrivial state. This is\nclearly confirmed by our explicit calculation of the edge\nstatesofthe corresponding1Dribbons with a width of70\nunits, as shown in Fig. 4(b). Furthermore, the nontrivial\nedge state is protected by the combination of the time\nreversal ( T) and space inversion ( P) symmetries ( T P),\nas demonstrated for CuMnAs in Ref. [50].\nC. Spin-gapless Semoconductors\nIt is reported that MXene Ti 2C can be a spin-gapless\nsemiconductor (SGS) under 2% strain.12We found two\nSGS candidates among i-MXenes in the hexagonal geom-\netry,e.g., (Hf2/3Fe1/3)2C and (Zr 2/3Fe1/3)2C with total\nmagnetic moments of 1.79 and 0.74 µB/f.u., respectively.\nSuch two SGSs are not within the classification of four[a]\n[b]\nFIG. 5: The band structures (left) and anomalous Hall con-\nductivity (right) for (Hf 2/3Fe1/3)2C [a] and (Zr 2/3Fe1/3)2C\n[b]. The horizontal dashed lines denote the Fermi energies.\ntypes SGSs defined in our previous work,34because the\nVBM and CBM have both spin characters as shown in\nFig. 5. For such tow SGSs at Fermi level, in the spin\nup channel the CBM and VBM touches each other di-\nrectly between K−Γ (in A region), while in the spin\ndown channel there is only a small gap between for the\nCBM and VBM between K−M(in region B). That is\nthe VBM and CBM of spin up and down channels can\nbe roughly seen at the same energy level. When SOC is\nswitched on, the touching bands open a local band gap,\nsuggesting a topologically nontrivial state. This can be\nmanifested by explicit evaluation of the anomalous Hall\nconductivity. Forinstance, itisobservedthatthe anoma-\nlous Hall conductivity is finite around the Fermi energy,\nwhich is almost quantized to 3 and 5 e2/hwith tiny band\ngaps of 3.8 meV and 0.3 meV for (Hf 2/3Fe1/3)2C and\n(Zr2/3Fe1/3)2C, respectively. That is, such i-MXenes are\nChern insulators with a nontrivial Chern number of 3\nand 5, though the band gaps are small.\nV. CONCLUSION\nIn conclusion, we have done a systematic study on\nthe magnetic and electronic properties of i-MXene com-\npounds, whichprovideaninterestingplaygroundformul-\ntifunctional applications. The magnetic ground states\nfor i-MXene are investigated for both rectangular and\nhexagonal lattices, where slight strain can be applied to\ntune the magnetic ground state. Due to the underly-\ning crystal fields, the magnetocrystalline anisotropy of i-\nMXenecanbesignificantlyenhanced, i.e., thereare7sys-6\ntems with amagneticcrystallineanisotropyenergylarger\nthan 0.5 meV/f.u.. Furthermore, investigation on the\nelectronic properties reveals that i-MXene can host fas-\ncinating transport properties, including significant ther-\nmoelectric effects, antiferromagnetic topological insula-\ntor state, and spin gapless semiconductors. Our calcu-\nlations suggest that i-MXene is a class of 2D materials\nwhich are promising for future applications, calling for\nfurther experimental exploration.ACKNOWLEDGMENTS\nQiangGaothanksthefinancialsupportfromtheChina\nScholarshipCouncil. 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Lacunar spinels with the formula GaM 4X8with M=V, Mo; X=S, Se are a convenient case\nstudy towards this goal as they are some of the \frst bulk systems suggested to host equilibrium\nchiral skyrmions far from the paramagnetic transition. We derive the magnetic phase diagrams\nlikely to be observed in these materials, accounting for all possible magnetic interactions, and prove\nthat skyrmion stability in the lacunar spinels is a general consequence of their crystal symmetry\nrather than the details of the material chemistry. Our results are consistent with all experimental\nreports in this space and demonstrate that the di\u000berences in the phase diagrams of particular spinel\nchemistries are determined by magnetocrystalline anisotropy, up to a normalization factor. We\nconclude that skyrmion formation over wide ranges of temperatures can be expected in all lacunar\nspinels, as well as in a wide range of uniaxial systems with low magnetocrystalline anisotropy.\nThe prediction and experimental demonstration of\ntopologically non{trivial magnetic structures, commonly\ncalled magnetic skyrmions[1{3], has sparked considerable\ninterest in the conditions required for the formation of\nthese phases, and their potential for applications in spin-\ntronic devices[4, 5]. Most reports of skyrmion formation\nin bulk systems have focused on cubic helimagnets, most\nfrequently with the B20 structure: MnSi[3], FeGe[6],\nCu2OSeO 3[7] and Co xZnyMnz[8], where skyrmions are\ntypically observed in a narrow range of \felds and tem-\nperatures near the paramagnetic transition. Much larger\nstability windows for skyrmions have been reported in\nthin{\flm systems[9], and recently in the bulk uniaxial\nhelimagnet GaV 4Se8[10, 11] and Heusler alloys such as\nMn1:4PtSn[12]. The realization of practical skyrmion{\nbased spintronic devices requires the wide stability win-\ndows observed in these materials, motivating a search\nfor general principles leading to formation of thermally{\nrobust skyrmion phases.\nThe lacunar spinels GaM 4X8are a convenient model\nsystem for studying mechanisms leading to robust\nskyrmion stability at all temperatures below the Curie\ntemperature. GaV 4S8[13] and GaV 4Se8[10, 11] have been\nreported as skyrmion hosts with unusually wide ther-\nmal stability windows, while GaMo 4S8has been sug-\ngested as a skyrmion host on the basis of computational\ndata[14]. Materials in this class exhibit signi\fcant metal{\nmetal bonding[15, 16], with electronic structure de\fned\nby isolated M 4molecular units. Their magnetic behav-\nior is well described by interactions between e\u000bective\nspins centered on the M 4clusters[17]. Furthermore, the\nR3m symmetry and strong Dzyaloshinskii{Moriya inter-\nactions (DMI) common to these systems guarantee that\nskyrmion formation can be treated with explicit spin\nmodels as a largely two{dimensional problem.\n\u0003dkitch@ucsb.edu\nyavdv@ucsb.eduHere, we demonstrate that skyrmion stability in\nGaM 4X8is a consequence of the symmetry of these ma-\nterials rather than speci\fcs of the magnetic interactions,\nwith the exception of magnetocrystalline anisotropy. We\nconstruct a \feld{temperature magnetic phase diagram\nfor the lacunar spinels based on a general cluster expan-\nsion Hamiltonian parametrized using density{functional\ntheory (DFT) data[18{23], which we \fnd to be in close\nagreement with experimental reports. By analyzing the\nsensitivity of the phase diagram to all symmetrically{\nallowed perturbations in the Hamiltonian, we \fnd that\nthe form of the phase diagram is largely controlled by\nuniaxial anisotropy, as well as higher{order in{plane\nanisotropy. In the low{anisotropy regime, skyrmion for-\nmation is guaranteed by the lack of a competing canted\nspin{wave phase magnetized along the high{symmetry\naxis, consistent with phenomenological predictions[24],\nwhich leads us to conclude that the phase behavior we\ncompute is likely to be broadly applicable to uniaxial\nmagnets with strong in{plane DMI.\nMETHODS\nCluster expansion generation and \ftting\nWe construct the magnetic cluster expansion following\na methodology similar to that described by Thomas and\nVan der Ven[23, 25, 26]. Our cluster expansion of the in-\nternal energy only includes magnetic degrees of freedom,\nwhere the moment on each M 4tetrahedron is represented\nby a 3{dimensional unit vector. To construct the cluster\nexpansion, we identify all site{clusters up to a target ra-\ndius and number of sites, and the symmetry operations\nwhich map each cluster to itself. Then, we generate all\npossible basis functions for spin interactions on each clus-\nter. Following previous derivations of cluster expansions\nfor orientational degrees of freedom[20, 21, 27], we usearXiv:1911.11297v2  [cond-mat.mtrl-sci]  22 Jan 20202\nproducts of spherical harmonics jl;mi=p\n4\u0019Yl\nm(\u001e;\u0012) as\na complete basis set for spin interactions, where ( \u001e;\u0012)\nare the spin vector orientation in spherical coordinates.\nWe additionally group these products according to the\ntotal symmetry of the interaction to form basis functions\nof the form:\njl1;l2;L;Mi= 4\u0019X\nm1;m2cl1;l2;L\nm1;m2;MYl1\nm1(\u001e1;\u00121)Yl2\nm2(\u001e2;\u00122)\nwherecl1;l2;L\nm1;m2;Mare Clebsch{Gordan coe\u000ecients. This\nprocedure isolates the basis functions corresponding to\nexchange (L= 0), DMI ( L > 0, odd), and anisotropy\n(L > 0, even). Finally, we \fnd the purely real compo-\nnent of each basis function invariant to the symmetry of\nthe cluster, and using Gram-Schmidt orthogonalization,\nobtain an orthonormal basis set for spin interactions on\neach cluster. The cluster expansion implementation re-\nlies on an in{house python code accelerated using the\nNumba package[28], while general structure processing,\ndata handling, and symmetry analysis rely on the pymat-\ngen package[29]. A detailed description of the cluster ex-\npansion and the procedure used to generate interaction\nfunctions is available in Supplementary Note 1.[30]\nTo obtain a \ft for the interaction coe\u000ecients in the\ncluster expansion from DFT data, we follow a standard\nmethodology designed for the automated generation of\nphase diagrams[19, 31]. First, we enumerate symmetri-\ncally distinct collinear and spin{wave con\fgurations com-\npatible with supercells up to size 4. We re\fne this dataset\nby identifying the least{constrained correlation vectors in\nthe input data as the eigenvectors of the correlation co-\nvariance matrix with the smallest eigenvalues. We then\nobtain spin{con\fgurations corresponding to these corre-\nlation vectors and add them to the \ftting dataset. Fi-\nnally, we \ft the cluster expansion interaction coe\u000ecients\nusing least{squares regression, while using a genetic algo-\nrithm to eliminate basis functions from the Hamiltonian\nso as to maximize the cross{validation (CV) score.\nMonte Carlo sampling and ground state search\nWe use a Hamiltonian Monte Carlo approach to sam-\nple the \fnite{temperature behavior given by the clus-\nter expansion Hamiltonian. Our Monte Carlo imple-\nmentation exactly follows the formalism described by\nWang et al [32], with trajectory sampling based on the\nNo U-Turn Sampler/Dynamic Multinomial Sampling\nmethods[33, 34]. All Monte Carlo runs reported here are\nconstant{\feld heating runs, where each temperature step\n\frst rejects 800 uncorrelated samples for equilibration,\nand then saves 2000 uncorrelated samples for production.\nTo ensure that the obtained samples are uncorrelated, we\nset the number of Monte Carlo passes between samples\nto exceed the estimated autocorrelation decay time.\nTo identify ground state spin con\fgurations, we \frst\ngenerate candidate structures using simulated annealing\nstarting from a random con\fguration and representativecon\fgurations of known phases. We then relax each con-\n\fguration to its local minimum using conjugate gradient\nminimization, and save the lowest energy structure.\nDensity functional theory calculations\nDFT calculations are performed using the Vienna\nAb-Initio Simulation Package (VASP) [35], using the\nprojector-augmented-wave method [36] with the Perdew-\nBurke-Ernzerhof (PBE) exchange{correlation functional\n[37]. We do not apply a Hubbard{ Ucorrection because\nour previous benchmarks on GaV 4Se8revealed that the\nstandard on{site Hubbard{ Uapproach leads to an incor-\nrect electronic con\fguration and magnetic behavior[15].\nAll calculations account for spin{orbit coupling and are\nconverged to 10\u00006eV in total energy. We use a recip-\nrocal space discretization of 100 k-points per \u0017A\u00003, and\nsmearing width of 0.05 eV based on a convergence of total\nenergy across all distinct supercells containing 2 formula\nunits of GaMo 4S8to 0.5 meV/f.u. To further reduce\nerror arising from changes in the k-point mesh across\ndi\u000berent supercells, we reference all magnetic con\fgura-\ntion energies to that of a c{axis ferromagnet computed\nusing the same supercell. In all cases, DFT calculations\nare done statically, based on the experimentally{observed\nlow{temperature structure.\nRESULTS\nMagnetic cluster expansion Hamiltonian\nWe begin by de\fning an e\u000bective spin Hamiltonian for\nthe magnetic behavior of a GaM 4X8lacunar spinel in the\nform of a cluster expansion, which is a summation over\ninteraction correlation functions 'with interaction coef-\n\fcientsJ. The correlation functions 'are determined by\nthe lattice type and symmetry of the material, while the\ninteraction coe\u000ecients Jare speci\fc to each chemistry.\nThus, we can systematically explore the magnetic be-\nhavior of GaM 4X8by establishing which magnetic phase\ndiagrams are likely to arise given the overall form of the\nHamiltonian, across possible choices of interaction pa-\nrametersJ.\nThe full form of a cluster expansion Hamiltonian is\nE=X\n\nX\n\u000bJ\n\u000bX\n!2\n^p!\u0002\n\u001e\n\u000b\u0003\n=X\niJi'i\nwhere\u001eare interaction basis functions and Jare interac-\ntion coe\u000ecients. Each interaction is de\fned with respect\nto a cluster of sites !, where symmetrically{equivalent\nclusters are grouped into orbits \n. The interaction ba-\nsis functions contain all spin{couplings consistent with\nthe symmetry of the cluster, which include conventional\nHeisenberg exchange, DMI, and anisotropy interactions,\nas well as any higher{order terms. The symmetry oper-\nation ^p!generates the cluster !from a reference cluster3\nFIG. 1. a.Low-temperature structure of a GaM 4X8lacunar spinel in the R3m conventional unit cell. b.Model for\nthe magnetic structure, treating M 4tetrahedra as magnetic units forming a face{centered{cubic lattice, where the magnetic\nHamiltonian consists of by single-site and nearest-neighbor interaction energies. Lattice vectors ( ap,bp,cp) shown form the\nprimitive lattice used to de\fne the cluster expansion Hamiltonian. c.Example con\fgurations of helimagnetic phases observed\nin lacunar spinels, where qdenotes a propagation wavevector and color corresponds to spin orientations.\nfor its orbit \n. The total contribution of a basis function\n\u001efor the symmetrically-equivalent clusters in \n de\fnes\nthe correlation function '.\nWe take the symmetry of the crystal to be R3m as\nshown in Figure 1a, which results from a low-temperature\ndistortion of the F\u001643mvacancy{ordered spinel structure\nalong theh111idirection. The magnetic sublattice con-\nsists of a distorted face{centered{cubic (FCC) arrange-\nment of M 4tetrahedral clusters, shown in Figure 1b,\nwhere each M 4tetrahedron can be treated as a single spin\nvector. As the distance between M 4clusters is large, we\napproximate the magnetic energy with only on-site and\nnearest-neighbor couplings. For the three symmetrically{\ndistinct couplings present (on-site, out-of-plane pair and\nin-plane pair), we derive spin{interaction basis functions\nconsistent with the symmetry of each cluster. The basis\nfunctions are polynomials of the spin{vector components,\nup to sixth order for the on-site term, and bilinear order\nfor the pair terms. These interactions, listed in Table I,\nform a complete basis set for the magnetic Hamiltonian\nthat is applicable to any material with a relatively sparse\nFCC magnetic sublattice and R3m symmetry.\nThe coe\u000ecients Jof each correlation function '\nparametrize the variation of the Hamiltonian across dif-\nferent chemistries. Thus, in order to understand the\nphase behavior of all GaM 4X8lacunar spinels with R3m\nsymmetry, it is su\u000ecient to evaluate how the \feld{\ntemperature phase diagram evolves with the components\nofJ. We limit ourselves to J{vectors appropriate for lo-\ncally ferromagnetic materials ( J7>0,J12>0), allowing\nfor strong spin-orbit coupling. This regime is character-\nistic of the behavior of skyrmion{hosting lacunar spinels\nwith M=V,Mo and X=S,Se, where the observed mag-\nnetic phases are ferromagnet, cycloid, canted cycloid, andskyrmion. Example con\fgurations of these phases are\nshown in Figure 1c.\nDerivation of an example phase diagram for GaM 4X8\nOur strategy for exploring phase behavior in this sys-\ntem is to construct a full \feld{temperature phase dia-\ngram for one choice of J{vector, and then calculate how\nperturbations in Jtranslate to changes in phase tran-\nsitions. While this approach is only strictly valid for\nsmall deviations from the initial choice of J, the degree\nto which extrapolation is valid is determined by whether\nthe form of the phase diagram is more determined by\nthe values of J, or by which correlation functions 'are\npresent in the Hamiltonian. In this case, we will argue\nthat given an appropriate normalization, the form of the\ncorrelation functions 'plays the more important role,\nleading to a universal behavior of the phase diagram.\nWe choose our initial J{vector,J(0), by \ftting the\nHamiltonian to reproduce the magnetic behavior of\nGaMo 4S8as computed from DFT. We choose GaMo 4S8\nas a convenient reference point as a recent report has sug-\ngested that this material exhibits cycloid and skyrmion\nphases with particularly short wavelengths[14], which al-\nlows us to directly study these phases with periodic-cell\nMonte Carlo. Furthermore, the electronic structure of\nGaMo 4S8appears to be reasonably captured by the stan-\ndard PBE functional, while the better-known V-based\nanalogs require more sophisticated, computationally ex-\npensive methods such as RPA[15]. Following a state{of{\nthe{art cluster expansion \ftting procedure, as well as a\nDFT calculation scheme designed to minimize spurious\nsources of error (details available in the methods), we ob-4\nFIG. 2. a.Energy of an ideal cycloid relative to that of a ferromagnet magnetized along the c-axis, as a function of n, which\nde\fnes the cycloid propagation wavevector qn= [1\nn00].b.,c.Formation energies of ground state con\fgurations as a function\nof average magnetization along the c-axis, constrained to a ( n,n,3) periodic supercell of the R3m unit cell shown in Fig. 1a.\nFormation energies are given with respect to the n= 12 cycloid and c-axis ferromagnet (b.), and the full n= 12 ground-state\nenergy pro\fle (c.). d.,e.Magnetic phase diagram arising from the J(0)parametrization of the cluster expansion Hamiltonian,\nas a function of magnetic \feld magnitude (d.) and total magnetization (e.). Color denotes the number of q-points for which\nthe structure factor S(q) is non-zero. Phase labels correspond to ferromagnet (FM), cycloid (Cyc), canted cycloid (C. Cyc),\nskyrmion (Sk), Brazovskii region (BR) and paramagnet (PM). The locations and orders of phase boundaries are drawn based\non Monte Carlo data to best agree with changes in S(q), the topological index (see Supplementary Figure 1 [30]), discontinuities\nin internal energy and magnetization, and peaks in \ructuation data. Tcdenotes the Curie temperature. Note that \\other\"\ndenotes a change in structure factor S(q) not accompanied by any discernible discontinuities in free energy.\ntain theJ(0)vector given in Table I with a RMSE of 0.3\nmeV/formula unit (f.u.) across a total energy range of 7\nmeV/f.u. The low absolute value of the error justi\fes our\nchoice of truncating the Hamiltonian at nearest-neighbor\ninteractions and bilinear pair couplings, as the inclusion\nof any additional basis functions would likely only be\ncapturing noise in the DFT data. While both the total\nerror and the uncertainty on the components of J(0)are\nsmall, the signi\fcance of these error bars in relation to\nphase behavior is not immediately clear. However, as our\nobjective is to obtain a reasonable initial J(0)for our per-\nturbative analysis, we proceed to characterize the phase\ndiagram given by this \ft and address the role of uncer-\ntainty, as well as general perturbations to J, in a later\nsection.\nWe \frst establish the ground states of the J(0)Hamil-tonian as a function of total magnetization, which in-\nclude cycloid, canted cycloid, skyrmion and ferromagnet\nphases. The dominant periodicity of a helimagnet is set\nby the competition between DMI and exchange, which\ntypically remains close to the period of the cycloid phase.\nThe lowest energy commensurate cycloid in this system\nhas wavevector q= [1\nn00] forn= 12, as shown in Fig-\nure 2a. We thereby choose a (12 ;12;3) supercell of the\nconventional unit cell (1296 M 4units) for a full ground\nstate enumeration, as this supercell is compatible with\nall low{energy cycloid variants, as well as the typical\n6{fold skyrmion lattice phase. We obtain the internal\nenergy pro\fle shown in Figure 2b, which as a function\nof magnetization along the c-axis proceeds through the\ncycloid phase at low magnetization, skyrmion phase at\nintermediate magnetization, and ferromagnet phase at5\nCluster type Basis function J(0)(meV)\nOn-site \u001eA\n1=p\n2j2;0i 0\nr= (0;0;0)\u001eA\n2=\u0000i(j4;\u00003i+j4;3i) 0.03(5)\n(1 equiv.) \u001eA\n3=p\n2j4;0i 0\n\u001eA\n4=j6;\u00006i+j6;6i 0\n\u001eA\n5=\u0000i(j6;\u00003i+j6;3i) 0\n\u001eA\n6=p\n2j6;0i 0\nOut-of-plane \u001eE\n7=p\n2\n3j1;1; 0;0i 0.62(5)\nr1= (0;0;0)\u001eD\n8=i\n3(j1;1; 1;1i\u0000j1;1; 1;\u00001i) 0.88(9)\nr2= (\u00001;0;0)\u001eA\n9=\u0000i\n3(j1;1; 2;1i+j1;1; 2;\u00001i) 0\n(3 equiv.) \u001eA\n10=p\n2\n3j1;1; 2;0i 0\n\u001eA\n11=1\n3(j1;1; 2;2i+j1;1; 2;\u00002i) 0\nIn-plane \u001eE\n12=p\n2\n3j1;1; 0;0i 1.07(5)\nr1= (0;0;0)\u001eD\n13= (1\n6+ip\n12)j1;1; 1;1i+ 0.20(9)\nr2= (\u00001;0;1) (1\n6\u0000ip\n12)j1;1; 1;\u00001i\n(3 equiv.) \u001eD\n14=\u0000ip\n2\n3j1;1; 1;0i 0.08(7)\n\u001eA\n15=p\n2\n3j1;1; 2;0i 0\n\u001eA\n16= (1p\n12\u0000i\n6)j1;1; 2;1i\u0000 0\n(1p\n12+i\n6)j1;1; 2;\u00001i\n\u001eA\n17= (1\n6\u0000ip\n12)j1;1; 2;2i+ 0.1(1)\n(1\n6+ip\n12)j1;1; 2;\u00002i\nTABLE I. Clusters and symmetrized basis functions for the\nGaM 4X8magnetic cluster expansion Hamiltonian, and the\nJ(0)vector \ftted to GaMo 4S8DFT data. Cluster site coordi-\nnates and basis functions are given for the reference cluster, in\nlattice coordinates with respect to the primitive lattice vectors\n(ap,bp,cp) given in Fig 1b. The number of equivalents for\neach cluster refers to the number of symmetrically{equivalent\nclusters of this type per primitive cell. Basis functions are\nde\fned in terms of spherical harmonics jl;mifor the on-\nsite terms and Clebsch{Gordan functions jl1;l2;L;Mifor pair\nclusters (r1;r2), as described in the methods. The cartesian\nform of the basis functions is available in Supplementary Table\n1.[30] Basis function superscripts denote whether the interac-\ntion corresponds to exchange (E), DMI (D), or anisotropy (A).\nParenthesis in the J(0)vector components denote uncertainty\nin the last digit.\nhigh magnetization. Repeating the ground state search\nfor other (n;n; 3) supercells, we \fnd that while the com-\nmensurate cycloid and canted cycloid phases are always\nminimized for n= 12, the skyrmion phase relaxes from\nn= 12 ton= 13 at high magnetization ( \u00198% change\nin wavelength), as shown in Figure 2c. This result is in-\ntriguing from the perspective of experimentally detecting\nskyrmions in di\u000braction data by means of a shift in mag-\nnetic structure q{vector away from that of a cycloid, and\nis consistent with observed changes in qvector in the\nskyrmion phase of GaV 4S8[13]. However, for the pur-\nposes of thermodynamic stability calculations, the di\u000ber-\nence in energy between the n= 12 andn= 13 skyrmion\nis small enough to be negligible.\nThe \fnite{temperature phase diagram of the J(0)Hamiltonian is shown in Figure 2d,e as a function of ap-\nplied \feld and observed magnetization respectively. The\nlocations and orders of phase transitions are estimated\nbased on changes in the magnetic structure factor, topo-\nlogical index, discontinuities in internal energy and mag-\nnetization, and peaks in \ructuation data.\nThe low{\feld phase up to the Curie temperature ( Tc)\nis a cycloid. Magnetization in the ( ab){plane leads to\na continuous transition into a canted cycloid phase, fol-\nlowed by a transition to a ferromagnet. At low tem-\nperatures (approximately T < 0:5Tc), the transition\nfrom canted cycloid to ferromagnet is \frst{order, while\nat higher temperatures this transition becomes second{\norder. We conclude that the order of the phase transi-\ntion changes because we observe a peak in the magnetic\nsusceptibility at this point at all temperatures, but the\ndiscontinuity in magnetization only exists below 0 :5Tc.\nMagnetization along the c{axis leads to the formation of\nthe skyrmion phase, followed by the ferromagnet phase.\nThe formation of the topologically{nontrivial skyrmion\nphase is also con\frmed by a change in the topological\nindex from 0 to -1 as the c{axis magnetization is in-\ncreased (see Supplementary Figure 1[30]). Both tran-\nsitions are \frst-order at most temperatures, becoming\nsecond{order only close to Tc(approximately T >0:8Tc).\nAt low temperatures, skyrmions are stabilized with re-\nspect to cycloids enthalpically, consistent with the behav-\nior of the ground{state con\fgurations. However, above\napproximately 0 :67Tcthe skyrmion region expands at\nthe expense of the cycloid region indicating that at el-\nevated temperatures, skyrmions are additionally stabi-\nlized entropically. Immediately above Tc, the cycloid,\ncanted cycloid and skyrmion phase regions extend into\na partially{disordered phase dominated by \ructuations\nin the (ab){plane at the cycloidal q{vectors, as a two{\ndimensional analog of the Brazovskii region described in\ncubic helimagnets[38, 39]. Note that despite having a\nsimilar structure factor to the skyrmion phase, the two{\ndimensional Brazovskii region is topologically trivial as\ncan be seen in Supplementary Figure 1.[30] Finally, at\nhigher temperatures, the system fully disorders to form\na paramagnet.\nVariation in phase stability with changes in the\nHamiltonian\nHaving established the \fnite temperature phase dia-\ngram for one parametrization of the Hamiltonian, J(0),\nwe evaluate how the phase diagram may change as the\nJcoe\u000ecients are varied. We \frst identify normalization\nfactors for the phase diagram to account for changes in\nthe Hamiltonian that amount to rescaling the \feld and\ntemperature axes. We then use generalized Clausius{\nClapeyron relationships to identify which components of\ntheJvector may alter the locations of \frst{order tran-\nsitions seen in the phase diagram.\nFigure 3a shows the phase diagram obtained for J(0),6\nFIG. 3. a.Normalized magnetic phase diagram derived using the J(0)Hamiltonian, marking three \frst-order phase transitions:\ncycloid{to{ferromagnet in the ( ab) plane (HC!F\n(ab)), and cycloid{to{skyrmion and skyrmion{to{ferromagnet along the caxis\n(HC!S\nc andHS!F\nc respectively). Phase labels are de\fned in the caption to Figure 2. b.Change in the normalized phase\ntransition \felds HC!F\n(ab),HC!S\nc andHS!F\nc upon variation of the Hamiltonian parameters J. Note that the \feld normalization\nfactor 2\u00192q2jJj=\u0016BgMsvaries with \u0001 J, while the units of \u0001 J[ (2\u00192q2jJj)\u0001J=0] are taken to be constant. c.Magnetocrystalline\nanisotropy functions '1;:::;6shown relative to a M 4tetrahedron, where the color and distance from the tetrahedron center along\na certain direction represent the value of 'ifor that spin orientation.\nin units of the characteristic temperature and \feld for\nthis system. The red dots highlight the three \frst{order\ntransitions that de\fne the low{temperature region of the\nphase diagram. The normalization factor for temper-\nature isjJj=kB, as any homogeneous rescaling of the\nHamiltonian must also rescale temperature. The nor-\nmalization factor for \feld is given by the characteristic\ndi\u000berence in energy between the low-\feld and high-\feld\nground states, which are the cycloid and ferromagnet\nphases respectively. In units of magnetic \feld, this fac-\ntor is 2\u00192q2jJj=\u0016BgMs, whereqis the magnitude of the\ncycloid wavevector in lattice coordinates and Msis the\nmagnetic moment per spin. A full derivation is avail-\nable in Supplementary Note 2.[30] Note that this factor\nis identical to the D2=Anormalization used in previous\nliterature where DandAare the e\u000bective DMI and ex-\nchange constants respectively.\nFigure 3b plots how the locations of the three low{\ntemperature \frst{order transitions change with variation\nin the components of J(0). The left panels account for\nthe correlation functions corresponding to conventional\nbilinear spin couplings, while the right panels illustrate\nthe e\u000bect of higher order on{site anisotropy terms, plot-\nted schematically in Figure 3c. Note that '1corresponds\nto quadratic single{spin anisotropy, while '10and'15areequivalent to XXZ anisotropy for out{of{plane and in{\nplane exchange respectively. In this case, these terms\nyield exactly the same behavior and are plotted as a\nsingle line. The change in the phase boundary loca-\ntionHy=2\u00192q2jJj\n\u0016BgMsis given by the generalized Clausius{\nClapeyron relation,\n2\u00192q2jJj\n\u0016BgMs@\u0010\nHy=2\u00192q2jJj\n\u0016BgMs\u0011\n@J=\u0001h'i\n\u0001M\u00002Hy\nq@q\n@J\u0000HyJ\nwhere \u0001h'iand \u0001Mare the change in the correlation\nfunctions and magnetization across the \frst{order phase\ntransition. Note that this expression explicitly accounts\nfor the variation in qandjJjfor the purposes of normal-\nizingHy, while the change in Jis expressed in units of\n(2\u00192q2jJj)\u0001J=0.\nThe perturbation data shown in the left panel of\nFigure 3b reveals that the low{temperature region of\nthe phase diagram is largely invariant to changes in\nmost of the J{coe\u000ecients in the Hamiltonian, save for\nchanges injJjandqwhich amount to rescaling the tem-\nperature and \feld axes. By far the most important\ncorrelation functions are those corresponding to uniax-\nial anisotropy ( '1;3;6;10;15), which have a qualitatively7\nFIG. 4. a.Variation in the normalized magnetic phase diagram with the uniaxial anisotropy parameter J1, focusing on\nthe cycloid and skyrmion phase transitions with applied \feld along the c-axis. Representative phase diagrams for the easy-\naxis, easy-plane, and isotropic cases ( J1=2\u00192q2jJj=\u00000:9;+0:9;0 respectively) are highlighted. Phase labels are de\fned in\nthe caption to Figure 2. b.Variation in skyrmion phase boundaries with \feld orientation, for the easy-axis, easy-plane and\nisotropic cases. The color of the phase boundaries denote the angle of the \feld with respect to the c-axis.\nsimilar impact on the phase boundaries, and higher{\norder in{plane anisotropy '4(right panels). The uni-\naxial terms '1;3;6;10;15shift the skyrmion/ferromagnet\nboundary, penalizing skyrmion formation in the easy{\naxis regime. The '4anisotropy function is unique in\nthat it only alters the phase boundaries left una\u000bected\nby the uniaxial terms, shifting the cycloid/skyrmion\nand canted cycloid/ferromagnet boundaries, but leaving\nthe skyrmion/ferromagnet boundary unchanged. For-\ntuitously, the only anisotropy function included in the\nJ(0)simulation is '2, which has no impact on the phase\nboundaries so that the J(0)results are equivalent to a\nzero{anisotropy regime where all helimagnetic phases of\ninterest appear at all temperatures. Finally, the only im-\npact of exchange and DMI terms beyond varying qand\njJjis to alter the \feld at which a canted cycloid in the\n(ab){plane transforms to a ferromagnet.Skyrmion stability determined by uniaxial\nanisotropy\nAt low temperature, the stability of skyrmions in this\nsystem is largely controlled by the presence of uniaxial\nanisotropy in the form of the '1;3;6;10;15correlation func-\ntions, as well as high{order in{plane anisotropy in the\nform of'4. As uniaxial anisotropy is the most common\nform of anisotropy observed in uniaxial magnets, we now\nderive the impact of this term on the skyrmion region\nat all temperatures, using '1as a proxy for all uniaxial\nanisotropy functions in the Hamiltonian. We note how-\never that in rare cases where higher-order anisotropies\n('2;:::;6) are strong, more complex phase behavior is pos-\nsible.\nThe evolution of the phase diagram with the coe\u000e-\ncient of'1,J1, as a function of \feld along the c{axis and\ntemperature, is shown in Figure 4a. The phase diagram\nexhibits three broad regions corresponding to easy{axis,\neasy{plane, and isotropic scenarios. The skyrmion and\ncycloid regions are highlighted in blue and red respec-\ntively for representative slices in each region. We obtain\nthis phase diagram using a linear extrapolation of the8\nHelmholtz free energy A:\nA=A(0)+@A\n@J1\u0001J1=A(0)+h'1i\u0001J1\nwhereA(0)is the Helmholtz free energy obtained for J(0).\nAs this extrapolation is only applicable at constant tem-\nperature, we neglect any changes in Tcdue to changes\ninJ1. Similarly, as variation in J1has a negligible im-\npact onjJjand no impact on q, the normalization factors\nfor \feld, temperature, and Jare taken to be constant.\nTo check the validity of this extrapolation, we con\frm\nthat the extrapolated phase diagrams highlighted in the\neasy{axis and easy{plane regions of Figure 4a agree with\nMonte Carlo data for the same conditions.\nThe impact of uniaxial anisotropy on skyrmion sta-\nbility is largely determined by the stabilization of the\ncompeting out{of{plane and in{plane ferromagnetic and\ncanted cycloid phases. Easy{axis anisotropy favors the\nout{of{plane ferromagnet con\fguration and thus sup-\npresses the skyrmion phase, and eventually the cycloid\nphase. Easy{plane anisotropy destabilizes the out{of{\nplane ferromagnet and thus enhances the skyrmion sta-\nbility region, up to the point where the easy{plane{\nanisotropy is su\u000ecient to stabilize canted cycloids and\nin{plane ferromagnetic con\fgurations at zero{\feld, at\nwhich point the skyrmion region remains stable only\nat highc{axis \felds, consistent with phenomenologi-\ncal solutions[40, 41]. Elevated temperatures suppress\nthe e\u000bect of anisotropy while preserving the qualitative\ntrends, leading to a much wider range of anisotropy\nconstants for which skyrmions and cycloids are stable\nthan in the low{temperature limit. The profound im-\npact of anisotropy constants on skyrmion stability at\nlow temperature is reminiscent of recent reports of low{\ntemperature skyrmion stabilization in Cu 2OSeO 3where\nthe role of anisotropy is to suppress the competing canted\nspin{wave phases.[42, 43]\nUniaxial anisotropy also varies in the range of \feld di-\nrections for which skyrmions may be observed, with easy{\naxis anisotropy favoring skyrmions over a wide range of\n\feld orientations and easy{plane allowing for skyrmions\nonly for \felds close to the c{axis. Figure 4b shows the\nevolution of the cycloid and skyrmion phase boundaries\nas a function of applied \feld direction. The color of the\nphase boundaries corresponds to the \feld angle, shown\non the legend in terms of angle with respect to the c{\naxis and high{symmetry directions in the structure. In\nthe isotropic case, the skyrmion region moves to slightly\nhigher \felds with increasing \u0012, up to\u0012max\u001940o, be-\nyond which only cycloid, canted cycloid and ferromagnet\nphases are stable. In the easy{plane case, the skyrmion\nregion quickly narrows with increasing \u0012, disappearing\nabove\u0012max\u001920o. In the easy{axis case, the skyrmion\nstability region is much smaller in \feld{magnitude, but\nwider in \feld{angle, with a high{temperature skyrmion\nphase appearing up to \u0012max\u001970o. These results, as\nwell as the broad impact of uniaxial anisotropy, con\frm\nthe conclusions reached by a phenomenological analysisTc(K)q J 1(\u0016eV)J1=2\u00192q2jJj\nGaV 4S8 13 1/26 -25 -1.00\nGaV 4Se8 18 1/27 2 0.06\nGaMo 4S8 23 1/12 -34 -0.17\nGaMo 4Se8 27 1/20 25 0.29\nTABLE II. Behavior of V and Mo{based lacunar spinels\nbased on the phase diagram shown in Figure 4. Tc,qand\nJ1data for the V systems is based on experimental data from\nrefs. [13] and [11], while data for the Mo systems is based on\nour DFT calculations.\nreported by Leonov and Kezsmarki[44].\nThe formation of skyrmions in easy{axis systems over\na wide range of \feld orientations is important in the\ncontext of resolving skyrmion formation experimentally.\nThe easy{axis scenario leads to skyrmion formation for\n\felds applied along the [101] family of miller indices of\nthe unit cell shown in Figure 1a. These directions corre-\nspond to theh001iaxes of variants of the lacunar spinel\ndistorted along directions equivalent to the h111iaxis of\nthe high{temperature F\u001643mphase. In a real material,\nwhere all symmetrically{equivalent variants of the h111i\ndistortion are likely to be present, we would thus expect\nskyrmion phase boundaries following both the black and\ngreen curves, even if the \feld is applied parallel to the c{\naxis of one of the variants. This is precisely the scenario\nobserved in GaV 4S8[13], which we estimate corresponds\nto the strongly easy{axis J1=2\u00192q2jJj=\u00001:00 slice of\nthe phase diagram.\nPredicted phase diagrams for V and Mo systems\nFinally, we use the phase diagram shown in Figure 4 to\nevaluate the magnetic phase diagrams of several V and\nMo{based spinels. Our estimates for the value of the\nreduced anisotropy J1=2\u00192q2jJjfor GaV 4S8, GaV 4Se8,\nGaMo 4S8and GaMo 4Se8are given in Table II, with\nphase diagrams given by \feld{temperature slices of Fig-\nure 4a at the given level of anisotropy. The magnetic\nphase diagrams of GaV 4S8and GaV 4Se8have been stud-\nied in{depth experimentally, and o\u000ber a direct compari-\nson to our results. We similarly rely on experimental data\nto obtainTc,qandJ1for the V systems[11, 13]. Experi-\nmental data on GaMo 4S8and GaMo 4Se8is more sparse,\nbut as these systems are well{represented by DFT, we use\ncomputational data to obtain estimates for the magnetic\nparameters. Given the small energy scale and critical im-\nportance of the anisotropy constants to phase behavior,\nwe obtain the value of the anisotropy constants indepen-\ndently of other terms in the magnetic Hamiltonian, using\ndata computed within the same unit cell so as to mini-\nmize numerical noise arising from k{point discretization\nerror. Speci\fcally, the energy of orienting a ferromag-\nnetic con\fguration along various crystallographic direc-9\ntions, computed using the primitive cell of the structure,\nfully constrains J1;:::;6independently of any exchange or\nDMI terms. Thus, we are able to re\fne the cluster ex-\npansion \ft by \frst \ftting J1;:::;6to this high{accuracy\nanisotropy data, and then \ftting all other terms to the\nfull dataset. As a result, we are able to reduce the \ft-\nting error of the anisotropy energy to the level of \u0016eV,\nas can be seen in Supplementary Figure 2[30], su\u000ecient\nto reliably resolve the anisotropy constants.\nDISCUSSION\nBroad applicability of the uniaxial skyrmion phase\ndiagram\nA direct comparison of our predicted phase diagram to\nexperimental results reported for GaV 4S8and GaV 4Se8\nlends credibility to our analysis. In both cases, the phase\ndiagrams obtained using the reduced anisotropy values\nfrom Table II are in qualitative agreement with experi-\nment. We quantitatively reproduce temperature behav-\nior, but predict phase transitions at \felds 1.5 to 2 times\nlarger than those observed. The most likely source for\nthis error is a deviation in \u0016BgMsaway from our as-\nsumed value of 2 \u0016B. We also neglect the impact of stray\n\felds, which destabilize the cycloid and N\u0013 eel skyrmion\nstates seen here in favor of a ferromagnetic con\fgura-\ntion with 180odomain walls[24]. In the strong DMI\nregime relevant to lacunar spinels, this error does not\nqualitatively change the results. However, the impact\nof stray \felds becomes pronounced for cycloidal systems\nwith weak DMI, limiting the applicability of Figure 4a\nto systems well away from the long{wavelength cycloid\nstability bounds derived by Bogdanov and Hubert[24].\nFurthermore, the agreement between the phase behavior\nwe derive in Figure 4 and that obtained phenomenologi-\ncally for the same symmetry[40, 41, 44] provides an im-\nportant consistency check for our atomistic model of the\nmagnetic behavior of these skyrmion{host materials.\nThe insensitivity of the skyrmion region of the phase\ndiagram to the values of DMI and exchange suggests that\nthe phase behavior seen in Figure 4a may generalize to\nother uniaxial systems. We observe that high{moment\nN\u0013 eel skyrmions form preferentially to canted cycloids\nwhen the magnetic \feld is orthogonal to the rotation\naxis of any cycloidal variant. This mechanism is inde-\npendent of any speci\fc interaction parameters and arises\nfrom the fact that canted cycloids only develop a mo-\nment parallel to their rotation axis as shown in Figure 1c.\nThus, we speculate that a similar mechanism may lead to\nN\u0013 eel skyrmion formation over wide \feld and temperature\nranges in other systems where cycloid variants have ro-\ntation axes constrained to a single plane. Based on sym-\nmetry arguments alone, this behavior is most likely in\nstrong{DMI, low{anisotropy systems whose point group\nis one of 6mm (C6v), 3m(C3v), 4mm (C4v), ormm2\n(C2v) [45]. If we assume that a similar mechanism is ap-plicable to the formation of canted helices and Bloch an-\ntiskyrmions, materials with \u001642m(D2d) and \u00164 (S 4) point\ngroup symmetry may also exhibit this behavior. Sev-\neral phenomenological analyses of skyrmion formation in\nCnv, D2dand S 4crystals[40, 41, 44] report similar results,\nlending support to a broad applicability of these trends.\nParametrization of magnetic cluster expansion\nHamiltonians\nAn important implication of our results is that un-\ncertainty quanti\fcation is essential to the construction\nof magnetic cluster expansion Hamiltonians. Conven-\ntional cluster expansion \ftting techniques rely on the\nminimization of total error against DFT energies. This\nmethodology works well when all interactions have simi-\nlar energy scales, and when working with discrete degrees\nof freedom such as atomic con\fgurations[46]. However,\nthis approach is not su\u000ecient when relatively small\nenergy terms, such as magnetocrystalline anisotropy\nexplored here, play a decisive role in determining which\nphases form. By analyzing the sensitivity of the phase\ndiagram to the values of the interaction coe\u000ecients\nthrough generalized Clausius-Clapeyron relations and\nlinear free energy extrapolation based on @A=@Ji=h'ii,\none can identify terms in the cluster expansion which\nplay an outsized role in determining phase behavior. In\nparticular, one can use this approach to evaluate the\nimportance of terms not included in the original Hamil-\ntonian, either due to basis set truncation, or elimination\nduring the \ftting process. Once all important terms\nare known, the \ftting procedure must be adjusted to\nensure these terms are \ftted accurately[47, 48], which\nmay increase the average error of the \ft but nonetheless\nyields more qualitatively correct predictions of phase\nbehavior.\nCONCLUSION\nWe have demonstrated that skyrmion stability across\na wide range of \felds and temperatures in the GaM 4X8\nlacunar spinels is a general consequence of the symme-\ntry of the material and the fact that magnetocrystalline\nanisotropy energy is typically small. We reproduce the\ncomplex magnetic phase diagrams of these materials, in-\ncluding long{wavelength magnetic order, without relying\non empirical parameters, and thus gain insight into the\nrelationship between spin{orbit coupling and skyrmion\nformation. We \fnd that magnetic cluster expansions\nparametrized using density functional theory data can\naccurately predict this magnetic phase diagram provided\nthat the \ftting procedure leads to a high{\fdelity form for\nthe anisotropy energy. More generally, reliable magnetic\nphase diagram prediction requires an evaluation of the\nimpact of \ftting error and uncertainty on phase stabil-10\nity. As we \fnd that the magnetic phase behavior here is\ndetermined by simple, transferable mechanisms dictated\nby point{group symmetry, we speculate that our obser-\nvations are likely broadly applicable to uniaxial systems\nwith Cnv, D2dand S 4symmetry.ACKNOWLEDGMENTS\nThe research reported here was supported by the Mate-\nrials Research Science and Engineering Center at UCSB\n(MRSEC NSF DMR 1720256) through IRG-1. 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Maccherozzi2\n1School of Physics and Astronomy, University of Nottingham,\nUniversity Park, Nottingham NG7 2RD, United Kingdom\n2Diamond Light Source, Harwell Science and Innovation Campus, Didcot OX11 0DE, United Kingdom\n3Institute of Physics, Polish Academy of Sciences,\nAleja Lotnikow 32/46, PL-02668 Warsaw, Poland\n4Institute of Physics, Czech Academy of Science,\nCukrovarnick\u0013 a 10, 162 00 Praha 6, Czech Republic\n5Faculty of Mathematics and Physics, Charles University,\nKe Karlovu 3, 121 16 Prague 2, Czech Republic\n6High Field Magnet Laboratory (HFML-EMFL), Radboud University,\nToernooiveld 7, 6525ED Nijmegen, The Netherlands\n(Dated: June 22, 2021)\nWe report magnetic \feld induced rotation of the antiferromagnetic N\u0013 eel vector in epitaxial CuM-\nnAs thin \flms. Firstly, using soft x-ray magnetic linear dichroism spectroscopy as well as magne-\ntometry, we demonstrate spin-\rop switching and continuous spin reorientation in \flms with uniaxial\nand biaxial magnetic anisotropies, respectively, for applied magnetic \felds of the order of 2 T. The\nremnant antiferromagnetic domain con\fgurations are determined using x-ray photoemission electron\nmicroscopy. Next, we show that the N\u0013 eel vector reorientations are manifested in the longitudinal and\ntransverse anisotropic magnetoresistance. Dependences of the electrical resistance on the orientation\nof the N\u0013 eel vector with respect to both the electrical current direction and the crystal symmetry\nare identi\fed, including a weak 4th order term evident at high magnetic \felds. The results provide\ncharacterization of key parameters including the anisotropic magnetoresistance coe\u000ecients, mag-\nnetocrystalline anisotropy and spin-\rop \feld in epitaxial \flms of tetragonal CuMnAs, a candidate\nmaterial for antiferromagnetic spintronics.\nI. INTRODUCTION\nIn antiferromagnetic (AF) materials, the atomic mag-\nnetic moments alternate in direction to produce zero\nnet magnetization. The consequent absence of magnetic\nstray \felds and relative insensitivity to external \felds of-\nfer advantages for certain memory applications [1, 2], but\nalso pose a challenge for the writing and reading of infor-\nmation. Recent works have shown that electric currents\ncan be used to set the AF spin axis in crystalline materi-\nals, including CuMnAs and Mn 2Au, where the spin sub-\nlattices are space-inversion partners [3{10]. While these\nobservations open up opportunities to use AF materials\nin magnetic memory devices, a better understanding of\nthe electrical readout mechanism is required. The rota-\ntion of the AF spin axis results in an anisotropic magne-\ntoresistance (AMR) e\u000bect where the resistance depends\non the relative orientation of the spin axis and current\ndirection.\nA collinear AF material is characterized by a N\u0013 eel vec-\ntorL= (m1\u0000m2), where m1andm2are the magne-\ntizations of each spin sublattice. An external magnetic\n\feldHapplied perpendicular to Lcompetes against the\nstrong internal exchange \feld Hewhich couples the sub-\nlattices. As a result, only a small canting of the magnetic\n\u0003Kevin.Edmonds@nottingham.ac.ukmoments into the \feld direction occurs when H\u001cHe,\nresulting in a small net magnetization M= (m1+m2).\nA more signi\fcant reorientation can occur at the so-called\nspin-\rop transition, from an AF state with Laligned par-\nallel to the external \feld, to a state with Lperpendic-\nular, but with individual moments canted into the \feld\ndirection. This is detected as a step-like increase in the\nmagnetic susceptibility in bulk materials with uniaxial\nanisotropy, but is hard to detect in thin \flms due to the\ntypically very small canting angle leading to a very weak\nmagnetic signal.\nAMR in AF materials has been explored only recently\n[11{21]. In hexagonal MnTe \flms, a clear cos 2 \u0012depen-\ndence on the angle \u0012between the current and the external\n\feld was demonstrated, due to the spin-\rop rotation of\nthe N\u0013 eel vector [15]. Studies of the canted AF semicon-\nductor Sr 2IrO 4have shown a crystalline contribution to\nAMR, with four-fold symmetry, originating from changes\nin the equilibrium electronic structure induced by the ro-\ntation of the magnetic moments [17, 18, 20]. In the case\nof the collinear antiferromagnet CuMnAs, understand-\ning of the anisotropic magnetoresistance and magnetic\nanisotropy is crucially important for the design of mem-\nory devices based on the reorientation of the N\u0013 eel vec-\ntor and interpretation of their electrical readout signal.\nMoreover, due to the predicted dependence of the elec-\ntronic band structure on N\u0013 eel vector orientation in CuM-\nnAs and related materials [7, 21], a signi\fcant crystallinearXiv:1911.12381v2  [cond-mat.mtrl-sci]  21 Jun 20212\nAMR e\u000bect may be expected, i.e.a dependence of the\nelectrical resistance on the directions of the current and\nthe N\u0013 eel vector with respect to crystalline symmetry [16].\nHere, we demonstrate magnetic \feld induced spin rota-\ntion and spin-\rop behavior in epitaxial \flms of tetragonal\nCuMnAs, using x-ray magnetic linear dichroism (XMLD)\nspectroscopy as well as integral magnetometry. XMLD is\nthe dependence of the x-ray absorption cross-section on\nthe relative orientation of the x-ray linear polarization\naxis and the magnetization axis [22]. Unlike x-ray mag-\nnetic circular dichroism (XMCD), XMLD is an even func-\ntion of the magnetic moment and can be detected equally\nin ferromagnets and antiferromagnets, and is widely used\nfor,e.g. AF domain imaging [23, 24],[5 ?, 6]. We then\nshow that the signature of the spin \rop or spin reorien-\ntation can also be detected electrically. This allows us\nto determine the crystalline and non-crystalline contri-\nbutions to the AMR in this material, of importance for\ninterpreting and optimizing the electrical readout in AF\nmemory devices.\nII. SAMPLE GROWTH AND MAGNETIC\nCHARACTERIZATION\nThe tetragonal CuMnAs \flms, with thickness between\n10 nm and 50 nm, were grown on GaP(001) substrates us-\ning molecular beam epitaxy. CuMnAs is lattice-matched\nto GaP through a 45\u000erotation, such that the CuMnAs\n[100] axis is aligned with the GaP [110] [25]. For thin ( \u0014\n20 nm) CuMnAs / GaP(001) layers, a uniaxial in-plane\neasy axis along the substrate [110] direction has been\ndemonstrated [26, 27], similar to well-known ferromag-\nnet / III-V systems such as Fe/GaAs(001) [28]. Thicker\n\flms show a biaxial AF domain structure with easy axes\nalong the CuMnAs [110] and [1 \u001610] axes [6].\nMnL2;3edge x-ray absorption spectroscopy measure-\nments were performed on beamline I06-1 at Diamond\nLight Source, at a sample temperature of 200 K. The\nsamples showed negligible XMCD even at the highest\n\feld applied of 6 T, indicating very small canting of\nthe magnetic moments due to their strong AF coupling.\nThe XMLD measurements were performed with the x-\nray beam and the external magnetic \feld both at 15\u000eto\nthe plane of the \flm [see Fig. 1(a), inset], with the x-\nray linear polarization along the CuMnAs [010] in-plane\ndirection ( i.e., the substrate [1 \u001610] direction). The ab-\nsorption spectra for a 20 nm \flm are shown in Fig. 1(a).\nA signi\fcant di\u000berence is observed between spectra ob-\ntained at low and high external magnetic \felds, as shown\nby the di\u000berence spectrum in Fig. 1(a).\nThe di\u000berence spectrum in Fig. 1(a) is of compara-\nble magnitude, relative to the absorption edge, to the\nXMLD spectrum previously observed in ferromagnetic\nMn compounds [29, 30]. Also, comparable spectra are\nobserved when the absorption is detected with both total\nelectron yield (probing depth \u00183 nm) and \ruorescence\nyield (probing depth >10 nm), ruling out a pure surface\nFIG. 1. (a) Mn L2;3x-ray absorption spectra for 20 nm CuM-\nnAs, in zero \feld (blue), 6 T \feld (red), and the di\u000berence\n(black), at 200 K. The x-ray incidence kand external mag-\nnetic \feld Hare at 15\u000eto the sample surface, while the x-ray\nlinear polarization Eis along the CuMnAs [010] in-plane axis,\nas illustrated in the inset. The CuMnAs [100] axis is projected\nalong the beam direction. (b,c) XMLD signal versus magnetic\n\feld for (b) 20 nm and (c) 50 nm CuMnAs \flms. Blue squares\nand orange circles are for the in-plane projection of the \feld,\nHx, along the [100] and [010] axes, respectively.\ne\u000bect. Figure 1(b,c) shows the \feld-dependence of the\nXMLD signal, de\fned here as the peak-to-peak of the dif-\nference between spectra obtained in a magnetic \feld and\nin zero \feld. Distinct behaviors are observed for 20 nm\nand 50 nm thick CuMnAs \flms. For 20 nm CuMnAs,\nthe XMLD signal shows a sharp onset at around 1.5 T\nand a plateau at around 2.5 T for magnetic \feld pro-\njected along the [100] direction, while negligible XMLD\nis observed for \feld projected along the [010] direction.\nFor the 50 nm \flm, similar behaviour is seen for both\n[100] and [010] directions, with an initial quadratic rise\nfollowed by saturation for \felds above around 2 T.\nThe observed behavior is consistent with the N\u0013 eel vec-\ntor reorientation expected for a system with competing\nuniaxial and biaxial magnetic anisotropies. For the thin-\nner \flm with dominant uniaxial anisotropy, the Lvector\nundergoes a spin-\rop transition into an axis perpendic-\nular to the applied magnetic \feld. In this experimental\ngeometry, Lis perpendicular to the x-ray polarization at3\nFIG. 2. (a) XMLD hysteresis loop for a 10 nm CuMnAs \flm\nat 200 K. (b) SQUID hysteresis loop for a 10 nm CuMnAs\n\flm at 2 K, after removal of the substrate diamagnetic signal.\n(c,d) XPEEM images, with 30 \u0016m \feld-of-view, of the anti-\nferromagnetic domain structure in a 50 nm CuMnAs \flm at\nroom temperature. The sample is at remanence after apply-\ning 7 T magnetic \felds in the directions B1 and B2 shown in\nthe inset. Dark / light domains correspond to a N\u0013 eel vector\nparallel / perpendicular to the x-ray polarization, which is\nalong the CuMnAs [110] axis.\nlow \felds, and parallel to it at high \felds, resulting in\na spectral change due to XMLD. For the thicker \flm,\nbiaxial magnetic anisotropy is dominant, resulting in a\nso-called continuous spin-\rop transition [31] as the Lvec-\ntor rotates continuously into an axis perpendicular to the\nexternal \feld.\nThe XMLD hysteresis loop for a 10 nm CuMnAs \flm is\nshown in Fig. 2(a). At high magnetic \felds, the XMLD\nsignal is symmetric with respect to the \feld, as expected\nfrom its dependence on the square of the sublattice mag-\nnetic moment [22]. The hysteresis behaviour observed in\nthe vicinity of the spin-\rop transition indicates the for-\nmation of multidomain states in the thin CuMnAs \flm\n[32].\nTo further con\frm and quantify the spin-\rop behavior,\nthe \feld dependence of the magnetization M(H) along\nthe uniaxial easy axis of a 10 nm CuMnAs \flm was de-\ntermined at 2 K using a Quantum Design MPMS super-\nconducting quantum interference device (SQUID) mag-\nnetometry system [33]. To in situ compensate the much\nlarger magnetic response of the GaP substrate, the sam-ple was mounted between two abutting 8 cm long strips\ncut from another (equivalent) 2\" GaP wafer, and careful\ncalibration procedures were performed [34]. The compen-\nsational sample holder assembled for these measurements\nis presented in Fig. 4(d) in ref. [34]. The results [Fig.\n2(b)] indicate a magnetization due to the spin canting\nof (2:0\u00060:5) emu/cm3under 2 T \feld, where the uncer-\ntainty mostly arises from the subtraction of the substrate\nbackground signal. Using the local magnetic moment of\n(3:6\u00060:2)\u0016Bper Mn atom in tetragonal CuMnAs ob-\ntained from neutron di\u000braction [25], this corresponds to a\ncanting angle of around 0.1-0.2\u000e. From this the exchange\n\feld can be estimated to be \u00160He= (700\u0006200) T. The\nspin-\rop \feld \u00160Hsf\u00192 T therefore corresponds to a\nmagnetic anisotropy \feld of the order of H2\nsf=He\u00195 mT.\nFigures 2(c) and (d) show images of the antiferromag-\nnetic domain con\fguration in a 50 nm CuMnAs \flm ob-\ntained using x-ray photoemission electron microscopy, in\nremanence after applying 7 T magnetic \felds at approx-\nimately 15\u000efrom the CuMnAs [110] [Fig. 2(c)] and [1 \u001610]\n[Fig. 2(d)] biaxial easy axes. In both cases, the \flm is\nin a multidomain state after removing the external mag-\nnetic \feld, with a domain structure which is consistent\nwith previous observations of similar CuMnAs \flms [6].\nSimilar features can be identi\fed in each image, albeit\nwith a signi\fcant remnant e\u000bect, i.e. a preponderance\nof light domains in Fig. 2(c) and dark domains in Fig.\n2(d). This indicates that the magnetization process in\nthe biaxial CuMnAs \flms likely proceeds through N\u0013 eel\nvector reorientation within the multiple domains as well\nas domain wall motion.\nIII. ANISOTROPIC MAGNETORESISTANCE\nDC magnetotransport measurements were performed\non Hall bar devices fabricated from the CuMnAs \flms.\nFigure 3 shows data for the 10 nm thick CuMnAs \flm at\n4 K, for devices with current channels along the [100],\n[010], [110] and [1 \u001610] crystalline directions. Magnetic\n\felds were applied in the plane of the \flm at an angle\n\u0012to the current direction. Figure 3(a,b) show the nor-\nmalized longitudinal component of the resistivity tensor\n[(Rxx\u0000Rxx(H= 0))=Rxx(H= 0), where Rxxis the lon-\ngitudinal resistance] as a function of magnetic \feld. A\nstep-like change in the resistance is observed between 1 T\nand 2 T for \felds applied along the easy axis [Fig. 3(a)],\nwhile a much weaker magnetoresistance is observed for\n\feld perpendicular to the easy axis [Fig. 3(b)].\nFigure 3(c,d) shows the normalized transverse compo-\nnent of the resistivity tensor [( Rxy\u0000Rxy(H= 0))=Rsq,\nwhereRsqis the sheet resistance of \u001920 \n per square\nat zero \feld]. Again, a step-like change is observed for\n\feld applied along the easy axis. For the longitudinal\nand transverse resistances, the step-like behavior is ob-\nserved for current parallel / perpendicular and at 45\u000e/\n225\u000eto the magnetic \feld, respectively. This is consistent\nwith the standard phenomenology of AMR [35]. Includ-4\nFIG. 3. Longitudinal (a,b) and transverse (c,d) magnetoresis-\ntances for 10 nm thick CuMnAs at 4 K, with magnetic \feld\napplied along the [100] (a,c) and [010] (b,d) crystal axes, for\nvarious current directions.\ning terms dependent on the crystalline anisotropy in a\nsingle crystal with twofold and fourfold symmetry com-\nponents, the longitudinal and transverse components of\nAMR can be written [36]:\nAMR xx= (Rxx\u0000Rave)=Rave\n=CIcos(2\u001e) +CUcos(2 )\n+CCcos(4 ) +CICcos(4 \u00002\u001e) (1)\nAMR xy=Rxy=Rsq\n=CIsin(2\u001e)\u0000CICsin(4 \u00002\u001e) (2)\nwhere\u001eand are the angles of the magnetization (for\na ferromagnet) or the N\u0013 eel vector (for an AF material)\nwith respect to the current direction and the [100] crys-\ntalline axis, respectively. Raveis the longitudinal resis-\ntance averaged over a full rotation in the plane of the \flm.\nCI,CU,CCandCICare phenomenological AMR coef-\n\fcients corresponding to a non-crystalline term, twofold\nand fourfold crystalline terms, and a crossed crystalline\n/ non-crystalline term, respectively.\nFIG. 4. Longitudinal magnetoresistances at 2 K, 100 K, 200 K\nand 300 K for (a-d) 20 nm CuMnAs and (e-h) 50 nm CuMnAs\n\flms. The current is along the [100] and the magnetic \feld is\napplied along the [100] (blue squares) and [010] (red circles)\ndirections.\nThe longitudinal magnetoresistance for 20 nm and\n50 nm CuMnAs \flms at various temperatures are shown\nin Fig. 4, for current along the [100] and magnetic \felds\nalong the [100] and [010] directions. Uniaxial spin-\rop\nbehavior and continuous spin reorientation are observed\nfor the 20 nm and the 50 nm \flms respectively, con-\nsistent with the XMLD \feld-dependence shown in Fig.\n1(b,c). For the 20 nm \flm, with decreasing temperature,\nthe spin-\rop \feld decreases and a small negative mag-\nnetoresistance is observed for \feld along [010]. This in-\ndicates an increasing importance of the biaxial magnetic\nanisotropy with decreasing temperature, consistent with\nthe expected strong dependence of biaxial anisotropy\n\felds on sublattice magnetization [37].\nThe longitudinal resistance versus angle \u0012is shown in\nFig. 5(a), for a 10 nm CuMnAs \flm at 4 K with magnetic\n\felds comparable or smaller than Hsf. The current is\nalong the [010] direction. With decreasing external \feld,\nthe oscillations in the resistance become smaller and more5\nFIG. 5. Resistance vs. angle \u0012between the magnetic \feld and\ncurrent directions at 4 K. (a) 10 nm CuMnAs, current along\nthe [010] direction, magnetic \felds of magnitude 1.0 T, 1.5 T,\n2.0 T and 3.0 T. (b) Calculated anisotropic magnetoresistance\nfor a uniaxial antiferromagnet with a Gaussian distribution of\nspin-\rop \felds centered on 1.6 T. (c) 10 nm CuMnAs, cur-\nrent along the [100], [010], [110] and [1 \u001610] crystal directions,\nmagnetic \feld of magnitude 5 T. (d) 50 nm CuMnAs, current\nalong the [100], [010], [110] and [1 \u001610] crystal directions, mag-\nnetic \feld of magnitude 5 T. The lines in (c) and (d) are \fts\nas described in the text.\nanharmonic. This can be ascribed to the increasing im-\nportance of the magnetic anisotropy, resulting in the N\u0013 eel\nvector only partially reorienting from its zero-\feld direc-\ntion. The observed behavior can be simulated within\na simple single-domain model including a uniaxial mag-\nnetic anisotropy. This simpli\fed model agrees well with\nthe observed dependence on both the magnitude and di-\nrection of the external magnetic \feld, including the ap-\npearance of additional features in the angle-dependence\nfor\u00160H=1 T, as shown in Fig. 5(b).At higher \felds, the longitudinal resistance follows a\ncos 2\u0012dependence, as expected from equation (1) with\nCI;CU;CIC\u001dCCwhen the N\u0013 eel vector aligns fully per-\npendicular to the \feld ( i.e.,\u001e=\u0012). However, as shown\nin Fig. 5(c,d) for 10 nm and 50 nm \flms, the observed\nRxx(\u0012) depends strongly on the direction of the current,\ndemonstrating the importance of the crystalline contri-\nbutions to the AMR. For currents along the [110] and\n[1\u001610] directions, the AMR is much reduced and for the\n10 nm \flm is phase shifted by around \u000625\u000e. By \ftting\nthe \feld rotation data we obtain the AMR coe\u000ecients\nCI= (4:1\u00060:1)\u000210\u00004,CU= (1:0\u00060:2)\u000210\u00004and\nCIC= (3:5\u00060:1)\u000210\u00004for the 10 nm \flm, and CI=\n(8:3\u00060:1)\u000210\u00004,CU\u00190 andCIC= (6:2\u00060:1)\u000210\u00004\nfor the 50 nm \flm. In both cases the CICterm is com-\nparable in magnitude to the CIterm. Therefore, these\ntwo terms largely cancel each other out for current along\nthe [110] and [1 \u001610] directions, and the AMR is strongly\nsuppressed. The non-negligible uniaxial term CUfor the\n10 nm \flm accounts for the di\u000berence in the AMR magni-\ntude between the [100] and [010] current directions, and\nthe phase shift for the [110] and [1 \u001610] directions. It is\nlikely that this uniaxial crystalline AMR has the same\norigin as the uniaxial magnetic anisotropy giving rise to\nthe spin \rop, e.g. anisotropic growth initiated at the\nIII-V semiconductor surface.\nThe magnetoresistance at higher magnetic \felds is\nshown in Fig. 6(a-c), for a 10 nm CuMnAs \flm at 4 K.\nIn addition to the AMR, a positive magnetoresistance is\nobserved for all directions of magnetic \feld and current.\nMoreover, the highest resistance occurs when the current\nand the magnetic \feld are perpendicular at high \felds.\nThis is opposite to the case at low \felds where the highest\nresistance is when the \feld and current are coaxial. We\nascribe the high \feld behaviour to a geometrical magne-\ntoresistance e\u000bect, not directly related to the magnetic\norder, which is commonly observed in thin conducting\n\flms [38]. Field rotation measurements at the crossover\nbetween low-\feld and high-\feld regimes, shown in Fig.\n6(d), indicate a weak AMR with a dominant fourfold\nsymmetry. For the rotation data at 5 T [Fig. 5(c)], the\nfourfold AMR term CCcould not be distinguished due to\nthe much larger two-fold AMR terms [see eq. (1)]. In the\ncrossover regime [Fig. 6(d)], the twofold AMR terms are\nnearly cancelled by the geometrical magnetoresistance ef-\nfect. The remaining fourfold AMR has resistance minima\nfor magnetic \felds along the h110icrystalline axes.\nIV. DISCUSSION AND SUMMARY\nThe magnetotransport e\u000bects shown in Figs. 3 to 6\nprovide a practical experimental tool for the determina-\ntion of spin reorientation in antiferromagnetic domains\nand magnetic anisotropies in thin CuMnAs AF \flms.\nThey also provide a direct measurement of the AMR\ncoe\u000ecients and point to the importance of crystalline\nAMR terms in these epitaxial AF materials. The mea-6\nFIG. 6. (a,b) Magnetoresistance for 10 nm CuMnAs under\nhigh magnetic \felds at 4 K, for current along the [100] and\n[010] directions, and applied magnetic \feld perpendicular and\nparallel to the current. (c) Di\u000berence between the magnetore-\nsistances for parallel and perpendicular orientations of current\nand magnetic \feld, for currents along the [100] and [010] di-\nrections. (d) Angle dependence of resistance for current along\n[010] at 13 T (squares) and for current along [100] at 14 T\n(circles). The lines are \fts of the form ( acos 2\u0012+bcos 4\u0012)\nwithaandbas \ft parameters.sured AMR corresponds to a fraction of a percent of the\nsheet resistance, comparable to the size of the electrical\nreadout signals observed in the \frst demonstrations of\ncurrent-induced switching in CuMnAs microdevices [4{\n6].\nMore recently, resistive signals of of the order of 10-\n100 percent have been observed in the same CuMnAs\n\flms as used in the present study. These signals are\nswitchable with unipolar current pulses or polarization-\nindependent optical pulses, even in magnetic \felds much\nlarger than Hsf[39]. The present study provides con-\n\frmation that the large switching signals are not linked\nto a net reorientation of the N\u0013 eel vector and the corre-\nsponding AMR. A recent magnetostatic imaging study\nhas instead linked the large unipolar switching signal to\na current-induced fragmentation to nanoscale AF domain\ntextures, and subsequent relaxation towards an equilib-\nrium domain con\fguration [40]. In principle, however,\nso far unidenti\fed structural e\u000bects may also contribute\nor even govern these large e\u000bects. Future systematic ex-\nperiments in magnetic \feld might, therefore, not only\ncontribute to our understanding of the current-induced\nN\u0013 eel vector switching and AMR but also of these high-\nresistive switching signals in CuMnAs.\nACKNOWLEDGMENTS\nThis work was supported by the Engineering\nand Physical Sciences Research Council Grant No.\nEP/P019749/1, the EU FET Open RIA Grant No.\n766566, the Ministry of Education of the Czech Re-\npublic Grants LM2015087 and LNSM-LNSpin, and the\nGrant Agency of the Czech Republic Grant No. 19-\n28375X. We thank the Diamond Light Source for the\nallocation of beam time under Proposals SI11846 and\nSI20793. The high magnetic \feld study was supported\nby HFML-RU/NWO-I, member of the European Mag-\nnetic Field Laboratory (EMFL) and by EPSRC via its\nmembership to the EMFL (grant no. EP/N01085X/1).\nWe thank Prof. Amalia Patane for providing access to a\nhigh \feld cryostat at the University of Nottingham. 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Paudyal3  1CNR-NANO Istituto Nanoscienze, Centro S3, I-41125 Modena, Italy 2Harish-Chandra Research Institute, HBNI, Jhunsi, Allahabad 211019, India 3The Ames Laboratory, U.S. Department of Energy, Iowa State University, Ames, IA 50011–3020, USA  We report magnetocrystalline anisotropy of pure and Fe/Si substituted SmCo5. The calculations were performed using the advanced density functional theory (DFT) including onsite electron-electron correlation and spin orbit coupling. Si substitution substantially reduces both uniaxial magnetic anisotropy and magnetic moment. Fe substitution with the selective site, on the other hand, enhances the magnetic moment with limited chemical stability. The magnetic hardness of SmCo5 is governed by Sm 4f localized orbital contributions, which get flatten and split with the substitution of Co (2c) with Si/Fe atoms, except the Fe substitution at 3g site. It is also confirmed that Si substitutions favor the thermodynamic stability on the contrary to diminishing the magnetic and anisotropic effect in SmCo5 at either site.  INTRODUCTION  Search for high-performance permanent magnets is always being a challenging task for modern technological applications as they are key driving components for propulsion motors, wind turbines and several other direct or indirect market consumer products. Curie temperature Tc, magnetic anisotropy, and saturation magnetization Ms, of the materials, are some basic fundamental properties used to classify the permanent magnets. At Curie temperature a material loses its ferromagnetic properties, hence higher the Tc, better is the magnets to be used under extreme conditions. Out of all rare earth based permanent magnets, SmCo5 is the champion magnet in terms of uniaxial magneto-crystalline anisotropy energy (MAE) with high Curie temperature. MAE is another important parameter, which defines the hardness with respect to the microscopic characteristics. In spite of the highest uniaxial MAE of ~24.2 MJ/m3 (12.96 meV/f.u.) in SmCo5,1 the world market is dominated by the Nd2Fe14B (MAE of ~4.9 MJ/m3) 2,3,4,5,6,7 permanent magnet because of its low energy product. Since after the formulation of SmCo5 in 1970, lots of research has been done to increase the magnetic moment of SmCo5 compared to that of Nd2Fe14B.   The high coercivity of samarium based magnets originates from the Sm sublattice anisotropy, whereas the transition metals such as Co sublattice yields a high curie temperature and thus stabilizes through inter-sublattice exchange.8 Past studies9,10 have also confirmed that Tc depends on the interaction of magnetic exchange between adjacent spins. Magnetic anisotropy is the energy required per unit volume to change the                                                 *rajivchouhan@gmail.com, rajiv.chouhan@nano.cnr.it orientation of the magnetic moments under the external magnetic field, which is required to achieve the high coercivity in permanent magnets. Magnetic anisotropy also depends on the intrinsic property of a material i.e. does not depend on the microstructure arrangement, where the spin-orbit coupling and its interaction to the crystal electric field created by their neighbor favors the energetically stable magnetic alignment in the specific direction of the lattice. This intrinsic behavior of magnetic anisotropy has the potential to enhance the characteristics of the microstructure within SmCo5. Hence, one can utilize grain boundaries developed through the site substitutions to increase the energy product.11 Further, the saturation magnetization Ms is the measure of the moment per unit volume defined12 as maximum energy product 𝐵𝐻max≈ 𝑀!!.  In this manuscript, we report the influence of Fe/Si substitutions in the non-equivalent Co (2c) and Co (3g) sites of SmCo5. We have investigated the effect of doping of a d block element iron (Fe) and a p block element silicon (Si). SQUID magnetometer measurements are performed to investigate the magnetic moment properties of SmCo5. We have also analyzed the adequacy of DFT that includes the onsite 4f electron correlation and spin orbit coupling calculations, which is very important for investigating the properties of rare earth based permanent magnets.  METHODS  We have performed the advanced density functional theory (DFT)13 based calculations to study electronic, magnetic, and MAE properties of pure and site substituted SmCo5. We have considered Fe and Si substitutions to reduce the Co content and to optimize the magnetocrystalline anisotropy and magnetic moment if possible.  We employed full-potential linearized augmented plane wave (FP-LAPW)14 the method within the framework of generalized gradient approximation, onsite electron correlation, and spin orbit coupling (GGA+U+SOC)15,16. To validate our calculations, we have employed a non-collinear plane-wave self-consistent field (PwScf)17 the method within the GGA+U+SOC framework when needed. We have used the optimized onsite electron correlation parameters U = 6.7 eV and J = 0.7 eV, additionally, it is known in the literature that U= 4 to 7 eV works well for rare earth systems without substantially changing the physical properties18. Here, MAE is calculated using the difference between the c-axis and the planar direction total energies.19 The k-space integrations have been performed with a Brillouin zone mesh measuring at least 13×13×15, which was sufficient for the convergence of total energies (10-6Ryd.), charges, and magnetic moments. For SmCo5 the atomic radii for Sm and Co are set as 2.5 and 2.19 with force minimization of 3%. In site-substituted systems, these radii are changed while following the force minimization criteria. As mentioned in the literature1, slightly higher values of plane-wave cutoff (RKmax = 9.0/𝑅!\"!\"# = 4.11 a.u.-1 and Gmax = 14𝑅𝑦) are required for rare earth element systems. Formation energies (△𝐻!\"#$) of the substitute compounds are calculated by taking the difference of the total energies and chemical potentials of the elements: △𝐻!\"#$=𝐸!\"#!$%$\"$%&'!\"#$%− (𝜇!\"+𝑛 Σ 𝜇!\"!#!$%), where 𝐸!\"#!$%$\"$%&'!\"#$% is the total energy of the substituted compound, n is number of atoms per element per f.u., and 𝜇!\" & 𝜇!\"!#!$% are the chemical potentials of the Sm, and substituted elements (Co, Si, and Fe), respectively. The bulk calculations for pure elements are done with the experimental parameters: Sm20 (SG = 166, a = 16.97 a.u., alpha = 23.31 degree); Co21 (SG = P63/mmc, a = 4.738 a.u., c = 7.691 a.u.); Si22 (SG = 227, a = 10.259 a.u.); Fe23 (bcc, a = 5.424 a.u.). Finally, to verify the theoretically predicted magnetic moments in SmCo5, we have prepared a sample (as cast) using the arc melting procedure and measured magnetization as a function of the magnetic field using a SQUID magnetometer.   RESULTS   SmCo5 forms in the hexagonal CaCu5-type structure (figure 1) with three non-equivalent sites: Sm (1a), Co (2c), and Co (3g).  Sm lies in the middle of the hexagonal layer of Co (2c) atoms, and this layer is sandwiched by the plane containing Co (3g) atoms.  This structural environment creates a crystal field that splits localized Sm 4f states and partially quenches the Sm 4f orbital moment giving rise to a large part of the magnetic anisotropy and net Sm 4f moment. We have performed volume relaxation by varying the lattice constants c and a. The equilibrium lattice constants came out to be very close to the experimental values (a = 9.45 a.u. and b= 7.48 a.u.)24 with a volume of 579.19 a.u.3  \n FIG. 1. Crystal structure of SmCo5 representing Co1 (2c site) and Co2 (3g site) of cobalt.  Similar volume relaxation procedure was performed for the Fe, and Si substituted compounds (figure 2). The optimized lattice parameter for SmCo2Fe3, SmFe2Si3, SmCo2Si3, and SmSi2Co3 are mentioned in Table.1. The optimized lattice parameter results indicate that the stability of the Fe substituted structure at 2c, and 3g sites are achieved with the increase in the unit cell volume by 3.9 %, and 5 % in SmCo5, whereas for Si substitution at 2c, and 3g sites the volume of the unit cell change by -1.5 %, and 1.9 %, respectively.   System a (in a.u.) c (in a.u.) ∆Volume w.r.t SmCo5 SmFe2Co3 9.638 7.362 3.9 % SmCo2Fe3 9.639 7.557 5 % SmSi2Co3 9.92 6.725 -1.5 % SmCo2Si3 9.771 7.128 1.9 % Table 1: Lattice parameters for various systems in atomic unit. \n  FIG. 2. Volume relaxation of the crystal geometry SmCo2(Fe/Si)3 and Sm(Fe/Si)2Co3.  The △𝐻!\"#$ for the substitute SmCo5 is plotted in figure 3. The data clearly shows that the silicon substitutions are energetically favorable whereas Fe substitution at 2c site is unstable compared to the 3g sites. The instability of iron substitution calculated for iron substitutions are also confirmed by previous results.25  \n FIG. 3. Formation energies of the Sm(Si/Fe)x Co(5-x) compounds  The calculated spin magnetic moment (employing GGA+U+SOC) for Sm in SmCo5 is 4.88 µB with the orbital contribution of -1.44 µB. The spin moments for Co (2c) (1.55 µB) \nand Co (3g) (1.58 µB) are aligned parallel to the spin moment of Sm, and similarly, the orbital moments of Co (2c) (0.14 µB) and of Co (3g) (0.12 µB) are aligned parallel to the spin moments of Sm. Henceforth, the total magnetic moment comes out to be 11.12 µB. This value is large because of the partial quenching of the Sm 4f orbital moments. The orbital moment also depends heavily on the atomic radius and the value of the onsite Sm 4f electron correlation. The GGA+SOC calculation shows more negative Sm 4f orbital moment (-2.29 µB), which further reduces the net moment to 10.54 µB which is comparable to the SQUID magnetometer measured the experimental value of ~ 8.02 µB (at 2K) shown in figure 4.  \n FIG. 4. Magnetization as a function of applied magnetic field at 2k and 300k for bulk SmCo5.  The partially filled Sm 4f orbitals play a key role in the single ion magnetic anisotropy in SmCo5. The calculated MAE for this compound is 26.06 meV/f.u. (uniaxial) which agrees well with the available experimental values (~13 to 16meV/f.u.).26,27,28,29,30 The GGA+SOC calculations also show uniaxial MAE of 20.82 meV/f.u., which agrees well with the experimental MAE. There is no sign problem as indicated by earlier studies31. Although the MAE with GGA+SOC is much closer to the experiment, however, the occupied and unoccupied 4f states are incorrectly located at the Fermi level leading to artificial 4f-4f bonding in SmCo5. Therefore, for better spectroscopy, we need to incorporate Hubbard U. The earlier work using GGA+SOC+U reported a MAE of 21.6 meV/cell,1 which is smaller than our findings. It is not surprising because the k-points, muffin-tin radii, and Hubbard parameters used in P. Larson et al.31 calculations are different than ours. To reveal the origin of high MAE, we further performed charge density calculations (figure 5) along the uniaxial direction as a side and top view to the plane. It is clearly seen from the charge densities plots that charges are distributed in the uniaxial direction around the Sm atoms and give rise to the magnetocrystalline anisotropy in the system. The origin of axial charge accumulation is because of the hexagonal Co (2c), and planar Co (3g) ring structures (Figures 1, and 5), which bind the Sm atoms between symmetric environments within the planes.  \n  FIG. 5. Charge densities for SmCo5 along c-axis side view (a, a’) and c-axis top view (b, b’) indicating the uniaxial anisotropy of Sm atoms.  Further, to see the effect of doping in SmCo5 we carried out Fe and Si substitution at Co (2c), and Co (3g) sites. First we doped the constituent atom at the 2c site and found that in SmFe2Co3 the spin (orbital) moments for Sm (1a), Fe (2c), and Co (3g) sites are 4.90 µB (-2.38 µB), 2.60 µB (0.05 µB), and 1.54 µB (0.09 µB), respectively. The total magnetic moment comes out to be 12.02 µB in SmFe2Co3. However in the case of Si doping at the 2c site the total magnetic moments drastically dropped to 3.03 µB making a spin contribution of 5.14 µB (orbital -1.39 µB), 0.01 µB, and 0.37 µB (zero orbital moments) for Sm (1a), Si (2c), and Co (3g), respectively. The calculated MAE values for SmFe2Co3 and SmSi2Co3 are 13.05 meV/f.u., and 16.82 meV/f.u., which are along the uniaxial direction similar to that for SmCo5 but with a lower magnitude. However, when we substituted Fe, and Si at the 3g sites the compound has a major doping effect. In the case of SmCo2Fe3 the spin moments for Sm (1a), Co (2c), and Fe (3g) sites are 4.87 µB, 1.60 µB, and 2.55 µB, whereas the orbital moments are -1.44 µB, 0.104 µB, and 0.05 µB, respectively. This results in the total magnetic moment of 14.02 µB in the SmCo2Fe3 system. Interestingly, the magnetocrystalline anisotropy for Fe at the 3g site is 28.84 meV/f.u., which is comparable to SmCo5 with a 26% large magnetic moment. But when we substitute Si at the 3g sites the SmCo2Si3 behaves like a non-magnetic system (total moment ~ 0.6 µB) by vanishing the individual magnetic contributions of the Sm, and Co \natoms. This is because of the Sm-Co layer is sandwiched and hybridized by the p orbital of the newly formed Si layers. This results in the MAE of the entire SmCo2Si3 system to be planar (-0.15 meV/f.u.).  The density of states plot in figure 6 shows the contribution of from the individual atoms in the compounds SmCo5, Sm(Fe/Si)2Co3, and SmCo2(Fe/Si)3. It is clearly visible that the inclusion of onsite correlation in the calculation shifts the Sm (4f) DOS to -6.0 eV energy making the split of three states compared to two states shown by GGA+SOC in SmCo5. Also, there is an extra crystal field splitting in one of the 4f states because of the spin-orbit coupling interaction within the orbitals. The splitting becomes more prominent with the U incorporation in the calculation going from GGA+SOC to GGA+U+SOC. This is one of the main reasons for such high magnetocrystalline anisotropy in SmCo5.  \n  FIG. 6. Individual atomic density of states (DOS) for SmCo5, Sm(Fe/Si)2Co3 and SmCo2(Fe/Si)3. Red line corresponds to the 4f DOS and blue and green correspond to the DOS of 2c and 3g sites.  \nThe Co (3g) site has a significant contribution to the magnetic moment in SmCo5 and it remains unaffected with the Fe substitution at the 2c site, whereas Si substitution at the 2c site reduces the spin moment contribution of Co (3g) by p-d hybridization. It can also be noticed that in the case of SmFe2Co3, we can see that there is a further splitting of 4f orbital peak (5 peaks) below the Fermi energy, which may be responsible for lowering of MAE compared to the four peaks in SmCo5. This is not the case in the SmCo2Fe3 because the anisotropy is comparable to SmCo5 and raising the total magnetic moment contribution by 26% per formula unit. In Si substitution, the magnetic moment and MAE both are affected drastically. At the 3g sites, the Si substitution kills all the contributions of Sm 4f orbitals. Similarly, in SmSi2Co3 the Sm 4f split states are distorted and kill the 3d spin contribution of Co atoms. As a result, it is suggested that Fe should be substituted at the 3g sites to gain the magnetic moment without compromising the anisotropy of the system.  Conclusion:   In conclusion, the substitution mechanism of magnetic Fe and non-magnetic Si atoms suggests that the hexagonal ring of 2c sites containing Sm atom in middle are responsible to maintain the magnetic hardness in the materials. This hardness is defined by the 4f localized orbital contribution of Sm atoms. This localized orbital get broaden when Co (2c) sites are substituted by the Si, and Fe atoms, which further decrease the anisotropy contribution significantly. Whereas the Co (3g) layer contributes the total magnetic moment of the compound and when it is substituted with Fe atoms, it not only increase the total magnetic moment to 14.02 µB but also boosts the anisotropy by 10% in the SmCo2Fe3 compared to SmCo5.  This PDOS analysis clearly demonstrates that Sm (4f) contribution is not affected in SmCo2Fe3, however, the extra increment of anisotropy comes from the increased PDOS contribution of Fe (3g) layer. Further, we confirm that the Si substitution favors the thermodynamic stability on the contrary to diminishing the magnetic as well as the anisotropic effect of SmCo5 significantly.   Acknowledgment:   The research was supported by the Critical Materials Institute, an Energy Innovation Hub funded by the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, Advanced Manufacturing Office. This work was performed at the Ames Laboratory, operated for DOE by Iowa State University under Contract No. DE-AC02-07CH11358.  R. K. Chouhan also thanks to HPC computational facilities of Harish-Chandra research institute, Prayagraj, India for performing few unfinished calculations. We acknowledge fruitful discussions with Professor Vitalij Pecharsky (group leader), and late Professor Karl A. Gschneidner Jr. (chief scientist, CMI) during the course of this research.                                                  1 Larson P and Mazin I I 2003 J. Appl. Phys. 93 6888. 2 Herbst J F 1991 Rev. Mod. Phys. 63 819.                                                                                                                                             3 Sagawa M, Fujimura S, Togawa N, et al. 1984 J. Appl. Phys. 55 2083–2087.  4 Buschow K H J 1991 Rep. Progr. Phys. 54, 1123-1213. 5 Givord D, Li H S, and Perrier de la Bthie R1984 Solid State Commun. 51 857. 6 Boge M et al. 1985 Solid State Commun. 55 295. 7 Chouhan Rajiv K, Pathak Arjun K, Paudyal D, Pecharsky V K 2018 arXiv:1806.01990. 8 Skomski R and Coey J M D 1999 Permanent Magnetism (IOP, Bristol). 9 Lamichhane T N et al. 2019 Phys. Rev. Applied 11 014052. 10 Choi Y et al. 2016 Phys. Rev. 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Harcourt Brace College Publishers, New York. 24 Wyckoff R W G 1986 Crystal Structures (Krieger, Melbourne, FL), Vol. 2. 25  Landa A et al. 2018 Journal of Alloys and Compounds 765 659e663. 26 Szpunar B 1981 Acta Phys. Pol. A 60 791. 27  Zhao T S, Jin H M, Grossinger R, Kou X C, and Kirchmayr H R 1991 J. Appl. Phys. 70 6134. 28  Leslie-Pelecky D L and Schalek R L 1999 Phys. Rev. B 59 457. 29  Chen C H, Walmer M S, Walmer M H, Gong W, and Ma B M 1999 J. Appl. Phys. 85 5669. 30  Al-Omari I A, Skomski R, Thomas R A, Leslie-Pelecky D, and Sellmyer D J 2001 IEEE Trans. Magn. Mag. 37 2534. 31 Larson P, Mazin I I, and Papaconstantopoulos D A 2003 Phys. Rev. B 67 214405. "    },    {        "title": "1912.03474v2.Role_of_higher_order_exchange_interactions_for_skyrmion_stability.pdf",        "content": "Role of higher-order exchange interactions for skyrmion stability\nSouvik Paul,1,\u0003Soumyajyoti Haldar,1Stephan von Malottki,1and Stefan Heinze1\n1Institute of Theoretical Physics and Astrophysics,\nUniversity of Kiel, Leibnizstrasse 15, 24098 Kiel, Germany\n(Dated: January 10, 2020)\nTransition-metal interfaces and multilayers are a promising class of systems to realize nanometer-\nsized, stable magnetic skyrmions for future spintronic devices. For room temperature applications,\nit is crucial to understand the interactions which control the stability of isolated skyrmions. Typ-\nically, skyrmion properties are explained by the interplay of pair-wise exchange interactions, the\nDzyaloshinskii-Moriya interaction and the magnetocrystalline anisotropy energy. Here, we demon-\nstrate that the higher-order exchange interactions { which have so far been neglected { can play\na key role for the stability of skyrmions. We use an atomistic spin model parametrized from \frst-\nprinciples and compare three di\u000berent ultrathin \flm systems. We consider all fourth order exchange\ninteractions and show that, in particular, the four-site four spin interaction has a giant e\u000bect on\nthe energy barrier preventing skyrmion and antiskyrmion collapse into the ferromagnetic state. Our\nwork opens new perspectives to enhance the stability of topological spin structures.\nMagnetic skyrmions { localized spin structures with a\ntopological charge [1] { have raised high hopes for fu-\nture magnetic memory and logic devices due to their\nnanoscale dimensions, stability and ultra-low energy\ndriven motion [2{6]. Skyrmion lattices have been \frst\nobserved in bulk magnets with a broken inversion sym-\nmetry in their crystal structure [7, 8]. The discovery\nof a skyrmion lattice in a single atomic layer of Fe on\nthe Ir(111) surface [9] has opened the door to a new\nclass of systems: transition-metal interfaces and mul-\ntilayers. Due to the possibility of varying \flm com-\nposition and structure, these systems allow to mod-\nify magnetic interactions and thereby the properties of\nskyrmions. At such transition-metal interfaces, individ-\nual magnetic skyrmions with diameters ranging from a\nfew 100 nanometers down to a few nanometers have been\nrealized as a metastable state in the \feld-polarized fer-\nromagnetic background [8, 10{16] as needed for applica-\ntions.\nA key challenge of skyrmion based data processing and\nstorage technology is the robustness of information carri-\ners, i.e., stability of the skyrmionic bits, against random\nthermal \ructuations at operating temperatures. At \fnite\ntemperature, the magnetic moments of skyrmions are\ncoupled to the environment, which induces \ructuations.\nOver time, a rare energy \ructuation can grow in excess of\nthe barrier height and can prompt the skyrmion to over-\ncome the barrier and collapse to the ferromagnetic back-\nground leading to a loss of topological charge. There-\nfore, an accurate assessment of barrier height is essential\nto determine the stability of skyrmions. To achieve data\nreading and writing capabilities of skyrmionic bits with\nhigh e\u000eciency, a control over the barrier height is also\nnecessary.\nThe existence of chiral magnetic skyrmions [17] is as-\ncribed to a competition of the Heisenberg pair-wise ex-\nchange interaction, the Dzyaloshinskii-Moriya interac-\ntion (DMI) [18, 19], the magnetocrystalline anisotropyand the dipole-dipole interactions. A prerequisite of DMI\n{ which provides a unique rotational sense to skyrmions {\nis the concerted action of spin-orbit coupling (SOC) and\nbroken inversion symmetry, which can be achieved at in-\nterfaces of transition-metals [20]. The DMI further stabi-\nlizes metastable isolated skyrmions against annihilation\ninto the ferromagnetic background [17, 21]. Often the\nexchange interactions are treated in a micromagnetic or\ne\u000bective nearest-neighbor approximation. However, the\nexchange interactions are long-range in itinerant mag-\nnets such as 3 dtransition-metals. This can lead to a\ncompetition between exchange interactions from di\u000ber-\nent shells of atoms resulting in an enhanced skyrmion\nstability [16, 22] even in the absence of DMI [23].\nThe itinerant character of 3 dtransition-metals lim-\nits the applicability of the Heisenberg model to describe\ntheir magnetic properties. Based on the spin-1 =2 Hub-\nbard model, it has been shown that the higher-order ex-\nchange interactions (HOI) beyond pair-wise Heisenberg\nexchange can arise such as the two-site four spin (bi-\nquadratic) or the four-site four spin interaction [24, 25].\nSuch higher-order terms can lead to intriguing magnetic\nground states due to a superposition of spin spirals { so-\ncalled multi- Qstates { which have been predicted based\non \frst-principles calculations [26]. The interplay of the\nfour-site four spin interaction and DMI is the origin of the\nnanoskyrmion lattice of the Fe monolayer on Ir(111) [9]\nand the e\u000bect of the biquadratic interaction on skyrmion\nlattice formation has been studied systematically [27]. It\nhas been further demonstrated that the HOI can com-\npete with the DMI and stabilize novel magnetic ground\nstates [28]. Based on a multi-band Hubbard model, a\nthree-site four spin interaction has recently been pro-\nposed for systems with a spin beyond S= 1=2 in addition\nto the biquadratic and the four-site four spin interaction\nin fourth order perturbation theory of the hopping pa-\nrametertwith respect to the Coulomb energy U[29].\nThis term has been attributed to stabilize a double- QorarXiv:1912.03474v2  [cond-mat.mes-hall]  9 Jan 20202\nso-called up-up-down-down ( uudd ) state in an Fe mono-\nlayer on Rh(111) [30]. Despite the compelling experi-\nmental evidence of the relevance of HOI, they have been\nneglected so far in the theoretical description of the prop-\nerties of isolated magnetic skyrmions at transition-metal\ninterfaces.\nHere, we reveal the intriguing role played by the HOI\nfor the stability of topologically non-trivial spin struc-\ntures such as skyrmions and antiskyrmions at transition-\nmetal interfaces. We use spin dynamics simulations\nbased on an atomistic spin model with all parameters\ncalculated via density functional theory (DFT). The en-\nergy barrier, preventing skyrmions and antiskyrmions\nto collapse into the ferromagnetic state, is obtained us-\ning the geodesic nudged elastic band (GNEB) method\n[31]. We consider three ultrathin \flm systems: (i) fcc-\nPd/Fe bilayer on Rh(111) (Fig. 1a) for which sub-10\nnm skyrmions have been predicted at low magnetic \feld\n[32], (ii) fcc-Pd/Fe bilayer on Ir(111), the most inten-\nsively studied ultrathin \flm system that hosts isolated\nskyrmions [10, 11, 22, 33{40] and an hcp-Fe/Rh bilayer\non Re(0001) with an in-plane easy magnetization axis\n[41].\nUpon including the HOI, the stability of skyrmions\nand antiskyrmions in all of these \flms is greatly modi-\n\fed. Surprisingly, the e\u000bect of the biquadratic and thethree-site four spin interaction concerning the energy bar-\nrier is to a good approximation already captured in the\nexchange constants obtained by mapping the DFT re-\nsults to a spin model neglecting HOI. The four-site four\nspin interaction has a large e\u000bect on the saddle point and\nit is responsible for the large change in energy barriers.\nWe \fnd a linear scaling of the barrier height with the\nfour-site four spin interaction. The barrier is enhanced\nor reduced depending on its sign. Even small values of\nthe four-spin four site interactions of 1 to 2 meV, typ-\nical for 3dtransition metals, modify the energy barrier\nby 50 to 120 meV. This leads to a huge enhancement or\nreduction of the skyrmion or antiskyrmion lifetime. We\nfurther show that the HOI can stabilize topological spin\nstructures in the absence of DMI.\nResults\nAtomistic spin model and DFT calculations. We\ndescribe the magnetic state of an ultrathin \flm by a set\nof classical magnetic moments fMiglocalized on each\natom siteiof a hexagonal lattice and their dynamics is\ngoverned by the following Hamiltonian:\nH=\u0000X\nijJij(mi\u0001mj)\u0000X\nijDij\u0001(mi\u0002mj)\u0000X\niK(mz\ni)2\u0000X\ni\u0016sB\u0001mi\u0000X\nijBij(mi\u0001mj)2\n\u00002X\nijkYijk(mi\u0001mj)(mj\u0001mk)\u0000X\nijklKijkl[(mi\u0001mj)(mk\u0001ml) + (mi\u0001ml)(mj\u0001mk)\u0000(mi\u0001mk)(mj\u0001ml)](1)\nwhere mi=Mi=Miis a unit vector. The exchange\nconstants ( Jij), the DMI vectors ( Dij), the mag-\nnetic moments ( \u0016s) and the uniaxial magnetocrystalline\nanisotropy energy (MAE) constant ( K) were calculated\nbased on DFT (see Refs. [22, 32, 37, 41]). We neglect\nthe dipole-dipole interaction since it is small in ultrathin\n\flms, which is of the order of 0.1 meV/atom, and it can\nbe e\u000bectively included into the MAE [42, 43].\nThe last three terms are the biquadratic interaction\n(Bij), the three-site four spin interaction ( Yijk) and the\nfour-site four spin interaction ( Kijkl), respectively. Since\nthese terms arise from the fourth-order perturbation the-\nory, we restrict ourselves to the nearest-neighbor approx-\nimation, i.e., up to the \frst term of these HOI. The cor-\nresponding constants are denoted as B1,Y1andK1[44].\nThe evaluation scheme of the HOI on a hexagonal two-\ndimensional (2D) lattice is illustrated in Fig. 1b. Within\nthe nearest-neighbor approximation, the minimal connec-\ntion among the three ( ijk) and four ( ijkl) adjacent lat-\ntice sites on a hexagonal lattice leads to a triangle and adiamond, respectively. A clusters of 12 diamonds and 6\ntriangles for each lattice site contributes to the four-site\nand three-site four spin interactions, respectively. Out\nof the 12 diamonds, two distinct types, each contain-\ning six diamonds, can be identi\fed (Fig. 1b). For the\nbiquadratic interaction, 6 nearest-neighbor pairs, analo-\ngous to the nearest-neighbor exchange constant ( J1), are\nsu\u000ecient.\nIn order to obtain the exchange constants, the energy\ndispersion of homogeneous \rat spin spirals is calculated\nusing DFT. A spin spiral is characterized by a wave vec-\ntorqin the two-dimensional Brillouin zone (Fig. 1d)\nand the magnetic moments of an atom at lattice site Ri\nis given by Mi=M(sin(qRi),cos( qRi),0) with the size\nof the magnetic moment M. Figure 1e shows the energy\ndispersionE(q) of spin spirals along two high symmetry\ndirections obtained via DFT for a fcc-Pd/Fe bilayer on\nthe Rh(111) surface [32]. At the high symmetry points\nof the two-dimensional Brillouin zone (2DBZ), we \fnd\nwell-known magnetic states: a ferromagnetic (FM) state3\nFigure 1. Higher-order exchange interactions and multi- Qstates .aIllustration of the Pd/Fe/Rh(111) ultrathin \flm\nwith a spin spiral propagating in the Fe layer. bFour sites ( ijkl) involved in the four-site four spin interaction form a diamond\n(red and brown) on a 2D hexagonal lattice and three sites ( ijk) involved in the three-site four spin interaction result in a\ntriangle (yellow). A total of 12 diamonds and 6 triangles are possible for site i. The 12 diamonds can be categorized into two\ngroups. One from each group is shown in red and brown. All the three sites j,kandlof the brown diamond are nearest\nneighbors to the central site i. The topmost site of the red diamond is a next-nearest neighbor to i, while the other two remain\nnearest neighbors. cFormation of an uudd state as a superposition of two 90\u000espin spirals with opposite rotational sense, i.e.,\nclockwise (CW) and counterclockwise (CCW). d2D Brillouin zone with two high symmetry direction \u0000K and \u0000M. The q\nvectors corresponding to the uudd state along \u0000K (pink \flled circle) and along \u0000M (blue \flled circle) as well as the 3 Qstate\nat the Mpoint (green \flled circle) are indicated. eEnergy dispersion E(q) of homogeneous spin spirals of Pd/Fe/Rh(111).\nThe \flled circles (red) are DFT total energies including spin-orbit coupling. The solid line is a \ft to the Heisenberg and the\nDzyaloshinskii-Moriya interaction. The energies of the uudd state along \u0000K (pink \flled circle), the uudd state along the \u0000M\ndirection (blue \flled circle) and the 3 Qstate (green \flled circle) are denoted at the qvalue of the corresponding single- Q\nstate. The spin structures of the uudd state along \u0000K (pink), which is formed by a superposition of two 90\u000espin spirals at\nq=\u0006(3=4)\u0000K, the uudd state along the \u0000M direction (blue), which is formed by a superposition of two 90\u000espin spirals at\nq=\u0006(1=2)\u0000M, and the 3 Qstate (green), which is a superposition of three spin spirals corresponding to three M points (green)\nin 2DBZ. Inset of eshows that the ground state of Pd/Fe/Rh(111) is a spin spiral with a wavelength of 4.8 nm.\nat the \u0000 point, a row-wise antiferromagnetic (AFM) state\nat the M point and a N\u0013 eel state with angles of 120\u000ebe-\ntween adjacent magnetic moments at the K point.\nClearly, the FM state is the energetically lowest among\nthese three states (Fig. 1e). Along both the symmetry\ndirections, the 90\u000espin spirals (Fig. 1c) are found at\nq= (1=2)\u0000M and at q= (3=4)\u0000K (Fig. 1d). The total\nenergy of homogeneous spin spirals without SOC is \ftted\nto functions obtained by expressing the Heisenberg model\nin reciprocal space. When SOC is included, the DMI\nlowers the energy of cylcoidal spin spirals with a clockwise\nrotational sense in the vicinity of the \u0000 point (see inset of\nFig. 1e). The MAE, on the other hand, shifts the energy\nof spin spirals by K/2 with respect to the FM state.\nSince the functional form of the three-site four spin andthe biquadratic interactions for homogeneous spin spi-\nrals resemble that of the \frst three exchange constants,\nwe cannot separate the exchange and the higher-order\nconstants by \fts (see Methods). Therefore, we calculate\nthe higher-order exchange constants from the energy dif-\nference between the spin spiral (single- Q) and multi- Q\nstates and modify the exchange constants obtained from\n\ft by neglecting the HOI accordingly.\nThe multi-Qstate is a superposition states of spin spi-\nrals corresponding to the symmetry equivalent qvec-\ntors (cf. Fig. 1(c,d)). Within the Heisenberg model of\npair-wise interaction, the multi- Qand single- Qstates\nare energetically degenerate. However, in the extended-\nHeisenberg model (Eq. (1)), the HOI lift the degeneracy\nwhich provides a way to compute their strengths. In DFT4\ncalculations, all the interactions are implicitly included\nthrough the exchange-correlation functional. Therefore,\nwe can obtain the HOI constants from total energy cal-\nculations of multi- Qand single-Qstates.\nWe consider two collinear states, the so-called uudd\nor double-row wise antiferromagnetic states [45] and\na three-dimensional non-collinear state, the so-called\n3Qstate [26], to uniquely determine three higher-order\nexchange constants (for spin structures see insets of\nFig. 1e). The resulting uudd state along \u0000K, represents\nrows of alternating spins in the nearest-neighbor direc-\ntion, is formed by superposition of q=\u0006(3=4)\u0000K, whereas\nthe other along \u0000M, represents rows of alternating spins\nin the next nearest-neighbor direction, is formed by su-\nperposition of q=\u0006(1=2)\u0000M (Fig. 1c). The 3 Qstate { a\nthree-dimensional spin structure on the two-dimensional\nhexagonal lattice (inset of Fig. 1e) { is constructed by a\nsuperposition of three spin spirals at the three symmetry\nequivalent M points of the 2DBZ [26] (Fig. 1d).\nThe biquadratic ( B1), the three-site four spin ( Y1)\nand the four-site four spin interaction ( K1) constants are\ncomputed from the energy di\u000berences between the multi-\nQand the corresponding single- Qstates by solving the\nfollowing equations [29]:\nE3Q\nM\u0000E1Q\nM=16\n3(2K1+B1\u0000Y1) (2a)\nEuudd\nM=2\u0000E1Q\nM=2= 4(2K1\u0000B1\u0000Y1) (2b)\nEuudd\n3K=4\u0000E1Q\n3K=4= 4(2K1\u0000B1+Y1) (2c)\nThe three multi- Qstates are higher in energy com-\npared to the corresponding spin spirals states for\nPd/Fe/Rh(111) (Fig. 1e) and they are energetically far\nabove the spin spiral ground state (energies are given in\nSupplementary Table 1). Nevertheless, they have a large\ne\u000bect on skyrmions in this \flm system as we show below.\nThe computed higher-order constants modify the \frst\nthree exchange constants, obtained from \fts of the spin\nspiral energy dispersion neglecting HOI, as follows (see\nMethods for a detailed derivation):\nJ0\n1=J1\u0000Y1 (3a)\nJ0\n2=J2\u0000Y1 (3b)\nJ0\n3=J3\u0000B1=2 (3c)\nwhere we denote the exchange parameters obtained from\n\fts neglecting HOI as unprimed and the modi\fed ones\nby considering the higher-order terms as primed. It is\nimportant to note that the four-site four spin interac-\ntion does not adjust any exchange parameter since its\ncontribution to the energy dispersion of spin spirals is a\nconstant value of \u000012K1independent of the spin spiral\nvector.\nThe exchange and the higher-order exchange constants\nof Pd/Fe/Rh(111) are displayed in Table I. We haveSystems J0\n1 J0\n2 J0\n3 B1 Y1K1\nPd/Fe/Rh(111) 11.73 \u00004:31\u00004.21 2.74 1.61 2.61\nPd/Fe/Ir(111) 13.60 \u00003:28\u00004:17 2.96 0.80 2.14\nFe/Rh/Re(0001) 8.85 \u00000:77 0.05 \u00000:39 1.00 \u00001:36\nTable I. Exchange constants including HOI. Exchange\nconstants for i-th neighbor spins ( J0\ni), biquadratic ( B1),\nthree-site four spin ( Y1) and four-site four spin interac-\ntion ( K1) constants for Pd/Fe/Rh(111), Pd/Fe/Ir(111) and\nFe/Rh/Re(0001). The exchange constants are modi\fed upon\nincluding the HOI according to Eqs. (3a-c). This table shows\nonly those exchange constants which are modi\fed upon in-\ncluding the HOI. The full sets of exchange constants as well\nas the DMI, the MAE and the total magnetic moments used\nin the atomistic spin dynamics simulations are listed in Sup-\nplementary Table 2 and Supplementary Table 3. All values\nare given in meV.\nperformed similar DFT calculations in order to obtain\nthese constants for the other two ultrathin \flm systems:\nPd/Fe/Ir(111) and Fe/Rh/Re(0001) (Table I and Sup-\nplementary Table 1). We \fnd that the Pd/Fe bilayer on\nRh(111) and on Ir(111) behaves similar in terms of ex-\nchange and higher-order exchange constants which can\nbe understood based on the fact that Rh and Ir are\nisoelectronic 4 dand 5dtransition metals. In these two\n\flm systems, the signs of the nearest-neighbor exchange\nconstant (J1) and the second and third nearest neigh-\nbors are opposite which leads to exchange frustration\n[22, 32]. The exchange interaction in Fe/Rh/Re(0001), in\ncontrast, is dominated by the nearest-neighbor exchange\nconstant. Note that the sign of the biquadratic ( B1)\nand the four-site four spin constants ( K1) is negative in\nFe/Rh/Re(0001), while it is positive for the other two\nsystems. As we will see below, the sign of the four-site\nfour spin interaction is essential for the skyrmion stability\nin these \flms.\nSpin dynamics simulations. We use atomistic spin\ndynamics simulations (see Methods) based on the Hamil-\ntonian of Eq. (1) with all parameters obtained from DFT\nincluding DMI, MAE and total magnetic moments, as\ndiscussed in the previous section, to calculate the zero\ntemperature phase diagram and the properties of isolated\nskyrmions and antiskyrmions in the three \flm systems.\nThe results for Pd/Fe/Rh(111) are shown in Fig. 2 (for\nthe other two \flm systems see Supplementary Figure 1\nand Supplementary Figure 2).\nWe \frst discuss the e\u000bect of the HOI on the zero\ntemperature phase diagram (Figs. 2a,b). The FM state\n(green line) and the homogeneous spin spiral states (solid\nblack line) remain almost una\u000bected by the higher-order\nterms. However, the skyrmion lattice (red line) loses a\nsmall amount of energy with respect to the homogeneous\nspin spirals which remains constant throughout the range\nof magnetic \felds. This leads to an expansion of the\nspin spiral (SS) and FM phases at the expense of the5\nFigure 2. Phase diagram, radius and barrier heights of Pd/Fe/Rh(111) including HOI. (a,b) Zero temperature\nphase diagram of Pd/Fe/Rh(111) obtained neglecting and including HOI, respectively. The energy of the relaxed spin spiral\n(SS), the skyrmion lattice (SkX) and ferromagnetic (FM) \feld-polarized state are shown with respect to the homogeneous spin\nspiral (black dashed line). The SS, SkX and FM phases are denoted by blue, red and green background color, respectively.\nc-eEquilibrium spin structures of a SS, a SkX, an isolated skyrmion and an antiskyrmion, respectively. gRadius of isolated\nskyrmions and antiskyrmions as a function applied magnetic \feld with respect to the critical \feld Bcwith and without taking\nHOI into account. Inset of gpro\fle of isolated skyrmions (ISk) and antiskyrmion (IASk) with HOI at B= 5.2 T. hBarrier\nheights of isolated skyrmions and antiskyrmions neglecting and including HOI as a function of magnetic \feld with respect to\nBc. Blue and green are used to distinguish two di\u000berent collapse mechanisms of isolated skyrmions. Above B\u0000Bc= 1.36 T,\nisolated skyrmions (including HOI) annihilate by the radial collapse mechanism (solid blue circles) and below B\u0000Bc= 1.36\nT, they annihilate via a chimera structure at the saddle point (solid green circle) and at 1.36 T, they can annihilate by both\nmechanisms (solid blue-green circle).\nskyrmion lattice (SkX) phase, which is squeezed. Since\nisolated skyrmions can be stabilized in the FM (\feld-\npolarized) phase it is of prime importance. The on-\nset of the FM phase, characterized by the critical \feld\nBc, has shifted from 2 :75 T to a lower value of 2 :25 T\ndue to the HOI. As expected from the magnetic inter-\naction constants, a similar trend of the phase diagram\nis obtained for Pd/Fe/Ir(111) (Supplementary Figure 1).\nHowever, since the higher-order constants are quite smallfor Fe/Rh/Re(0001) (cf. Table I), the phase diagram is\nbasically unchanged (Supplementary Figure 2).\nIn our spin dynamics simulations, we have created iso-\nlated skyrmions and antiskyrmions in the \feld-polarized\nbackground following the theoretical pro\fle [46] and re-\nlaxed the spin structures with full set of DFT parameters.\nThe radius of skyrmions and antiskyrmions { de\fned as\nin Ref. [46] { increases with HOI for Pd/Fe/Rh(111) on\naverage by about 35% and 30 %, respectively (Fig. 2g).6\nThe skyrmion and antiskyrmion pro\fles at 5.2 T (see\ninset of Fig. 2g) can be \ft by the standard skyrmion pro-\n\fle. The antiskyrmion exhibits a steeper pro\fle which\nre\rects that it has a smaller radius than the skyrmion.\nSimilar trends of the skyrmion and antiskyrmion radii\nare found for Pd/Fe/Ir(111) (Supplementary Figure 1).\nDue to relatively small HOI, the skyrmion radii remain\nalmost unchanged for low magnetic \felds above Bcfor\nFe/Rh/Re(0001) (Supplementary Figure 2 and Supple-\nmentary Note 1).\nTo study the stability of metastable isolated skyrmions\nand antiskyrmions, we employ the geodesic nudged elas-\ntic band (GNEB) method [47, 48] which allows to calcu-\nlate the minimum energy path (MEP) for the collapse of\na single skyrmion and antiskyrmion into the FM back-\nground (see Methods). The point of maximum energy\non this path, known as the saddle point, with respect\nto the initial state (skyrmion or antiskyrmion) is a mea-\nsure of the barrier height. As seen in Fig. 2h, the HOI\nincrease the energy barrier for skyrmion annihilation in\nPd/Fe/Rh(111) by more than a factor of two at small\nmagnetic \felds above Bc. For antiskyrmions, the barrier\nheight is even increased by a factor of 5.\nThe energy barriers of skyrmions vary nonlinearly at\nsmall magnetic \felds, and, thereafter, reduce almost lin-\nearly with increasing magnetic \felds [22]. On an average,\nwe notice an increase in barrier height of nearly 100 meV\nfor skyrmions upon including the HOI. At low \felds up\ntoB\u0000Bc= 1:36 T, there is a transition from the nor-\nmal radial collapse mechanism [22, 32] without HOI to\na chimera collapse mechanism [16, 49] with HOI. Above\nB\u0000Bc= 1:36 T, the skyrmions merge into the FM back-\nground through the normal radial collapse without HOI\nwhich remains unchanged after including HOI. The bar-\nrier heights of antiskyrmions with HOI exhibit a similar\nvariation with \feld as that of skyrmions. The energy bar-\nriers of antiskyrmions without HOI are extremely small\n(\u001820 meV) which implies that they are almost unsta-\nble. However, after including HOI, the energy barriers\nbecome\u0018100 meV at small \felds (up to 1.5 T above Bc),\nwhich suggests that a metastable antiskyrmion could be\nrealized. The annihilation mechanism of antiskyrmions\nis via the radial collapse [22], which is una\u000bected upon\nincluding HOI.\nIn Pd/Fe/Ir(111), the HOI increase the energy barriers\nof isolated skyrmions and antiskyrmions by similar values\nof\u0018100 meV and\u001890 meV, respectively (Supplemen-\ntary Figure 1). On the other hand, in Fe/Rh/Re(0001),\nthe stability of isolated skyrmions is reduced on an aver-\nage by 70 meV upon including the HOI (Supplementary\nFigure 2). This large barrier reduction shows that even\nsmall values of the higher-order constants (cf. table I)\ncan have signi\fcant e\u000bects.\nAnalysis of collapse mechanisms. Now we focus\non the question which of the HOI is responsible for the\nlarge changes of the energy barriers for skyrmion or an-tiskyrmion collapse. We consider both collapse mech-\nanisms for skyrmions, i.e., the chimera collapse at low\n\felds and the radial collapse at higher \felds and the ra-\ndial collapse mechanism for antiskyrmions (cf. Fig. 2h).\nIn Fig. 3, the energy decomposition of three represen-\ntative minimum energy paths are displayed at selected\nmagnetic \felds for Pd/Fe/Rh(111) with and without tak-\ning HOI into account (for the other two \flm systems sim-\nilar plots are displayed in Supplementary Figure 3 and\nSupplementary Figure 4).\nThe total energy rises along the minimum energy path\nas one moves from the initial (skyrmion) state to the\nsaddle point and descend thereafter to the \fnal (ferro-\nmagnetic) state (Fig. 3a). In the simulation neglecting\nthe HOI, we \fnd that the energy barrier is dominated by\nthe energy contribution from the DMI which favors the\nskyrmion state. Due to exchange frustration, there is also\na small energy contribution to the barrier from the ex-\nchange energy. Naturally, the energy due to the Zeeman\nterm and the magnetocrystalline anisotropy decrease in\nthe ferromagnetic state.\nUpon including the HOI (Fig. 3b), we \fnd the large in-\ncrease of the energy barrier as discussed in the previous\nsection. In addition, the annihilation mechanism changes\nat this magnetic \feld from the radial collapse without\nHOI (Fig. 3a) to the chimera collapse mechanism with\nHOI (cf. Figs. 3j,k) show the spin structures in the vicin-\nity of the saddle point of the two types of annihilation\nmechanisms). Interestingly, the chimera collapse mech-\nanism has been previously discussed in ultrathin \flms\nwith very strong exchange frustration [16, 49, 50], which\nsuggests that HOI acts in a similar way. The energy de-\ncomposition shows that the DMI contribution is of sim-\nilar magnitude at the saddle point of the path with and\nwithout HOI (Fig. 3c). However, one cannot compare\nthe exchange interactions before and after the HOI are\nincluded in the simulations, since the higher-order terms\nmodify the exchange constants according to Eqs. 3(a-c).\nTherefore, we add the contributions due to the exchange,\nthe three-site four spin interaction and the biquadratic\nterms which we denote as combined exchange. The com-\nparison of Fig. 3a and Figs. 3b shows that the exchange\nand the combined exchange behave qualitatively quite\nsimilar along the path { which is also true for the other\ntwo collapse mechanisms (Figs. 3d,e and Figs. 3g,h). The\nabsolute energy change from exchange to combined ex-\nchange at the saddle point is relatively small (Fig. 3c,f,i).\nThe four-site four spin interaction acts in a qualita-\ntively di\u000berent way compared to all other terms. For all\nconsidered paths (Figs. 3b,e,h), it gains in energy slowly\nas one approaches the saddle point, in the vicinity of the\nsaddle point it becomes very steep, reaches a maximum\nat the saddle point and drops quickly thereafter. The en-\nergy contributions at the saddle point (Figs. 3c,f,i) show\nthat it provides by far the largest di\u000berence between the\nsimulations with and without HOI, irrespective of the7\nFigure 3. Minumum energy paths of skyrmion and antiskyrmion collapse in Pd/Fe/Rh(111) . Total and individual\nenergy contributions along minimum energy paths neglecting and including HOI. Combined exchange for simulations with HOI\ndenotes the sum of exchange, biquadratic, and three-site four spin interaction. aRadial collapse of an isolated skyrmion at\nB= 3 T neglecting HOI. bChimera collapse of an isolated skyrmion at B= 3 T including HOI. cEnergy decomposition at the\nsaddle point of the path shown in a,bwith respect to the initial (skyrmion) state. d,eRadial collapse of an isolated skyrmion\natB= 5:2 T neglecting and including HOI, respectively. fEnergy decomposition at the saddle point of the path shown in\nd,ewith respect to the initial (skyrmion) state. g,hRadial collapse of an isolated antiskyrmion at B= 5:2 T neglecting and\nincluding HOI, respectively. iEnergy decomposition at the saddle point of the path shown in g,hwith respect to the initial\n(antiskyrmion) state. Inset of b,e,h The exchange, biquadratic and three-site four spin (3-spin) terms. j,kSpin structures\nbefore (SP-1), after (SP+1) and at the saddle point (SP) for the chimera collapse in band the radial collapse in e, respectively.\nFor brevity, the two-site, three-site, and four-site four spin interactions are denoted as Biq, 3-spin and 4-spin, respectively.8\ncollapse mechanism or the initial state.\nThe energy contribution from the three-site four spin\nand the biquadratic interactions decreases along all col-\nlapse processes and the energy drop escalates after the\nsaddle point (see insets of Figs. 3b,e,h). The di\u000berence\nin energy pro\fle of the DMI and MAE is an associated ef-\nfect of the HOI caused by the changes in relative spin an-\ngles during the collapse process. Figs. 3(c,f,i) show that\nthe combined exchange can provide a tiny contribution\nto the energy barrier depending on the collapse mech-\nanism, while the DMI and Zeeman terms assert only a\nlittle weight if not compensated by each other. There-\nfore, the four-site four spin interaction mainly controls\nchange of the barrier height.\nThe spin structure in the vicinity of the saddle point is\nshown in Figs. 3(j,k) for the chimera and the radial col-\nlapse of the isolated skyrmion including the e\u000bect of HOI.\nWe see that the radial collapse of an isolated skyrmion is\nvery similar to that found by neglecting HOI in Ref. [22].\nHowever, at the saddle point there are four spins pointing\ntowards each other while previously a three-spin struc-\nture was reported. The unusual saddle point including\nHOI is obtained throughout the studied \feld range and\nfor annihilation of skyrmions in Pd/Fe/Ir(111). How-\never, for the skyrmion collapse in Fe/Rh/Re(0001), a\nthree-spin structure at the saddle point similar to that\nof Ref. [22] occurs. The chimera skyrmion collapse and\nthe radial antiskyrmion collapse are similar to that found\nin simulations neglecting HOI [16, 22, 49].\nAnalysis of the four-site four spin interaction . To\nunderstand the prominent e\u000bect of the four-site four spin\ninteraction on the energy barrier, we present its site-\nresolved energy at the saddle point with respect to the\ninitial state (skyrmions or antiskyrmions) for the three\nminimum paths of Figs. 3(b,e,h) in Figs. 4(a-c). We no-\ntice that a group of only 14 spins around the core provide\ncontributions to the four-site four spin interaction, while\nthe surrounding spins do not add any signi\fcant value.\nThis \fnding is independent of whether we consider the\nsaddle point of the chimera collapse (Fig. 4a), the radial\nskyrmion collapse (Fig. 4b) or the radial collapse of the\nantiskyrmion (Fig. 4c). Similar observations are made\nfor the other \flm systems (Supplementary Figures 5 to\nSupplementary Figures 8). Therefore, the four-site four\nspin interaction at the saddle point exhibits a general be-\nhavior irrespective of the type of collapse mechanism or\nthe initial spin con\fguration.\nIn order to explain this localized energy gain at the\nsaddle point, we use a simpli\fed model in which we con-\nsider only the site with the largest contribution at the\norigin. To evaluate the four-site four spin interaction, we\nneed to consider at least 6 nearest neighbors and the 6\nnext-nearest neighbors of the central site [cf. Fig. 4(d-f)].\nTo simplify the discussion, we slightly symmetrize the\nspin structure. For the radial skyrmion and antiskyrmion\ncollapse, the 12 neighboring spins are nearly all in-plane(Figs. 4e,f). We neglect any out-of-plane component as\nshown in Figs. 4(h,i) to calculate the contributions from\nthe 12 diamonds for the four-site four spin interaction.\nFor the saddle point of the radial skyrmion collapse\n(Fig. 4h), we \fnd three distinct types of diamonds which\ncontribute to the four-site four spin interaction. We \fnd a\npair of diamonds with values + K1and\u0000K1, two pairs of\ndiamonds with values +1\n2K1and\u00001\n2K1, which cancel out\nmutually. Out of the six remaining diamonds, there are\nthree groups each containing two diamonds with values\n\u0000p\n3\n2K1,\u0000K1and +1\n2K1, which results in a total energy\nat the saddle point of EISk\nSP=\u00002:73K1.\nFor the saddle point of the radial antiskyrmion col-\nlapse (Fig. 4i), we identify three types of diamonds with\nthe same magnitude as for the skyrmion saddle point\n(Fig. 4h). However, two uncompensated diamonds with\n+1\n2K1and\u0000p\n3\n2K1and two diamonds with \u0000K1each\nlead to a total energy of EIASk\nSP=\u00002:37K1.\nThe spin structure at the saddle point of the chimera\ncollapse (Fig. 4d) is more complex. There are non-\nnegligible out-of-plane components of the spins surround-\ning the central spin which we take into account in the\nsymmetrization (Fig. 4g). As a consequence, we \fnd six\ndistinct types of diamonds (Fig. 4g). Similar to the other\ntwo saddle points, there is a mutual cancellation of many\nterms which leads to a total energy contribution of the\nfour-site four spin interaction of Echimera\nSP =\u00002:37K1.\nNote that the values of these three energies taking\nthe exact spin structure at the saddle points are EISk\nSP=\n\u00002:0K1,EIASk\nSP=\u00002:2K1andEchimera\nSP =\u00002:14K1, which\nare very close to those obtained using the simpli\fed spin\nstructures.\nWe obtain similar values for Pd/Fe/Ir(111) at the\nsaddle points corresponding to the skyrmion and anti-\nskyrmion initial states and at the chimera saddle point,\nthe value is only slightly di\u000berent Echimera\nSP =\u00002:58K1\n(Supplementary Figures 5 to 7). For Fe/Rh/Re(0001),\nwe \fndEISk\nSP=\u0000p\n3K1(Supplementary Figure 8).\nThe energy contribution per site of the four-site four\nspin interaction for the ferromagnetic state or any \rat\nspin spiral is\u000012K1, which is also relatively close to the\nenergy in the skyrmion state (cf. Fig. 3b and Supple-\nmentary Figures 3b,e and 4b). Therefore, we obtain an\nenergy di\u000berence of ESP\u0000EFM\u001910K1for the two sym-\nmetric central sites of the saddle point. The surrounding\nsites provide smaller contributions, however, they still\nscale withK1. In total, we \fnd an energy contribution\nof the four-site four spin interaction to the energy bar-\nrier of roughly 40 K1. Due to the linear dependence on\nK1, it is also clear that the sign of the four-site four spin\ninteraction determines whether there is an energy gain\n(K1>0) or loss ( K1<0) at the saddle point as ob-\nserved for the two types of systems: Pd/Fe on Rh(111)\nand on Ir(111) vs. Fe/Rh/Re(0001) (cf. Table I).\nDiscussion. Our simpli\fed model states that the barrier9\nFigure 4. Four-site four spin energy at the saddle points of Pd/Fe/Rh(111). a-c Atomic-site resolved energy\ncontribution of the four-site four spin interaction at the saddle points of the minimum energy paths of Figs. 3( b,e,h ), i.e.,\nfor the chimera skyrmion collapse, the radial skyrmion collapse and the radial antiskyrmion collapse. Energies are given with\nrespect to the initial state. d-fSaddle point spin structures of Figs. 3( b,e,h ).g-iSpin structure around the core spin at the\norigin, i.e., at (0 ;0), of the saddle point is highlighted in d-fby the gray shaded star. Note that all spins of h,iare in-plane\nwhile most of the spins in gare not in-plane. The sign and the value of the four-site four spin interaction is shown for di\u000berent\ndiamonds.\nheightESP\u0000EISk=IASkvaries linearly with the magnitude\nof the four-site four spin constant K1. To verify this pre-\ndiction, we have carried out spin dynamics simulations\nfor three ultrathin \flm systems at a given magnetic \feld\nby changing only the four-site four spin interaction con-\nstant while leaving all other interactions the same. Note\nthat the four-site four spin interaction does not a\u000bect the\nenergy dispersion of spin spirals which is essential for theequilibrium properties of skyrmions and antiskyrmions.\nFigure 5 shows that { as expected from our model { the\nbarrier heights for skyrmions and antiskyrmions exhibit a\nlinear scaling with the four-site four spin constant. Only\nfor Pd/Fe/Rh(111), we \fnd a slight deviations from the\nlinear dependence. The scaling constant \u000b, de\fned as\nthe ratio of the change in energy barrier to the change\nin four-site four spin constant, is the same for skyrmions10\nFigure 5. Variation of energy barriers with four-site\nfour spin constant . Energy barriers for skyrmions (\flled\ncircles and solid lines) and antisykrmions (open circles and\ndashed lines) in the three considered ultrathin \flm systems\nas a function of the strength and sign of the four-site four\nspin interaction constant. A constant magnetic \feld is cho-\nsen for each of the systems: B= 5:2 T for Pd/Fe/Rh(111),\nB= 6 T for skyrmions and B= 4 T for antiskyrmions in\nPd/Fe/Ir(111) and B= 3 T for Fe/Rh/Re(0001).\nand antiskyrmions consistent with our discussion of the\nenergy contributions at the saddle point. We \fnd a value\nof the scaling constant \u000bof about 40 to 60 depending on\nthe system.\nFigure 3 implies that the HOI can stabilize skyrmions\nand antiskyrmions by themselves. To test this notion, we\nhave performed spin dynamics simulations by completely\nswitching o\u000b the DMI while keeping all other magnetic\ninteractions as before. As shown in Supplementary Fig-\nure 9, we \fnd stable skyrmions and antiskyrmions for\nPd/Fe bilayers on Rh(111) and Ir(111) with radii reduced\nto 1.4 and 1.6 nm just above Bc, respectively. Signi\fcant\nenergy barriers of the order of 80 to 90 meV are obtained\ndue to the four-site four spin interaction (Supplementary\nFigure 10), which are identical for skyrmions and anti-\nskyrmions as the DMI is zero. This suggests that it is\npossible to stabilize both types of topological states with\nopposite rotational sense due to the HOI at inversion\nsymmetric transition-metal interfaces.\nThe lifetime \u001cof skyrmions is given by the Arrhenius\nlaw\u001c=\u001c0exp(\u0001E/kBT), where \u0001 Eis the energy bar-rier and\u001c0is the prefactor. An enhancement of only the\nbarrier height, \u0001 E, by 100 meV, as observed for an iso-\nlated skyrmion in Pd/Fe/Rh(111), leads to an increase\nof skyrmions lifetime by orders of magnitude because of\nthe exponential factor. For example, at a temperature of\nT= 10 K, at which the spin-polarized scanning tunnel-\ning microscopy experiments on such ultrathin \flms are\ntypically performed [10, 11], we \fnd an enhancement by\n50 orders of magnitude, at 100 K, it is still a factor of\nabout 105, and even at room temperature, it is a factor\nof about 50. One can also discuss the e\u000bect of the HOI\nin terms of the temperature dependent phase diagram of\nPd/Fe/Rh(111) [32]. Without the HOI, skyrmions can\nbe stable for an hour up to temperatures of 25 K [32],\nwhich is increased to a temperature of about 50 K upon\nincluding the HOI.\nConclusion. We have demonstrated that the HOI\nbeyond pair-wise Heisenberg exchange can play a key\nrole for the stability of skyrmions or antiskyrmions at\ntransition-metal interfaces. Depending on the sign of\nthe four-site four spin interaction, the energy barrier\npreventing the collapse of a metastable topological spin\nstructure can be greatly enhanced or reduced. Even for\nthe small values of HOI typical for 3 dtransition met-\nals, we \fnd large changes of the energy barriers and\ntherefore giant e\u000bects on the lifetime, which means that\nthese interactions cannot be neglected. The energy bar-\nriers are so enhanced with HOI that it can stabilize\nisolated skyrmions and antiskyrmions in the absence of\nDzyaloshinskii-Moriya interaction. Our study opens up\na new avenue to stabilize topological spin structures at\ntransition-metal interfaces.\nMethods\nFirst-principles calculations. The computational\ndetails for calculating the exchange, the DMI, the\nMAE constants and the magnetic moments are shown\nin Ref. [32] for Pd/Fe/Rh(111), in Ref. [22] for\nPd/Fe/Ir(111) and in Ref. [41] for Fe/Rh/Re(0001).\nHere, we have evaluated the higher-order exchange con-\nstants for all three systems. The electronic structure was\ncalculated using a spin-polarized DFT code based on the\nprojected augmented wave (PAW) scheme [51] as imple-\nmented in the Vienna ab initio simulation package ( vasp )\n[52]. It ranks among the best available DFT codes in\nterms of accuracy and e\u000eciency [53]. We use the same\nstructural parameters as mentioned in the above refer-\nences. We have used two atomic overlayers on top of\nnine substrate layers to mimic the surfaces. To maintain\nconsistency with spin spiral calculations, we have chosen\nlocal density approximation (LDA) for the exchange and\ncorrelation part of potential [54]. A high energy cut-o\u000b of\n400 eV was used to precisely calculation the energy of the11\nmulti-Qstates. The 2DBZ was sampled by a Monkhorst-\nPack [55] mesh of 22 \u000228\u00021k-points for the uudd state\nin the \u0000K direction, of 14 \u000244\u00021k-points for the uudd\nstate in the \u0000M direction and of 15 \u000215\u00021k-points for\nthe 3Qstate at the M point. The total energy conver-\ngence criteria were set to 10\u00006eV for all calculations.\nFitting function for HOI. The spin spiral is the exact\nsolution of the classical Heisenberg model for a periodic\nlattice. The spin spiral, which is characterized by a wave\nvector qin the 2DBZ and the magnetic moments of an\natom at lattice site Ri, is given by [56],\nMi= 2(Rqcos(q\u0001Ri)\u0000Iqsin(q\u0001Ri)) (4)\nwhere RqandIqare two vectors that span the xy-\nplane. They obey the following relation,R2\nq=I2\nq=M2=2;R2\nq\u0001I2\nq= 0 (5)\nwhereMis the magnitude of Miand without loss of\ngenerality, we set its norm to unity. The spins of a spin\nspiral rotate around the z-axis in the xy-plane as one\nmoves from one lattice site to another in the direction of\nq. Using the above two equations, the scalar product of\na pair of spins can be written as [56],\nMi\u0001Mj= cos( q\u0001R) (6)\nIn reciprocal space, qis de\fned as q=qxb1+qyb2, with\nbbeing the reciprocal lattice vectors. In our case,\nb1=(2\u0019=a)(1,-1/p\n3) and b2=(2\u0019=a)(1,1/p\n3), hereais\nthe in-plane lattice constant. For a spin spiral propagat-\ning along \u0000KM, q=(2\u0019/a)q(1,0) with q2[0,1] and along\n\u0000Mq=(2\u0019/a)q(p\n3/2,-1/2) with q2[0,1/p\n3].\nUsing Eq. (6), we calculate the energies of the exchange\ninteractions up to third nearest-neighbor, the HOI on a\nhexagonal lattice along \u0000K direction as,\nE\u0000K(J0\n1;J0\n2;J0\n3;Y1;B1;K1) =\u00002[cos(2\u0019q) + 2cos(\u0019q)][J0\n1+Y1]\u00002[1 + 2cos(3 \u0019q)][J0\n2+Y1]\n\u00002[cos(4\u0019q) + 2cos(\u0019q)][J0\n3+B1=2]\u000012K1(7)\nand along \u0000M direction as,\nE\u0000M(J0\n1;J0\n2;J0\n3;Y1;B1;K1) =\u00002[1 + 2cos(p\n3\u0019q)][J0\n1+Y1]\u00002[cos(2p\n3\u0019q) + 2cos(p\n3\u0019q)][J0\n2+Y1]\n\u00002[1 + 2cos(2p\n3\u0019q)][J0\n3+B1=2]\u000012K1(8)\nIt is clear from Eqs. (7) and (8) that we can at most\nobtain the combined terms J0\n1+Y1,J0\n2+Y1andJ0\n3+B1=2\nby \fts. Therefore, we evaluate the higher-order exchange\nconstants from the total energy di\u000berence between the\nmulti-Qstates and single- Qstates according to Eqs. 2(a-\nc) and modify the exchange constants obtained assuming\nvanishing high-order contributions using Eqs. 3(a-c) of\nthe main text. The four-site four spin energy is constant,\ni.e,\u000012K1, for all the spin spiral vector q, which implies\nthat it does not a\u000bect the exchange constants.\nAtomistic spin dynamics simulations. We study the\ntime evolution of atomistic spins as described by Landau-\nLifshitz equation, where the dynamics is expressed as a\ncombination of the precession and the damping terms:\n}dmi\ndt=@H\n@mi\u0002mi\u0000\u000b\u0012@H\n@mi\u0002mi\u0013\n\u0002mi(9)where }is the reduced Planck constant, \u000bis the damp-\ning parameter and the Hamiltonian is Hde\fned in Eq.\n(1). In the simulation, \u000bhas been varied from 0.05 to\n0.1, while we have chosen a time step of 0.1 fs and sim-\nulated over 2\u00003 millions steps to ensure relaxation of\nthe spin structures. We employed a semi-implicit scheme\nproposed by Mentink et al. [57] to accomplish a time\nintegration of Eq. (9).\nGeodesic nudged elastic band method. We calcu-\nlate the annihilation energy barrier of isolated skyrmions\nand anitiskyrmions and their collapse mechanism via the\ngeodesic nudged elastic band method (GNEB) [47, 48].\nThe objective of GNEB is to \fnd a minimum energy\npath (MEP) connecting initial state (IS), in this case,\nskyrmion or antiskyrmion, and \fnal state (FS), i.e., FM\nstate, on an energy surface. The GNEB is a chain-of-\nstate method, in which a string of images (spin con\fgu-\nrations of the system) is used to discretize the MEP. The12\nmethod selects an initial path connecting IS and FS and\nsystematically brings it to MEP by relaxing the interme-\ndiate images. The image relaxation is performed through\na force projection scheme, in which the e\u000bective \feld acts\nperpendicular and spring force acts along the path. The\nmaximum energy on the MEP corresponds to a saddle\npoint (SP) which de\fnes energy barrier separating IS and\nFS. 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J.\nPhys. 58, 1200{1211 (1980).\n[55] Monkhorst, H. J. & Pack, J. D. Special points for\nBrillouin-zone integrations. Phys. Rev. B 13, 5188{5192\n(1976).\n[56] Kurz, P. Non-collinear magnetism at surfaces and in\nultrathin \flms . Ph.D. thesis, Institute f ur Festk orper-\nforschung (2000).\n[57] H. Mentink, J., V. Tretyakov, M., Fasolino, A., I. Kat-\nsnelson, M. & Rasing, T. Stable and fast semi-implicit\nintegration of the stochastic Landau{Lifshitz equation.\nJ. Phys.: Condens. Matter 22, 176001 (2010).\nAcknowledgements. We gratefully acknowledge com-\nputing time at the supercomputer of the North-German\nSupercomputing Alliance (HLRN) and \fnancial support\nfrom the Deutsche Forschungsgemeinschaft (DFG, Ger-\nman Research Foundation) via project HE3292/13-1. We\nthank Pavel F. Bessarab for valuable discussions.\nAuthor contributions. S.He. devised the project. S.P.\nimplemented the three-site four spin interaction into the\natomistic spin dynamics code and tested the code in-\ncluding HOI. S.Ha. and S.P. performed the DFT calcu-\nlations. S.Ha., S.P., and S.v.M. performed the spin dy-\nnamics and GNEB simulations. S.Ha. and S.P. prepared\nthe \fgures. S.P. and S.He. wrote the manuscript. All\nauthors discussed the data and contributed to preparing\nthe manuscript.\nAdditional information. Supplementary Information\naccompanies this paper.\nCompeting interests. The authors declare no compet-\ning \fnancial interests."    },    {        "title": "2001.03902v3.Magnetic_Skyrmions_in_FePt_Square_Based_Nanoparticles_Around_Room_Temperature.pdf",        "content": "Magnetic Skyrmions in FePt Square-Based Nanoparticles Around\nRoom-Temperature\nChristos Tyrpenou1, Vasileios D. Stavrou1, and Leonidas N. Gergidis∗1\n1Department of Materials Science and Engineering, University of Ioannina, 45110 Ioannina,\nGreece\nSeptember 13, 2022\nAbstract\nMagnetic skyrmions formed at temperatures around room temperature in square-based parallelepiped\nmagnetic FePt nanoparticles with perpendicular magnetocrystalline anisotropy (MCA) were studied during\nthe magnetization reversal using micromagnetic simulations. Finite Differences (FD) method were used for\nthe solution of the Landau-Lifshitz-Gilbert equation. Magnetic configurations exhibiting Néel skyrmionic\nformations were detected. The magnetic skyrmions can be created in different systems generated by the\nvariation of external field, side length and width of the squared-based parallelepiped magnetic nanoparticles.\nMicromagnetic configurations revealed a variety of states which include skyrmionic textures with one distinct\nskyrmion formed and being stable for a range of external fields around room-temperature. The size of the\nformed Néel skyrmion is calculated as a function of the external field, temperature, MCA and nanoparticle’s\ngeometrical characteristic lengths which can be adjusted to produce Néel type skyrmions on demand having\ndiameters down to 12 nm. The micromagnetic simulations revealed that stable skyrmions at the temperature\nrange 270 - 330 K can be created for FePt magnetic nanoparticle systems lacking of chiral interactions such\nas Dzyaloshinskii-Moriya.\n1 Introduction\nMagnetic skyrmions are considered by many researchers as dynamic candidates for next generation high density\nefficient information encoding providing enhanced capabilities in magnetic writing and storage [1, 2, 3, 4, 5].\nTheir small size and the low current density needed for their manipulation and control [6] pave the way for\npotentialengineeringapplications. Overthelastdecademagneticskyrmionshavebeenunderintensetheoretical,\ncomputational and experimental investigation [7, 8, 9, 10, 11, 12, 13, 14, 15]. It is of significant importance\nto understand how the confined nature of geometry can affect skyrmion creation-stabilization [16, 17, 18] and\nconsequently the size and energetics of the skyrmionic states.\nRecently, Fert and his collaborators [19] combined concomitant magnetic force microscopy and Hall resistiv-\nity measurements to demonstrate the electrical detection of sub-100 nm skyrmions in a multilayered thin film at\nroom-temperature. Additionally,Legrandetal. [20]showedthatroomtemperatureantiferromagneticskyrmions\ncan be stabilized in synthetic antiferromagnets (SAFs), in which perpendicular magnetic anisotropy, antifer-\nromagnetic coupling and chiral order can be adjusted concurrently. In up to date reports room-temperature\nmagnetic skyrmions in ultrathin magnetic multilayer structures of Co/Pd [21] , Pt/ Co/MgO [22] and Co/Ni [23]\nwere reported. Brãndao et al. [24] reported on the evidence of skyrmions in unpatterned symmetric Pd/Co/Pd\nmultilayers at room temperature without prior application of neither electric current nor magnetic field. Hu-\nsain et al. [25] observed stable skyrmions in unpatterned Ta/Co 2FeAl(CFA)/MgO thin film heterostructures\nat room temperature in remnant state employing magnetic force microscopy showing that these skyrmions\nconsisting of ultrathin ferromagnetic CFA Heusler alloy result from strong interfacial Dzyaloshinskii-Moriya\ninteraction (i-DMI). Ma et al. [26] employed atomistic stochastic Landau-Lifshitz-Gilbert simulations to inves-\ntigate skyrmions in amorphous ferrimagnetic GdCo revealing that a significant reduction in DMI below that of\nPt is sufficient to stabilize ultrasmall skyrmions even in films as thick as 15 nm.\nIn the present simulation work, the formation of magnetic skyrmions around room-temperature in the exter-\nnal field range used for the magnetization reversal in a single FePt nanoparticle having parallelepiped geometry\nofsquarebaseisstudied. Thecurrentresearcheffortwasinspired-initiatedbythevarietyofskyrmionicmagnetic\ntextures detected in FePt triangular and reuleaux prismatic nanoparticles encapsulating multiple skyrmions at\n∗lgergidi@uoi.gr; lgergidis@gmail.comarXiv:2001.03902v3  [cond-mat.mes-hall]  10 Sep 20222 MICROMAGNETIC MODELING\n0 K using Finite Elements micromagnetics simulations [27, 28, 29]. It was shown that for magnetocrystalline\nanisotropy (MCA) values between Ku= 200−500 kJ/m3three distinct skyrmions are formed and persist for a\nrange of external fields along with rich magnetic structures in the bulk of the FePt nanoparticle similar to those\nobserved in DMI stabilized systems [30]. Néel type skyrmionic textures were detected on the Reuleaux geome-\ntry’s bases coexisting with Bloch-type textures in the bulk of the FePt nanoelement. It was also shown that the\nsize of skyrmions depends linearly on the field value Bextand that the slope of the linear curve can be controlled\nby MCA value [29]. Single layered square-based FePt nanoparticles in the absence of chiral interactions like\nDzyaloshinskii-Moriya (DM) or its interfacial analogue (iDM) (present in multi-layered hetero-structures with\nspin-orbit coupling) are investigated. Skyrmionic textures and formations are explored along with the under-\nlying physical mechanisms for temperatures around 300 K and for different geometrical characteristics of the\nnanoparticle.\n2 Micromagnetic modeling\nThe Landau-Lifshitz-Gilbert (LLG) equation governs the rate of change of the dynamical magnetization field\nMand is given by the relation\ndM\ndt=γ\n1 +α2(M×Beff)−αγ\n(1 +α2)|M|M×(M×Beff). (1)\nThe quantity α >0is a phenomenological dimensionless damping constant that depends on the material and\nγis the electron gyromagnetic ratio. The effective field that governs the dynamical behavior of the system\nhas contributions from various effects that are of very different nature and can be expressed as Beff=Bext+\nBexch+Banis+Bdemag +Bthermal. Respectively, these field contributions are the external field Bext, the\nexchange field Bexch, the anisotropy field Banis, the demagnetizing field Bdemagand the thermal field Bthermal.\nIn particular, the thermal field Bthermalwhich incorporates implicitly the temperature can be expressed by\nBthermal (t) =n(t)/radicalBig\n2µ0αkBT\nBsatγ∆V∆taccording to Brown [31]. In thermal field expression αis the aforementioned\ndamping constant, µ0is the vacuum permeability constant, kBis the Boltzmann constant, Tis the temperature\n(throughoutmanuscriptslightlyslanted Tisusedfortemperature), Bsatthesaturationmagnetizationexpressed\nin Tesla (T) , ∆Vthe cell volume, ∆tthe time step and na random vector generated from a standard normal\ndistribution whose value is changed every time step. For the solution of the LLG equation micromagnetic Finite\nDifferences (FD) calculations have been conducted using Mumax3 Finite Differences software [32, 33]. The\ndimensionless damping constant αwas set to 1 in order to achieve fast damping and reach convergence quickly\nas we are interested in static magnetization configurations. The time step used for the integration of the LLG\nequation was equal to ∆t= 1 fs.\nThe square-based nanoparticle having side length a= 150 nm and width w= 36 nm will be considered\nas the reference nanoparticle and will be used for the multi-parametric investigation conducted in the present\nwork. It is chosen since it encapsulates the most stable skyrmion with respect to the magnetic field range and\nskyrmion type as it will be made clear in the sequel. The particular width value w= 36nm is selected following\nprevious studies [27, 28, 29] which focus on FePt triangular and reuleaux prismatic nanoelements. In addition,\nthe 36 nm thickness of nanoparticle matches that reported in [34]. For the objectives of the multi-parametric\nstudy additional magnetic samples-nanoparticles with parallelepiped geometry of square base with varying side\nlength afrom 40 to 180 nm and wranging from 6 to 36 nm were also used. The following frame of reference axes\nassignment convention was used: x,yalong the square’s edges, and zperpendicular to the nanoparticle’s square\nbase. The mesh used for the discrete representation of the rectangular parallelepiped nanoparticle under study\nwas a regular 3D mesh with characteristic discretization lengths ∆x= ∆y=2 nm, ∆z=1 nm inx,y,z-directions\nrespectively. The lengths ∆x,∆y,∆zused for the discretization of the rectangular domain under investigation\nwere lower than the exchange length lex=/radicalBig\n2A\nµ0M2s≈3.5nm for the FePt magnetic material.\nThe material parameters used in this study have been reduced accordingly in order to follow their temper-\nature variation. Since the exact temperature dependence is not exactly known the reduction of the anisotropy\nparameter and micromagnetic exchange takes place through the approximate relations K(T)∼[me(T)]3,\nAexch(T)∼[me(T)]2[35, 36]. The spontaneous equilibrium magnetization me(T)is taken from the atom-\nistic FePt model presented in [37]. The dependence of saturation magnetization on temperature was modeled\nusing the findings of Okamoto et al. [38]. The MCA was oriented perpendicular to the nanoparticle’s square\nbase and the MCA constant was varied. The Ku= 250 kJ/m3in our previous micromagnetic numerical endeav-\nors [28, 29] at 0 K was capable of creating interesting skyrmionic textures in triangular and Reuleaux based\nFePt nanoelements.\nThe Finite Differences (FD) micromagnetic simulations were conducted on Nvidia GTX 1080 GPU. The\nmagnetization curves for every production run were investigated by applying external magnetic fields Bextwith\nfixed orientation running parallel to z-direction (the normal to the nanoparticle’s square base). The range values\n23 RESULTS\nofBextwere +1 T (maximum) and -1 T (minimum) having an external magnetic field step of δBext= 0.01 T\nfor the actual magnetic reversal process.\n3 Results\n3.1 Skyrmion formation and stabilization for the reference nanoparticle\nSkyrmion formation and stabilization as a function of the external field, the geometric characteristics of the\nnanoparticle and of the temperature can be quantitatively characterized by the calculation of the topological\ninvariantS, widely known as skyrmion or winding number [39, 8, 40]. The skyrmion number Sis computed\nusing the relation S=1\n4π/integraltext\nAm·(∂m\n∂x×∂m\n∂y)dA. The quantity mis the unit vector of the local magnetization\ndefined as m=M/MswithMbeing the magnetization and Msthe saturation magnetization. Magnetization\nMis provided by the FD numerical solution of the LLG equation. The skyrmion number Sis a physical and\ntopological quantity that measures how many times mwraps the unit sphere [8, 41, 42]. Surface Ais the surface\ndomain of integration and corresponds to the square base of the FePt nanoparticles under investigation.\nThe skyrmion number Shas been computed during the magnetization reversal process as a function of\nBextand is shown in Fig. 1a for the reference nanoparticle at T=300 K and for Ku=130.4 kJ/m3. As\nthe external field decreases the magnetic system departs from saturation.The magnetization reversal process is\nclosely followed by the external field step of δBext=0.01 T. At first glance for fields down to 0.2 T skyrmion\nnumber attains very low values fluctuating around zero. Below 0.2 T the skyrmion number remains fluctuating\nbut attains only positive values indicative for possible starting generation process for a specific micromagnetic\nconfiguration. The decrease of the field below 0 T activates a gradual increase of the skyrmion number from\nS= 0.1toS= 0.3when the external field reaches the Bext=−0.1T value. Infinitesimal decrease of the\nFigure1: Skyrmionnumber Sasafunctionof Bextforthereferencenanoparticleat300K(panela). Thepseudo\ncolor which is shown in the depicted micromagnetic configurations refers to the z-component of magnetization\n(mz). Relative energy differences as a function of the external field (panel b).\nmagnetic field below Bext=−0.1 Ttriggers a jump-like discontinuity on skyrmion number. The value of S\nabruptly increases from S≈0.3toS≈1. Further decrease of the external field does not affect the skyrmion\nnumber which develops an extended plateau region with S≈1for external field values down to Bext=−0.63 T.\nFor magnetic field value below Bext=−0.63 Ta new abrupt jump-like reduction of skyrmion number Ssignals\n33.2 Skyrmion dependence on nanoparticles’s width 3 RESULTS\nthe skyrmion annihilation and the finalization the magnetization reversal process. It is worth mentioning that\nthe existence of thermal field is reflected on the oscillatory behavior of S(Bext). These low amplitude oscillations\nalso captured by the micromagnetic configurations visualizations are indicative of the noisy character caused\nby the thermal field.\nThe numerical solution of the LLG equation allows the direct representation and inspection of the mi-\ncromagnetic configurations. These configurations are also shown in Fig. 1a for representative values of the\nexternal field. It should be noted that the pseudo color used for the micromagnetic configurations refers to\nz-component of magnetization ( mz). At the initial states of the reversal process the magnetization vectors are\naligned parallel to the external field (red colored square in Fig. 1a ). The micromagnetic configurations for\nfieldsBext∈[−0.14,0.2]T have a donut-like shape where the core magnetizations point upwards ( +z) and\nthe magnetizations on the distinct peripheral (rim) domain are tilted having mz= 0(white color domains in\nFigure 1a ) or reversed ( mz<0) (light blue color spots also in Figure 1a ). The aforementioned magnetization\nconfiguration is a magnetic skyrmionium which is a non-topological soliton which has a donut-like out of plane\nspin texture in magnetic nanoparticles and thin films. It can be viewed as a coalition of two magnetic skyrmions\nwith opposite skyrmion numbers giving a zero total ( Sskyrmionium = 0) [43, 44]. The skyrmionium is consid-\nered as a distinct skyrmionic state with particular characteristics [43] heralds the abrupt transition to a clear\nskyrmionic state ( S= 1). The formed skyrmion which has a perfect circular shape is a Néel-type skyrmion\nand as mentioned earlier remains stable for a significant external field range. In the one skyrmion plateau\nregion as the field value decreases from Bext=−0.16 TtoBext=−0.63 Tthe actual diameter of the skyrmion\nalso decreases as it can be seen from the representative micromagnetic configurations shown in Fig. 1a. The\nmagnetization reversal process is complete when all magnetization vectors are aligned parallel to z−direction\n(blue colored square in Fig. 1a).\nThe jump discontinuities of skyrmion number as the external field decreases were evident in Fig. 1a, albeit\nthedetailedmechanismbehindthisbehaviorisnotclear. Thecomplexphenomenarelatedtoskyrmionformation\nas well as with the skyrmion number discontinuities could be associated with the rich energetic environment\nhaving contributions from demagnetization Edemag, exchange Eexchand anisotropy Eanisenergies. In order\nto shed light on the skyrmion formation the individual energetic contributions [45] were quantified during the\nmagnetization reversal process by computing the absolute relative energy difference ∆Erel,i\ntype=|Ei+1\ntype−Ei\ntype\nEi\ntype|×\n100(%)(wheretypestands foranis,exch,demag ) between the consecutive ( iandi+ 1) external magnetic field\nvaluesBi\next,Bi+1\nextwith (i= 0,199). The values of the relative differences of anisotropy, demagnetization and\nexchange energies are also shown in Fig. 1b as functions of BextforKu= 130.4 kJ/m3.\nInitially, the relative energy differences have very low fluctuating values as external field decreases down to\nBext= 0.5 Twhere a discontinuity can be observed for all relative anisotropy and demagnetization energies and\nclearly can be associated with the creation process of the aforementioned skyrmionium texture. The relative\ndifference values are 6%, 6.5% for ∆Erel\nanis,∆Erel\ndemag, respectively. Gradual decrease of the external field is\nfollowed by demagnetization and exchange energy fluctuations for field values down to -0.14 T. Further decrease\nof the field triggers the skyrmion formation followed by abrupt jump discontinuities in ∆Erel\nanis,∆Erel\ndemagen-\nergies at -0.14 T. The formed skyrmion ( S≈1) at -0.14 T remains stable as the field further decreases with\n∆Erel\nanis,∆Erel\ndemagretaining their fluctuating character showing a clear tendency to attain lower values. At the\nfield value of -0.63 T a new jump discontinuity but with significantly reduced jump amplitude is evident on the\nrelative energy differences shown in Fig. 1b which signals the skyrmion annihilation. The relative difference\nvalues at the skyrmion annihilation discontinuity are 2%, 3% for anisotropy, demagnetization, respectively. It\nis worth noting that ∆Erel\nexch(< 0.7%) remains practically constant with weak fluctuations during the magne-\ntization reversal process while the demagnetization and anisotropy energies play the crucial role to skyrmion\nformation.\n3.2 Skyrmion dependence on nanoparticles’s width\nFor nanoparticles having a= 150 nm individual micromagnetic simulations have been conducted at different\nwidth (w) values in order to investigate the effect of the nanoparticle’s width on the actual skyrmion formation\nduring the magnetization reversal process at the temperature of 300 K. The skyrmion number Sas a function\nofBextis shown for width values w=6 - 36 nm in Fig. 2. Skyrmion can be produced in substantial ranges\neven for widths down to 12 nm. The present FD micromagnetic simulations reveal that the 12 nm width for\nnanoparticles having a= 150 nm is critical for the creation of skyrmions. Nanoparticles with width values\nbelow 12 nm cannot generate complete skyrmions. The S(Bext)line shapes for the nanoparticle with w= 6 nm\nsubstantially deviate from the respective line shapes for higher widths. A smooth increase (still fluctuating)\nof the skyrmion number as a function of the reducing external magnetic field during the skyrmion’s creation\nprocess is evident. The nanoparticles with w= 12−36 nmexpose a discontinuous jump-like character for\nexternal fields around 0 T. The higher the width the higher the jump-like discontinuity. For all widths the\nskyrmion annihilation which occurs at different field values is an abrupt process. The skyrmion number reduces\n43.3 Skyrmion dependence on MCA 3 RESULTS\nFigure 2: Skyrmion number Sas a function of Bextfor side value a= 150 nm at 300K for geometries having\ndifferent widths w= 6,12,18,24,30,36nm.\nfromS≈1toS≈0with no intermediate transition magnetic states.\n3.3 Skyrmion dependence on MCA\nFor the reference nanoparticle ( a= 150 nm ,w= 36 nm ) simulations have been conducted with varying MCA\nvaluesKu= 78.2−234.7 kJ/m3while its orientation is set parallel to external field and z-direction. It is\ninteresting the fact that MCA mainly affects the magnetic field range of stabilization of the skyrmion formed and\nnot the skyrmionic state of S= +1. Two external fields during the creation ( BcreationSK ) and the annihilation\n(BannihSK) of skyrmion were recorded and presented in Fig. 3. TheBcreationSK values of the magnetic\nfield follow a rather weak decrease from -0.12 T to -0.25 T with the gradual increase of the MCA from Ku=\n78.2 kJ/m3toKu=234.7kJ/m3. Theskyrmionannihilationfieldvalueof BannihSKexposesaweakbutgradual\nincrease from -0.7 T to -0.6 T as the MCA value increases.\nThe external magnetic field skyrmion’s stabilization range |BannihSK−BcreationSK|is also monitored in\nFig. 3exposing a monotonic decrease as MCA value increases from Ku=104.3 and 234.7 kJ/m3. The switching\nfield which is related to initiation of the reversal process is affected by the MCA value showing a clear reduction\nfrom 0.5 to 0.3 T with the increase of Ku.\n3.4 Skyrmionic states\nThe topological invariant Sreported along with the actual magnetic configurations can provide valuable quali-\ntative information toward the detection and characterization of skyrmionic states. It is very interesting the fact\nthat a variety of micromagnetic skyrmionic states emerges for the square-based FePt nanoparticles. The most\ncommonly detected micromagnetic states are presented along with the respective skyrmion number SinFig.\n4and can be assigned to the following categories:\na, f.Uniform states : States where the magnetization is uniform and the actual magnetization vectors point all\nupwards (red color) or all downwards (blue color) [46, 13].\nb, c.Skyrmionium : A magnetic skyrmionium is a non-topological soliton of a doughnut shape [43, 44].\nd.Domain wall : A domain wall is a gradual reorientation of individual magnetic moments across a finite\ndistance undergoing an angular displacement of 90oor 180o[47].\ne.Skyrmion with S= 1: State where one skyrmion is formed [40, 28].\nFromtheresultspresentedandtheconclusionsdrawnsofaritisclearthatskyrmionictexturescanbecreated\n53.4 Skyrmionic states 3 RESULTS\nFigure 3: External fields relative to the formation ( BcreationSK ) and annihilation ( BannihSK) as a function of\nMCA value ( Ku= 78.2−234.7 kJ/m3) for the reference nanoparticle at 300 K.\nand stabilized by adjusting the width of the nanoparticle, the MCA value and the temperature of the magnetic\nsystem. In addition, the skyrmion number Shas been computed during the magnetization reversal for FePt\nnanoparticles with variable side length. The Fig. 5depicts for the temperature of 300 K, Ku= 130.4 kJ/m3\nandw= 36nm a skyrmionic state diagram constructed by the different values of external field Bextand of the\nsquare side length a.\nNanoparticles with side length a≤50 nm are not capable of hosting stable skyrmionic entities irrespective\nof the applied external field as can be seen in Fig. 5. The restricted surface area affects also the precursor\nskyrmionium state which is absent in the aforementioned nanoparticles. The precursor skyrmionium states for\npositive external field values start to appear assisting the development of Néel skyrmions for square bases with\na≥60nm. The skyrmionium states are present and their development is intimately related to the side length\na. The higher the side length the higher the value of the characteristic positive external field value in which they\nappear. It is very interesting the fact that for the square based parallelepiped geometry the magnetic skyrmions\nare created for Bext<0values and in particular for -0.1 T. The detected skyrmions are Néel skyrmions having\nskyrmion number values in the [0.79,0.96] interval.\nMagnetic nanoparticles that promote not only the creation but also the stabilization of skyrmions in a\nconsiderable range of external fields are the nanoparticles with square side length a≥90 nm. For instance the\nskyrmions created on the nanoparticles with a= 90,100 nmcan be persistent for the field [-0.4,-0.1] T range\nwhile the a= 100,120,130 nmanda= 140−180 nmthe annihilation field is in the vicinity of -0.5 and -0.6\nT, respectively. At this point it should be noted that the micromagnetic states represented in Fig. 5reveal a\nFigure 4: Different states detected for the square base FePt nanoparticle (reference geometry). The states\nare: (a, f) uniform, (b, c) skyrmionium, (d) domain wall, (e)skyrmion with S= +1. MCA value was\nKu= 130.4 kJ/m3\n63.4 Skyrmionic states 3 RESULTS\nFigure 5: Micromagnetic states revealed at 300 K for the FePt nanoparticle. The color bar refers to the z-component of magnetization ( mz).\n73.5 Nanoparticle’s internal magnetic structure 3 RESULTS\nsignificant symmetry with respect to -0.1 T magnetic field. In the majority of the cases the skyrmionic textures\nof well developed skyrmioniums at positive fields have their mirror -0.1 T axis skyrmion analogue created at\nnegative fields.\n3.5 Nanoparticle’s internal magnetic structure\nFinite Difference micromagnetic simulations can provide rich and detailed information about the magnetization\nbehavior on the surface and in the bulk of nanoparticle. The detected skyrmions on the surface are inevitably\nrelated to the magnetization formations on the internal domain of the nanoparticle. The existence of magnetic\nstructure in the internal domain of the magnetic nanoparticle can be revealed by monitoring the magnetization\nvectors Mforgridpointslocatedatdifferent z-levels(xy-crosssection)andfor yz,xz-crosssections. Thesecross\nsections were chosen for micromagnetic systems hosting one Néel skyrmion in order to describe the qualitative\ncharacteristics of the actual skyrmionic texture. Néel skyrmion is shown in Fig. 6for the reference geometry\nand forBext=−0.3T and MCA value Ku= 130.4 kJ/m3.\nFigure 6: Micromagnetic configurations of a Néel skyrmion at different z-levels (z=0, 9, 18, 27, 36 nm)\nforBext=−0.3Tand for the reference nanoparticle at 300K. In addition, xzandyz-cross sections sliced\naty= 75,x= 75nm respectively are depicted for the square-base parallelepiped geometry of the magnetic\nnanoparticle. In particular, a magnified domain of the yz-cross section is also shown. The color bar represents\nthez-component of the magnetization ( mz).\nAt first glance the presence of magnetic structure is evident for the Néel skyrmion ( Fig. 6) in all cross\nsections visualized. In particular, xy-cross sections at z= 0,9,18,27,36 nmreveal skyrmionic configurations\nappearing in each square z-level and are presented in Fig. 6. Interesting is the fact that micromagnetic\nconfigurations in the non-surface z-values expose different magnetic characteristics with respect to skyrmions\nobserved on the square bases of the FePt nanoparticle. On the top or bottom surface Néel skyrmions are formed\nwhile in the proximity of central bulk region magnetization vectors deviate ( z= 9,27nm of Fig. 6) from the\nNéel skyrmion pattern adopting a more Bloch-type skyrmionic magnetic texture at z= 18nm with vortex type\ncirculating magnetizations. The size of the skyrmionic regions at different z-levels is different with the skyrmion\ndiameter being increased from the surface to the bulk domain of the nanoparticle. The higher diameter can be\nobserved for z= 18nm.\nMagnetization configurations represented by the magnetization vector on the nodal points at z=0 nm and\nz=36 nm are different as can be observed in Fig. 6. It should be pointed out that although the two aforemen-\ntionedz-levels where the magnetization configurations hosting skyrmions look different they are topologically\nequivalent. This can be justified by the fact that the calculated integrals of the topological density on the two\nsquare base areas give the same S.\nThe present simulation results support the depth dependence of helicity which is an interesting result that\nhas been recently observed experimentally by tomography in systems with bulk skyrmions reported by Zhang\net al. [30]. It should be pointed out that although the physics behind skyrmion formation in this system\nmay differ the depth dependence is obviously related to surface effects in both systems. In the present case\nwhere skyrmions are created in thin nanoparticles the depth dependence of the demagnetizing field (which has\nincreasedz-component near the surfaces [48]) can force the vortex to acquire a Néel character at the surfaces\n83.6 Skyrmion’s size-diameter 3 RESULTS\nwhile maintaining its typical chiral nature at the bulk. The skyrmionic structures observed here depend on the\ndemagnetizing effects which are proportional to the thickness w(as the easy axis remains in-plane) and vanish\nfor low thickness values ( <6 nm) as it was shown from the presentation of the current simulation results and\ndiscussion. Inevitably, the rich skyrmionic textures and physical phenomena related to the depth dependence\nof helicity are being suppressed as the thickness becomes comparable to the exchange length ( lex).\nAtxz-cross section (sliced at y=75 nm) which hosts the Néel skyrmion the magnetization vectors develop\nthree distinct regions along the x-direction as exposed in Fig. 6 with two clear vortex-like magnetization\ncirculations on the interfaces between the regions. In the regions (blue-shaded) located at the right and left\ncorners ofxz-cross section magnetizations are pointing downwards with the dimensionless magnetization z-\ncomponent attaining values mz=-1. Moving away form the corners toward the skyrmionic bulk regions the\ntilting of magnetizations starts producing the aforementioned vortex formations at the interfaces. At the\ncentral bulk region the magnetizations start to align parallel to z-direction pointing upwards ( mz=+1).\nTheyz-cross section is also shown for the Néel in Fig. 6. A region hosting magnetization circulations is\nlocatedatthecenterofthe yz-crosssection. Distinctregionshostingparallelmagnetizationspointingdownwards\n(blue-shaded region with mz=-1) and upwards (red-shaded region with mz=+1) are evident. Their position\nfollows an alternating fashion blue-red-blue with the interface regions (white region having mz=0) hosting the\ncenter of vortex formations.\n3.6 Skyrmion’s size-diameter\nThe isolated Néel skyrmions that have been generated on the FePt nanoparticle of square parallelepiped geome-\ntries during the reversal process have circular geometry with varying diameter dsk. Their size can be controlled\nby the magnitude of external field as reported in the recent literature [49] and shown by Finite Element mi-\ncromagnetics simulations for FePt nanoparticles having reuleaux geometries at 0 K [29]. The skyrmion’s size\ndependence on the external field and on the square base side length ais computed and quantified.\nFigure 7: Skyrmion’s diameters dskatT=300 K as a function of the applied external field Bextalong with\nrepresentative micromagnetic configurations for the reference nanoparticle ( a= 150 nm, w= 36 nm).\nInFig. 7thecalculatedskyrmion diameter dskisshown asa functionofthe external fieldin the [-0.64, -0.16]\nT range where S≈1. Thedskdepends more or less linearly on Bextand a linear regression of the simulation\npoints give dsk(Bext) = 87.074Bext+ 78.18with a correlation coefficient R= 0.9817879. The formed skyrmion\natBext=−0.16T has its maximum diameter value close to 62 nm. Further decrease of the magnetic field\ncauses the gradual decrease of the skyrmion diameter. Just before its annihilation at Bext=-0.64 T skyrmion\n93.6 Skyrmion’s size-diameter 3 RESULTS\nattains its minimum diameter which is 15 nm. It is evident that the external magnetic field plays a dominant\nrole on the actual size of the Néel-skyrmion.\nFigure8: Skyrmiondiameters dskatT= 300K,Ku= 130.4 kJ/m3,Bext=-0.4Tasafunctionofnanoparticle’s\nwidthwin blue color (side length was set to a= 150 nm ) and as a function of the side length ( a) in red color\n(width length was set to w= 36 nm).\nTheeffectsofthegeometricalcharacteristicsofthenanoparticleontheformedskyrmionsizearealsostudied.\nNéel skyrmion diameters dskforKu= 130.4 kJ/m3, external field value Bext=−0.4T atT= 300K were\ncalculated as functions of nanoparticle’s width ( w) and square’s side length ( a). In Fig. 8dsk(w)functional\nform is shown in blue color with the side length set to a= 150 nm anddsk(a)is presented in red color having\nconstant width w= 36 nm . The increase of wis followed by a monotonic increase of dskvalues from 17\nto 46 nm. This monotonic decrease consists of two different linear regions. The first one for nanoparticles\nwith width between 12 and 24 nm ( dsk(w) = 0.78583w+ 8.9417andR= 0.9639279). A second linear region\n(dsk(w) = 1.6342w−12.162andR= 0.9999989) with higher slope is evident for nanoparticles having 24 to 36\nnm width. The width value w= 24nm represents a critical value for the creation of skyrmions with intermediate\nsizes having diameters close to 27 nm. Regarding the effect of the square side length aof the nanoparticle on\nthe skyrmion diameter also reveals two regimes as can be seen in Fig. 8. The first one is linear and follows the\nincrease of dskfrom 15 nm to 45 nm as the side length aincreases from 90 nm to 150 nm. The second one is\na plateau regime in which the skyrmion diameter remains almost constant at 45 nm for square side lengths up\nto 180 nm.\nThe skyrmion diameter-size has been calculated at different MCA values for the reference nanoparticle and\nthe results are given in Fig. 9 for the specific value of Bext=−0.4T. At first glance two regimes exist\ndescribing the skyrmion’s size dependence on the MCA value. Lower MCA values have a stronger effect on dsk.\nIn particular, as MCA value increases linearly ( dsk= 0.4148Ku−8.8949) from 78.2, to 104.3 and finally to\n130.4 kJ/m3the diameter of skyrmion increases attaining the values dsk≈25 , 31, 47 nm, respectively. Further\nincrease of the MCA to values up to Ku= 234.7 kJ/m3does not affect the diameter of the skyrmion and a\nplateau region is evident with the created skyrmion having dsk≈47.5nm.\nFinally, the skyrmion diameter for the reference nanoparticle and for Ku= 130.4 kJ/m3,Bext=−0.4 Thas\n103.6 Skyrmion’s size-diameter 3 RESULTS\nFigure 9: Skyrmion diameters dskfor magnetic field value Bext=−0.4 Tas a function of temperature T(red\nhorizontal axis) and of MCA Ku(blue horizontal axis) at T= 300K. The reference square-based nanoparticle\nwas used.\nbeen calculated for different temperatures ( dsk(T)) around 300 K as can be seen also in Fig. 9. It is clear that\nthe temperature does not affect the size of the created skyrmion even for temperatures 30 K above or below the\nroom-temperature.\nConclusions\nThe skyrmion formation in FePt nanoparticles were studied using FD micromagnetic simulations. The adopted\nmicromagnetic model takes into account thermal effects in the form of a Brownian term in the effective field\ninvolved in LLG equation and the magnetic material properties were reduced to the temperature range of\n270 - 330 K. MCA and external field were set normal to nanoparticle’s surface in all conducted numerical\nsimulations. Magnetic skyrmions have been detected during the magnetization reversal process for FePt square-\nbased parallelepiped nanoparticles having different side lengths and widths with MCA value kept constant\natKu= 130.4kJ/m3andT= 300K. Computation of the topological invariant of skyrmion number S\naccompaniedbythevisualizationoftheactualmicromagneticconfigurationshaveprovideddetailedquantitative\nand qualitative information relative to skyrmion formation and stabilization. Interesting magnetic structures\nwere revealed for particular external magnetic field value ranges. Néel type skyrmionic textures were detected\non the outer surfaces coexisting with Bloch-type textures in the bulk of the FePt square-based parallelepiped\nnanoelements.\nThe computed sizes of the created Néel skyrmions showed a linear dependence with respect to the external\nfield. At different temperatures and for the external magnetic field value of Bext=−0.4T skyrmions can be\ngenerated with a diameter around 45 nm for the reference nanoparticle having a= 150 nm . Variation of MCA\nhave a significant effect on the skyrmion diameter only for Ku∈[78.2, 130.4] kJ/m3while for the higher MCA\n11REFERENCES REFERENCES\ndoes not affect the diameter which attains values close to 47.5 nm. The nanoparticles’ side length ( a) and width\n(w) affect the skyrmion diameter. The increase of wis followed by an increase of the skyrmion diameter values\nfrom 17 to 46 nm. The side length increase can give birth to skyrmions with diameters close to 50 nm. In\nconjunction with previous finite element studies at 0 K it is clear that magnetic skyrmions can be produced\naround 300 K in a wide range of external fields and for nanoparticles with different geometrical characteristic\ndimensions even in the absence of chiral interactions such as Dzyaloshinskii-Moriya for FePt nanoparticles.\nAcknowledgments\nC. Tyrpenou is supported through the project ”Dioni: Computing Infrastructure for Big-Data Processing and\nAnalysis.” (MIS No. 5047222) which is implemented under the Action ”Reinforcement of the Research and\nInnovation Infrastructure”, funded by the Operational Programme ”Competitiveness, Entrepreneurship and\nInnovation” (NSRF 2014-2020) and co-financed by Greece and the European Union (European Regional De-\nvelopment Fund). V. D. Stavrou was supported through the Operational Programme ”Human Resources De-\nvelopment, Education and Lifelong Learning” in the context of the project ”Strengthening Human Resources\nResearch Potential via Doctorate Research” (MIS-5000432), implemented by the State Scholarships Foundation\n(IKY) co-financed by Greece and the European Union (European Social Fund-ESF). We would like to thank\nMr. K. Dimakopoulos for technical support.\nReferences\n[1] N. S. Kiselev, A. N. Bogdanov, and R. Schafer abd U. K. Robler. Chiral skyrmions in thin magnetic films:\nnew objects for magnetic storage technologies. Journal of Applied Physics D , 44:392001, 2011.\n[2] N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and\nR. Wiesendanger. Writing and deleting single magnetic skyrmions. Science, 341:636–639, 2013.\n[3] G. Yu, P. Upadhyaya, Q. Shao, H. Wu, G. Yin, X. Li, C. He, W. Jiang, X. Han, P. K. Amiri, and Kang L.\nW. 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Shukla2,∗∗\n1Department of Physics, Aligarh Muslim University, Aligarh 202002, India\n2UGC-DAE Consortium for Scientific Research, Indore 452001, India\n3Institute of Physics, Academia Sinica, Taipei 11529, Taiwan\n4Indus Synchrotrons Utilization Division,\nRaja Ramanna Center for Advanced Technology, Indore 452013, India\n5School of Physics, University of Hyderabad, Hyderabad 500046, India\nAbstract\nAttempts to unravel the nature of magnetic ordering in LaSrCoO 4(Co3+), a compound intermediate\nbetween antiferromagnetic (AFM) La 2CoO 4(Co2+) and ferromagnetic (FM) Sr 2CoO 4(Co4+), have met\nwith a limited success so far. In this report, the results of a thorough investigation of dc magnetization\nand ac susceptibility (ACS) in single-phase LaSrCoO 4provide clinching evidence for a thermodynamic\nparamagnetic (PM) - ferromagnetic (FM) phase transition at T c= 220.5 K, followed at lower temperature\n(Tg= 7.7 K) by a transition to the cluster spin glass (CSG) state. Analysis of the low-field Arrott plot\nisotherms, in the critical region near T c, in terms of the Aharony-Pytte scaling equation of state clearly\nestablishes that the PM-FM transition is basically driven by random magnetic anisotropy (RMA). For\ntemperatures below ≈30 K, large enough RMA destroys long-range FM order by breaking up the infinite\nFM network into FM clusters of finite size and leads to the formation of a CSG state at temperatures\nT/lessorsimilar8 K by promoting freezing of finite FM clusters in random orientations. Increasing strength of the\nsingle-ion magnetocrystalline anisotropy (and hence RMA) with decreasing temperature is taken to reflect\nan increase in the number of low-spin (LS) Co3+ions at the expense of that of high-spin (HS) Co3+ions.\nAt intermediate temperatures (30 K /lessorsimilarT/lessorsimilar180 K), spin dynamics has contributions from the infinite FM\nnetwork (fast relaxation governed by a single anisotropy energy barrier) and finite FM clusters (extremely\nslow stretched exponential relaxation due to hierarchical energy barriers).\n∗abdul.ahad@ntu.edu.sg; Present Address: School of Physical and Mathematical Sciences, Nanyang Technological\nUniversity, Singapore\n†Present Address: RIKEN Center for Emergent Matter Science, Saitama, Japan\n1arXiv:2002.00135v3  [cond-mat.mtrl-sci]  16 May 2023I. INTRODUCTION\nLow-dimensional magnetic systems are susceptible to a variety of external perturbations, such\nas doping, magnetic field, and strain, etc. [1]. Their higher sensitivity to external stimuli com-\npared to the 3D counterparts opens up room for new applications, e.g., control of exchange bias\nin a single unit cell [2]. Moreover, in low-dimensional systems, control of spin moment can pro-\nvide tremendous application opportunities for spintronics devices (layered cobaltates are potential\ncandidates). LaCoO 3(Co3+), an often studied 3D compound, exhibits peculiar properties such\nas a metal-insulator transition, high-spin (HS) state to low-spin (LS) state transitions, magneto-\nelectronic phase separation and ferromagnetic (FM) correlations [3]. A fascinating cobaltate and\nits 2D analog, La 2CoO 4(Co2+) is an antiferromagnetic insulator (AFI) [4], but not explored\nenough mainly due to its topotactic oxidation [5]. Its counterpart Sr 2CoO 4(Co4+) has a simi-\nlar structure and shows ferromagnetism and half-metallicity [6]. In LaSrCoO 4(an intermediate\ncomposition of the above-mentioned end compounds), La/SrCoO 33D blocks are separated by\nrock-salt La/SrO layers.\nDepending upon how the energy-splitting ∆CFbetween the t2gandegorbitals, caused by the\ncrystal-field, compares in magnitude with the exchange energy Jexassociated with the Hund’s rule\ncoupling, the Co3+ion can exist in three different spin states [7]: the magnetically-active Co3+\nhigh-spin (HS; t4\n2ge2\ng, S = 2) state when ∆CF<Jex, intermediate-spin (IS; t5\n2ge1\ng, S = 1) state when\n∆CF∼Jexand non-magnetic low-spin (LS; t6\n2ge0\ng, S = 0) state when ∆CF/greatermuchJex. In the LaSrCoO 4\n(LSCO) compound, the spin-state of Co3+ions has been a controversial issue. For instance,\nLSCO is reported [8–12] to have a mixture of LS and (thermally-activated) HS states of the Co3+\nions. Presence of homogeneous IS state of Co3+ions has also been claimed [13] and subsequently\nchallenged [14]. In our previous study [7], we probed the spin-states in La 2−xSrxCoO 4(0.5≤\nx≤1) compounds and found existence of the HS and LS states together at room temperature,\ndiscarding the presence of IS state. First-principles calculations [9] support the view that mixed\nspin-states (HS and LS) of Co3+ions inhabit the ground state of LaSrCoO 4.\nExistence of mixed spin-states of the Co3+ions in LSCO is expected to have an important\nbearing on the magnetic and transport properties, since the super-exchange interaction between\n‡Present Address: National Institute of Technology Hazratbal Srinagar J & K, India\n§Present address: Department of Physics, School of Engineering, University of Petroleum and energy studies,\nDehradun, India\n¶sn.kaul@uohyd.ac.in\n∗∗dkshukla@csr.res.in\n2the HS Co3+ions mediated by the intervening LS Co3+ion can induce long-range FM order,\nwhile the superexchange interaction between the adjacent HS Co3+neighbors can give rise to\nantiferromagnetic (AFM) order. The competing FM and AFM exchange interactions, in turn,\ncould result in a spin glass (SG) state. Conflicting reports [15–20] about the nature of magnetism\nin LSCO, however, render the existing magnetic data inconclusive, as elucidated below. While a\nparamagnetic (PM)-SG transition at TSG∼20 K [15] (TSG∼8 K [16]) is construed as a transition\nfromtheHStoISstate,noPM-SGtransitionisobserveddownto2K(thetemperatureuptowhich\nan insulating PM state persists [18]) and a broad hump in the thermomagnetic data, observed\nat/similarequal250 K, is attributed to the ferromagnetic (La,Sr)CoO 3impurity [18]. This insulating PM\nground state is taken to reflect [18] the IS spin state of Co3+ions. In sharp contrast, two magnetic\ntransitions, PM-FM transition at T c/similarequal250 K [19, 20] and the SG-like transition at TSG∼12\nK [19] or at TSG∼60 K [20], have been reported. Considering that the spin and valence states\nof Co are inextricably linked in cobaltates, presence of valence states (Co2+and/or Co4+) of Co,\narising from the varying degree of oxygen off-stoichiometry in the LSCO samples used in such\ninvestigations, could be at the root of these apparent discrepancies.\nIn this work, the intrinsic magnetic behavior near the PM - FM and SG transitions is unrav-\neled by ensuring close to a pure Co3+state in our LaSrCoO 4sample. DC magnetization and\nfrequency-dependent ac susceptibility (ACS) data, taken on the LaSrCoO 4compound, unambigu-\nouslyconfirmtheexistenceoftwomagnetictransitions: athermodynamic( frequency-independent )\nPM - FM phase transition at T c= 220.5 K and a frequency-dependent cluster spin glass (SG)\ntransition, approaching the value T g= 7.7 K in the zero-frequency limit. Both of these transitions\nare basically driven by random magnetic anisotropy (RMA). Increasing strength of the single-ion\nmagnetocrystalline anisotropy (and hence RMA) with decreasing temperature is shown to be a\nmanifestation of an increase in the number of low-spin (LS) Co3+ions at the expense of that of\nhigh-spin (HS) Co3+ions.\nII. EXPERIMENTAL METHOD\nThedetailsofsamplesynthesisarereportedelsewhere[7]. SynchrotronX-raydiffraction(XRD)\npattern was measured at room temperature using the BL-12 ADXRD beamline of RRCAT, India.\nTo avoid possible fluorescence from Co, an x-ray photon energy of 7690 eV ( λXRD= 1.612 ˚A),\nwhichliesbelowtheabsorptionedge,wasusedforXRDmeasurements. X-rayabsorptionnearedge\n3spectra (XANES) were recorded in the fluorescence mode at the same beamline. Energy resolution\nat the Co K-edge energy was ∼0.7 eV. ‘Zero-field’ (H = 0) and ‘in-field’ (H = 80 kOe) electrical\nresistivities as functions of temperature were measured by four-probe method, using the Oxford\ncryostat. Magnetization measurements were carried out utilizing the 7 Tesla Quantum design\nMPMS 3 magnetometer. M(H) hysteresis loops at fixed temperatures were recorded following the\n‘field-cooled’ (FC) protocol. Magnetization was measured as a function of temperature, M(T),\nover the temperature range extending from 5 K to 300 K at fixed fields, employing the FC and\n‘zero-field-cooled’ (ZFC) protocols. The virgin M(H) curves were recorded after demagnetizing\nthe sample and the superconducting magnet by setting field to zero in the oscillatory mode.\nFor constructing the Arrott plots from the virgin M(H) isotherms, the applied field has been\ncorrected for the demagnetizing field to arrive at the internal effective magnetic field [21] H eff,\nin accordance with the relation, H eff= Happlied- (4πN) M, where N is the demagnetizing factor,\nwhich was determined from the low-field portions of the M(H) isotherms taken at T <Tc. ZFC\nrelaxation measurement has been performed by cooling the sample down to 150 K in zero field and\nafter waiting for 100 s (t w), a static magnetic field of 50 Oe was applied and the time evolution of\nmagnetizationat150Kwasmeasuredupto ∼8000s. Thereal( χ/prime(ω,T))andimaginary( χ/prime/prime(ω,T))\ncomponents of ac magnetic susceptibility were measured in the temperature range of 2 - 300 K, at\nfixed frequencies (1.1 Hz - 941 Hz) of the ac driving field of rms amplitude, hac= 3 Oe employing\nthe Quantum Design MPMS 3 system equipped with an ac susceptibility option.\nIII. RESULTS AND DISCUSSION\nFig. 1 shows room temperature synchrotron XRD pattern of the LaSrCoO 4(LSCO) compound\ntogether with the Rietveld fit. Absence of any extra Bragg peaks, attributable to the impu-\nrity phases such as La 2CoO 4(Co2+) and Sr 2CoO 4(Co4+), confirms the single-phase nature of\nthe LaSrCoO 4sample. Rietveld refinement yielded the lattice parameters, corresponding to the\ntetragonal space group I4/mmm, asa= 3.8025(3) ˚A andc= 12.4985(2) ˚A . Inset (a) compares\nthe Co K-edge XANES data of LaSrCoO 4with those of the reference compounds CoO and Co 3O4.\nLinear Co-valency versus Co K-edge photon energy plot in the inset (i) yields the value +2.98(2)\nfor the Co valence and thereby confirms the Co-valency of +3 in the LSCO sample. Data points,\ndepicted by circles in the linear plot shown in inset (i), correspond to the first derivative of the\nabsorption edge.\n4253 03 54 04 55 05 5770077207740771677191\n.82.12.42.73.01\n802703601021041060\n.270.308124\n68812(\ni)(ii)( iii)(\nb) \nYobs \nYcalc \nYobs-Ycalc \nBragg PositionsIntensity (arb. units)2\nθ (degrees)(a) CoO \nLaSrCoO4 \nCo3O4 \nNorm. Abs. (arb. units)P\nhoton Energy (eV)Co K-edgeλXRD = 1.612 Å   \nC\no valency \n0 kOe \n80 kOeρ (Ω cm)T\n (K) \nT\n-1/4 (K-1/4) ln ρ1\n03 T-1 (K-1)FIG. 1. Room temperature synchrotron X-ray diffraction pattern of LaSrCoO 4(LSCO) compound, mea-\nsured at the x-ray photon energy of 7690 eV, along with the Rietveld fit. A Schematic of the unit cell\nshows that the structure consists of blocks of La/SrCoO 3separated by the La/SrO layers. Inset (a)\ncompares the Co K-edge XANES of LaSrCoO 4with those of the reference compounds CoO and Co 3O4.\nLinear Co-valency versus Co K-edge photon energy plot in the inset (i) extrapolates to the Co-valency\nof +3 in the LSCO compound. Inset (b) displays the temperature variations of the ‘zero-field’ ( ρ(H =\n0)) and ‘in-field’ ( ρ(H = 80 kOe)) resistivity. The insets (ii) and (iii) depict the lnρversusT−1andlnρ\nversusT−1/4plots, respectively.\nInset (b) of Fig. 1 clearly demonstrates semiconducting-like temperature variation of the ‘zero-\nfield’ (ρ(H = 0)) and ‘in-field’ ( ρ(H = 80 kOe)) resistivity in the LSCO sample. If, apart from\nCo3+ions, Co2+and Co4+ions were also present in the sample, double exchange, involving charge\nhopping, would have resulted in a metallic state (characterized by low resistivity, typically ∼10\n- 100µΩcm, andρincreasing with temperature) as opposed to the observed semiconducting-\nlike state with very large resistivity decreasing from 106Ωcmat 130 K to 102Ωcmat 310 K.\n5Consistent with the conclusion drawn from the Co K-edge XANES data, this finding completely\nrules out the presence of Co2+and Co4+ions in our LSCO sample. Furthermore, negligibly\nsmall magnetoresistance (MR), [( ρ(T,H = 80 kOe) - ρ(T,H = 0)]/ ρ(T,H = 0), observed in the\npresent case, sharply contrasts the reasonably large negative MR, expected in double-exchange\nferromagnets. The insets (ii) and (iii) of Fig. 1(b) bear out clearly that the processes such as the\nthermal activation of charge carriers across the band gap (described by expression ρ∼eE/kBT)\nand Mott variable-range polaron hopping (represented by the expression ρ∼e(T0/T)1/4) [22–24]\noperate within the temperature ranges 225 K - 300 K (T >Tc) and 130 K - 220 K (T ≤Tc),\nrespectively.\nAfter ensuring that the sample is of very good quality (free from the impurity phases), in the\nfollowing, we attempt to unravel the intrinsic magnetic behavior of the LaSrCoO 4compound.\nFig. 2(a) displays the ‘field-cooled’ (FC) and ‘zero-field-cooled’ (ZFC) thermomagnetic curves,\nM(T), at fixed magnetic fields in the range 10 Oe - 10 kOe. A clear bifurcation between FC\nand ZFC M(T) is observed at low fields. Such bifurcations are indicative of either a spin-glass\n(SG) phase [25] or cluster spin-glass (CG) phase or magnetic anisotropy energy barriers or super-\nparamagnetism (SPM) [26]. A closer look at the low-temperature ZFC M(T) data shows a clear\ncusp (inset (b) of Fig. 2) at T g∼8 K. Similar feature was observed previously [16, 17, 19] and\ndenoted as a freezing temperature. An interesting point to note is that on cooling below T Irr\n(the temperature at which FC and ZFC M(T) bifurcate), M FCincreases, which is taken to be a\ncharacteristic feature of cluster spin glass behavior in various systems [27]. The increase in M FC\nwith decreasing temperature, even below T g, points to the presence of a cluster spin glass (CSG)\nstate[28]becauseM FCnormallyexhibitsaplateauforcanonicalspinglass[29]. Itisnoteworthyto\nmention that, for a typical re-entrant SG systems, the irreversibility occurs far below T cwhile for\nCSG TIrr≤Tc[30]. In this connection, it is also reported [3] that if the bifurcation temperature\nTIrr>>Tg, the compound exhibits cluster spin glass type behavior while, for canonical SG [31],\nthe TIrrcoincides with the T g. Thus, we conclude that the signatures discussed above basically\nreflect the CSG state in LaSrCoO 4at low temperatures.\nNext, we discuss the role of magnetic anisotropy as a possible source of bifurcation in M(T).\nBecause of the prevalent elongated octahedra in the crystal structures similar to that of LaSrCoO 4,\nthe pseudo orbital moment of Co3+HS, ˜L= 1 [32], prefers to lie in-plane and forces the spin\nmoment also to lie within the plane [33] due to the spin-orbit coupling. Neutron diffraction\nstudiesoniso-structuralcompoundsLaSrFeO 4[34]andhalf-dopedLa 1.5Sr0.5CoO 4[35]haveclearly\n601002003000.00.51.01.52.06\n8101264.465.165.80\n1230\n10\n.00.51.01.50\n10\n1002000.00.51.00\n100200015 kOeT\nIrrM (10-3 emu / g)T\n (K) FC \nZFC \n M (emu / g)χ'FCχZFCT\ngT\n (K) 1\n00 Oe \n (\nb)1\n0 Oe \n  \n50 Oe(d)1\n00 Oe5\n0 Oe10 kOe7\n50 Oe5\n00 Oe2\n00 Oe100 OeT\n (K)   \n1\n0 Oe(e)(a) \n  \n200 Oe(f)    \n500 Oe(c)(\ng)χ (10-3 emu / g-Oe) \n  \n750 Oe(h)FIG. 2. (a) Thermomagnetic M(T) curves taken at fixed magnetic fields. Inset (b) shows the enlarged\nview of ZFC M(T) at low temperatures which facilitates the observation of the spin glass transition at T g.\n(c-h) Temperature variations of the susceptibilities measured (solid circles) and calculated (open circles)\nat the different fixed magnetic fields.\nborne out that, in these systems, magnetic moments are confined to the abplane but can rotate\nwithin the plane. This is indicative of an XY-type anisotropy and hence c-axis may not be the\neasy axis of magnetization. To ascertain whether or not magnetic anisotropy plays a role, we\nadopted the approach, proposed by Joy et al.[26], according to which, if the decrease in M ZFC\nat low temperatures is due to magnetocrystalline anisotropy, M ZFCshould follow the relation,\nMFC\nHapp+Hc≈MZFC\nHapp. That this is indeed the case for the fields H≥100 Oe, is evident from Fig. 2(e -\nh). However, at H≤50 Oe,χZFC(=MZFC\nHapp) strongly deviates from χ/prime\nFC(=MFC\nHapp+Hc) (see Fig. 2(c,\nd)) in the low temperature region. The demagnetization-limited behavior of χZFCatT <Tcfor\nH=10 Oe, suggests that the shape anisotropy also becomes important at low fields H≤50 Oe.\nNote that, in the calculation of χ/prime\nFC, we have used the coercive field, H c(T), data shown in the\n7-9- 6- 30 3 6 9 -3-2-101230\n801602400.00.40.8 \n5 K  8 K \n15 K  30 K \n225 K  240 KT (K)M (emu / g)H\n (kOe) Hc (kOe) \n \n FIG. 3. ‘Field-cooled’ M(H) hysteresis loops. Inset shows the variation of H cwith temperature.\ninset of Fig. 3.\nFig. 3 presents the M(H) hysteresis loops at fixed temperatures, taken in the ‘field-cooled’ (H =\n70 kOe) mode at temperatures below and above the PM-FM transition temperature, T c= 220.5\nK, (corresponding to the dip in dM ZFC(T)/dT at low fields H = 10 - 100 Oe, not shown here).\nWith temperature decreasing from T /greatermuchTc, where the sample is in the PM state and as such the\nM(H) isotherms are linear, the coercive field, H c, increases from zero at T = T c= 220.5 K to\nabout 800 Oe at 5 K; the rate of increase in H cpicks up for temperatures below 30 K, as is evident\nfrom the inset of Fig. 3. A steep increase in H cas the temperature falls below 30 K is indicative of\nan increase in the random magnetic anisotropy (RMA). The observation that magnetization does\nnot saturate even in fields as high as 70 kOe at low temperatures (Fig.5(a)) and the presence of\nRMA (as inferred from the critical-point analysis in the following section) support the existence\nof a CSG state at low temperatures.\nThepanels(a)and(b)ofFig.4displaythetemperaturevariationsofthereal( χ/prime)andimaginary\n80501001502002503003500510150123455\n1 0151617180\n.080.120.160.21001011021032\n242322401232\n242282320.51.01.52.0(ii)χ'' (10-6 emu / g-Oe)T\n (K)(i)d\nχ'/dTω (Hz)(\nb)χ' (10-4 emu / g-Oe)(a)T\n (K)T (K) \n1.1 Hz \n11 Hz \n110 Hz \n941 HzT (K)T\nC = 220.5 K215220225220.5 KT\ng = 7.68(2) Kz\nν = 7.98(2)(\nTp(ω) - Tg) / Tgχ\n'-1 (104 g-Oe / emu)γ\n = 1.261(5)Tc = 220.5(1) K(\niv)(\niii)FIG. 4. Temperature dependence of (a) ‘in-phase’ component, χ/prime, and (b) ‘out-of-phase’ component, χ/prime/prime\nofχacat the different frequencies (1.1 to 941 Hz) measured at an ac driving field of rms amplitude 3 Oe.\nInset (i) of panel (a) shows a zoomed view of the frequency dependence of the peak in χ/prime(T) at around\n∼8 K while the inset (ii) depicts the d χ/prime/dTvs.T at different frequencies. The linear fit through the ω\nversus (Tp(ω)−Tg)/Tgdata (solid circles) in the inset (iii) testifies to the validity of the critical slowing\ndown model. Inset (iv) displays the fit (red curve) to the inverse ac susceptibility, χ/prime−1(T), data (open\ncircles) above Tc= 220.5 K in the range 221 K - 225 K, based on Eq.(2). This fit, when extended to 230\nK (dashed curve), demonstrates that the χ/prime−1(T) data start deviating from the fit for T >225 K.\n9(χ/prime/prime) parts of ac magnetic susceptibility, respectively. While the inset (i) of panel (a) highlights the\nfrequency-induced shift in the peak in χ/prime(T) at T = Tp/similarequal8 K, the inset (ii) clearly brings out the\nfrequency-independence of the dip in the d χ/prime/dTvs.T plots atTc= 220.5 K, which corresponds\nto the inflection point in χ/prime(T). The frequency-independent value of Tc= 220.5 K confirms the\ntrue thermodynamic nature of the FM - PM phase transition at Tc.\nThe shift in Tpper decade of frequency ( ∆Tp/{Tp∆(log 10ω)}= 3.2×10−2) is an order of\nmagnitude greater than that ( /similarequal10−3)) observed [31] in canonical spin glasses but is typical of\nthe cluster spin glasses [31, 36]. The frequency dependence of the peak temperature, Tp(ω), is\nanalyzed in terms of the following expression given by the ‘critical slowing down’ model [31, 36]\nω=ω0/bracketleftbiggTp(ω)−Tg\nTg/bracketrightbiggzν\n(1)\nwhereτo= 2π/ω 0is the relaxation time due to the correlated spin dynamics, Tgis the spin-\nglass transition temperature in the zero-frequency limit, νis the spin-spin correlation length\ncritical exponent and zis the dynamical critical exponent. The linear fit through the ωversus\n(Tp(ω)−Tg)/Tgdata (solid circles) in the inset (iii) of Fig. 4, based on the Eq. (1) with the\nparameter values τo= 2.49(1)×10−8s,Tg= 7.68(2) K and the product zν= 7.98(2), testifies to\nthe validity of the critical slowing down model. Note that, in the canonical spin glasses, relaxation\ntime,τo, is typically of the order of 10−12s [31, 36]. Several orders of magnitude larger τo(i.e.,\nmuch slower spin dynamics), in the present case, further confirms the existence of a cluster spin\nglass ground state.\nNext, we focus our attention on the FM - PM transition at Tc= 220.5 K. In the critical region\naboveTc, the inverse of measured ac susceptibility, χ−1\nac(T)≡χ/prime−1(T), is related to the inverse\nintrinsic susceptibility, χ−1\nint(T) asχ−1\nac(T) = 4πN +χ−1\nint(T), where N is the demagnetizing factor.\nAs the temperature approaches Tcfrom above, χintdiverges (or equivalently, χ−1\nintgoes to zero) at\nTcin accordance with the relation\nχ−1\nac(T) = 4πN+A/bracketleftbiggT−Tc\nTc/bracketrightbiggγ\n(2)\nwhereAandγare the critical amplitude and critical exponent, respectively. Inset (iv) of Fig. 4\ndepicts the best theoretical fit (continuous curve) to the χ−1\nac(T)data, based on the Eq. (2), yields\nTc= 220.5 K, A= 5.9(1)×105andγ= 1.261(5). This value of γwill be put into a proper\ncontext later in the text.\n1001020304050600123402 4 6 8 0.00.51.01.5(\nb) \n \n  (a)β\n = 0.5γ\n = 1.0Mean field(\nHeff/M)1/γ (103 Oe-g / emu)1/γM (emu / g) \n M1/β (emu / g)1/β               180K 190K \n200K 210K 215K \n220K 225K 230K \n240K 250K 260KH\neff (104 Oe)FIG. 5. (a) M(H) isotherms at and around Tc. (b) The modified Arrott ( M1/βvs.(H/M )1/γ) plots with\nthe choice of the critical exponents βandγgiven by the mean-field theory.\nIt is well-known that the critical-point (CP) analysis of magnetization, M(T,H), data taken\nin the critical region near T cprovides a powerful means to unravel the true nature of magnetic\nordering prevalent in a spin system. As elucidated below, the CP analysis makes use of the critical\nexponentsβ,γandδfor the spontaneous magnetization (order parameter), ‘zero-field’ suscepti-\nbility and the critical M(H) isotherm, (characterizing a continuous FM-PM phase transition at\nTc), that are defined as [37]\nMs=B|/epsilon1|βT <Tc, H→0 (3)\nχ0=A /epsilon1−γT >Tc, H→0 (4)\n11M=M0H1/δT=Tc (5)\nwhere/epsilon1= (T−Tc)/Tc, A, B and M 0are the critical amplitudes [37, 38]. Diverse systems\nhaving exactly the same values for the critical exponents β,γandδ, fall into a single universality\nclass. Universality class, in turn, is governed by the order parameter (spin) dimensionality ( n) and\nspatial dimensionality ( d) [37, 39] as long as the interaction coupling the spins is of short range.\nThe presence of different valence and spin states in the present system is expected to have a\nprofound effect on the critical behavior. This expectation called for a detailed study of the critical\nbehavior of LaSrCoO 4at temperatures in the vicinity of the FM-PM phase transition. To this\nend, we have analyzed the virgin M(H) isotherms in the critical region (shown in Fig. 5(a)) using\nthe scaling equation of state (SES) method, detailed in the reference [37]. Instead of following\nthe customary practice of using, at first, the Arrott [40] SES, we use the generalized form of the\nArrott SES, given by Arrott and Noakes [41] (AN), i.e.,\n(H/M )1/γ=a[(T−Tc)/Tc] +bM1/β(6)\nto arrive at the correct choice of the critical exponents that makes the AN, M1/βversus\n(H/M )1/γ, plots a set of parallel straight lines in the critical region near T c. The temperature at\n02 4 6 8 1 00.00.40.81.2Tc ~ 220 Kβ\n = 0.796γ\n = 1.051M1/β (emu / g)1/β(\nHeff/M)1/γ (103 Oe-g / emu)1/γ \n  \n180K 190K \n200K 210K \n215K 220K \n225K 230K\nFIG. 6. Arrott plots constructed using the critical exponent values β= 0.796 and γ= 1.051.\n12180190200210220234567\n8 9 -2.0-1.5-1.0-0.50.00246810120.00.30.60.90\n.00.30.60.91.20.000.050.100.150.200.25 \nδ\n = 2.33 \nδT\n (K)Tcδ1 = 5.0ln M (emu / g)l\nn Heff (Oe)M1/β (emu / g)1/β \n   (\nd)( c)(b)( a)β = 0.5γ\n = 1.01\n80 K1\n90 K2\n00 K2\n10 K2\n15 K2\n20 K(Heff/M)1/γ (103 Oe-g / emu)1/γM1/β (emu / g)1/β  T\nc  \n  T\nc = 220 KFIG. 7. (a) Theoretical fits to the Arrott plot ( i.e.,M1/βversus (H/M )1/γplot with mean-field exponents\nβ= 0.5andγ= 1) isotherms, based on the scaling equation of state, Eq. (8), predicted by the random\nanisotropy model. (b) Enlarged view of the data and fits at fields nearing zero. (c) The linear log M\nversus logHeffplots at T≤Tc. (d) Critical exponent δ1at different temperatures below T cand the\ncritical exponent δat Tc, obtained from the linear fits to the log Mversus logHeffdata shown in (c).\nwhich the linear AN plot isotherm passes through the origin marks the T c. To begin with, we\nascertain if the values of critical exponents βandγ, theoretically predicted for any universality\nclass, make the AN plot isotherms at temperatures close to T clinear. For example, in Fig. 5 (b)\nit is evident that the mean-field (MF) critical exponent values β= 0.5 andγ= 1 do not yield\nlinear AN isotherm in the critical region. The data presented in Fig. S1 of the supplementary\nmaterial demonstrate that, like the MF critical exponent values, none of the universality class\n13critical exponent choices: β= 0.365,γ= 1.386 for three-dimensional (3D) Heisenberg; β= 0.345,\nγ= 1.316 for 3D-XY; β= 0.325,γ= 1.241 for 3D Ising and β= 0.25,γ= 1.0 for mean-field\ntricritical, results in the linear AN plot isotherms.\nFurthermore, regardless of the choice of the critical exponents βandγ, extrapolation of the\nhigh-field portions of the AN isotherms to H= 0does not give any intercept on the M1/βaxis.\nNo intercept on the ordinate axis implies no spontaneous magnetization and hence absence of\nlong-range FM order. This result prompted us to look for even the non-universal exponent values\nthat could lead to the desired linear AN plot isotherms.\nBy varying the values of βandγin Eq. (6), we succeeded in making the AN plot isotherms\nnearly straight over the field range 1.5 kOe /lessorsimilarH/lessorsimilar10 kOe with the critical exponent values,\nβ= 0.796,γ= 1.051. Note that the field range for the linear AN isotherms is much wider\nin the immediate vicinity of T c. From the modified Arrott plot (Fig. 6), it is evident that the\nextrapolation of the linear portions of the AN isotherms to H = 0 yields finite spontaneous\nmagnetization, Ms, (inverse susceptibility, χ−1) below (above) T cand bothMsandχ−1are zero\nat the transition temperature T c/similarequal220 K. This value of T cis very close to that deduced from\nthe ac susceptibility data. Using the critical exponent values, β= 0.796 and γ= 1.051, in the\nWidom scaling relation [42] δ= 1 +γ/β, we obtained the critical isotherm critical exponent δas\n2.32. To crosscheck this value, the relation (Eq. (5)) is used to fit the virgin M(H) isotherm at T c\n/similarequal220 K. Fig. 7(c) shows the critical M(H) isotherm plotted as the ln M versus ln H effplot. As\nexpected from Eq. (5), this log-log plot is linear with inverse slope δ= 2.35(5). This value of δis\nin excellent agreement with that obtained using the Widom scaling relation.\nDespite a close agreement between the value of δdirectly determined from the critical isotherm\nand that from the exponent values β= 0.796 and γ= 1.051 via the Widom scaling relation,\ntwo basic questions arise: (i) how does one reconcile with the non-universal critical exponent\nvalues, and (ii) what causes the deviations from the linear AN isotherms at sufficiently low fields.\nConsidering that the random magnetic anisotropy (RMA) model [43] yields nonlinear mean-field\nAN isotherms at low fields, it is worth investigating if the SES form given by Aharony and\nPytte [43] helps resolve the above issues. Note that nonlinear mean-field isotherms have been\nobserved in the past in a number of amorphous rare earth - transition metal systems and explained\n[44] in terms of the Aharony and Pytte model.\nNeglecting the critical fluctuations, Aharony and Pytte (AP) make the following predictions\nfor a (d,n) spin system with RMA. (I) The M(H) isotherms follow the relation H∼M1/δatTc\n1402 4 6 8 1.0051.0201.0351.0501.0650369121507142128355\n0 Oe(a)(\nb)M (t)/M(0)t\n (103 sec)x = 0.45(1)30 K M2 (emu / g)2H\n/M (103 Oe-g / emu)5 K1\n50 KFIG. 8. (a) M2versus (H/M) plots in the low (5-30 K) temperature region demonstrate that no spon-\ntaneous magnetization, M s, exists at such temperatures. (b) Time evolution of the ZFC magnetization,\nM(t), at 150 K measured in a static magnetic field of 50 Oe switched on with a time delay of 100 s. Note\nthat M(t) is normalized to its value at the time when the field was applied. The solid red curve through\nthe data points (open circles) represents the fit to the stretched exponential function.\nandH∼M1/δ1at all temperatures below Tc, with the exponents δandδ1given by\nδ=10−d\n6−d, δ 1=8−d\n4−d(7)\nFor a three-dimensional ( d= 3) system, Eq. (7) gives δ= 7/3 = 2.33 and δ1= 5.0. While\nthe theoretical estimate δ= 2.33 is in perfect agreement with the presently determined value δ\n= 2.35(5), the observed δ1values do not reproduce the theoretically-predicted abrupt jump in δ1\nat T/similarequalTcto the value δ1= 5.0, which remains constant for T < Tc(as evidenced in Fig. 7(d)).\nThis discrepancy between theory and experiment can be traced back to a total neglect of critical\nfluctuationsoftheorderparameter, leadingtoEq.(7). (II)AccordingtotheAPmodel, thescaling\n15equation of state has the following form [43] for a d= 3,n= 3 system with random uniaxial and\ncubic anisotropies\nM1/β=/bracketleftBigg/parenleftbiggH\nM/parenrightbigg1/γ\n−a(T−Tc)\nTc/bracketrightBigg/slashBigg\nb/bracketleftBigg\n1 +c/parenleftbiggH\nM+ 4κ M2/parenrightbigg−1/2/bracketrightBigg\n(8)\nwhereβ= 0.5,γ= 1.0,c∼(D/J)2,Dandκare a measure of random uniaxial and cubic\nanisotropies, respectively, and Jsignifies the exchange interaction. Eq. (8) permits a clear distinc-\ntion between the cases: (i) κ= 0 when only the random uniaxial anisotropy exists, and (ii) κ/negationslash=\n0 when, in addition to the random uniaxial anisotropy, cubic anisotropy is present. The low-field\nMT(H)data, plotted in the form of the Arrott plot (i.e., M1/βvs.(H/M )1/γplot with mean-field\nexponentsβ= 0.5andγ= 1) isotherms, and the theoretical fits based on the AP SES, Eq. (8), for\nthe casesκ= 0 (dashed curves) and κ/negationslash=0 (continuous curves), are displayed in Fig. 7 (a). Fig. 7\n(b) provides an enlarged view of the data and fits at fields not very far from zero. Evidently, the\nAP SES correctly reproduces the observed change in the curvature of the Arrott plot isotherms\nfrom concave-downward to concave-upward at T c. Furthermore, superiority of the κ/negationslash=0 fits over\nthose corresponding to κ= 0, particularly at fields close to zero, asserts that the inclusion of cubic\nanisotropy is necessary. Consequently, in conformity with the observed behavior, an extrapolation\ntoH= 0 gives finite intercepts on the ordinate (finite M s) for T<Tcand on the abscissa (finite\nχ−1) for T>Tc; both M sandχ−1go to zero at T = T c. However, due to sizable gradients of the\nAP isotherms at very low fields ( H→0) for temperatures on either side of T c, the intercepts on\nthe ordinate and abscissa depend on the range of Hchosen for extrapolation. Thus, the M s(T)\nandχ−1(T), so obtained, are not accurate enough for the determination of critical exponents β\nandγ. In sharp contrast, the ‘zero-field’ ac susceptibility (ACS) does not suffer from such extrap-\nolation errors. The value γ= 1.261(5) of the susceptibility critical exponent, determined from\nthe ACS data, is thus reliable. The deviation of this value of γfrom the mean-field value, γ=\n1.0, underscores the importance of critical fluctuations (neglected in the AP RMA model). Thus,\nthe RMA model, due to Aharony and Pytte [43], captures the main physical essence in that the\ncurvature in the mean-field Arrott isotherms at low fields has its origin in the random uniaxial\nand cubic anisotropies, but fails to yield correct values for the critical exponents βandγ.\nThe evidence presented so far asserts that (i) at temperatures in range 180 K ≤T≤Tc= 220.5\nK, RMA is too weak to destroy long-range FM order, and (ii) the transition to a cluster spin glass\n(CSG) state occurs at T g= 7.7 K. It remains to be verified if, in the intermediate temperature\n16range, a mixed state is formed in which long-range FM order coexists with the CSG order. In\nthis connection, Fig. 8(a) makes it obvious that, in the low temperature region (T /lessorsimilar30 K), the\nArrott (M2versus H/M) plots, when extrapolated to H = 0, do not yield any intercept on the\nordinate axis regardless of the field-range chosen for extrapolation. It immediately follows that\nno spontaneous magnetization, and hence no long-range FM order, exists at these temperatures.\nConcurrently, in the same temperature range T /lessorsimilar30 K, a steep rise in RMA (reflected in a sharp\nincrease in the coercive field, as noticed in the inset of Fig. 3) occurs. Within the framework\nof the RMA models [43, 45], sizable magnitude of RMA at T /lessorsimilar30 K can account for both the\nbreakdown of long-range FM order [43] and the freezing of spin clusters in random orientations\nat T = T g, leading to the formation of cluster (correlated) spin glass state (with a finite FM\ncorrelation length [45]) at temperatures below T g.\nWith a view to gain more insight into the nature of magnetism at temperatures intermediate\nbetween the spin glass transition temperature T g/similarequal8 K and T c/similarequal220.5 K, the time (t) evolution\nof the ZFC magnetization, MZFC(t), at 150 K was measured in a static magnetic field of 50 Oe\nswitched on after a fixed waiting time, tw, ranging between 102and104s. TheMZFC(t) data, so\nobtained, were fitted to the stretched exponential function given by expression\nM(t) =M0−Mrexp(−(t/tr)x) (9)\nwhereM0represents the FM component while Mrandtrare the spin-glass magnetization\ncomponent and the characteristic SG relaxation time, respectively [46]. The value of exponent x\nreflects the type of energy barriers present. For instance, x= 1 describes the relaxation behavior\nof a FM system with a single anisotropy energy barrier separating the ZFC and FC states [47].\nx/similarequal0.5, on the other hand, is associated with the stretched exponential relaxation, caused by\nhierarchical energy barriers in spin glasses [31]. From the fit (solid red curve through the data\npoints), based on the above expression (Eq. (9)), to the M(t)/M(0)data (Fig. 8(b)), we obtain\nx= 0.45(1),M0/M(0)= 1.083(1), Mr/M(0)= 0.086(1) and tr= 2252 sec, where M(0) = 1.0348\nis magnetization value, recorded at the time twwhen the field was applied. The determined\nvalue ofxis in good agreement with those generally reported for spin glass systems [31]. This\nresult provides an evidence for the coexistence of a glassy (a cluster spin glass-like) phase with\nferromagnetic order at intermediate temperatures. Note that, within the uncertainty limits, the\nabove-mentioned parameter values seem to be insensitive to the choice of twpresumably because\nMrconstitutes a tiny fraction of the total magnetization at the measurement temperature of 150\n17FIG. 9. Schematic showing the two interchangeable spin state configurations involving correlated hopping\noft2gandegelectrons between the HS and LS states in the ground state and HS and IS states in the\nexcited state.\nK, which lies well above T g/similarequal8 K.\nFinally, an attempt is made to ascertain how the present results can be understood in terms\nof the HS/IS/LS spin states of Co3+ions in the LaSrCoO 4compound. At first, we recall that\nthe single-ion magneto-crystalline anisotropy (MCA) has its origin in the combined action of the\ncrystal-field interaction (CFI) and spin-orbit coupling (SOC). Since the LS state is associated with\nS = 0, the LS Co3+ions, neither in isolation nor as neighbors, contribute to spin magnetic moment\nand MCA (due to the lack of SOC). However, in the case of LS Co3+ions with HS neighbors,\nthe spin-state configurations, fluctuating [11, 12] between the Co3+HS - O2−- Co3+LS - O2−\n- Co3+HS and Co3+IS - O2−- Co3+HS - O2−- Co3+IS configurations (refer to the schematic\ndiagram shown in Fig. 9), can lead to FM superexchange interaction between the spins of Co3+\nHS ions. This process involves correlated hopping [11, 12] of t2gandegelectrons between Co3+\nLS and its Co3+HS neighbors without any charge transfer as shown in Fig. 9 and results in a\nfinite effective S at the LS Co3+ion site. Finite S, in turn, enables SOC and ensures a significant\ncontribution to spin magnetic moment. In conformity with the observed steep rise in MCA as\nthe temperature drops below ∼80 K, MCA increases with decreasing temperature because LS\nCo3+ions with SOC and much stronger CFI (and hence MCA) increase [11, 12, 48] in number\nat the expense of HS Co3+ions. In sharp contrast, arguments based on the mixed HS-IS or IS\n18ground state (similar to those presented above), at best, lead to a much slower increase in MCA\nwith lowering temperature because of their low CFI relative to LS. A competition between the\nabove-mentioned FM interaction and the AFM interaction between the spins of neighboring Co3+\nHS ions, caused by the Co3+HS - O2−- Co3+HS superexchange, give rise to a spin glass (SG)\nground state.\nIV. SUMMARY AND CONCLUSION\nIn the LaSrCoO 4compound, as functions of temperature, ‘zero-field-cooled’ (ZFC) dc suscep-\ntibility,χdc=M/H, at low fields (H ≤100 Oe), and ac susceptibility, χac, (ACS), at different\nfrequencies of the driving ac field of rms amplitude hac= 3 Oe, exhibit peaks at around 200 K\nand 8 K. While the low-temperature peak in χac(T) shifts to higher temperatures with increasing\nfrequency, theinflection-pointin χac(T)(orequivalently, adipin dχac/dT)at220.5Kisfrequency-\nindependent. Note that the dip in dχdc/dTalso occurs at exactly the same temperature. This\nobservation provides strong experimental evidence for the existence of a thermodynamic PM -\nFM phase transition at T c= 220.5 K. The frequency-induced shift in the low-temperature peak is\ndescribed well by the critical slowing down model for a cluster spin glass (CSG). According to this\nmodel, the transition to the CSG state takes place at T g= 7.7 K in the zero-frequency limit. The\nAharony-Pytte (AP) model for a ( d = 3, n = 3 ) Heisenberg spin system with random magnetic\nanisotropy (RMA) yields the value ( δ= 7/3) of the critical exponent for the critical isotherm at\nTc= 220.5 K, which agrees closely with the observed value of δ= 2.35(5). Moreover, the AP\nscaling equation of state correctly describes the observed crossover from the concave-downward to\nconcave-upwards curvature in the low-field Arrott plot isotherms, as the temperature is increased\nthrough T c. These findings clearly establish that the PM-FM transition is basically driven by\nRMA.\nFor temperatures below ≈30 K, large enough RMA destroys long-range FM order by break-\ning up the infinite FM network into FM clusters of finite size and leads to the formation of a\nCSG state at temperatures T /lessorsimilar8 K by promoting freezing of finite FM clusters in random ori-\nentations. Increasing strength of the single-ion magnetocrystalline anisotropy (and hence RMA)\nwith decreasing temperature reflects an increase in the number of low-spin (LS) Co3+ions at the\nexpense of that of high-spin (HS) Co3+ions. At intermediate temperatures (30 K /lessorsimilarT/lessorsimilar180\nK), spin dynamics has contributions from the infinite FM network (fast relaxation governed by\n19a single anisotropy energy barrier) and finite FM clusters (extremely slow stretched exponential\nrelaxation due to hierarchical energy barriers).\nACKNOWLEDGEMENTS\nThe authors acknowledge Kranti Sharma for AC susceptibility measurements. S.S.M. acknowl-\nedges the financial support from SERB, India, in the form of the national post doctoral fellowship\n(NPDF) award (PDF/2021/002137). R.S. acknowledge the financial support provided by the Min-\nistry of Science and Technology in Taiwan under project numbers MOST-111-2124-M-001-009 &\nMOST-110-2112-M-001-065-MY3 & AS- iMATE-111-12.\n[1] C. 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Tripathy,2F. Rahman,1\nR. J. Choudhary,2R. Sankar,3A.K. Sinha,4,§S. N. Kaul,5,¶and D. K. Shukla2,∗∗\n1Department of Physics, Aligarh Muslim University, Aligarh 202002, India\n2UGC-DAE Consortium for Scientific Research, Indore 452001, India\n3Institute of Physics, Academia Sinica, Taipei 11529, Taiwan\n4Indus Synchrotrons Utilization Division,\nRaja Ramanna Center for Advanced Technology, Indore 452013, India\n5School of Physics, University of Hyderabad, Hyderabad 500046, India\n∗abdul.ahad@ntu.edu.sg; Present Address: School of Physical and Mathematical Sciences, Nanyang Tech-\nnological University, Singapore\n†Present Address: RIKEN Center for Emergent Matter Science, Saitama, Japan\n‡Present Address: National Institute of Technology Hazratbal Srinagar J & K, India\n§Present address: Department of Physics, School of Engineering, University of Petroleum and energy\nstudies, Dehradun, India\n¶sn.kaul@uohyd.ac.in\n∗∗dkshukla@csr.res.in\n1arXiv:2002.00135v3  [cond-mat.mtrl-sci]  16 May 2023I. THE MODIFIED ARROTT (M1/βVS. (H/M)1/γ) PLOTS FOR DIFFERENT\nUNIVERSALITY CLASSES:\nUsing Arrott-Noakes scaling equation of state [ ?], the data presented in Fig. S1 (a-d)\nclearly demonstrate that none of the universality class critical exponent choices: (a) β=\n0.365,γ= 1.386 for the three-dimensional (3D) Heisenberg, (b) β= 0.345,γ= 1.316 for\n3D-XY, (c) β= 0.325,γ= 1.241 for 3D Ising, and (d) β= 0.25,γ= 1.0 for mean-field\ntricritical, yields linear AN plot isotherms.\n0.00.51.01.52.02.502460\n123402460\n1234567024680\n102030405060036912   3D Heisenbergβ  = 0.365γ\n = 1.386β = 0.345γ\n = 1.316 \n 3\nD XY(\nc)MFTβ  = 0.325γ\n = 1.24 \n \n  3D Ising(a) 180K 190K  \n200K 210K \n215K 220K \n225K 230K \n240K 250K \n260K(\nd)β = 0.25γ\n = 1.0 M1/β (emu / g)1/β(\nHeff/M)1/γ (103 Oe-g / emu)1/γ(b)\nFIG. S1. The modified Arrott ( M1/βvs.(H/M )1/γ) plots for the choices of the critical exponents\nβandγgiven by different universality classes.\n2II. ISOTHERMAL MAGNETIC ENTROPY CHANGE ∆SM:\nThe isothermal magnetic entropy change, ∆S M, as a function of temperature at fixed mag-\nnetic fields was computed from the magnetization, M(T,H), data within the temperature\nranges in the vicinity of T cand Tg, using the Maxwell relation [ ?], ∆SM=/integraltextb\na/parenleftbigdM\ndT/parenrightbig\nHdH.\nFig. S2 displays ∆S M(T) at different fields. Regardless of the magnitude of H, a\nwell-defined dip in ∆S M(T), characteristic of a second-order (continuous) ferromagnetic-\nparamagnetic phase transition, occurs at Tc/similarequal220 K, as is evident from the sub-figure (a).\nThis value of Tcis in perfect agreement with that determined from the scaling equation of\nstate analysis. In addition, in the low temperature region (see Fig. S2 (b)), a sign change\nin ∆SMoccurs at the spin glass transition temperature ∼8 K.\n1802 002 202 40-1.2-0.8-0.40.08\n1 21 6-15-10-501 kOe(\nb)(a)T\n (K) \n 1\n00 OeT\nc=220 KΔSM (10-3 x J/Kg-K) \n T\n (K)Tg~8.8 K\nFIG. S2. ∆S Mas a function of temperature at different fields in the temperature ranges near (a)\nTcand (b)Tg.\n3"    },    {        "title": "2002.03940v1.On_the_anisotropies_of_magnetization_and_electronic_transport_of_magnetic_Weyl_semimetal_Co3Sn2S2.pdf",        "content": "1 \n On the anisotropies of magnetization and electronic \ntransport of magnetic Weyl semimetal Co 3Sn2S2 \n \nJianlei Shen ,1,2 Qingqi Zeng ,1,2 Shen Zhang ,1,2 Wei Tong ,3 Langsheng Ling ,3 \nChuanying Xi ,3 Zhaosheng Wang ,3 Enke Liu ,1,4* Wenhong Wang ,1,4 Guangheng Wu,1  \nand Baogen Shen1 \n \n1. State Key Laboratory for Magnetism, Institute of Physics, Chinese Academy of \nSciences, Beijing 100190, China  \n2. University of Chinese  Academy of Sciences, Beijing 100049, China  \n3. Anhui Province Key Laboratory of Condensed Matter Physics at Extreme \nConditions, High Magnetic Field Laboratory of the Chinese Academy of Sciences, \nHefei  230031, China   \n4. Songshan Lake Materials Laboratory, D ongguan, Guangdong 523808, China  \n  \n                                                             \n  Corresponding author.  \nE-mail addresses: ekliu@iphy.ac.cn  \n 2 \n Abstract  \n \nCo3Sn2S2, a quasi -two-dimensional system with kagome lattice, has been found \nas a magnetic Weyl semimetal recently. In this work, the anisotropies of \nmagnetization  and transport properties of Co 3Sn2S2 were investig ated. The high field \nmeasurements reveal a giant  magnetocrystalline anisotropy with an out -of-plane \nsaturation field of 0. 9 kOe and an in -plane saturation field of 2 30 kOe at 2 K , showing \na magnetocrystalline anisotropy coefficient  Ku up to 8.3 ×  105 J m-3, which indicates \nthat it is extremely difficult to align the small moment of 0.29 μ B/Co on the kagome \nlattice from c axis to ab plane . The out -of-plane angular dependences of Hall \nconductivity further reveal strong anisotropies in Berry cur vature and ferromagnetism, \nand the vector directions of both are always  parallel with each other. For in -plane \nsituation,  the longitudinal and transverse measurements for both  I // a and I  a cases \nshow that the transport on the kagome lattice is isotropic . These results provide \nessential understanding on the magnetization and transport behaviors for the magnetic \nWeyl semimetal Co 3Sn2S2. \n  3 \n Magnetic Weyl semimetals that can combine the topology and magnetism are \nhighly desired.1,2 Till now, the expe rimental realization of magnetic Weyl semimetals \nis still on the way.  Very r ecently,  the Shandite compound Co3Sn2S2 was predicted as a \nmagnetic Weyl semimetal, showing an intrinsic giant anomalous Hall conductivity \nand anomalous Hall angle o wing to the topologically enhanced Berry curvature .3-5 \nBoth angle -resolved photoemission spectroscop y (ARPES ) and scanning tunneling \nspectroscopy  (STM ) were performed to detect the topological state and surface Fermi \narcs, which spectroscopically  confirmed the predict ion of magnetic Weyl semimetal \nphase in this system.6,7 Nowadays, the anomalous Nernst effect,8 negative orbital \nmagnetism,9 exchange bias,10 electronic correlations,11 spin-transfer torque,12,13 even \ntopological catalysis14 have been found  subsequently. Moreover, due to its long -range \nout-of-plane ferromagnetic order and topological band structure, this semimetal  \nbecomes an ideal candidate for developing a quantum anomalous Hall state15 and an \nexcellent platform for comprehensive studies on  topological electronic behaviours.  \nThe ternary Shandite compounds with the general formula  T3M2X2 , where T = Ni, \nCo, Rh or Pd; M = Sn, In, or Pb; and X = S or Se, have been investigated extensively \nin recent decades.16-22 Among of them, Co 3Sn2S2 is the only compound showing \nferromagnetism . The crystal structure  of Co 3Sn2S2 (space group R -3m) possesses a \nquasi -2D Co -Sn kagome layers sandwiched between S atoms, and stacks in ABC \nfashion along the c-axis.23,24 The magnetic moment of Co atoms are fixe d on kagome \nlayer in ab-plane and along the c-axis, as shown in  Fig. 1(a) . Although the basic \nproperties including the magnetic anisotropy in this system  have been addressed ,25 the \nin-depth  studies on anisotropies of both magnetization  and transport behaviors  remain  \nrare. In this work, we systematically investigated the an isotropies of magnetic  and \ntransport properties of Co 3Sn2S2 single crystal. The experimental results show that the \nmagnetization presents a giant  magnetocrystalline anisotropy,  and angular (θ)  \ndependence of Hall conductivity with different magnetic fields indicates strong \nmagnetization and Berry curvature anisotropies, and the vector directions of both are \nalways  parallel with each other. Meanwhile  the electronic transport  on the kagome \nlattice  is isotropic.  4 \n The single crystals of Co 3Sn2S2 can be grown by slowly cooling  the melts with \ncongruent composition  (Co : S n : S = 3 : 2 : 2).3 The room temperature X -ray \ndiffraction (XRD) with Cu -Kα radiation shows only the (000 l) Bragg peaks, which \nindicates that the measur ed plane surface  of the crystal  is ab-plane. The inset shows a \nnaturally dissociated flake single crystal with a size of about 0.5× 3× 5 mm3, as shown \nin Fig. 1(b).  Magnetic properties  were measured by supercondu cting quantum \ninterference device (SQUID) magnetometer and  the Cell -5 Magnet of the High \nMagnetic Field Laboratory, Chinese Academy of Sciences (CHMFL). E lectrical \ntransport  was measured by  the physical property measurement system (PPMS).  \nFigures 2(a) and (b) show the t emperature dependence of magnetization with \nZFC and FCC for B // c and B // ab at B = 200 Oe  (left ordinate). A sharp magnetic \ntransition can be observed near TC ≈ 175 K. Below TC ≈ 175 K, The magnetization  for \nB // c is abo ut one-hundred  times larger than that of B // ab, which indicates the strong \nmagnetocrystalline  anisotropy in Co 3Sn2S2. The 1/χ (T) curves from 200 to 300 K can \nbe fitted by a modified Curie –Weiss law χ ( T) = C/(T-θp), where C is the Curie \nconstant and θp is the Weiss temperature. The fitting over the temperature ranging \nfrom 200 to 300 K are shown as solid yellow curves in Figs. 2(a) and (b)  (right \nordinate). The meff is calculated as 1.05 μB/Co for B // c and 1.14 μB/Co for B // ab, \nwhich are consistent with reported  values.26 The positive θp for two directions \nindicates ferromagnetic interaction in Co3Sn2S2. \n Figures 2(c)  shows the f ield dependence of magnetization at 2 K for both B // c \nand B // ab. For B // c, the magnetization quickly reaches the saturation at about 0. 9 \nkOe and the saturation magnetization Ms is 10 emu/g , while for B // ab the \nmagnetization exhibits a s aturation until 2 30 kOe, which is about three orders of \nmagnitude higher than that of B // c. These results further indicate the quite strong \nmagnetocrystalline anisotropy in Co3Sn2S2. Furthermore, the magnetocrystalline \nanisotropy coefficient ( Ku) can  be obtained by Ku = μ0HkMs/2, where  μ0 is the vacuum \npermeability and Hk is the anisotropy field  defined as the critical field above which \nthe difference in magnetization between the two magnetic field directions ( B // ab and \nB // c) becomes smaller than 2%.27 The value of Ku is about 8. 3× 105 J m-3 at 2 K , 5 \n which is larger than those of some magnetic quasi -two-dimensional materials, such as \nFe3Sn2,27 Fe3-xGeTe 2,28 CrBr 3 and CrI 3,29 as shown in Table 1. In order to observe the \nmagnetocrystalline  anisotropy  more clearly, we measured the angular dependent \nmagnetization from out -of-plane to in -plane at 2 K in B = 500 Oe, as shown in Fig. \n2(d).  \nIn order to further characterize the magnetocrystalline anisotropy, the anisotropic  \nmagnetocaloric effect was measure d (also see supplementary materials Fig. S1 ). A set \nof magnetic isotherms M( H) curves of B // c and B // ab were measured from 160 to \n190 K. The M( H) curves at 175 K are given in Fig. 2(e)  for clarity. Based on these \nmeasurements, the magnetic entropy changes ( -∆Sm) for B // c and B // ab are \ncalculated using Maxwell relation. We show -∆Sm at 50 kOe, as shown in Fig. 2(f) . A \nmaximum -∆Sm value of 1.13 J kg-1 K-1 is obtained for B // c at 175  K, which is \nconsistent with the previous report s.30,31 For B // ab, it is only 0.38 J kg-1 K-1, which \nindicates that there is a strong anisotropy in magnetocaloric effect. The anisotropic \nmagnetocaloric effect further demonstrates the large  magnetocrystalline anisotropy  in \nCo3Sn2S2. \nIn order t o probe the relation of  anisotropies between the Berry curvature and \nferromagnetism, the angular (θ)  dependence s of Hall conductivity\nyx with different \nmagnetic fields rotating around y-axis in ac-plane have been measured and shown in \nFig. 3(a). The configuration of current and magnetic field is shown in  Fig. 3(b). Wang \net al. reported\nyx changed by only 1% when the magnetic field is tilted  away from the \nc-axis up to ± 30° .5 In our work, the  \nyx remains unchanged in a wide angle range  \nindeed . In the case of 10 kOe, the stable range is as wide as more than 170° .  \nTo understand this robust behaviour against the rotation of magnetic field, we \nextracted the in-plane ( Ba) and out -of-plane ( Bc) components of the applied mag netic \nfield, as well as\nyx in different magnetic field at θ  = 5° (we take θ  = 5°  as an example \nfor the clarity), as shown in Figs. 3(c) and (d). It can be clearly seen that\nyx increases \nfirst and then decreases, reaching its maximum at 10 kOe.  For B = 10 kOe , Ba and Bc 6 \n at θ = 5°  are 9.96 kOe and 0.87 kOe, respectively. At this moment, Ba is much  lower \nthan the in -plane saturation field 2 30 kOe, while Bc is basically equivalent to \nout-of-plane saturation field. So the magnetic moment is basically along the c-axis \nand saturated. The maximum value of\nyx at θ = 5° is ~ 1170 Ω-1 cm-1, which is \nconsistent with the result of calculation based on the Berry curvature along  c axis.3 \nFor B < 10 kOe, the out -of-plane field cannot make the magnetic moment saturated \nand the single domain cannot form, so that\nyx is less than that of 10 kOe. Although  Bc \nis higher than out -of-plane saturation field, the increase of Ba causes the magnetic \nmoment to deviate away from the c-axis and the decrease of\nyx at the sam e time. The \nvalue of\nyx depends on the strength of Berry curvature along  c axis.3 The decrease of\nyx\nindicates that the direction of Berry curvature  tilts from the  c axis and follows the \ndirection of magnetic moment . All these results suggest that  Berry curvature  has \nstrong anisotropy as magnetic moment  and the vector directions of both are keep the \nsame and locked together.  In additi on, the φ M between magnetic moment and easy \nc-axis can be approximately estimated as φ M = B a × 90° /2 30 kOe = Bcosθ(90°/2 30 \nkOe) during the field rotation, accordingly,  all the magnetic moments are aligned \ncompletely in ab plane  for B = 230 kOe applied in ab plane, as shown in Fig. 3(d).   \nFigure 4(a) shows the temperature dependence s of the longitudinal resistivity\nxx  \nin zero field and 90 kOe for both I // a and I  a. The configuration of current and \nmagnetic field is shown in the inset in  Fig. 4(a).  It can be clearly seen that\nxx\ncompletely coincide in the whole temperature range for two orientations  at zero field \nand 90 kOe, respectively, which indicates that it is isotropic in transport for two \norientations. The field -dependence of magnetoresistance also shows  the consistent \nbehavior between two orientations, as shown in Fig. 4(b).  \nFigure 4(c) shows the dependence s of Hall resistivity\nxy on temperature for I // a \nand I  a under B = 1 kOe. With the decrease of temperature, the largest\nxy  for both \norientations appear at 150 K with the same value. T he dependence of the  Hall 7 \n resistivity\nxy on magnetic field at 2 and 150 K  were measured and shown in Figs. 4(d) \nand (e), respectively. The\nxy of both orientations  is consistent with each other for 2 \nand 150 K. The notable n onlinear field dependence of the\nxy can be clearly observed \nand further indicates the coexistence of hole and electron carriers at 2 K for both \norientations, while the single hole carriers are observed at 150 K for both orientations, \nwhich are in good agreement with previous report.3 The\nA\nyx at 2 and 150 K  are then \nobtained a t the zero field . They  are 0.98 and 44.6 μΩ cm for I // a, and 1.08 and 45.6 \nμΩ cm  I a. The dependence of\nyx on magnetic field at 2 and 150 K for both \norientations  are shown in  Fig. 4(f). The \nA\nyx at 2 K reaches up 1230 and 1300 Ω-1 cm-1 \nfor I // a and I  a, respectively , which are attributed to the large Berry curvature  \ninduced by Weyl nodes and gapped nodal line near Fermi energy . In this system, there \nare three pairs of Weyl nodes  and they are all in the same energy position, only 60 \nmeV above the Fermi level .3,6,7 This produces the coexistence of the Fermi surfaces of \nholes and electrons . Moreover, the maximum value of the anomalous Hall angles (\nA\nyx\n/\nxx) are 18.5% and 19% at 150 K, respectively , as shown Table 2.  It is evidenced \nthat both anomalous Hall conductivity  and anomalous Hall angle  are isotropic on the \nkagome lattice.  \nAccording to band structure calculations and ARPES confir mation,3,6 the \nrelatively small Fermi surfaces of holes and electrons indicate  the coexistence of holes \nand electrons, which are consistent with t he notable nonlinear field dependence of the  \nxy\n. Furthermore, we extracted the carrier densities  and mobilities at 2 K using the \nsemiclassical two -band model  (see supplementary materials Fig. S2 ), as shown in \nTable 2. The carrier densities and mobilities for I // a and I  a are basically the same \nto each other , which indicates that the transport is isotropic on the kagome lattice  in \nthis magnetic Weyl semimetal . These results further reveal  a near compensation of \ncharge carriers  of electrons and holes . It is considered that a B2-dependence of MR \ndescribed by power law MR = A Bα is the signature  of a compensated metal.32 For 8 \n in-plane magnetic field cases,3,25 α can be close to 2 in Co3Sn2S2. However, in the \ncase of  I // a and I  a under  B // c (out-of-plane field) , the α is only about 1.6, as \nshown in Fig. 4(b). One possible reason for this deviation  may be the negative MR \ndue to the spin -dependent scattering in ferromagnetic materials, which will balance \nout a part of the positive MR, resulting in a reduced exponent  from  2. Owing to the \nstrong magnetocrystalline anisotropy , in contrast, this negative MR effect shows weak \ninfluence on the total MR when field is applied in -plane.  Therefore, a near \ncompensation  behavior can be expected in Co3Sn2S2. \nIn conclusion, we investigated the anisotropies of magnetic and transport \nproperties of magnetic Weyl semimetal Co 3Sn2S2. This semimetal exhibits strong \nmagnetocrystalline  anisotropy with out-of-plane saturation field of 0. 9 kOe and \nin-plane saturation field of 2 30 kOe, and the value of Ku up to 8. 3× 105 J m-3. In \naddition, angular (θ)  dependence of Hall conductivity\nxy with different magnetic field \nindicates strong magnetic and Berry curvature anisotropies and the vector directions \nof both are locked together . For in -plane situation,  the longitudinal and transverse \nmeasurements for both  I // a and I  a cases show that the transport on the kagome \nlattice is isotropic . These results provide essential understanding on the magnetization \nand transpo rt behaviors for the magnetic Weyl semimetal Co 3Sn2S2, which will \nbenefit  the potential applications of the magnetic topological quantum materials.  \nSee supplementary material for  magnetization  curves and magnetic entropy \nchange  measured  from  160 to 190  K for B // c and B // ab, respectively  (Fig. S1) . The \nfitting of longitudinal and transverse  conductivities  at 2 K based on two -band model  \nto obtain carrier densities  and mobilities  of I // a and I  a (Fig. S2 and Table S1)  . \nThis work was supported by National Natural Science Foundation of China ( Nos. \n51722106 and 11974394 ), Beijing Natural Science Foundation  (No. Z19J00024), \nUsers with Excellence Program of Hefei Science Center CAS ( No. 2019HSC -UE009), \nand Fujian Institute of Innovation, Chinese Academ y of Sciences . A portion of this \nwork was performed on the Steady High Magnetic Field Facilities, High Magnetic \nField Laboratory, Chinese Academy of Sciences.  9 \n References  \n1B. Yan and C. Felser, Annu. Rev. Condens. Matter Phys. 8, 337 (2017)  \n2Z. P. Guo, P . C. Lu, T . Chen, J . F.Wu, J . Sun, and D . Y. Xing,  Sci. China Phys. Mech. \nAstron. 61, 038211  (2018).   \n3E. Liu, Y . Sun, N. Kumar, L. Muchler, A. Sun, L. Jiao, S. Y . Yang, D. Liu, A. Liang, \nQ. Xu, J. Kroder, V . Suss, H. 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(a) Crystal structure  of Co 3Sn2S2 and kagome layer composed of Co atoms . (b) \nXRD pattern of Co 3Sn2S2 single crystal at room temperature. The inset shows a \nphotograph of Co 3Sn2S2 single crystals.  \n \n13 \n  \nFig. 2. (a) and (b) Temperature dependence of magnetization with ZFC and FCC \nmeasurements for B // c and B // ab at B = 200 Oe ( left) and 1/χ vs T behavior ( right ) \nfor B // c and B // ab, respectively. (c) Field dependence of magnetization at 2 K for B \n// c and B // ab, respectively.  The inset is the low field part  for B // c. (d) Angular \ndependent magnetization from out -of-plane to in -plane at 2 K measured in B = 500 \nOe. (e) Field dependence of magnetization at 175 K for B // c and B // ab, respectively. \n(d) Temperature  dependence of magnetic entropy change -∆Sm at B = 50 kOe for B // \nc and B // ab, respectively.   \n  \n14 \n  \nFig. 3. (a) Angular (θ)  dependence of Hall conductivity\nyx with different magnetic \nfield rotating around y-axis in ac-plane at 2 K. Where θ is the angle between the \nmagnetic field and a axis. (b) The configuration of current and magnetic field.  (c)    \nHall conductivity\nyx with different magnetic field at θ = 0 -5° . (d) Components of \nmagnetic field in -plane ( Ba) and out -of-plane ( Bc), Hall conductivity\nyx and angle φM \nbetween magnetic moment and c-axis at θ = 5°.  \n15 \n  \nFig. 4 (a) Temperature dependence s of longitudinal resistivity\nxx at zero field and 90 \nkOe for I // a and I  a, respectively. (b) Magnetoresistance as a function of magnetic \nfield B at 2 and 150 K for I // a and I  a with B // c, respectively . The fitting of \npositive MR at 2 K based on MR = A Bα for I // a and I  a. (c) Hall resisti vities\nxy as \na function of temperature for I // a and I  a with B = 1 kOe, respectively. (d) and (e) \nHall resistivit ies\nxy as a function of magnetic field B at 2 and 150 K for I // a and I  \na with B // c, respectively. (f) Hall conductivit ies\nyx as a function of magnetic field B \nat 2 and 150 K for I // a and I  a with B // c, respectively.  \n  \n16 \n Table 1 . Magnetocrystalline anisotropy coefficient ( Ku) of some materials  \nSystems  Fe3Sn2 Fe3GeTe 2 CrI 3 CrBr 3 Co3Sn2S2 \nKu (105J m-3) 3.3 0.69 0.86 3.0 8.3 \n \n \nTable 2. Anomalous  Hall resistivity  (\nA\nyx ), carrier densit y of hole  (nh), carrier densit y \nof electron  (ne), mobilit y of holes  (μh), mobilit y of electron  (μe) of 2 K and maximal \nanomalous Hall angle  (\nA\nyx /\nxx) of Co3Sn2S2 for I // a and I  a . \nI \nA\nyx Ω-1 cm-1 nh \n1020 cm-3\n ne \n1020 cm-3\n μh \ncm2 V-1 s-1\n μe \ncm2 V-1 s-1\n \nA\nyx/\nxx  \nI // a 1230  1.31 1.14 845 943 18.5%  \nI  a 1300  1.24 1.2 870 918 19% \n "    },    {        "title": "2002.11517v1.Extreme_narrow_magnetic_domain_walls_in_U_ferromagnets__The_UCoGa_case.pdf",        "content": "1 \n Extreme narrow magnetic domain walls in U ferromagnets: The UCoGa case  \n \nPetr Opletal, Klára Uhlířová, Ivo Kalabis, Vladimír Sechovský and Jan Prokleška* \n \nCharles University, Faculty of Mathematics and Physics, Department of Condensed Matter Physics, \nKe Karlovu 5, 121 16 Prague 2, Czech Republic  \n \n \nAbstract  \nSurface magnetic domains of a UCoGa single crystal during magnetization/demagnetization processes \nin increasing/decreasing magnetic fi elds were investigated by means of  magnetic -force -microscopy  \n(MFM ) images at low temperatures. The observed domain structure is typical for a ferromagnet with \nstrong uniaxial anisotropy. The evolution of magnetic domains during cooling of the crystal below  TC \nhas also been manifested by MFM images. Analysis of the available data reveals that the high uniaxial \nmagnetocrystalline energy in combination with the relatively small ferromagnetic exchange \ninteraction in UCoGa gives rise to the formation of very nar row domain walls formed by the pairs of \nthe nearest U neighbor ions with antiparallel magnetic moments within the basal plane.  Since the very \nhigh anisotropy energy is a common feature of the majority of the uniaxial U ferromagnets, analogous \ndomain -wall properties are expected for all these materials.    \n \n \nKeywords: uranium intermetallics; magnetic force microscopy; magnetism; UTX; anisotropy  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n*corresponding author: prokles@mag.mff.cuni.cz   2 \n Introduction  \nMagnetocrystalline anisotropy (MA) is manifested by locking the macroscopic magnetic moment in a \ncertain direction of a crystal, the easy magnetization axis (easy axis). The strength of MA is \ncharacterized by a magnetic field (anisotropy field Ha) which sh ould be applied in the perpendicular \ndirection (hard axis) to rotate the moment from the easy axis to the hard axis. The anisotropy energy  Ea \nis equal to the energy difference between the magnetic moment directed along the hard and the easy \naxis.  \nThe key prerequisites of MA are the orbital moment, the spin -orbit (s -o) interaction and the \nanisotropic interactions of a magnetic ion with neighboring ions [1,2] . The s -o interaction, as a \nrelativistic effect, becomes stronger in heavier ions. Consequently, the MA dominates the magnetism \nin f-electron materials.   \nThe crystalline -electric -field (CEF) interaction is a mechanism of the MA in lanthanide \ncompounds wit h well -localized 4 f-electron magnetic moments [3–5]. The electron orbitals adopt \norientations in the crystal lattice that minimize the energy of the CEF interaction. In contrast to the 4 f-\norbitals, deeply buried in the core electron density of lanthanide ions, the U 5 f-electron wave functions \nare spatially extended. Consequently, they overlap with the 5 f-orbitals of the neighbor U ions (5 f-5f \noverlap) and with the orbitals of the valence electrons of other ligands which leads to hybridization of \nthe 5 f-electron states with the ligand valence -electron states (5 f-ligand hybridization)  [6].  \nThe 5 f-5f overlap allows for direct U -U exchange interaction (EI) whereas the 5 f-ligand \nhybridization mediates the indirect 5 f-ligand -5f exchange. These mechanisms cause that the 5 f wave \nfunction s may lose their atomic character, the 5 f-electrons become delocalized and the 5 f-moments \nnaturally reduced. Despite the 5 f-electron delocalization, the strong s -o coupling induces a \npredominantly orbital magnetic moment antiparallel to the spin moment in the spin -polarized 5 f-\nelectron energy bands [7]. This gives rise to strong MA even in U intermetallics with strongly \ndelocalized 5 f electrons. The itinerant 5 f-electron ferromagnet UNi 2 with a U mag netic moment of \nonly few hundredths  of a μB, which exhibits a very strong MA with Ha >> 35  T at 4.2 K, serves as an \nexcellent example [8–11].  \nA mu ch stronger MA than observed for lanthanide ions in an analogous crystal environment is \ninherent to U magnetism. The typical values of Ha of most U intermetallic compounds so far studied \nare of t he order of hundreds of teslas [12]. The interaction of the spatially e xtended U 5 f-orbitals with \nligands implies an essentially different mechanism of the MA, based on a two -ion (U -U) interaction. \nA relatively simple model of the two -ion interaction has be en formulated by Cooper et al. [13,14]  \nbased on the Coqblin -Schrieffer approach to the mixing of ionic f -states and cond uction -electron states \n[15]. The theory has been further extended so that each partially delocalized f-electron ion is coupled \nby the hybridization to the band electron sea leading to a hybridization -mediated anisotropic  two-ion \ninteraction causing MA  [16].  \nWhen cooled in zero magnetic field, the bulk of a ferromagnetic (FM) material decomposes into \nmutually antiparallel FM domains (with the exception of flux closure domains) in order to minimize \nthe magnetostatic energy. Neigh boring domains are separated by domain walls of a thickness \ndetermined by the balance between Ea and the exchange -interaction (EI) energy ( Eex). The EI favors a \nslow rotation of the magnetization while the MA prefers a sudden magnetization reversal.  In the 3d-\nelectron ferromagnets with a typically weak MA and strong EI ( Eex >> Ea) the domain -wall thickness 3 \n may be even hundreds of distance between nearest magnetic ions. On the other hand, the strong MA \n(Ea >> Eex) in f-electron ferromagnets may cause reduction of the domain wall width down to a few \ninteratomic distances [17,18] .  \nA number of U TX compounds with transition metals ( T) and p-electron metals ( X) crystallize in \nthe hexagonal ZrNiAl -type structure  [12] (space group P -62m, No. 189) , shown in FIG. 1a). The  \nstructure is characterized by stacking of U-T1 and T2-X basal -plane layers along the c-axis, where T1 \nand T2 are two different positions for the transition metal T. The U ions in the U -T1 basal plane are \narranged in a slightly distorted kagome lattice  (Fig. 1b )) in which each U ion has four U nearest \nneighbors at a shortest U -U distance. This arrangement leads to considerable 5 f-5f overlap and a 5f(U) -\nd(T) ligand hybridization which leads to a compre ssion of the 5 f-electron density towards the basal \nplane. This phenomenon assisted by the orbital -moment polarization results in a strong uniaxial MA \nwith the easy axis perpendicular to the basal plane, i.e. along the c-axis. This suit s the more general \nscenario of the easy axis oriented perpendicular to the nearest U -U bonding links in the lattice [19]. \nHuge Ha values of the order of hundreds of teslas have been estimated for the isostructural U TX \nferromagnets [20–22] from the intercepts of the easy - and hard -magnetization curves extrapolated to \nhigh magnetic field s. The rather low values of the Curie temperature ( Tc) of the order of tens of kelvins \n[12] indicate that Ea >> Eex , which leads to the occurrence of very narrow domain walls.  \n \n \nFIG. 1. a) Hexagonal ZrNiAl - type structure of U TX compounds. Two different position of the transition \nmetal T are shown as T1 and T2 . The c -direction is perpendicular to the plane, as a result U -T1 and T2-X \nplanes a re shown together . b) Schematic representation of magnetic U ions creating a distorted kagome lattice \nwith the non -distorted kagome lattice for comparison. The c -direction is perpendicular to the plane.  \nThe direct imaging and investigation of domain structures in U compounds have been done on \npolycrystalline samples at room temperature (RT) in materials with a high concentration of 3 d \ntransition metals [23,24]  with magnetic moments which are ferromagnetically coupled by strong EI \nresulting in  high TC-values (>> RT).  The only report on the single crystalline material containing \nuranium has been recently published [25] on the UMn 2Ge2. Although this material has comparable \n[26] anisotropy (roughly ¼) in compari son to the compound under study [27], the formation of \nmagnetic domains is determined by the ordering of Mn moment at higher (above RT) temperatures.  \nIn this paper, we present the direct low -temperature investigation of a domain structure in a U 5 f-\nelectron moment ferromagnet with TC << RT. UCoGa, a member of the family of U TX compounds \nwith ZrNiAl -type of struc ture, has been selected as a typical U ferromagnet with huge uniaxial MA \n[20,27 –30] having the easy axis along the c -axis of the hexagonal structure. First, the UCoGa single \na) \n b) 4 \n crystal has been characterized by magnetization measurements in magnetic fields applied along the \nmain crystallographic axes, a and c. The domain -structure imaging has been accomplished by using a \nlow-temperature atomic force microscope - attoAFM/MFM Ixs (attocube). Analysis of the available \ndata has led to the conclusion that the huge uniaxial MA, together with the relatively wea k EI in \nUCoGa and possibly in most other U ferromagnets, causes extremely narrow magnetic domain walls \nwith width equal to the distance between the nearest magnetic U neighbors with a ntiparallel magnetic \nmoments.  \n \nExperimental  \nThe growth and thermal treatm ent of the UCoGa single crystal are described in [29]. An approximately \n1 mm thick disk has been cut perpendicular to the crystallographic c-axis from an annealed ingot. The \nflat surface was polished with diamond paste of 1 µm grade. The surface was then washed in acetone \nand isopropanol. The surface to pography, measured by atomic force microscopy (AFM)  in tapping \nmode  at 50 K, is presented in FIG. 2. From the same area, MFM images were acquired. The \nconcentration of scratches and dirt particles  was below the acceptable limit for measurements in the \nlift-mode. These objects served as marke rs on the surface.  \n \n \nFIG. 2. Topography image (AFM) of the sample surface at 50 K. The MFM data were collected from the \nsame area. The dirt particle in the red circle is used as a position marker in all images.  \nMagnetization measurements were perfo rmed in a SQUID magnetometer MP MS-7-XL and a \nPPMS -14T (both Quantum Design) on the same sample as used for the MFM studies. The AFM and \nMFM measurements were performed using a low -temperature atomic force microscope - \nattoAFM/MFM Ixs  inserted in a PPMS -14T.  PPP -MFMR probes  by NANOSENSORS  with hard -\nmagnetic coating (coercivity of about 300 Oe) were used. The MFM measurements were done in the \nconstant -height mode at a lift -height of 80 -100 nm.  The frequency modulation mode was applied  in \nwhich the cantilever oscillates at its resonance frequency and the change of resonance frequency  df \n5 \n represents the magnetic -force contrast. The residual field of the PPMS -14T superconducting magnet \nwas minimized prior to the MFM measurement. This allowed to measure zero -field-cooled images \nwith (almost) equal distribution of up - and down -magnetized domains.  \nThe width of the surface domains was determined by the stereological method in which it is \ncalculated from the number of intersections of  an arbitrary test line  with domain walls  in the MFM \nimage ( Ws = 2l/πn, where l is the total test -line length and n the number of intersections) [31], 40 lines \nper MFM image were evaluated. The error was calculated as three standard deviations of a normal \ndistribution.    \n \nResults  \nThe temperature dependence of the magnetization of UCoGa in a magnetic field of 10 mT applied \nalong the c-axis is shown in  FIG. 3. TC = 45 K has been estimated as the temperature of the middle of \nthe sharp decrease of the M vs T dependence .  \n0 10 20 30 40 50 600.000.020.040.060.080.100.12\n  M (B/f.u.)\nT (K)TC\n \nFIG. 3. Temperature dependence of the magnetization in an external magnetic field of 10 mT applied along \nthe c-axis.  \n 6 \n \n0 2 4 6 8 10 12 140.00.10.20.30.40.50.60.7\n-0.2 -0.1 0.0 0.1 0.2-0.6-0.4-0.20.00.20.40.6\n M (B/f.u.)\n0H (T) H||c-axis\n Hc-axis\n  M (B/f.u.)\nH (T) \nFIG. 4. Magnetization isotherms measured at 2 K in fields applied parallel and perpendicular to  the c-axis. The \ninset shows the low -field zoom of a magnetic isotherm (showing the details of the hysteresis) measured at 20 \nK in a field applied along the c-axis. \n \nThe magnetization isotherms measured at 2 K in fields applied parallel and perpendicular to the \nc-axis sho wn in FIG. 4 clearly demonstrate the huge uniax ial MA of UCoGa. The c-axis magnetization \nwhich is equal to 0.61  μB/f.u. at 1 T slowly saturates with increasing the field and reaches the value of \n0.75 μB/f.u. The a-axis magnetization shows a very weak linear (paramagnetic) response to the \nmagnetic field and reaches only a value of 0.03 μB/f.u. in 14 T. When the a-axis M vs. H straight line \nis extrapolated to high magnetic fields it reaches 0.75  μB/f.u. at μ0H = 350 T. This number can be \ntentatively considered as the minimum estimated Ha value. It is considerably larger than Ha ≥ 150 T \nderived from magnetization data collected  at 4.2 K in fields up to 35 T [20]. \nThe inset of F ig. 4 shows a low -field detail of the hysteresis loop measured in field along the c-\naxis field at 20 K. At this temperature, the coercive field is about 0.3 T. The coercivity collapses upon \nheating to temperatures above 26 K. This finding is in agreement wit h the results reported in Refs [20] \nand [29]. \nIn the case of a ferromagnet with strong uniaxial anisotropy, a domain pattern created by domain \nbranching can be observed  at a surface  perpendicular to the easy axis  [31]. The evolution of magnetic \ndomains during the magnetization process in UCoGa, studied at 20 K by means of MFM is presented \nin Fig. 5. Starting with the zero-field-cooled (ZFC) image (5 a), one can see a labyrinth -like domain \nstructure. At 0.01 T (5 b), a ve ry similar domain pattern is observed. The “dark” domains (magnetized \nin the same direction as the probe) expand with further increasing magnetic field. A partial \nmagnetization is observed, e.g. at 0.05 T ( 5c). For μ0H = 0.1 and 0.3 T (5d and 5 e, respectiv ely), the \ndomains disappear which is consistent with the saturation of magnet ization in the inset of Fig. 4 . New \ndomains appear for  μ0H < 0.05 T (5f and 5 h) when the sample becomes demagnetized from the \nsaturated state when sweeping the fiel d back down (se e inset of Fig. 4 ). These domains have a different \nshape than those observed in the ZFC state (Fig. 5a). Nevertheless, some of them are pinned to the \nsame surface point where a lattice defect can be expected. The lattice defects, which serve as centers  \nof pinning  of the magnetic domain walls, are usually unaffected by the applied magnetic field and, 7 \n therefore, they are fixed in the lattice in the magnetization/demagnetization process. An analogous \nmagnetization process was observed in a negative applied magnet ic field.  \nWeak parallel lines, forming a diamond like pattern (indicated by the green dashed lines), persist \nin the magnetic contrast ( 5d-f) and cannot be removed by an applied magnetic field up to 14 T (not \nshown). The pattern lines cross with angles of 1 20° and are probably related to the structure defects \nmentioned above. In the partially magnetized state, some of the magnetic domain walls are pinned to \nthese lines.  \nThe MFM images in Fig. 6 show the evolution of magnetic domains upon ZFC cooling. The \nmagnetic domains are formed just below TC. The d f contrast increases upon cooling as expected for \nincreasing magnetic moments of the domains. A gradually developing domain -branching pattern is \nobserved upon cooling from 40 K to lower temperatures. The shape of the domains at the surface \nchanges only slightly, with decreasing temperature emphasizing the morphological details. The \nmagnetic domains resemble the ZFC domains at 20 K ( 5a), only the con trast is inverted due to the \nopposite magnetization of the probe (due to the magnetic history of the previous measurements).  \nThe width of the surface domains Ws decreases with temperature  decrease  as can be seen in Fig. \n7. This decrease reflects the narrow ing of the magnetic domains inside the sample with increasing \nmagnetic moment.  \n \na) b) c) d)  \ne) f) g) h)  \nFIG. 5. MFM scans at 20 K in increasing and decreasing magnetic field. All scans from a) to h) were made \nduring increasing, followed by decreasing, of the applied magnetic field. The scan in a) was done at zero field \nafter cooling from high temperatures. The scans in a) 0 T (ZFC state), b) 0.01 T, c) 0.05 T, d) 0.1 T and e) 0.3 \nT show the process in increasing magnetic field. T he process in the decreasing field is shown in the scans f) \n0.05 T, g) 0.01 T and h) 0 T. The MFM probe was magnetized in a positive magnetic field.   \n \n8 \n a) b) c) d)  \nFIG. 6. Magnetic domains of UCoGa at a) 40 K, b) 30 K, c) 20 K and c)  5 K. \n \nFIG. 7. Evolution of the domain width with temperature upon heating and cooling.  \nThe domain -wall energy γW in a ferromagnet with a strong uniaxial anisotropy can be calculated \nby using the equation  \n \n 𝛾𝑊=𝑊𝑠𝑚𝑠2\n49µ0, (1) \n \nwhere ms is the spontaneous magnetization and μ 0 the permeability of vacuum  [31–33]. A γW value of \n1.1 mJ/m2 is obtained with the Ws and ms values, determined at the lowest temperature of measurement \n(5 K) where the thermal effects are minimized.  \n9 \n Most FM materials are characterized by Bloch domain walls. However, a simple calculation \nshows that this is not the case for UCoGa. The width of a Bloch domain wall in UCoGa can be \ncalculated from the equation  \n \n 𝛿=𝛾𝑊\n𝐾1 , (2) \n \nwhere K1 is the anisotropy constant  [34]. Usi ng the value K1 = 88 MJ/m3 reported by Prokes et al. \n[27] and the above given domain -wall energy of 1.1 mJ/m2, a Bloch domain wall thickness δ  = 12.5 \npm is obtained. This value is about 30 times smaller than the distance between two  nearest magnetic \nU ions (350 pm  [35]) in UCoG a. The failure of the Bloch -domain -wall model is not surprising. In this \nmodel, the magnetic moment rotates between neighbouring magnetic ions by an angle of about 1° in \ncase of weak MA and/or strong EI, which is typical for the 3d -electron ferromagnets. I n contrast, in \nthe case of UCoGa and most U 5 f-electron ferromagnets, the MA is very strong while EI is rather \nweak.  \nIf we consider a large angle of rotation between neighbouring magnetic moments, 180° in \nparticular, the domain wall is a slab (oriented along the c -axis) consisting of pairs of nearest U \nneighbors (within the basal plane) having antiparallel magnetic momen ts aligned along the c-axis. The \ndomain wall thickness is than equal to the distance between the neighboring magnetic ions. In such a \ncase, the contribution of the uniaxial MA to the domain -wall energy is zero because both directions of \nmagnetic moments in  the wall are energetically equal. The energy needed to create a domain is then \nsimply equal to the EI energy between the nearest magnetic moments.   \nThe results of our recent study of the critical exponents suggest that UCoGa belongs to the \nuniversality c lass of the 2D Ising system with long -range magnetic order [36]. The 2D Ising character \nhas recently been reported also for th e isostructural compound URhAl [37]. The crystal structure of \nUCoGa is characterized by a distorted kagome lattice of U ions as schematically shown in Fig. 1b). \nThe EI energy can be derived from TC using the relation for the 2D Ising ferromagnet kagome lattice \n[38]. The smallest element of a domain wall in UCoGa is a trio of U ions  forming an equilateral \ntriangle with two U ions with parallel moments and one U ion with an antiparallel moment. The EI \nenergy such a trio of moments is equal to  \n \n 𝐽=𝑘𝐵𝑇𝐶𝑙𝑛(3+2√3)\n2, (3) \n  \nand when divided by  the area of the smallest element of a domain wall, it leads to γW = 1.4 mJ/m2. \nThis value is in a reasonable agreement with the value of  1.1 mJ/m2 determined above from the surface \ndomain width. The difference between the two γW values may be also due to neglecting some \ncontributions to the EI.  \nFor simpl icity, only the short -range direct FM 5 f-5f interaction has been considered in our \nscenario.  The role of hybridization of the magnetic U 5 f-electron states with the Co 3 d states has been \nneglected. Small magnetic moments on Co sites may be expected as a natural consequence of the 5 f-\n3d ligand hybridization. No relevant microscopic study in UCoGa has been done to prove this \nexpectation. Nevertheless, the detailed polarized -neutron -diffraction experiments on the isostructural \nUTX compounds, UCoAl in the met amagnetic state  [39] and the ferromagnets URhAl [40] and UPtAl 10 \n [41] below TC have unambiguously revealed the existence of small magnetic moments (para llel to the \nU moments) at the T ions on the inequivalent crystallographic positions in the U -T1 and T2-X basal -\nplane layers, respectively. These findings corroborate that the expectation of small induced Co \nmoments in UCoGa is justified. The responsible 5 f-3d hybridization also mediates additional indirect \nFM U -Co-U components to the hierarchy of exchange interactions in UCoGa. Relevant neutron and \nX-ray magnetic -scattering experiments would help to confirm this scenario.  \nConclusions  \nUCoGa exhibits a surface -domain structure typical for a strongly anisotropic uniaxial ferromagnet \nwhich is consistent with the huge uniaxial MA observed in this compound. The MFM images show \nthe appearance/disappearance of magnetic domains during the magne tization/demagnetization \nprocesses with increasing/decreasing applied magnetic field. In the partially magnetized state, the \nmagnetic domain walls are pinned to lattice defects which have been identified in the AFM and MFM \nimages. The evolution of magnetic  domains during cooling the UCoGa crystal below TC has been also \ndemonstrated by MFM images. Analysis of available data leads to the conclusion that the high energy \nof the uniaxial magnetocrystalline energy assisted by the relatively low energy of ferromag netic \nexchange interaction in UCoGa causes that the domain walls are extremely narrow, equal to the \ndistance between the nearest magnetic U neighbor ions with antipa rallel magnetic moments in the U-\nT1 basal plane of the hexagonal structure. 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B. 67 (2003).  \n "    },    {        "title": "2003.00907v2.Traps_for_pinning_and_scattering_of_antiferromagnetic_skyrmions_via_magnetic_properties_engineering.pdf",        "content": "Traps for pinning and scattering of antiferromagnetic skyrmions via magnetic\nproperties engineering\nD. Toscano,1,a)I. A. Santece,1R. C. O. Guedes,1H. S. Assis,1A. L. S. Miranda,1C. I. L. de Araujo,2F. Sato,1\nP. Z. Coura,1and S. A. Leonel1\n1)Departamento de F\u0013 \u0010sica, Laborat\u0013 orio de Simula\u0018 c~ ao Computacional, Universidade Federal de Juiz de Fora,\nJuiz de Fora, Minas Gerais 36036-330, Brazil\n2)Departamento de F\u0013 \u0010sica, Laborat\u0013 orio de Spintr^ onica e Nanomagnetismo, Universidade Federal de Vi\u0018 cosa, Vi\u0018 cosa,\nMinas Gerais 36570-900, Brazil\n(Dated: May 19, 2020)\nMicromagnetic simulations have been performed to investigate the controllability of the skyrmion position in\nantiferromagnetic nanotracks with their magnetic properties modi\fed spatially. In this study we have modeled\nmagnetic defects as local variations on the material parameters, such as the exchange sti\u000bness, saturation\nmagnetization, perpendicular magnetocrystalline anisotropy and Dzyaloshinskii-Moriya constant. Thus, we\nhave observed not only pinning (potential well) but also scattering (potential barrier) of antiferromagnetic\nskyrmions, when adjusting either a local increase or a local reduction for each material parameter. In order to\ncontrol of the skyrmion motion it is very important to impose certain positions along the nanotrack where the\nskyrmion can stop. Magnetic defects incorporated intentionally in antiferromagnetic racetracks can be useful\nfor such purpose. In order to provide guidelines for experimental studies, we vary both material parameters\nand size of the modi\fed region. The found results show that the e\u000eciency of skyrmion trap depends on a\nsuitable combination of magnetic defect parameters. Furthermore, we discuss the reason why skyrmions are\neither attracted or repelled by a region magnetically modi\fed.\nI. INTRODUCTION\nNanoscaled topological spin textures known as mag-\nnetic skyrmions have attracted a lot of attention recently,\nsince they behave as quasiparticles and have high poten-\ntial of being information carriers in novel spintronic tech-\nnologies1{3. In the beginning, ferromagnetic skyrmions\nhave been experimentally observed only at low temper-\natures and under the in\ruence of large external mag-\nnetic \felds4{9. Nowadays, these quasiparticles have been\nstabilized at room temperature in magnetic multilayer\nsystems with interfacial Dzyaloshinskii-Moriya couplings\nand high perpendicular magnetic anisotropy10{16.\nIn the last few years, much e\u000bort has been dedicated\nto control the nucleation and the transport of skyrmions\nin ferromagnetic nanotracks10,17{29. Understanding and\ncontrolling the propagation of skyrmions in magnetic\nnanotrack is crucial for the development and realization\nof spintronic devices30{33. Due to the peculiar character-\nistics of skyrmions, such as nanoscaled sizes, topological\nprotection and e\u000ecient electric manipulation, there is a\nhuge interest in replacing domain walls with skyrmions to\nperform logical operations34,35and/or to encode informa-\ntion data storage devices3,36. The reference33describes\nthe current state of the art of Skyrmionics (skyrmion-\nelectronics). Unlike domain walls that are restricted\nto the unidirectional movement along the nanotrack37,\nskyrmions can not be driven by electric currents with-\nout being moved away from the longest axis of the nan-\notrack. The skyrmion transport in a ferromagnetic nan-\notrack is hampered by the skyrmion accumulation at\na)Electronic mail: dtoscano@ice.ufjf.brthe nanotrack edges, or even by the information loss\nin the case of the skyrmions are expelled from the nan-\notrack. As a consequence of the topological charge, the\nskyrmion Hall e\u000bect38was predicted theoretically2,39and\nwas observed experimentally by two groups simultane-\nously40,41. Although the skyrmion Hall e\u000bect is an is-\nsue for the ferromagnetic spintronics, varied strategies\nhave been proposed to suppress this undesirable phe-\nnomenon, as follow: (1) Via engineering of magnetic\nproperties42{48. The skyrmion Hall e\u000bect can be sup-\npressed in magnetic nanotracks with strategically mod-\ni\fed magnetic properties46. Magnetic strips presenting\nspatial variations of the material parameters can be in-\ntentionally incorporated into the nanotrack to gererate\nattractive or repulsive interactions, and it can be used\nto modify the dynamics of ferromagnetic skyrmion. (2)\nThrough the manipulation of skyrmions in synthetic an-\ntiferromagnetic nanotracks49{54, that is, a ferromagnetic\nbilayer coupled antiferromagnetically. In this strategy,\nthe RKKY interaction is responsible for the coupling be-\ntween the two skyrmions located on the top and bottom\nlayers55. By tuning the spacer thickness (non-magnetic\nmetal layer) it is possible to obtain an interlayer anti-\nferromagnetic coupling. Thus, the motion of skyrmion\npairs with opposite topological charges can be useful to\nsuppress the skyrmion Hall e\u000bect in synthetic antiferro-\nmagnetic nanotracks. (3) Via nucleation of a skyrmio-\nnium56{58or a Resonant Magnetic Soliton (RMS)59in-\nstead of a ferromagnetic skyrmion. The nontopological\nquasiparticle known as skyrmionium is characterized by\na zero topological charge as well as the antiferromagnetic\nskyrmion, whereas the RMS is a spin texture stabilized\nby current and characterized by topological charge os-\ncillating around the averaged value of Q= 0. Quasi-arXiv:2003.00907v2  [physics.comp-ph]  16 May 20202\nparticles with Q= 0 are totally free of the skyrmion\nHall e\u000bect. (4) By replacing the nanotrack material by\nan antiferromagnetic material60{62, that is, using a real\nantiferromagnetic nanotrack. Besides the issue of the\nskyrmion Hall e\u000bect be automatically suppressed in an\nantiferromagnetic medium, it was reported62other ad-\nvantages of antiferromagnetic skyrmions over ferromag-\nnetic skyrmions. For example, high mobility under small\nvalues of the applied current density.\nIn order to create traps for quasiparticles (such as vor-\ntices63,64, skyrmions65,66and domain walls67,68), changes\nin geometry or in material magnetic properties can be\nintentionally incorporated in nanoscaled magnetic thin\n\flms. As a result, dynamics of quasiparticle can be ma-\nnipulated and used to engineer spintronic devices69,70.\nFor instance, it has been recently proposed a transis-\ntor that employs antiferromagnetic skyrmions71. Vari-\nous spintronic technologies require traps to stabilize the\nquasiparticle at prede\fned positions along the magnetic\nnanotrack. The interaction between an antiferromagnetic\nskyrmion and a non-magnetic defect (a hole in the nan-\notrack) has been investigated in Ref.72. There, the au-\nthors reported that the skyrmion can be captured, scat-\ntered or completely destroyed by the hole, depending on\nthe skyrmion velocity and the type of collision (frontal\nor lateral). Recent works have focused on the dynam-\nics of an antiferromagnetic skyrmion in racetracks with\na magnetic defect, consisting in a local variation of the\nperpendicular magnetocrystalline anisotropy62,73. In this\npaper, we study the interaction between an antiferro-\nmagnetic skyrmion and a magnetic defect generated not\nonly by the local variation in the perpendicular magnetic\nanisotropy, but also in other material parameters, such\nas the exchange sti\u000bness, saturation magnetization and\nDzyaloshinskii-Moriya constant. Such inhomogeneities\ncan be intrinsic (impurities generated during the fabri-\ncation processes) or induced (intentionally incorporated\nimperfections). Chappert et al.74pioneered the located\nmodi\fcation of the magnetic properties by employing the\nion irradiation in magnetic thin \flms and multilayers, for\na review see Ref.75. The modi\fcation of the the magnetic\nmagnetic parameters, such perpendicular anisotropy and\nthe strength of the Dzyaloshinskii-Moriya interaction,\ncan be also achieved by tuning thicknesses in magnetic\nmultilayer, which is easy to be experimentally controlled.\nThe present work demonstrates that the skyrmion-\ndefect interaction is a short range interaction. Our\nmethodology consists in computing energy di\u000berences\nwith the Hamiltonian of the system. Thus, we obtain\nboth well and barrier potentials, which identify the kind\nof the interaction that we are dealing with, either attrac-\ntive or repulsive. Our predictions are veri\fed through re-\nlaxation micromagnetic simulations, that is, solving the\nLandau-Lifshitz-Gilbert equation without any external\nagent. Here, we answer the following question, mag-\nnetic defects in antiferromagnetic nanotracks: traps for\nskyrmions in the Antiferromagnetic Spintronics?II. MODEL AND METHODOLOGY\nIn order to describe the antiferromagnetic nanotrack,\nwe have considered Heisenberg exchange, Dzyaloshinskii-\nMoriya, dipole-dipole interactions, and perpendicular\nmagnetic anisotropy. Such magnetic interactions have\nbeen included in the following Hamiltonian model:\nH=\u0000X\n<i;j>Jij[ ^mi\u0001^mj] +\n\u0000X\n<i;j>Dijh\n^dij\u0001( ^mi\u0002^mj)i\n+\n\u0000X\niKi[ ^mi\u0001^n]2+\n\u0000X\ni;jMij\u00143( ^mi\u0001^rij)( ^mj\u0001^rij)\u0000^mi\u0001^mj\n(rij=a)3\u0015\n(1)\nwhere ^mk\u0011(mx\nk;my\nk;mz\nk) is a versor along the direc-\ntion of the magnetic moment located at the site kof\nthe lattice. Once Jij<0, the \frst term in Eq. (1)\ndescribes the antiferromagnetic coupling. Due to the\nshort range of the exchange interaction, we have consid-\nered the summation over the nearest magnetic moment\npairs< i;j > . The second term in Eq. (1) describe\nthe Dzyaloshinskii-Moriya interactions, and the versor\n^dijdepends on the considered magnetic system. For\na magnetic multilayer system with the interfacially in-\nduced Dzyaloshinskii-Moriya interactions, ^dij= ^uij\u0002^z,\nwhere ^zis a versor perpendicular to the multilayer sur-\nface and ^uijis unit vector joining the sites iandjin\nthe same layer20,76. Such magnetic systems favor the\nnucleation of N\u0013 eel skyrmions (hedgehog-type con\fgura-\ntion). On the other hand, Bloch skyrmions (vortex-type\ncon\fguration) arise in magnetically ordered systems with\nintrinsic Dzyaloshinskii-Moriya interactions, ^dij= ^uijfor\nbulk materials. The third term in Eq. (1) describes the\nuniaxial magnetocrystalline anisotropy, since Ki>0 and\n^n= ^z. The last term in Eq. (1) represents the dipolar\ncoupling. Due to the long range of this interaction, we\nconsider all dipole-dipole interactions. The unit vector\n^mithat represents the magnetic moment located at the\nisite is located by position vector ~ ri= (xi;yi;zi). The\nversor ^rij=~ ri\u0000~ rj\nj~ ri\u0000~ rjjis directed along the direction join-\ning the sites iandj, beingrij=j~ ri\u0000~ rjjthe relative\ndistance between them. The lattice parameter is repre-\nsented bya. The parameter of the dipolar interaction\nis always positive ( Mij>0), thus one can see that its\nlast term tends to align the magnetic moments antifer-\nromagnetically, whereas the \frst one tends to align the\nmagnetic moments along the direction coupling them ^ rij.\nThe dipole-dipole interactions are responsible for the ori-\ngin of the shape anisotropy in nanoscaled magnetic sys-\ntems. The strength of the magnetic interactions have the\nsame dimension, that is, Jij,Dij,KiandMijin units of\nenergy (J).3\nExperimental results about the magnetization dynam-\nics in nanomagnets have been simulated through the nu-\nmerical solution of the famous Landau-Lifshitz-Gilbert\n(LLG) equation77,78. In the literature the LLG equation\nis written in terms of the e\u000bective magnetization vectors\n~Mi=MS^mias well as in terms of the e\u000bective mag-\nnetic moment vectors ~ mi= (MSVcel) ^mi, whereVcelis\nthe volume of the micromagnetic cell. In practice how-\never, what really matters is the direction of the unit vec-\ntors ^mi, which describe the average spatial orientation\nof the atomic moments within each micromagnetic cell.\nTypical parameters for KMnF 3/Pt bilayer system have\nbeen used in our simulations, the values are as follow61,62:\nexchange sti\u000bness constant A=\u00006:59\u000210\u000012J/m,\nDzyaloshinskii-Moriya constant D= 8:0\u000210\u00004J/m2,\nmagnetocrystalline anisotropy constant K= 1:16\u0002105\nJ/m3and saturation magnetization MS= 3:76\u0002105\nA/m. Unless otherwise stated, these values for the ma-\nterial parameters were used in most of our simulations.\nIn the micromagnetic approach, the renormalization of\nmagnetic interaction constants depend not only on the\nmaterial parameters, but also on the manner in which\nthe system is partitioned into cells. According to mi-\ncromagnetic formulation, there is an upper limit for the\nwork-cell size. Thus, the volume of the micromagnetic\ncellVcelhas to be taken very carefully. In order to choose\na suitable size for the work cell we need to estimate the\ncharacteristic lengths, which are relevant to the micro-\nmagnetic problem. Such characteristic lengths are func-\ntions of the material parameters and we have estimated:\nthe exchange length \u0015=q\n2jAj\n\u00160M2s\u00198:61 nm, the wall\nwidth parameter \u0001 =q\njAj\nK\u00197:54 nm, and the length\nassociated20with the Dzyaloshinskii-Moriya interaction\n\u0018=2jAj\nD\u001916:48 nm. Thus, size of the work cell can\nbea= 1:0 nm<\u0001, which is lower than the small-\nest characteristic length. As in many micromagnetic\nsimulations, we consider a single layer of micromagnetic\ncells. All micromagnetic simulations were performed us-\ningVcel= (1\u00021\u00021) nm3, which is accurate enough for\nthe current study. For the case in which the magnetic\nsystem is discretized into cubic cells Vcel=a3, the pos-\nsible values for the magnetic interactions strength are\ngiven by:\nJij= 2a8\n<\n:A\nA0\nA00(2)\nDij=a28\n<\n:D\nD0\nD00(3)\nKi= 2a3\u001a\nK\nK00 (4)Mij=\u00160a3\n4\u00198\n<\n:MSMS\nMSM00\nS\nM00\nSM00\nS(5)\nwhereA;D;K;M Sare parameters of the host material,\nwhereasA00;D00;K00;M00\nSare the parameters of the guest\nmaterial.A0andD0are parameters which describe in-\nteractions at the interface between two antiferromagnetic\nmaterials. In order to allow the magnetic parameters to\nvary gradually from a magnetic medium to the other,\nthe geometric mean was adopted for the interface pa-\nrameters:A0=p\nA\u0001A00andD0=p\nD\u0001D00.\nFor the geometric parameters of the nanotracks, we\nhave considered length Lx= 300 nm, width Ly= 100 nm\nand thickness Lz= 1nm, di\u000bering one to another only in\nthe parameters of magnetic defect: the local variation of\nmagnetic property into a region of area S=L\u0002L. The\nspatial variations on the magnetic parameters were con-\nsidered individually. Thus, we have studied four possible\nsources of magnetic defects: Type A, TypeD, TypeK\nand TypeMS. TypeAmagnetic defects are those charac-\nterized only by spatial variations in the exchange sti\u000bness\nconstant (other parameters of the magnetic material were\nunchanged in the defect region), Type Dmagnetic de-\nfects are those characterized only by spatial variations in\nthe Dzyaloshinskii-Moriya constant, and so on. We have\nconsidered magnetic defects in the shape of squares with\ndi\u000berent sides, where Lranging from 5 to 23 nm. Guest\nmaterial parameters, A00;D00,K00andM00\nSwere regarded\nas tuning parameters. We can generate traps for the\nskyrmion by adjusting either local reduction ( X00< X)\nor local increase ( X00> X ) of a given magnetic prop-\nerty. The target parameter of the magnetic modi\fcation\nXcan beA;D;K orMS.\nThe magnetization dynamics is governed by LLG equa-\ntion, whose discrete and dimensionless version can be\nwritten as following:\nd^mi\nd\u001c=\u00001\n1 +\u000b2h\n^mi\u0002~bi+\u000b^mi\u0002( ^mi\u0002~bi)i\n(6)\nwhere~bi=\u0000\u00001\n2aA\u0001@H\n@^miis the dimensionless e\u000bec-\ntive \feld at lattice site i, which can contain individual\ncontributions from the exchange, Dzyaloshinskii-Moriya,\nanisotropy, dipolar, and Zeeman \felds. The Gilbert\ndamping parameter is \fxed at \u000b= 0:1. The connection\nbetween the time and its dimensionless corresponding is\ngiven byd\u001c=\u0017 dt, where\u0017=\u0012\u0015\na\u00132\n\r \u00160MS, being\n\r= 1:76\u00021011(T.s)\u00001the electron gyromagnetic ratio.\nThe LLG equation was integrated by using a fourth-order\npredictor-corrector scheme with time step \u0001 \u001c= 0:01.\nIn order to obtain the remanent magnetization of a\nnanotrack with a single antiferromagnetic skyrmion, we\nhave chosen as initial condition an analytical solution in\nwhich a N\u0013 eel skyrmion is placed exactly at the geomet-\nric center of the nanotrack. Speci\fcally, we start from a4\n  \n(a)\n(b)\n+1.0 -1.0 +0.5 -0.5 0mz\nFigure 1. (Color online). Schematic view of an antiferromagnetic nanotrack hosting a single skyrmion. Fig (a) highlights that\nthe majority of the nanotrack's magnetic moments is going out of the \fgure plane, except in the region of the ring shape (in\ncyan), where they are con\fned to the plane. One can see that the ring region surrounds the core of the skyrmion. Fig (b)\nshows a portion of the vector \feld ^ mk\u0011(mx\nk; my\nk; mz\nk) close to the skyrmion core; a N\u0013 eel's antiferromagnetic skyrmion.\nsolution of a ferromagnetic skyrmion65, so we reverse the\nmagnetization of a sublattice. If no external agent (mag-\nnetic \feld or current) are present, the integration of the\nLLG equation, Eq. (6), leads the magnetic system to the\nminimum energy con\fguration. Such approach makes\npossible the adjustment of the skyrmion size. An exam-\nple in which we obtain numerically the relaxed micro-\nmagnetic state of an antiferromagnetic nanotrack host-\ning a single skyrmion is shown in Fig. (1). Equilibrium\ncon\fgurations obtained in this way have been used as ini-\ntial con\fgurations in other simulations, as follow: (1) we\nhave considered the skyrmion located at the interface be-\ntween two antiferromagnetic media, and (2) a magnetic\ndefect with a square shape was inserted into the planar\nnanowire. We perform also a study about the skyrmion\nsize as a function of the medium magnetic properties,\nconsidering nanotracks free of magnetic defects. Suchpreliminary study will be of paramount importance to\nunderstand why skyrmions are attracted or repelled by\nmagnetically modi\fed regions.\nIII. RESULTS AND DISCUSSION\nA. Nanotracks made of a single magnetic material\nThe nucleation of antiferromagnetic skyrmions has al-\nready been demonstrated through micromagnetic simu-\nlations with a spin-polarized perpendicular current pulse\ninjected locally60,62. Here, we restrict our study to re-\nlaxation micromagnetic simulations, which contributed\nequally to understand the remanent magnetization of a\nnanotrack with a single antiferromagnetic skyrmion.5\n21Skyrmion diameter, d (nm)\n01020304050\nExchange stiffness, A (10-12 J/m)−2−3−4−5−6−7−8D = 8.0 x 10-4 (J/m2);\nK = 1.16 x 105 (J/m3);\nMs = 3.76 x 105(A/m).\n(a)\nSkyrmion diameter, d (nm)\n0102030405060\nDzyaloshinskii-Moriya, D (10-4 J/m2)0.7 0.8 0.9 1.0 1.1 1.2 1.3A = -6.59 x 10-12 (J/m);\nK = 1.16 x 105 (J/m3);\nMs = 3.76 x 105(A/m).\n(b)Skyrmion diameter, d (nm)\n0102030405060\nPerpendicular anisotropy, K (105 J/m3)0.2 0.4 0.6 0.8 1.0 1.2 1.4A = -6.59 x 10-12 (J/m);\nD = 8.0 x 10-4 (J/m2);\nMs = 3.76 x 105(A/m).\n(c)\nSkyrmion diameter, d (nm)\n6810121416\nSaturation magnetization, M s (105 A/m)0 1 2 3 4 5A = -6.59 x 10-12 (J/m);\nD = 8.0 x 10-4 (J/m2);\nK = 1.16 x 105 (J/m3).\n(d)\nFigura 1.4– global variation - skyrmion diameter - Crop 20 15 10 49\nFigure 2. (Color online). Diameter of the antiferromagnetic skyrmion as a function of the medium magnetic properties:\n(a) exchange sti\u000bness constant, (b) Dzyaloshinskii-Moriya constant, (c) perpendicular anisotropy constant and (d) saturation\nmagnetization constant. In all graphs, the magnetic property strength increases from the left to the right.\nAs we have a single magnetic material, the magnetic in-\nteractions strength are given by: Jij= 2aA,Dij=a2D,\nKi= 2a3K, andMij=\u00160a3\n4\u0019M2\nS, see the Eq. (1). An an-\ntiferromagnetic skyrmion can be considered as two cou-\npled ferromagnetic skyrmions with opposite topological\ncharges. When decomposing the lattice of the antifer-\nromagnetic material in two sublattices of anti-parallelly\naligned magnetic moments, we estimated the antiferro-\nmagnetic skyrmion size as the circle diameter at which\nthe out-of-plane magnetization changes of sign ( mz= 0)\nin a sublattice. Our study about the dependence of the\nskyrmion size on the nanotrack magnetic properties is\nshown in Fig. (2). We observe that the skyrmion col-\nlapses to the antiferromagnetic state, whenever the bal-\nance of magnetic interactions was not strong enough to\nstop the gradual decreasing of the skyrmion. On the\nother hand, whenever the skyrmion diameter was en-\nlarged with no control, we observe a deformed skyrmion\n(the skyrmion size was bigger than the nanotrack width)\nor even, its transformation to exotic spin textures (worm-\nlike magnetic domain). Such observations agree with\nthose reported a in previous study60, where the authors\npresent phase diagrams for the stability of antiferromag-\nnetic skyrmions as functions of the A,D,KandMS\nparameters. Therefore, the skyrmion size is governed bythe balance of the magnetic interactions.\nA few remarks are in order. In contrast to ferromag-\nnetic skyrmions65, the role of the dipolar coupling is to\ndecrease the size of the antiferromagnetic skyrmion, as\ntheMSincreases. From Fig. 2(a), one can see that\nthe skyrmion size decreases as the antiferromagnetic cou-\npling strength increases. Thus, we believe that the above-\nmentioned observation is related to the antiferromagnetic\ncharacter of the dipolar coupling; see Eq. (1). According\nto Fig. 2(b), the antiferromagnetic skyrmion size shrinks\nwith respect to decreasing the Dparameter. This behav-\nior is similar to its ferromagnetic counterpart24,65. If the\nDzyaloshinskii-Moriya interaction is not strong enough,\nthe skyrmion collapses to an ordered magnetic state (fer-\nromagnetic or antiferromagnetic) via gradual decreasing\nof the skyrmion size. In light of this fact, it is reason-\nable to expect that the skyrmion size grows as the D\nparameter increases, see Fig. 2(b).\nFrom the technological point of view, it is not interest-\ning a distorted skyrmion or even a particle that touches\nthe edges of the nanotrack during its motion. Therefore,\nthe racetrack width must be wide enough to host the\nskyrmion that will be injected. On the other hand, the\nmaterial parameters manipulation in nanotracks can be\nuseful to control the skyrmion size. For example, in nar-\nrow racetracks is desirable that the skyrmion size be as6\nsmaller as possible. Thus, the skyrmion behaves truly as\na quasiparticle.\nB. Nanotracks made of two magnetic materials\nIn this study, we have considered antiferromagnetic\nnanotracks compose by two magnetic media, as shown\nin Fig. 3(a). At the left side of the nanotrack is the\nmedium 1, which is characterized by the magnetic param-\neters:A;D;K;MSand at the right side of the nanotrack\nis the medium 2, which is characterized by the magnetic\nparameters: A00;D00;K00;M00\nS. The magnetic properties\nof two media are very close. More speci\fcally, the pa-\nrameters of the medium 2 are \u000610% the parameters of\nthe medium 1. The skyrmion was located exactly at the\ninterface between two magnetic media. Using this initial\ncondition, we numerically calculated the relaxed micro-\nmagnetic state of 100-nm-wide nanotracks in zero \feld\nfor di\u000berent values of the magnetic parameters of the\nmedium 2. Results of these simulations show that the\nskyrmion is stabilized either in the medium 1 or in the\nmedium 2, see Fig. 3. In \fgures (b) and (c) the medium\n2 di\u000bers from medium 1 only in the exchange sti\u000bness\nconstant,A00= 1:1AandA00= 0:9A, respectively. In\n\fgures (d) and (e) the medium 2 di\u000bers from medium 1\nonly in the Dzyaloshinskii-Moriya constant, D00= 0:9D\nandD00= 1:1D, respectively. In \fgures (f) and (g) the\nmedium 2 di\u000bers from medium 1 only in the perpendic-\nular anisotropy constant, K00= 1:1KandK00= 0:9K,\nrespectively. In \fgures (h) and (i) the medium 2 dif-\nfers from medium 1 only in the saturation magnetization\nconstant,M00\nS= 1:1MSandM00\nS= 0:9MS, respectively.\nFrom the relaxed micromagnetic states shown in \fgures\n(c), (e), (g) and (i), one can see that the magnetic sys-\ntem decreases its energy by moving the skyrmion to the\nmedium 2, which is the region of A,K, andMSreduced\norDincreased. On the other hand, one can see that the\nmedium 2 is avoided in \fgures (b), (d), (f) and (h). In\nthese simulations, the medium 2 is the region of A,K\nandMSincreased or Dreduced. To understand why the\nskyrmion is either attracted or repelled by the medium\n2, we remember the reader of our previous results about\nthe balance of the magnetic interactions, which de\fnes\nthe skyrmion size. In Fig. (2) it is shown that the\nskyrmion diameter is reduced by increasing of the ex-\nchange sti\u000bness, perpendicular anisotropy and saturation\nmagnetization parameters, whereas it is increased by in-\ncreasing of the Dzyaloshinskii-Moriya constant. Thus,\nwe can summarize the results of the Fig. (3) saying that\nthe skyrmion prefers the magnetic medium which tends\nto enlarge its diameter, because it is not only minimiz-\ning the system energy, but also ensuring the skyrmion\nsurvival. In other words, the system energy increases as\nthe skyrmion diameter decreases. Once the skyrmion can\ndisappear collapsing to the antiferromagnetic state, the\nskyrmion avoids the magnetic medium which tends to\nshrink its size.\n29\nInterface: A' and D'\nMedium 1: A, D, K and MsMedium 2: A'', D'', K'' and Ms''\n(a)Initial condition\n(b)A��=1.1A\n (c)A��=0.9A\n(d)D��=0.9D\n (e)D��=1.1D\n(f)K��=1.1K\n (g)K��=0.9K\n(h)M��\nS=1.1M S\n (i)M��\nS=0.9M S\nFigura 1.12– Skyrmion at an interface Crop L20 R15 T11 B8Figure 3. (Color online). (a) Schematic view shows a portion\nof the nanotrack that hosts the skyrmion located at the in-\nterface between two antiferromagnetic media. At the left side\nof the nanowire is the medium 1 and at the right side of the\nnanowire is the medium 2 (darkened region). Using this con-\n\fguration as initial condition, we integrate the LLG equation\nand obtain the \fnal con\fgurations shown in \fgures (b) to (i),\nwhich show that the skyrmion choose a medium. Figures (b),\n(d), (f) and (h) show that the skyrmion moves to the medium\n1, thus, it is repelled from the medium 2. On the other hand,\n\fgures (c), (e), (g) and (i) show that the skyrmion moves to\nthe medium 2, thus, it is attracted to the medium 2.7\nC. Nanotracks with a single magnetic defect\nConsidering our previous results, we investigate\nthe possibility of building traps for antiferromagnetic\nskyrmions. As shown in Fig. (4), a trap consists in a\nmagnetic defect which is incorporated into the nanotrack.\n  \nFigure 4. (Color online). Schematic view shows an antiferro-\nmagnetic nanotrack, which contains a skyrmion and a mag-\nnetic defect. The magnetic defect presents the shape of a\nsquare (darkened region) and it consists in the local modi\f-\ncation of the material parameters. Attractive and repulsive\ninteractions can be generated from the engineering of mag-\nnetic properties, when tuning either a local reduction or a lo-\ncal increase for some magnetic property. The center-to-center\nseparation between the skyrmion and the magnetic defect is\ns= 50 nm. The magnetic defect has an area of S= 529 nm2.\nIn order to measure the interaction energy U(s) be-\ntween the skyrmion and the magnetic defect as a func-\ntion of the center-to-center separation s, we have \fxed\nthe skyrmion at the center of the nanowire and only var-\nied the defect position along the nanowire axis. For each\nseparation s, the total energy E(s) of the system was\ncomputed using the Eq. (1), and the interaction energy\nhas been estimated using the following expression:\nU(s) =E(s)\u0000E(s!1 ) (7)\nOur study revealed two kinds of traps. In a pinning\ntrap, the skyrmion moves towards the magnetic defect,\nindicating an e\u000bective attractive potential of interaction\nbetween the skyrmion and the magnetic defect (poten-\ntial well). In a scattering trap, the skyrmion core moves\naway from the magnetic defect, indicating an e\u000bective\nrepulsive potential of interaction between the skyrmion\nand the magnetic defect (potential barrier). Fig. 5 (a)\nshows the interaction energy as a function of skyrmion-\ndefect separation, which indicates a repulsive potential\nforA00> A . On the other hand, Fig. 5 (b) shows\nan attractive potential for A00< A. When considering\nthe same spatial variations, replacing Type Amagnetic\ndefects with Type Dmagnetic defects, we observe that\npinning and scattering behaviors are interchanged. More\nspeci\fcally, Fig. 6 (a) shows a repulsive potential for\nD00<D, whereas Fig. 6 (b) shows an attractive poten-\ntial forD00> D . TypeA, TypeKandMSmagneticdefects are similar to each other. Fig. 7 (a) shows a re-\npulsive potential for K00>K, whereas Fig. 7 (b) shows\nan attractive potential for K00< K. Fig. 8 (a) shows a\nrepulsive potential when M00\nS> M S, whereas Fig. 8 (b)\nshows an attractive potential when M00\nS< M S. On the\nwhole, Figs. (5), (6), (7), (8) show that both pinning and\nscattering traps can be individually originated by a suit-\nable local variation of A;D;K;M S; it is enough to modify\na selected region to present either a local increase or a lo-\ncal reduction of a given magnetic property. The pinning\nand scattering strengths can be controlled by adjusting\nthe spatial variation of the magnetic property and the\ndefect area simultaneously.\nIt is interesting to discuss the in\ruence of the defect\nsize on the strength of the skyrmion-defect interaction.\nFrom Figs. (7) and (8), one can see that potential wells\nand potential barriers arise as the defect area increases.\nThe interaction potential can present a double-well po-\ntential (or double-barrier potential) instead of a single\npotential well (or a single potential barrier). Essentially,\nit occurs for small areas of the magnetic defect, thus\nskyrmion-defect interaction is extremely weak. As pre-\nviously reported65,73, the presence of a double-well po-\ntential (or double-barrier potential) is a signature of the\ntrap ine\u000eciency.\nIn Table (I) we summarize four ways of building\ntraps for pinning and scattering of antiferromagnetic\nskyrmions. Although the discussions have been held with\nthe N\u0013 eel skyrmion, we would like to emphasize that the\nqualitative results of the Table (I) remain unchanged for\nBloch skyrmions; it has been checked through micromag-\nnetic simulations. Moreover, we have veri\fed that the\nresults remain essentially the same for both skyrmion\nchiralities ( D> 0 andD< 0).\nNow we verify the predictions of Fig. (5) by solving\nthe LLG equation, see Figs. (9) and (10). By considering\na magnetic defect with a suitable size, we show that the\nskyrmion can be either attracted by a reduced Aregion\nor repelled by an increased Aregion. Although the re-\nsults presented in \fgures (9) and (10) are for the case of\nTypeAmagnetic defects, we have observed similar re-\nsults (both pinning and scattering traps) for other types\nof magnetic defects that were approached in this work.\nTable I. Two types of traps for antiferromagnetic skyrmions\ncan be originated in located variations of the magnetic prop-\nerties when tuning either a local increase ( X00> X ) or a\nlocal reduction ( X00< X ), where Xcan be A; D; K orMS. A\npinning trap corresponds to a potential well for the skyrmion,\nwhereas a scattering trap corresponds to a potential barrier.\nPinning Trap Scattering Trap\nA00< A A00> A\nD00> D D00< D\nK00< K K00> K\nM00\nS< M S M00\nS> M S8\n21Interaction energy, U (10-21 J)\n−1001020304050\nSkyrmion-defect separation, s (nm)−40−20 0 20 40A'' = 1.25 A\n529\n361\n225\n169\n121\n81\n49\n25S (nm2)\n(a)Interaction energy, U (10-21 J)\n−50−40−30−20−10010\nSkyrmion-defect separation, s (nm)−40−20 0 20 40529\n361\n225\n169\n121\n81\n49\n25S (nm2)\nA'' = 0.75 A\n(b)Interaction energy, U (10-21 J)\n−150−100−50050100150\nSkyrmion-defect separation, s (nm)−30−20−10 0 10 20 30A'' = 1.75 A\nA'' = 1.50 A\nA'' = 1.25 A\nA'' = 0.75 A\nA'' = 0.50 A\nA'' = 0.25 AS = 225 nm2\n(c)\nFigura 1.4– I am here\nFigure 5. (Color online). Type Amagnetic defects: local\nvariations in the exchange sti\u000bness constant. Figs. (a), (b)\nand (c) show the interaction energy between the skyrmion\nand the magnetic defect as a function of the center-to-center\nseparation. In Fig. (a) is shown a local increase of 25% in\nA, whereas in Fig. (b) is shown a local reduction of 25% in\nA. Both behaviors are shown for di\u000berent areas of a type A\nmagnetic defect. In Fig. (c) the area of the magnetic defect is\n\fxed at S= 225nm2and it is shown the behavior for di\u000berent\nvalues of the ratio A00=A.\n22Interaction energy, U (10-21 J)\n−20246810121416\nSkyrmion-defect separation, s (nm)−40−20 0 20 40529\n361\n225\n169\n121\n81\n49\n25S (nm2) D'' = 0.75 D\n(a)Interaction energy, U (10-21 J)\n−16−14−12−10−8−6−4−202\nSkyrmion-defect separation, s (nm)−40−20 0 20 40529\n361\n225\n169\n121\n81\n49\n25S (nm2)\nD'' = 1.25 D\n(b)Interaction energy, U (x 10-21 J)\n−40−30−20−10010203040\nSkyrmion-magnetic defect separation, s (nm)−30−20−10 0 10 20 30D'' = 1.75 D\nD'' = 1.50 D\nD'' = 1.25 D\nD'' = 0.75 D\nD'' = 0.50 D\nD'' = 0.25 DS = 225 nm2\n(c)\nFigura 1.5– I am hereFigure 6. (Color online). Type Dmagnetic defects: local\nvariations in the Dzyaloshinskii-Moriya constant. Figs. (a),\n(b) and (c) show the interaction energy between the skyrmion\nand the magnetic defect as a function of the center-to-center\nseparation. In Fig. (a) is shown a local reduction of 25% in\nD, whereas in Fig. (b) is shown a local increase of 25% in\nD. Both behaviors are shown for di\u000berent areas of a type D\nmagnetic defect. In Fig. (c) the area of the magnetic defect is\n\fxed at S= 225nm2and it is shown the behavior for di\u000berent\nvalues of the ratio D00=D.9\n23Interaction energy, U (10-21 J)\n−1012345678\nSkyrmion-defect separation, s (nm)−40−20 0 20 40529\n361\n225\n169\n121\n81\n49\n25S (nm2) K'' = 1.25 K\n(a)Interaction energy, U (10-21 J)\n−8−7−6−5−4−3−2−101\nSkyrmion-defect separation, s (nm)−40−20 0 20 40529\n361\n225\n169\n121\n81\n49\n25S (nm2)\nK'' = 0.75 K\n(b)Interaction energy, U (x 10-21 J)\n−15−10−5051015\nSkyrmion-magnetic defect separation, s (nm)−30−20−10 0 10 20 30K'' = 1.75 K\nK'' = 1.50 K\nK'' = 1.25 K\nK'' = 0.75 K\nK'' = 0.50 K\nK'' = 0.25 KS = 225 nm2\n(c)\nFigura 1.6– I am here\nFigure 7. (Color online). Type Kmagnetic defects: local vari-\nations in the perpendicular anisotropy constant. Figs. (a),\n(b) and (c) show the interaction energy between the skyrmion\nand the magnetic defect as a function of the center-to-center\nseparation. In Fig. (a) is shown a local increase of 25% in\nK, whereas in Fig. (b) is shown a local reduction of 25% in\nK. Both behaviors are shown for di\u000berent areas of a type K\nmagnetic defect. In Fig. (c) the area of the magnetic defect is\n\fxed at S= 225nm2and it is shown the behavior for di\u000berent\nvalues of the ratio K00=K.\n24Interaction energy, U (10-21 J)\n−0.500.511.522.533.54\nSkyrmion-defect separation, s (nm)−40−20 0 20 40529\n361\n225\n169\n121\n81\n49\n25S (nm2) Ms'' = 1.25 M s\n(a)Interaction energy, U (10-21 J)\n−3−2.5−2−1.5−1−0.500.5\nSkyrmion-defect separation, s (nm)−40−20 0 20 40529\n361\n225\n169\n121\n81\n49\n25S (nm2)\nMs'' = 0.75 M s\n(b)Interaction energy, U (x 10-21 J)\n−6−4−20246810\nSkyrmion-magnetic defect separation, s (nm)−30−20−10 0 10 20 30Ms'' = 1.75 Ms\nMs'' = 1.50 Ms\nMs'' = 1.25 Ms\nMs'' = 0.75 Ms\nMs'' = 0.50 Ms\nMs'' = 0.25 Ms\nS = 225 nm2\n(c)\nFigura 1.7– I am hereFigure 8. (Color online). Type MSmagnetic defects: local\nvariations in the saturation magnetization constant. Figs.\n(a), (b) and (c) show the interaction energy between the\nskyrmion and the magnetic defect as a function of the center-\nto-center separation. In Fig. (a) is shown a local increase of\n25% in MS, whereas in Fig. (b) is shown a local reduction of\n25% in MS. Both behaviors are shown for di\u000berent areas of a\ntypeMSmagnetic defect. In Fig. (c) the area of the magnetic\ndefect is \fxed at S= 225 nm2and it is shown the behavior\nfor di\u000berent values of the ratio M00\nS=MS.10\n  (a)\n(b)\n(c)  \n(d)  \n(e)   \nFigure 9. (Color online). E\u000bect of a pinning trap. Succes-\nsive snapshots show the skyrmion being attracted to the trap,\nwhich is characterized by a local reduction in the exchange\nsti\u000bness constant A00= 0:5A(a local reduction of 50 % in A)\ninto an area of 529 nm2. (a) Initial con\fguration at t = 0,\nwhere the center-to-center separation between the skyrmion\nand the magnetic defect is 30 nm. (b) Con\fguration after\n0.19 ns. (c) Con\fguration after 0.20 ns. (d) Con\fguration\nafter 0.23 ns, where the skyrmion is captured by the pinning\ntrap. (e) Con\fguration after 0.50 ns and later show that the\nskyrmion remains \fxed and/or centered at the trap.\n  \n(a)\n(b)\n(c)  \n(d)  \n(e)   \nFigure 10. (Color online). E\u000bect of a scattering trap. Succes-\nsive snapshots show the skyrmion being repelled from a trap,\nwhich is characterized by a local increase in the exchange sti\u000b-\nness constant A00= 1:5A(a local increase of 50 % in A) into\nan area of 529 nm2. (a) Initial con\fguration at t = 0, where\nthe center-to-center separation between the skyrmion and the\nmagnetic defect is 20 nm. (b) Con\fguration after 0.04 ns. (c)\nCon\fguration after 0.10 ns. (d) Con\fguration after 0.30 ns,\nwhere the separation between the skyrmion and the magnetic\ndefect is about 30 nm. (e) Con\fguration after 0.54 ns and\nlater show that the skyrmion remains distant from the trap.11\n27Interaction energy, U (10-21 J)\n−150−100−50050100150\nSkyrmion-defect separation, s (nm)−30−20−10 0 10 20 30A'' = 1.75 A\nA'' = 1.50 A\nA'' = 1.25 A\nA'' = 0.75 A\nA'' = 0.50 A\nA'' = 0.25 AS = 529 nm2\n(a)TypeAmagnetic defects\nInteraction energy, U (10-21 J)\n−60−40−200204060\nSkyrmion-defect separation, s (nm)−30−20−10 0 10 20 30D'' = 1.75 D\nD'' = 1.50 D\nD'' = 1.25 D\nD'' = 0.75 D\nD'' = 0.50 D\nD'' = 0.25 DS = 529 nm2\n(b)TypeDmagnetic defectsInteraction energy, U (10-21 J)\n−30−20−100102030\nSkyrmion-defect separation, s (nm)−30−20−10 0 10 20 30K'' = 1.75 K\nK'' = 1.50 K\nK'' = 1.25 K\nK'' = 0.75 K\nK'' = 0.50 K\nK'' = 0.25 KS = 529 nm2\n(c)TypeKmagnetic defects\nInteraction energy, U (10-21 J)\n−15−10−5051015\nSkyrmion-defect separation, s (nm)−30−20−10 0 10 20 30Ms'' = 1.75 Ms\nMs'' = 1.50 Ms\nMs'' = 1.25 Ms\nMs'' = 0.75 Ms\nMs'' = 0.50 Ms\nMs'' = 0.25 MsS = 529 nm2\n(d)TypeM Smagnetic defects\nFigura 1.10– Crop 20 15 10 49\nFigure 11. (Color online). Local variations of the magnetic properties considered individually in a magnetic defect of area\nS= 529 nm2. Figures (a) to (d) show the interaction energy between the skyrmion and the magnetic defect as a function of\nthe center-to-center separation. The behavior is shown for di\u000berent types of magnetic defects.\nFrom the technological point of view, the reader can\nwonder whether magnetic defects considered here will be\nwork as traps for skyrmions in antiferromagnetic nan-\notrack at room temperature? By analyzing our results for\nskyrmion-defect interaction energy, one can see that the\nheights of the potential barriers as well as the depths of\nthe potential wells depend strongly on the type of defect\nbeing considered, on the strength of the modi\fed mag-\nnetic property, and on the defect size (skyrmion-defect\naspect ratio). Thus, characteristics of trap can be exper-\nimentally controlled. For instance, the skyrmion-defect\ninteraction strength can be tuned in one or more order\nof magnitude larger than the thermal energy at room\ntemperature, that is, Ethermal =kBTroom\u001810\u000021[J].\nIt is worth mentioning that the region with modi\fed\nmagnetic properties can be designed to present a single\nor a combination of tuned magnetic parameters. The bal-\nance of the magnetic defect parameters will be responsi-\nble for adjusting the \fnality of the trap; either to capture\nor to blockade the skyrmions. Thus, the kind of the trapdepends on the suitable choice of material parameters\nthat will be modi\fed within the selected area. Maybe,\nthe a\u000bected region by ion beam irradiation contains more\nthan one modi\fed magnetic property. However, the mag-\nnitude of the a\u000bected magnetic parameters will not be\nthe same. For example, they can di\u000ber in the order of\nmagnitude. In Fig (11), we have equally varied the tar-\nget parameter of the magnetic modi\fcation, considering\nmagnetic defects of the same size. From this \fgure, one\ncan compare the individual contribution of each material\nparameter. In order of decreasing magnitude, we have\nmagnetic defects: Type A> TypeD> TypeK > Type\nMS. This makes evident the MSparameter modi\fca-\ntion results in the weakest skyrmion-defect interaction.\nIt is plausible, since MSparameter is linked to the dipo-\nlar coupling, which is the weakest magnetic interaction.\nThus, it is necessary to enlarge defect area as well as a\nsharp local variation in MSto improve the e\u000eciency of\nthe trap. On the other hand, the accidental modi\fcation\nof theMSparameter should be insigni\fcant compared to12\nthe target parameter of the magnetic modi\fcation.\nBy analyzing the graphs of Fig. (11), one can see that\nthe skyrmion-defect interaction energies employing local\nvariations in A,DandKdo not change upon undergo-\ning a re\rection across the horizontal axis. Note that it\ndoes not occur for Type MSmagnetic defects. We believe\nthat it is related to the strength of dipole-dipole interac-\ntions, which is directly proportional to the square of MS,\nwhereas the strength of other magnetic interactions vary\nlinearly with the parameters A,DandK; see Eqs. (2),\n(3), (4) and (5).\nIV. CONCLUSION\nIn summary, a crucial requirement for the development\nand realization of many spintronic devices is the control\nof the skyrmion position in nanotracks. Magnetic defects\nincorporated intentionally in racetracks can be useful to\nimpose prede\fned positions where skyrmions can stop.\nChanges in nanotrack geometry characterized by absence\nof magnetic material, such as notches or holes intention-\nally incorporated in nanotracks can be also useful for\nthis purpose. However, skyrmions can be completely de-\nstroyed by such non-magnetic defects62,72.\nIn this work, we have investigated the interaction be-\ntween the antiferromagnetic skyrmion and magnetic de-\nfects. Magnetic defects have been modeled as local vari-\nations on the nanotrack material parameters, such as\nA;D;K;MS. Thus, we have studied four sources of mag-\nnetic defects: Type A, TypeD, TypeKand TypeMS.\nBoth attractive and repulsive interactions have been ob-\nserved by adjusting either a local increase or a local re-\nduction for a given material parameter. For example, we\nshowed that the skyrmion is attracted to a trap char-\nacterized by a local reduction in the exchange sti\u000bness\nconstant (A00< A ), whereas the skyrmion is repelled\nfrom a trap characterized by a local increase in the ex-\nchange sti\u000bness constant ( A00> A ). Several strate-\ngies to pin and to scatter antiferromagnetic skyrmions\nwere investigated and they are summarized in Table (I).\nFurthermore, we have pointed that the e\u000eciency of the\ntrap is compromised if the defect size is smaller than the\nskyrmion size. This result is reasonable and it should be\nexpected. Due to the property of topological protection,\nskyrmions have the ability of overcoming obstacles like\npinning sites. Thus, the skyrmion-defect aspect ratio is\na crucial parameter to design traps for skyrmions. From\nthe technological point of view, experiments at room tem-\nperature should take into account not only the local vari-\nation strength of the magnetic properties, but also the as-\npect ratio between the skyrmion and the magnetic defect.\nIt is important to highlight that the behavior of attrac-\ntion and repulsion interactions involving the parameter\nMSis the opposite of that observed for ferromagnetic\nskyrmions65. The attraction and repulsion interactions\ninvolving parameter others ( A;D;K ) remains the same\nfor both ferromagnetic and antiferromagnetic skyrmions.Our results could guide the design of experimental\nworks that intend to investigate the realistic incorpo-\nration of such magnetic defects into antiferromagnetic\nnanotracks. In real experiments, the change of mag-\nnetic properties can be smooth. However, it can be com-\npensated by increasing the magnetic defect area. Since\nthe a\u000bected region is large enough, smooth modi\fcations\non the magnetic properties would still work as traps for\nskyrmions. In practice, circular defects should be gener-\nated by ion irradiation in magnetic thin \flms and multi-\nlayers. Although the results presented here are for mag-\nnetic defects in the shape of a square, we believe that\nsimilar results can be expected for defects with di\u000berent\nshapes, provided they present practically the same area.\nRecently, it was reported the interaction between circu-\nlar magnetic defects and antiferromagnetic skyrmions73.\nThe found results show that skyrmions will be repelled by\na local increase in Kand attracted by a local reduction in\nK. The skyrmion-defect repulsion that originates from a\nlocal increase in the perpendicular anisotropy can be also\nachieved by proper use of a triangular defect inserted at\nthe boundary of the nanotrack62. The results presented\nhere are trustworthy, since our study reproduces and ex-\ntends the predictions of other groups62,73.\nWhen considering the skyrmion located at the inter-\nface between two antiferromagnetic media, we discover\nthe reason why skyrmions are either attracted or re-\npelled by a region magnetically modi\fed. The basic\nphysics behind the mechanisms of pinning and scattering\nskyrmions is related to the energy minimization, that is,\nthe magnetic system minimizes its energy when moving\nthe skyrmion towards the magnetic medium that tends\nto maximize its diameter. Such observation is valid not\nonly for antiferromagnetic skyrmions but also for ferro-\nmagnetic skyrmions65. The dependence of the skyrmion\nsize on the nanotrack magnetic properties has been also\ninvestigated here, and we highlight that the manipulation\nof nanotrack material parameters can be used to control\nthe skyrmion size. When tuning the skyrmion size to be\nas smaller as possible, we ensure that the skyrmion be-\nhaves truly as a quasiparticle, avoiding that the skyrmion\ntouches the nanotrack edges, or even that, a distorted\nskyrmion is transported.\nWe believe that the presented results are promising\nfor potential applications in Antiferromagnetic Spintron-\nics. Although the discussions have been held with a sin-\ngle skyrmion, our results for traps of pinning and scat-\ntering of antiferromagnetic skyrmions can be planned\nand extended to skyrmion arrangements, once multiple\nskyrmions can coexist in the same magnetic thin \flm.\nDue to the skyrmion-skyrmion repulsion79,80, it is known\nthat skyrmions occupy sites of a honeycomb lattice8,81.\nFrom the perspective of potential applications, it can be\ndesirable to manipulate the parameters of the skyrmion\ncrystal, such as the spacing between skyrmions or even\nthe lattice type. Following the idea of using a 2D periodic\narrays of pinning sites previously proposed by Reichhardt\net al.82, a square lattice of magnetic defects considered13\nin this work can be useful to pin skyrmions at prede\fned\npositions along the antiferromagnetic thin \flm. From\nthe experimental point of view, the skyrmion individual\ndetection should be improved when increasing the spac-\ning between skyrmions. Besides, we can setup the min-\nimum spacing between particles in which the skyrmion-\nskyrmion repulsion is negligible. In practice, we can build\na non-interacting skyrmion array, which could be inter-\nesting for some skyrmionic devices. Another direction for\nfuture studies would be explore the usage of magnetic\nstrips46or an asymmetric environment83incorporated\ninto an antiferromagnetic medium to isolate transmis-\nsion channels of a multilane skyrmion racetrack84. We\nhope this study encourages experimental works to inves-\ntigate the controllability of the skyrmion position in anti-\nferromagnetic nanotracks with their magnetic properties\nmodi\fed spatially. In light of the skyrmion racetrack\nmemory concept70, a distribution of equally-spaced mag-\nnetic defect in a nanotrack can be useful to de\fne the bit\nlength, where the region between two magnetic defects\ncan host (1) or not (0) an antiferromagnetic skyrmion.\nIn a future work, we intend to study the current-induced\nskyrmion dynamics in such nanotrack to demonstrate\nthat the skyrmion motion can be controlled from a trap\nto another.\nACKNOWLEDGMENTS\nThe authors would like to thank CAPES, CNPq,\nFAPEMIG and FINEP (Brazilian Agencies) for the sup-\nport. Numerical simulations were performed at the Lab-\norat\u0013 orio de Simula\u0018 c~ ao Computacional do Departamento\nde F\u0013 \u0010sica da UFJF.\nREFERENCES\n1N. S. Kiselev, A. N. Bogdanov, R. Sch afer, and U. K. 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Shukla2,†\n1Department of Physics, Aligarh Muslim University, Aligarh 202002, India\n2UGC-DAE Consortium for Scientific Research, Indore 452001, India\n3Department of Physics, Central University of Punjab, Bathinda Punjab 151001, India\n4LSNCM-NFA, Institute of Physics, UnB, Brasilia DF 70910 900, Brazil\n5ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, United Kingdom\n6Deutsches Elektronen-Synchrotron, Notkestrasse 85, D-22607 Hamburg, Germany\n(Dated: October 20, 2020)\n[Ca 2CoO 3]0.62[CoO 2], a two dimensional misfit metallic compound, is famous for its rich phases ac-\ncessed by temperature, i.e.high temperature spin-state transition, metal-insulator transition (MIT)\nat intermediate temperature ( ∼100 K) and low temperature spin density wave (SDW). It enters\ninto SDW phase below T MIT which becomes long range at 27 K. Information on the independent\nrole of misfit layers (rocksalt/Ca 2CoO 3& triangular/CoO 2) in these phases is scarce. By combining\na set of complementary macroscopic (DC magnetization and resistivity) and microscopic (neutron\ndiffraction and X-ray absorption fine structure spectroscopy) measurements on pure (CCO) and Tb\nsubstituted in the rocksalt layer of CCO (CCO1), magnetic correlations in both subsystems of this\nmisfit compound are unraveled. CCO is found to exhibit glassiness, as well as exchange bias (EB)\neffects, while CCO1 does not exhibit glassiness, albeit it shows weaker EB effect. By combining local\nstructure investigations from extended X-ray absorption fine structure (EXAFS) spectroscopy and\nneutron diffraction results on CCO, we confirm that the SDW arises in the CoO 2layer. Our results\nshow that the magnetocrystalline anisotropy associated with the rocksalt layer acts as a source of\npinning, which is responsible for EB effect. Ferromagnetic clusters in the Ca 2CoO 3affects SDW in\nCoO 2and ultimately glassiness arises.\nMagnetism in misfit cobaltates is a debated topic of\ninvestigation although interesting1. The misfit struc-\nture makes the physics of these systems complex. For\nexample, a famous misfit structure, sodium cobaltate\n(NaxCoO 2), offers superconductivity in hydrated form\nand thermoelectricity with the metallic conductivity2,3.\nMoreover, the existence of cobalt ion (having spin state\nvariants4) in such misfit cobaltates makes the task daunt-\ning for the magnetic structure prediction. Besides, an-\nother ingredient of complexity is geometric frustration\ndue to triangular lattice CoO 2, having edge shared Co\nions octahedra in D 3dsymmetry5. In NaxCoO 2, sodium\ncontent decides the valency of Co ions in the triangu-\nlar lattice (CoO 2) and it shows rich phases with dif-\nferent concentrations of Na, e.g., the extreme member,\nNaxCoO 2(x = 1) is a non-magnetic insulator6and for\nx∼0.62, the compound shows the boundary in be-\ntween the anti-ferromagnetic (AFM) and ferromagnetic\n(FM) correlations dominant compositions7. In the crys-\ntal structure of Na xCoO 2, the CoO 2layers are separated\nby the layers of Na atoms and even with the two dimen-\nsional structure it has been found that for such structures\ninterlayer and intralayer magnetic interaction have com-\nparable strength1.\nFamous for its thermoelectricity the Ca 3Co4O9, more\nprecisely [Ca 2CoO 3]0.62[CoO 2] (hereinafter abbreviated\nas CCO), has two subsystems as intergrowth of one\non the other aperiodically. According to the chemical\nformula, it is comparable with x ∼0.6 composition of\nNaxCoO 2. One can roughly compare the magnetism of\nthe CoO 2layer in both structures, however, in CCO therole of the [Ca 2CoO 3] layer (having stack of CaO-CoO-\nCaO with rocksalt structure) is significant, therefore the\noverall magnetic behavior is unique.\nCCO exhibits the onset of SDW below TMIT∼100\nK which become long range at TSDW∼27 K followed\nby ferrimagnetic ordering at TFerri∼19 K. Many re-\nsearchers have tried to alter its properties by doping. For\nexample, it is reported that Sr doping at the Ca site weak-\nens the ferrimagnetism and shows AFM correlations8.\nThe electron doping at the Co site of the rocksalt layer\nby the trivalent ion doping at Ca site (Y3+& Bi3+) di-\nminishes the ferrimagnetism and affects the TSDW which\nhighlights the role of Co valency in the rocksalt layer8.\nIt is also reported that the SDW in the CoO 2subsystem\nhas oscillating moments in the cdirection and motion\nin theabplane and, by comparing the results with the\ndoped CCO, it is suggested that SDW can be tuned by\ndoping in the rocksalt layer at the Ca site9.\nHere we report on the drastic alteration in magnetic\nproperties of the CCO by electron doping at the Co site\nin the rocksalt layer by Tb doping at the Ca site. Doping\nconcentration of Tb is decided on the basis of earlier stud-\nies of CCO10. We have utilized the exchange bias, present\nin both CCO and Tb substituted CCO, as a tool to dis-\ncern the role of different magnetic lattices. Competition\nbetween the rocksalt layer caxis magnetism and trian-\ngular layer itinerant magnetism has been found as the\ncause of ferrimagnetism. Magnetocrystalline anisotropy\nassociated with the rocksalt layer has been identified as\nthe cause of pinning for exchange bias. Concomitant with\nthe broadness of neutron diffraction peaks, an anomalyarXiv:2004.08319v2  [cond-mat.mtrl-sci]  17 Oct 20202\nFIG. 1. Field cooled susceptibility as a function of tempera-\nture, measured at different magnetic fields (a) for CCO and\n(c) for CCO1. Insets show the 3D plot ( χ-H-T), showing the\nbifurcation between FC and ZFC. ZFC relaxation curves mea-\nsured at 5 K and 30 K under the same magnetic field (50 Oe),\n(b) for CCO and (d) for CCO1. (e) Dependence of T Irron\nmagnetic field for CCO. Blue solid line shows the fitting to\ndata (see text).\nin the spin phonon coupling in the CoO 2layer (observed\nvia EXAFS) confirms the truncation of long range in-\ncommensurate SDW (ISDW) into a glassy phase. The\nTb doping has been found to change the effect of rock-\nsalt on the SDW in CoO 2by screening the rocksalt field,\nand consequently CCO1 shows no glassiness and less EB\nin CCO1.\nPure (CCO) and Tb doped Ca 2.9Tb0.1Co4O9(CCO1)\nhave been synthesized using solid state route, as reported\nelsewhere11. Phase purity of the samples have been con-\nfirmed using X-ray diffraction12. X-ray photoemission\nspectroscopy (XPS) has been performed using an Omi-\ncron energy analyzer (EA-125) with Al Kα(1486.6 eV)\nX-ray source. Magnetization measurements were done\nusing a 7T Quantum Design magnetometer (MPMS-3).\nIsotherms, virgin and full loop M (H) have been recorded\nat various temperatures across the mentioned transitions,\ni.e., TSDW and TFerri in FC and ZFC modes. Magne-\ntization as a function of temperature M (T) at different\napplied magnetic fields were recorded in FC and ZFC pro-\ntocols. ZFC relaxation measurements have been done at\n5 and 30 K by cooling the sample in zero field down to the\ndesired temperature and, after a 100 s delay, magnetiza-\ntion have been recorded at 50 Oe for upto 8000 s. Neu-\ntron diffraction patterns have been collected at General\nMaterials Diffractometer (GEM), ISIS facility, UK, in the\ntemperature range 6-110 K. JANA 200613was used for\nfitting the neutron diffraction patterns. Extended X-ray\nFIG. 2. For CCO, (a) M(H) isotherm loops measured at 5 K\nin ZFC and FC (+70 kOe and -70 kOe), showing the hysteresis\nand EB (in field cooled cases). (b) M(H) loop at 5 K measured\nafter cooling under various fields. Inset shows the successive\nshifting of loop on the field axis. (c) M(H) loops measured at\ndifferent temperatures in +70 kOe field cooled condition. (d)\nTemperature dependence of H EBand H c. (e) Cooling field\ndependence of H EBand H c.\nabsorption fine structure spectroscopy (EXAFS) mea-\nsurements have been performed at beamline P65 at PE-\nTRA III, DESY, Germany. The EXAFS measurements\nwere done in fluorescence and transmission mode at Co\nK edge (7.7 keV). The sample amount was calculated for\none absorption length and homogeneously mixed with\nboron nitride and pressed into a pellet shape. A liq-\nuid helium flow cryostat has been used for low temper-\nature EXAFS measurements. Athena has been utilized\nfor data processing. In Artemis , the FEFF and IFEF-\nFIT codes were used to calculate theoretical scattering\npaths and to fit the experimental spectra, respectively.\nFirst we will discuss the results of CCO. FC magnetic\nsusceptibility ( χ) (see Fig. 1 (a)) with the bifurcation in\nFC and ZFC in low field (see inset) indicates presence\nof magnetic glassiness of some type or the presence of\nmagnetocrystalline anisotropy or both together14. The\nupturn in the χhas been attributed as T Ferri , in lit-\nerature, while in magnetic entropy (∆ SM) it is visible\nas a first derivative of M at ∼10 K (see Fig. S1 (a)\nin the supplementary file15). Also, there is no sponta-\nneous magnetization observed via Arrott plot at all mea-\nsurement temperatures (see Fig. S2 (a) of supplemen-\ntary15). ZFC relaxation measurements (see Fig. 1 (b))\nshow the time dependence of magnetization at 5 K but\nnot at 30 K. To confirm the glassiness16we have fitted\nthe ZFC relaxation curve with the stretched exponen-3\nFIG. 3. For CCO1, (a) M(H) isotherm loop measured at 5 K\nin ZFC and FC (+70 kOe and -70 kOe), showing the hysteresis\n(see inset) and EB (in field cooled cases). (b) M(H) loop at 5\nK measured after cooling under various fields. Inset shows the\nsuccessive shifting of loop on the field axis. (c) M(H) loops\nmeasured at different temperatures in +70 kOe field cooled\ncondition. (d) Temperature dependence of H EBand H c. (e)\nCooling field dependence of H EBand H c.\ntial function M(t) =Mo−Mrexp(−(t/tr)β) where the\nvalue ofβtells the distribution of barrier and is found\nto be∼0.37, which is close to the value for canonical\nspin glass16(∼0.42). From the inset of Fig. 1 (a) it is\nobserved that the bifurcation exists below ∼20 K for\nmagnetic fields up to ∼40 kOe. The bifurcation temper-\nature (TIrr) followed a trend with magnetic field, unique\nfor spin-glass (see Fig. 1 (e)). Fitting with the equation\nTIrr(H) =TIrr(0)(1−AHn) reveals the exponent n ∼\n0.66, which is typical for Almeida-Thouless (AT) line,\npredicted theoretically for spin-glass17,18. Based on pre-\nliminary analysis for now, we designate the bifurcation\nas related to the glassiness.\nInterestingly, we have observed the exchange bias (EB)\nin CCO at 5 K, +70 kOe FC with magnitude H EB∼-1.7\nkOe and coercivity H c∼5 kOe, calculated using HEB=\n(Hc1+Hc2)/2 andHc= (|Hc1|+|Hc2|)/2, respectively.\nHere Hc1and Hc2are the coercive fields in the negative\nand positive field side, respectively. The magnitude is\nconsiderably large, however one has to authenticate the\nexistence of it. Fig. 2 (a) shows the M (H) hysteresis\nmeasured in ZFC, +70 kOe FC and -70 kOe FC at 5 K.\nThe loop shifted to negative and positive directions for\ncooling in positive and negative fields, respectively. This\nis according to conventional EB system19. The cooling\nfield dependence (see Fig. 2 (b)) at 5 K and temperature\ndependence at +70 kOe has been observed (see Fig. 2(c)). These trends also match with the conventional EB\ncases19,20(see Fig. 2 (d & e)).\nFor conventional EB systems with AFM and FM lay-\ners with the strong interfacial coupling, the H EBis de-\nfined as21,22HEB=−JSAFMSFM\nµotFMMFMwhereJrepresents\nthe coupling strength across the interface, SFM/AFM is\nthe interface magnetization of FM/AFM phase, and tFM\n&MFMare the thickness and bulk magnetization of the\nFM layer. From this relation, it is clear that H EBwill\nincrease with the increase in interfacial FM, which in-\ncreases with the cooling field (H CF) due to spin align-\nment in field direction. Although, enhancement in the\nHCFresults in increase in cluster size (decreases SFM)\nand enhances the bulk magnetization, MFMtherefore\nreducing22the HEB. Moreover, for the phase separated\nsystems20, with FM clusters in the SG matrix, the above\nsituation is also observed, but the effect of magnetic field\non the glassy phase has to be considered, which usually\ndiminishes with the applied field.\nThe important and unusual observations in the present\ncase are the suppression of EB (Fig. 2 (d)) and bifurca-\ntion (Fig. 1 (a)) for temperature /greaterorsimilar15 K, the non satura-\ntion behavior of H EBup to 70 kOe (Fig. 2 (e)) and the\nsuppression of bifurcation in a field above ∼40 kOe (see\nFig. 4 and related texts). These open the question about\nthe origin of EB, because if the glassiness is considered\nas the origin of pinning, then it should vanish for a field\nabove∼40 kOe, which is not the case here.\nBefore making any comment on the origin of EB, we\nwill discuss the results of CCO1 (Tb doped CCO). For\nthe doping at the Ca site, the chemical formula can be\nwritten as [Ca 1.959Tb0.041CoO 3]0.62[CoO 2]. Tb3+is a\nmagnetic ion with the total spin moment S = 3 (4 f8)\nwith the theoretical Ising moment23,24mz∼9.72µB.\nFigs. 1 (c) and (d) display the χ(T) and isothermal ZFC\nrelaxation, respectively. The magnitude of moment is\nlarger for CCO1 because of the additional paramagnetic\ncontribution from the Tb. No transition of any type is\nobserved in the ∆ SM(see Fig. S1 (b)15), which means\nthat the Tb destabilizes the ferrimagnetism. Arrott plot\nis similar to that for CCO, i.e.Ms= 0 (see Fig. S2 (b) of\nsupplymentry15). These observations indicate that there\nis no glassiness in the CCO1 and it should not behave\nlike CCO,i.e., it should not possess EB. We have carried\nout the same set of magnetization measurements as done\nfor CCO, shown in Fig. 3 (a-e). Counterintuitive, this\nsystem, CCO1, also exhibits the EB albeit with lower\nstrength (H EB) and with lower coercivity (H c). In HEB\nplot as a function of cooling field H CF, no saturation or\ndecreasing trend has been observed up to 70 kOe, similar\nto CCO.\nIn order to know what is happening at microscopic\nscale, photoemission spectroscopy measurements were\ncarried out on both the samples which clearly indicates\nthat the Tb substitution increases the Co3+(see Fig.\nS415). A similar observation has been made by other\ngroups for high concentration of Tb doping10.\nThe observations from CCO1 indicate that the origin4\nFIG. 4. (a)-(i) Comparison between χFC(black symbols),\nχZFC (filled symbols) and calculated χ/prime\nFC(empty circles) for\nthe CCO (see text).\nof bifurcation in CCO may come from the magnetocrys-\ntalline anisotropy and therefore it should follow the equa-\ntion suggested by Joy et al.14,16given asMFC\nHappl+Hc(=\nχ/prime\nFC)≈MZFC\nHappl(=χZFC). Figs. 4 (a-i) show that large\nbifurcation appears because of the glassy phase, which\nis suppressed for higher fields ( >5 kOe), and for higher\nfield the magnetocrystalline anisotropy model is satis-\nfied. This explains the reason for large hysteresis in CCO,\ni.e., glassy phase, while CCO1 does not exhibit it. How-\never, the existence of EB in both samples indicates that\nthe origin of EB is not the glassiness. Therefore, it is\nproposed that the Tb doping affects the magnetocrys-\ntalline anisotropy, which indirectly suppresses the glassi-\nness. For, ingredient of EB19, we have AFM SDW in the\nCoO 2layer25, which is common in both the samples.\nThe SDW generally appears as an AFM ordering in\nlow dimensional metallic systems. This is simple fact by\nwhich one can distinguish the AFM (localized insulating)\nand SDW (metallic systems). SDW is itinerant but can\nshow the similar behavior as showed by localized helical\nor cycloidal ordered systems. These orderings can be de-\nscribed by the orientation of spins S in all directions (S x,\nSyand Sz) having same magnitude |S|, while for the\nSDW the direction remains the same (S x, Syor Sz) but\nthe magnitude has an oscillatory behavior26(Scos(2q.r)).\nIn CCO the rocksalt layer possesses short range FM be-\ncause of clustering of mixed valency (Co3+and Co4+)12.\nAnd the low spin state (LSS) Co4+(S = 1/2, ˜L=1) can of-\nfer the magnetocrystalline anisotropy through spin-orbit\ncoupling (SOC)27,28. In CCO1, as a result of Tb doping\nat the Ca site, the interlayer coupling ( J⊥) decreases, as\nwell as the amount of Co4+in the rocksalt layer. Thiscan explain the low temperature shifting of T Ferri or its\nabsence, as a result of the decrease in the J⊥value asso-\nciated with the interlayer coupling ( J⊥∝kBT). The\nschematic shown in Fig. 5 represents the above mentioned\nhypothesis that CoO 2exhibits AFM SDW with spins ar-\nranged in a wave pattern (red arrow). In the rocksalt\nlayer the FM clusters (made up of Co4+- Co3+) provide\nthe short range ferromagnetism along the caxis; black\narrows shows their effective strength and direction. The\neffective moments, as a sum of these two contributions,\nresult into overall ferrimagnetism. The Ca layer in be-\ntween the CoO 2and Ca 2CoO 3controls their coupling\nand the EB which arises as a result of coupling between\nthese two layers.\nFIG. 5. Schematic showing the AFM SDW in the CoO 2layer,\nhaving sinusoidal behavior (red arrows), resulting in a linear\nM(H) behavior. In the rocksalt layer the green clusters show\nshort range ferromagnetic arrangement of spins with strong\nspin orbit coupling (SOC), which restricts the magnetism in\nthe crystal caxis, resulting in a hysteretic M(H) loop. The\nresultant of these two gives rise to overall ferrimagnetism (or-\nange arrows), resulting in a non-saturating hysteretic M(H).\nThe layer of Ca in between these two magnetic layers controls\nthe coupling.\nFig. 6 shows the profile matched neutron diffraction\npattern measured at 17 K. The crystal structure of CCO\ncan be indexed using superspace group C2/m(0q0)00\nwhich is marked as phase 1 having unit cell parameters29\nasa= 4.8309 ˚A,b= 4.5615 ˚A,c= 10.8360 ˚A,β= 98.134◦\nandq= (0, 1.612, 0). Phases 2 and 3 represent the indi-\nvidual subsystems ([Ca 2CoO 3] & [CoO 2], respectively),\neach of them modulated with the magnetic propagation\nvector,qmag= (0.481, 0.377, 0.0015), obtained using the\nk-search software30. Similar three components modula-\ntion was observed for the well known SDW material31\n(TMTSF) 2PF6. Phase 2 has lattice parameters as a=\n4.8309 ˚A,b1= 4.5615 ˚A,c= 10.8360 ˚A andβ= 98.134◦\nwithq1=qmag= (0.481, 0.377, 0.015), and phase 3 has a\n= 4.8309 ˚A,b2= 4.5615/q= 2.8297 ˚A,c= 10.8360 ˚A and\nβ= 98.134◦withq2= (0.481, 0.377* q, 0.015). The mag-\nnetic modulation is quite complex, as from C2/msym-\nmetry only the P1 space group is allowed (found using\nMAXMAGN program32). From the propagation vector\nit is clear that the moments are propagating in the ab\nplane. Interestingly, from the contour plot it is observed\nthat the intensity of magnetic peaks are considerable at5\nFIG. 6. Neutron diffraction pattern of BANK 3 of the GEM\ndiffractometer, measured at 17 K, profile fitted using three\nphases (see text). Magnetic peaks are indicated by the verti-\ncal arrows. Inset show the temperature dependence of mag-\nnetic peak intensities, which become diffusive below ∼16 K.\nFIG. 7. Temperature dependence of selected reflections from\nneutron diffraction patterns of BANK 3 (a) and BANK 2 (b).\nAt low temperature broadness of magnetic peaks shows the\nshort or glassy magnetic correlations, shown in yellow block.\n∼16 K and become diffusive at lower temperatures (see\ninset (a)). Figs. 7 (a & b) show the evolution of magnetic\npeaks with temperature. As evident from the broadness\nof magnetic peaks, short range correlations appear be-\nlow∼13 K, which is a direct signature of glassiness33as\nobserved also in DC magnetization.\nIt has been reported34that the Co-Co correlation in\nthe CoO 2results in an anomaly in the mean square\nrelative displacement (MSRD) related to this pair, i.e.,\nσ2\nCo−Co. We have fitted the EXAFS data using a stan-\ndard protocol35by assuming the first shell parameters\nas temperature independent, as observed previously34,\nFIG. 8. (a) MSRD of Co-Co pair in CoO 2layeri.e.,σ2\nCo−Co,\nshowing an anomaly at the susceptibility upturn temperature.\nBlack solid lines are guide to the eye. (b) Resistivity as a\nfunction of temperature for CCO and CCO1, showing shift a\nin TMIT with Tb doping and magnitude change hints towards\nthe change in SDW gap.\nand only iterate the second shell parameters. We ob-\nserved an anomaly in the σ2\nCo−Coat temperature∼15\nK, as shown in Fig. 8 (a), which matches with the mag-\nnetization upturn. This result supports the spin-phonon\ncoupling in the CoO 2layer. This type of observation has\nbeen made earlier also by temperature dependent Raman\nscattering36. This type of strong spin-phonon coupling\nobservation and the AFM ordering via neutron diffrac-\ntion confirm that the AFM SDW is originating from the\nCoO 2only. Our NPD data clearly show that below 15\nK, the SDW in CoO 2tends to become short range and\ndirected along arbitrary directions. This truncated SDW\nis the reason of glassiness and result into the features of\nbulk spin glass. The interlayer coupling ( J⊥) between\nCoO 2and Ca 2CoO 3becomes stronger below ∼15 K and\nis responsible for truncation of SDW in CoO 2. The layer\nof Ca which can control the interlayer coupling and the\nSDW is altered via Tb doping (having high caxis Ising\nspins) and hence absence of glassiness in CCO1 ( i.e., Tb\nscreens the effect of rocksalt layer).\nTo further investigate the effects of Tb substitution, it\nwill be informative to look at the transport results. Fig. 8\n(b) comprises temperature dependent resistivity of both\nthe samples, showing the shift of T MIT towards higher\ntemperature with the Tb doping. The magnitude of re-\nsistivity of CCO1 is found larger compared to CCO. We\nhave fitted the curves with the activated behavior (not\nshown here) using the relation ρ=ρoexp(∆/kBT), as\nsuggested in earlier reports8,37and found a significant en-\nhancement in the band gap ∆. This shows that the dop-\ning in the rocksalt significantly affects the overall band\nstructure and hence also to the SDW gap. The TMIT\nin general is directly related with the correlation via the\nrelationkBTMIT = 1.14/epsilon1oe−1\nλe, whereλe=Un(EF)\nis the electron-electron coupling constant38. Assuming\nthe same density of states n(EF), for CCO and CCO1,\none can see the relatively large Uin case of CCO1. It is\nto be noted that large Ulinks to more localization and\nless AFM exchange ( J/bardbl∝t2\nU) between spins. Interest-\ningly, this scenario is in accordance with the low value6\nofθPobtained for CCO1 (Fig. S315). However, the\nabove mentioned argument is based on the assumption\nthat both samples (CCO & CCO1) have the same nest-\ning vector/magnetic vector ( q) (i.e., same DOS), while\nin reality the position of the Fermi level controls the q.\nIn conclusion, we have studied the pure and Tb sub-\nstituted CCO by means of DC magnetization, neutron\ndiffraction, XPS, EXAFS and resistivity measurements.\nExchange bias has been observed in both the samples.\nGlassiness has been found as the origin of larger hystere-\nsis in CCO than in CCO1. Interlayer coupling between\ntriangular (CoO 2) and rocksalt (Ca 2CoO 3) has been at-\ntributed as the reason behind ferrimagnetism. Magne-\ntocrystalline anisotropy in the rocksalt layer acts as pin-\nning for EB. Neutron diffraction and EXAFS results com-\nbinedly hints that incommensurate SDW is present in\nthe triangular layer, which tends to short ranged be-\nlow TFerri and ultimately turn into glassy phase due to\nstronger interlayer coupling with cluster ferromagnetismin the rocksalt layer. Weaker AFM correlation observed\nin CCO1 is substantiated by increased correlation effects\nas manifested from electrical transport data, highlighting\nthe intricate relation between magnetism and electron\ncorrelations in these samples.\nThe authors are thankful to Sharad Karwal and A.\nWadikar for help during XPS measurements. AA ac-\nknowledges UGC, New Delhi for providing financial sup-\nport in form of the MANF scheme (Grant No. 2016-\n17/MANF-2015-17-UTT-53853). DKS acknowledges fi-\nnancial support from DST-New Delhi through Grant No.\nINT/RUS/RFBR/P-269 and SERB-New Delhi through\nECR award grant. DST-DESY project is acknowledged\nfor financial support for performing synchrotron EXAFS\nstudies. 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Shukla2,†\n1Department of Physics, Aligarh Muslim University, Aligarh 202002, India\n2UGC-DAE Consortium for Scientific Research, Indore 452001, India\n3Department of Physics, Central University of Punjab, Bathinda Punjab 151001, India\n4LSNCM-NFA, Institute of Physics,\nUnB, Brasilia DF 70910 900, Brazil\n5ISIS Facility, Rutherford Appleton Laboratory,\nChilton, Didcot OX11 0QX, United Kingdom\n6Deutsches Elektronen-Synchrotron, Notkestrasse 85, D-22607 Hamburg, Germany\n(Dated: October 20, 2020)\n∗Present address: Optical Physics Lab, Institute of Physics, Academia Sinica, Taipei, Taiwan.\n†Corresponding Author: dkshukla@csr.res.in\n1arXiv:2004.08319v2  [cond-mat.mtrl-sci]  17 Oct 2020I. MAGNETIC ENTROPY\nFig. S1 (a & b) displays the change in magnetic entropy (∆ SM) with respect to tempera-\nture, for CCO and CCO1, respectively, calculated for different applied fields using Maxwell’s\nequation, ∆ SM=/integraltextb\na/parenleftbigdM\ndT/parenrightbig\nHdH. A transition is found at ∼10 K which we denote as the\nTFerri. This reflects the first derivative of M (T) while in literature T Ferri was calculated\nby the upturn of M(T). For CCO1, no such transition observed in ∆ SM.\nFIG. S1. Temperature dependent ∆ SMfor different fields. Inset shows the 3D contour plot (∆ SM-\nH-T) (a) for CCO and (b) for CCO1.\n2II. ARROTT PLOTS\nFig. S2 (a & b) shows the Arrott plot (5 to 101 K), representing absence of spontaneous\nmagnetization (no intercept at y-axis) in both, CCO and CCO1.\nFIG. S2. Arrott plot between 5 to 101 K showing absence of spontaneous magnetization (a) for\nCCO, and (b) for CCO1.\n3III. CURIE-WEISS FITTING\nCurie-Weiss fitting to χ−1vs T data measured at 10 Oe shows the negative Weiss tem-\nperature (θp) indicating the AFM correlations but weaker in case of CCO1 than in CCO.\nFIG. S3. Curie-Weiss fits to the temperature dependent inverse susceptibility ( χ−1(T)) data of (a)\nCCO and (b) CCO1.\n4IV. X-RAY PHOTOEMISSION SPECTROSCOPY\nFig. S4 (a & b) display the Co 2p and O 1s XPS results for CCO and (c & d) show the\ncorresponding for CCO1. The fraction of Co3+(calculated from the peak ratios) in CCO is\n∼68 % while for CCO1 it is ∼69.5 %.\nFIG. S4. (a & b) Co 2p and O 1s XPS spectra showing the mixed valency of Co (+3 & +4) and\noxygen vacancy in the lattice, for CCO (c & d) for CCO1.\n5"    },    {        "title": "2004.11591v1.Observation_of_magnetic_domain_and_bubble_structures_in_magnetoelectric_Sr__3_Co__2_Fe___24__O___41__.pdf",        "content": "Observation of m agnetic  domain  and bubble structures in magnetoelectric  \nSr3Co 2Fe24O41 \n \nH. Nakajima1, 2, H. Kawase3, K. Kurushima4, A. Kotani1, T. Kimura5, and S. Mori1 \n1Department of Materials Science, Osaka Prefecture University, Sakai, Osaka 599 -8531, Japan  \n2Department of Applied Quantum Physics and Nuclear Engineering, Kyushu University, Fukuoka 819 -0395, Japan  \n3Division of Materials Physics, Gra duate School of Engineering Science, Osaka University, Toyonaka, Osaka, 560 -\n8531, Japan  \n4Toray Research Center, Ohtsu, Shiga 520 -8567, Japan  \n5Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277 -8561, Japan  \n \nThe m agnetic  domain  and bubble  structures  in the Z-type hexaferrite Sr 3Co2Fe24O41 were investigated using  Lorentz \nmicroscopy . This hexaferrite  exhibit s a room -temperature magnetoelectric effect  that is attributed to its transverse conical  \nspin structure  (TC phase) . Upon heating , the TC phase transforms into  a ferrimagnetic phase  with magnetic moments in the \nhexagonal  ab plane  between 410 and 480 K  (FM2  phase ) and into another ferr imagnetic phase with moments parallel  to the c \naxis between  490 and 680 K  (FM1  phase ). Accordingly,  in this study,  the magnetic  domain  structure s in Sr3Co2Fe24O41 were \nobserved to change  dramatically with temperature.  In the TC phase , irregular fine magnetic domains were observed  after \ncooling the specimen from the FM2 to TC  phase . In the FM1 phase , stripe d magnetic domain  walls with p airs of bright and \ndark contrast  were formed parallel to  the c axis. Upon applying an external magnetic field, the stripe d magnetic domain walls \ntransformed into  magnetic  bubbles.  The topology of the  magnetic bubbles was dependent on  the angle between the external \nmagnetic field ( H) direction and the easy c axis. Namely, m agnetic bubbles with the topological number N = 1 (type  I) were \ncreated for H//c, whereas magnetic bubbles with N = 0 (type  II) were created when the magnetic field was tilted from the c \naxis by 5 °. We attribute the high magnetocrystalline  anisotropy  of Sr 3Co2Fe24O41 to the em ergence of magnetic bubbles in  the \nFM1  phase . \n \n \nI. INTRODUCTION  \nMagnetoelectric  (ME)  materials  have been the subject of intensive studies from both scientific and \ntechnol ogical viewpoints because their  ferroelectricity  (magnetism)  can be controlled with the application of  a \nmagnetic  (electric)  field [1]. Helimagnets ar e of particular interest because they exhibit  gigantic  ME responses \ndue to  the inverse Dzyaloshinskii –Moriya (DM) interaction  or spin –current mechanism  [2, 3]. Among this class \nof materials , hexagonal ferrites (hexaferrites)  showing helimagnetic orders are promising candidates for future \nspintronic devices  and multibit memory  because many hexaferrites exhibit the ME effect , and some of them 2 \n exhibit  a significant ME effect with the application of  relatively low magnetic fields  [4]. For example,  Y-type \nhexaferrite s, such as  Ba0.5Sr1.5Zn2Fe12O22 and Ba2Ni2Fe12O22, and M-type hexaferrites , such as \nBaFe12−x−0.05ScxMg 0.05O19, exhibit  proper screw or longitudinal conical  structure s in zero magnetic field , and upon \napplication of  a magnetic field, they exhibit  ferroelectricity  and the ME effect  via the spin–current mechanis m [5–\n7]. Although some of these hexaferrites  have helimagnetic transition temperature s above room temperature , their \nME activity is far below the transition temperature (~100 K) because of  their low resistivity.  By contrast , Z-type \nSr3Co2Fe24O41 has been reported to exhibit  high resistivity (~ 1.3 × 109 Ω cm ) and a low -field ME effect at room \ntemperature  [8]. Electric polarization in Z-type hexaferrites is induced by a magnetic field of several mil li-tesla, \nand the polarization does not vanish  during magnetic -field cycling.   \nZ-type hexaferrites  belong  to the space group P63/mmc, and their crystal lographic unit cell s contain 30 \ntransition  metal ions, Fe3+ and Co2+, with octahedral, tetrahedral,  and fivefold  coordination  [9, 10]. However,  the \nmagnetic unit cell  in a Z-type hexaferrite can be described by alternate stack s of large and small magnetic  \nmoments.  The magnetic structure in zero magnetic field  consists  of a transverse conical structure  whose wave \nvector is expressed as q = (0, 0, 1) in the hexagonal setting  [11]. The transverse conical structure can induce  \nelectric polarization perpendicular to both the net magnetization and c axis because of  the cycloi dal component of \nthe magnetic structure.  Moreover,  the Z-type hexaferrite  Ba0.52Sr2.48Co2Fe24O41 exhibits a large ME susceptibility \nof 3200 ps/m at room temperature , and the magnetization can be controlled by application of an electric field  [12]. \nIn ME measurements using single crystals [12], the electric polarization was almost zero in zero magnetic field \ndespite the transverse conical structure  formed . This result  indicates  that the transverse conical domains were \nrandomly distributed in the speci men. However, the magnetic domain s in a Z -type hexaferrite  have not yet been \nexamined  even though several studies on magnetic domains in Y-type and M -type ME hexaferrites  have been \nreported  [13–18]. Furthermore , previous magnetization measurements have reveal ed that Z-type hexaferrite s have  \nthree magnetic phase transition s and show magnetization along the  easy axis (c axis) in a ferri magnetic phase  \nabove the transverse  conical  phase  [8, 12 ]. These results  suggest a high magnetocrystalline anisotropy  and the \nemergence of magnetic bubbles  in the ferrimagnetic phase  of Z-type Sr3Co2Fe24O41. Therefore, it is important  to 3 \n determine the magnetic domain structures in both the transverse conical and ferrimagnetic phases of \nSr3Co2Fe24O41. \nIn this study , we evaluated  the magnetic domain structures in magnetoelectric  Sr3Co2Fe24O41 using Lorentz \nmicroscopy.  We identif ied the structural characteristics  of the magnetic domains in the transverse conical  and \nferrimagnetic phases  and observed  the formation of magnetic bubbles  in the ferrimagnetic phase. Depending on \nthe orientation of the applied magnetic field with respect to the crystallographic axis, t wo types of magnetic \nbubbles were formed by tilting the c axis of the specimen  from the ma gnetic fields. We discuss the evolution of \nthese magnetic domains and the emer gence of  magnetic bubble s by considering the magnetic anisotropy in the \nhexaferrite . \n \nII. EXPERIMENTAL METHOD  \nSingle crystals of Sr3Co2Fe24O41 (hereafter SCFO) were grown using a flux method  (Fe 2O3–Na2O flux) . The \ncrystal s had typical dimension s of ~1.3  mm × 1.3 mm × 0.5 mm . The m agnetization  of the crystals was measured  \nas a function of temperature T and magnetic field H using  a vibrating sample magnetometer with an oven option. \nThin specimens  were prepared for transmission electron microscopy  analysis using a focused ion beam or Ar -ion \nmilling after  mechanical polishing . High -angle annular dark field scanning tr ansmission elec tron microscopy \n(HAADF STEM) and energy -dispersive X-ray spectroscopy (EDS) were  performed using  a transmission electron \nmicroscope equipped with a spherical aberration corrector ( JEM -ARM 200CF). The STEM –EDS system was \nequipped with double silicon drift detectors , and the solid angle for the entire  collection system was \napproximately 1 .9 sr. The  energy of the EDS mappings was  set as follows for each element: Fe K-edge , 6263–\n6533 eV ; Co K-edge , 6789–7069 eV; and Sr K-edge , 1396 8–14358 eV.  Lorentz images were obtained using \nanother  transmission electron microscope (JEM -2100F) operated at 200 kV , and the objective lens of the \nmicroscope was used to apply external magnetic fields perpendicular to the widest face of the thin specimens . The \nFresnel method  (out-of-focus method)  was used to examine the magnetic domain walls  [19, 20].  Here  we briefly \ndescribe the pri nciple of imaging using the Fresnel method (Fig. 1) . When the i ncident electrons pass through  \nmagnetic domain s, the electrons are d eflected by the Lorentz force due to the magnetization , causing the electrons 4 \n to diverge or converge at the domain walls . If the  focus of the imaging lens is overfocused,  dark and bright  lines \nappear at the  left and right  domain wall positions , respectively . The contrast  at the domain walls  in the \noverfocused  image  is revers ed if the imaging lens is underfocused . Further,  the contrast disappear s for an in -focus \ncondition.  \n \nIII. RESULTS  \nA. Atomic structure  \nThe atomic structure of t he single crystal was identified using HAADF STEM  and EDS . Figure 2 presents  an \nHAADF image  [(a)] and chemical maps  [(b)–(e)] at atomic resolution  along the a direction. Figure 2(f) is a \nstructural model of a Z -type hexaferrite based on Rietveld  analysis using neutron diffraction [ 9, 10 ]. The model \nmatches well  with the STEM and EDS images ; therefore, the grown single crystal was confirmed to be a Z -type \nhexaferrite.  Note that  although the intensity in HAADF STEM is roughly proportion al to the atomic number Z2 \n[21], some Fe sites are brighter than Sr sites because  several  Fe atoms are stacked along the a axis. The Co atoms \nshowed a clear  a preference for the Me4(12k) site near Sr atoms. Structural analysis of SCFO  based on neutron \ndiffraction [9] previously revealed  that the Co site occupanc ies in the transition  metal sites Me1(2a) and Me4(12k) \nwere  higher than those  in other transition  metal sites. Comparing the EDS maps of Co [Fig. 2(c)] and the \nstructural model  [Fig. 2(f)], the Me4(12k) site  is dominantly occupied by Co atoms , which is consistent with the  \nprevious  structural analysis  [9]. \n \nB. Magnetic phase transitions  \nThe m agnetization M was measured to determine the magnetic phase transitions . Figure  3 shows the \nmagnetization measured at 10 mT as a function of temperature. The temperature dependence of the magnetization \nshows two clear anomalies at approximately 4 80 and 6 80 K. The rise of the magnetization at ~6 80 K corresponds \nto a transition from a paramagnetic to a ferrimagnetic phase.  The net magnetization was oriented parallel to the c \naxis between 480  and 680 K, whereas below 480 K , the direction of the magnetization was parallel to the  ab plane . 5 \n These results are  consistent with  those reported in  a previous neutron diffraction study [ 10]. The small jump at \n~410 K in the M–T curve  for H//c indicates  a transition from the ferrimagnetic to transverse conical  phase  [see \ninset of Fig. 3(a)]. We define the ferrimagnetic phase at 480 K ≤ T ≤ 680 K as the FM1  phase , the ferrimagnetic \nphase at 410 K ≤ T ≤ 480 K as the FM2  phase , and the transverse conical  phase below 410 K as the TC phase . \nFigure 3(b) shows the magnetic -field dependence of the magnetization in the respective phases ( at 300 K for TC, \n430 K for  FM2, and 520 K for FM1) . Metamagnetic -like stepwise transitions were only observed in the TC phase \nwhen  H was applied perpendicular to the c axis, which  can be explained in terms of successive chang es of the \ntransverse conical  axis [ 22]. The first increase of the magnetization corresponds to the reorientation that causes \nthe helical axis to point in the magnetic -field direction.  The second broad increase of the magnetizat ion indicates \nthat the cone angle of the helical structure decrease s with increasing  magnetic field. Although the magnetization \nperpendicular to the c axis showed metamagnetism, the magnetization change parallel to the c axis was less \npronounced because the ab plane is the easy plane  below 480 K. At 430 K  (FM2) , the saturation magnetic field \nfor H//c was smaller  than that for H ⊥ c, indicating that the easy axis of magnetization was perpendicular to the c \naxis. Conversely, the  magnetic  easy axis became parallel to the c axis in the FM1 phase at 520 K , which  is \nconsistent with the temperature dependence of the magnetization.  \n \nC. Examination of magnetic domain structures  \nNext , we investigated  the effect of these magnetic phase transitions on the magnetic domain  structures \nusing the Fresnel method.  Figure 4 shows the observed magnetic domain structures projected in the direction s \nperpendicular [Fig s. 4(a)–(c)] and parallel [Fig s. 4(d)–(e)] to the c axis in the respective phase s. To examine  the \nformation process of the magnetic domains, the observation  was performed during cooling and heating for the \ndata presented in Figs. 4(a)–(c) and 4(d)–(e), respectively , because no magnetic domain was observed in Figs. \n4(c) and 4(d).  This process eliminate d the possibility that the observed magnetic domains were  formed during the \nthinning stage of the specimen preparation.  Figures 4(a) and 4(d) present  magnetic domain images projected in \ndirection s perpendicular and parallel to the c axis, respectively, in the TC phase. Basing on these result s, one can \nconclude that the  TC phase contains  numerous  fine stripe -shaped  domains with typical domain width s of ~20 nm 6 \n and that the domain boundary is perpendicular to  the c axis. Because t he stripe -shaped  domain size is  small and \nthe spatial resolution is not high in the Fresnel image  because of  the defocus, the domain walls  (the bright and \ndark contrast) appear  to be combined.  The transverse conical domains in Fig. 4(a) indicate that the conical axes in \nthe domains change their directions at a nanometer scale because the bright and dark contrast depend on the \ndirection of the macroscopic magnetization in each domain.  In the FM2  phase  [Figs. 4(b) and (e) ], 180 ° magnetic \ndomains with their boundar ies perpendicular to the c axis were formed , and the magnetization direction was \nperpendicular to the c axis. This direction is consistent with the easy axis revealed by the magnetization \nmeasurements displayed in Fig. 3. The 180° magnetic domain width was ~400 nm. The left and right magnetized \ndomains were almost  equally populated , and t hese domains were separated by  magnetic domain walls showing \nbright or dark contrast . Note that the type of domain wall (Bloch or N éel wall) cannot be determined from the \nimage because the contrast at the domain  walls was caused by the magnetization in  the 180° domains , and no \ndifference is observed  between Bloch and N éel walls [ 23]. During the phase transition from the FM2 to TC phase , \nsome domain structure s remain ed unchanged ; however, the fine striped domains disappear ed in the FM2  phase  \n(as observed by comparing the areas in Figs. 4(a) and 4(b) marked by white and black lines) . The contrast of the \nstripe -shaped  domains was caused by the convergence or divergence of  the electron s deflected by the \nmagnetization s between  the domains. Thus, we consider  the directions of magnetization in the stripe -shaped \ndomains to be opposite those in the FM2 phase , as illustrated in Fig s. 4(g) and (h) . In the FM1  phase , further \ndrastic change in the magnetic domain structure occur red. Unlike in the TC and FM2  phases , no magnetic contrast \nwas observed  in the image projected in the direction perpendicular to the c axis [Fig. 4(c)], whereas pairs of bright \nand dark contrast were formed in the FM1  phase  [Fig. 4(f)]. This result indicates  that up - and down -magnetized \ndomains  (M//c) were formed  and that Bloch walls were located between th e domains  in the FM1 phase , as \nillustrated  in Fig. 4(i). A Bloch wall can be visualized as a pair of bright and dark contrast when the magnetization  \nin the domains  is parallel to the incident beam  (the out -of-plane magnetization) , and the Bloch wall has an in-\nplane component  [24]. The magnetization was parallel t o the c axis, which  agrees with the macroscopic \nmagnetization  measurements [Fig. 3(a)]. The w avy patterns  in Fig. 4(f) demonstrate  that the magnetic anisotropy \nis equivalent within the ab plane.   7 \n  \nD. Observation of magnetic bubbles  \nThe magnetic -field dependence of the magnetization [Fig. 3(b)] show ed a strong uniaxial magnetic \nbehavior  in the FM1 phase . Recent Lorentz microscopy experiments demonstrated that magnetic bubbles are \nformed in magnetic materials with a high magnetocrystalline  anisotropy ( Ku) [25–27]. Therefore , magnetic \nbubbles would be expected to form in the FM1 phase upon applying a magnetic field  to a thin specimen  along the \nc axis with the widest face perpendicular to the c axis. The result s of such an experiment  are displayed in Fig. 5. \nNote that the magnetic -field direction was downward  with respect to the magnetization  in these experiments . \nWhen a magnetic field  of 95 mT  was applied  at 520 K  along the c axis, the down -magnetized domains expanded \nand the up-magnetized domains shrank. As a result , circularly rotating magnetic domain walls  were created , as \nobserved  in Fig. 5(a). The magnetic bubbles were all type I, in which  the magnetic domain walls were magnetized \ncontinuously clockwise or counterclockwise , as shown in  Fig. 5 (d). The magnetic domain structure  had the same \ntopological number  [N =∬1\n4𝜋𝒏∙(𝜕𝒏\n𝜕𝑥×𝜕𝒏\n𝜕𝑦)𝑑𝑆 = 1] as that in  magnetic skyrmions , where n = M/|M| is the unit of \nthe magnetization vector  [14]. The diam eter of the magnetic bubbles was approximately  300 nm, which is a \ntypical size for magnetic bubble s [28, 29 ]. \nWe also investigated the relation  between the easy axis and magnetic -field dir ection. Figure s 5(b) and 5(c) \nshow  the magnetic -field dependence of the magnetic domains when the c axis of the s pecimen  was tilted by 5° \nfrom the magnetic -field direction. The magnetic domains were  pinched off from the edges to create type-II \nmagnetic bubbles with increasing magnetic fields . The type-II magnetic bubble consisted of two parallel domain \nwalls with a pair of Bloch lines , as illustrated in Fig. 5(e). Unlike the type-I structure , the type-II structure did not \nhave a topological number ( N = 0). The type -II magnetic bubble had a higher domain wall energy than the  type-I \nmagnetic bubble because the Bloch line contained in-plane magnetic components. The energy loss was \ncompensated by the Zeeman energy caused by the in -plane external magnetic field, and thus , the type-II magnetic \nbubble s were stabilized in thi s configuration. The change in the type of  magnetic bubble s achieved by tilting a  \nspecimen  was also recently  reported in  a uniaxial magnet,  BaFe12−x−0.05ScxMg 0.05O19 [13]. 8 \n  \nIV. DISCUSSION   \nA. Transverse conical  domain structure  \nThe fine stripe-shaped domain structures observed at 300 K (TC) develop ed when the specimen  underwent a \nphase transition  from  the FM2 phase with the 180 ° domain s. Therefo re, we consider these fine domains to be \nassociated with  transverse conical  domain s. Soda and coworkers  reported that the propagation vector of the \ntransverse conical  structure is  q = (0, 0, 1)  based on their neutron diffraction  study  [11]. Thus , the helical period is \n~5.2 nm  (the lattice constant of the c axis) . The fine irregular  domains of ~20 nm [Fig. 4(a)] correspond to \napproximately four periods  of the unit cell . We believe that this  short coheren ce length of the transverse conical \ndomains originates from the microscopic mechanism that stabilize s the helical structure because the observed fine \ndomains differ  from helical domains induced by the DM interaction.  In chiral magnets , because of  the DM \ninteraction, helical domains are long wavelength and continuous throughout a specimen even when def ects or \nedges are present [30, 31 ]. This finding is observed because the DM interaction depends on the crystal structure, \nwhich determines the helical structure. Unlike in chiral magnets, i n Z-type hexaferrite, the transverse conical \nstructure is caused by the superexchange interaction in an Fe –O–Fe bond  [4, 11]. The Fe atoms are located across \nthe boundary between the blocks that form  the large and small magnetic moments , and Ba and Sr ions are located \nnear the boundary . Therefore, the interaction between the block s along the c axis is considered small compared \nwith the ferromagnetic exchange interaction within  the ab plane. Consequently , the transverse conical domains \nhave a short period along the c axis, whereas the interaction  perpendicular to the c axis is long , producing the \nirregular stripy  domains observed in Fig. 4(a).  \nWe also comp are the helical  domains in SCFO with those in MnP  because MnP has similar characteristics to \nSCFO . MnP is ferromagnetic between 47 and 291 K and exhibits helimagnetism below 47 K. The helimagnetism \nis thought to be realized  through several superexchange interactions , and t he helical period is 9 a ~ 5.3 nm ( a is the \nlattice constant)  [32], although the DM interaction has been reported to modify  the spin configuration  [33]. The \nferromagnetic domains are 180 ° domains separated by the Bloch walls  in the ferromagnetic phase  [34]. The chiral \ndomain s (left- and right -handed helical domains)  of MnP have been  previously  examined using polarized neutron 9 \n diffraction topography  by Pat terson and coworkers [35]. The chiral domain patterns  in the helical phase  had \nirregular striped shape s, and the domains were elongated along the easy axis . They  reported that the first-order \nferro -helimagnetic transition of the stripe -type domains started  in the Bloch walls . The nucleation in the helical \nphase starts in a Bloch wall, and its growth occurs mainly along the easy axis to minimize its magnetostatic \nenergy. Although the type and  size of helical domains  differ , we speculate that the same discussion can be applied \nto SCFO . That is,  the helical domains grow  along the ab plane (the easy plane) , and thus , the strip ed domains are \nformed  to reduce  the magnetostatic energy , as observed in Fig. 4(a). We note that the helical domains were \nnanosize d in this study because the domain size tends to decrease in a thin foil compared with that  in the bulk to \nminimize  the magnetostatic energy (the magnetic flux density generated from the specimen  to vacuum). \nFurthermore , a recent resonant X -ray diffraction  study of a Y -type hexaferrite suggested  that stripe -type domains \nwith flat domain boundaries  extend  perpendicular  to the c axis [17]. The origin of this behavior is considered to \nbe the first -order ferro(fan) -helimagnetic transition starting in Bloch walls , as explained in the domains of MnP. \nConsidering the similarity of the crystal and magnetic structures of Y-type and Z -type hexaferrites, the same \nphenomena could be possible in the transition from the ferrimagnetic to helical phases in SCFO.  \n \nB. Formation of magnetic bubbles  \nIn the FM1 phase, w e observed magnetic bubbles whose  type (type  I or II) depend ed on the magnetic -field \ndirection. To form  magnetic bubbles, the anisotropy  field Hk (= 2Ku/Ms) must  be larger than the demagnetizing \nfield 4Ms, where Ku is the uniaxial anisotropy coefficient and Ms is the sat uration magnetization [ 25, 26, 29 ]. The \nanisotropy field Hk is defined  as the critical value  at which  the magnetizations parallel and perpendicular to the c-\naxis are the same , as indicated by the arrowheads in Fig . 3(b). If this requirement  is not satisfied, the \nmagnetization points parallel to the surface of a thin -film sample because of  the magnetic shape anisotropy ; the \nmagnetocrystalline anisotropy does not determine the magnetization direction.  Figure 3(b) shows that t he \nanisotropy field Hk was 0.47 T (4700 G)  and that the sa turation magnetization Ms was 13 B/f.u. (179 G)  at 520 K. \nThus, the requirement for  the formation of  magnetic bubbles Q = Hk/4Ms = Ku/2Ms2 ~ 2.0 > 1 was fulfilled  in \nthe FM1  phase . Here, the ratio of the anisotropy and demagnetizing fields Q is called the quality factor . The 10 \n uniaxial anisotropy coefficient Ku (=HkMs/2) obtained from Hk and Ms was 1.4 × 104 J/m3, which is  relatively \nsmaller than  that of other magnetic oxides  exhibiting magnetic bubbles  [28]. However, the magnetization Ms was \nalso smaller  because of the ferrimagnetism  in SCF O, resulting in satisfaction of the requirement . Note  that \nmagnetic bubbles were not produced in the FM2  phase . In the FM2 phase, Hk was also large , as indicated by the \nblack arrowheads  in Fig. 3(b). However, the magnetization can rotate  within the ab plane  in the FM2 phase \nbecause  the easy plane is the ab plane. Therefore, the magnetic domains were 180° domains in the FM2 phase \nbecause of  the magnetic shape anisotropy, as observed in Fig. 4(b). Thus,  the striped magnetic domains  observed  \nin the FM1 phase were not formed  in the FM2 phase . \nIn addition,  the magnetic bubbles  observed  in this paper are  the typical structure whose magnetizations \ngradually  point in the out -of-plane direction from the wall position . However, it has been reported that another \nmagnetoelectric hexaferrite (Sc -substituted M -type hexaferrite) forms magnetic bubbles with helicity reversals  \n[13]. The bubble  structure s were  observed  in M -type hexaferrite with small uniaxial anisotropy  because of  the Sc \nsubstitution  (Q ~ 1) and thin -film thickness ( h ~ 30 nm).  In the study, t he helicity reversals  were reproduced by \nnumerical simulation  based on the Garel –Doniach model  [36], which describe s ferromagnetic states  (stripes and \nbubbles)  in thin films using two parameters (the uniaxial anisotropy and film thickness) . In our study , the quality \nfactor (Q = 2.0)  was higher than that of  M-type hexaferrite  and the thickness was likely thicker than  that of the \nprevious study , which explains the difference s between the bubble  structures.  \nV. SUMMARY  \nIn summary, we studied the evolution of magnetic  domain  structures in  a ME hexaferrite,  Sr3Co2Fe24O41, \nforming various magnetical ly ordered phases by changing  the temperature and magnetic fields. Using Lorentz \nmicroscopy , we observed  that this hexaferrite contained characteristic domain structures  such as  fine stripe d \nmagnetic domain s in a ME transverse conical phase  as well as  magnetic  bubbles upon application of  magnetic \nfields  in a high -temperature ferrimagnetic phase . In addition, we reveal ed the uniaxial behavior  and change of the \neasy axis  in the ferrimagnetic phases  based on magnetization measurements.  Z-type hexaferrites exhibit high \nresistivity , and thus , the electric -field-driven motion of magnetic bubbles, which depends on the topological 11 \n number  (N = 0 or 1) , is expected  [37, 38 ]. Our findings demonstrate the versatility of a room -temperature ME Z-\ntype hexaferrite for spintronic memory  devices  that use the magnetic bubble properties  and magnetoelectric effect . \nACKNOWLEDGEMENTS  \nWe thank K. Shimada and K. Harada ( RIKEN CEMS) for the sample preparations using the focused ion \nbeam. 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Atomic-resolution EDS chemical maps showing the (b) Fe -\nK, (c) Co -K, and (d) Sr -K edges . (e) M erged image of (b) –(d). The incident electron beam is along the a \naxis, and the white arrow in panel (c) shows the direction of  the c axis. (f) Schematic of the crystal \nstructure  of SCFO  along the a axis. The b lue, red, and yellow balls represent Fe, Sr, and O atoms , \nrespectively . The metal site s, Me4(12k), are marked in green.  \n16 \n   \nFIG. 3. (a) Temperature dependence of the magnetization parallel and perpendicular to the c axis of \nSCFO . The inset show s the derivative of the magnetization (dM/dT) at approximately 4 00 K.  \nSchematics of the magnetization  directions are indicated  by red arrows.  PM, FM, and TC represent the \nparamagnetic, ferrimagnetic, and transverse conical phases, respectively. (b) Magnetic -field dependence \nof the m agnetization parallel and perpendicular to the c axis at various temperatures. The a rrowheads \nmark  the anisotropy field Hk. \n17 \n  \n \nFIG. 4. Magnetic domain structures in the TC [(a) and (d)], FM2 [(b) and (e)], and FM1 [(c) and (f)] \nphase s observed using the Fresnel method.  The widest face s of the thin sample s used for these \nexperiments were parallel [ (a)–(c)] and perpendicular [ (d)–(f)] to the c axis. The w hite and black lines \nin (a) and (b) represent  bright and dark contrast , respectively . The defocus value was −5 m. (g)–(i) \nSchematics of the magnetic domains in panel s (a), (b), and (f), respectively. The a rrowheads mark the  \ndomain walls in the FM1 and FM2 phases. The r ed arrows represent the magnetization directions in the \nrespective  domain s and domain wall s. \n  \n18 \n  \n \n \nFIG. 5. (a) Fresnel  image of clockwise and counter clockwise type -I magnetic bubbles  at 95 mT and 520 \nK (FM1 phase) . The plane is perpendicular to the c axis, and t he magnetic field was applied parallel to \nthe c axis. The defocus value was −2 m. (b), (c) Formation process of type -II magnetic bubbles at 520 \nK (FM1 phase). The specimen  (the c axis) was tilted by 5° from the magnetic -field direction. The \ndefocus value was −5 m. The inset of (c) presents a magnified image of the magnetic bubble marked \nby a red dashed rectang le. Schematics of magnetic bubbles of (d) type I and (e) type II and their \ncorresponding Fresnel images in an underfocus ed condition . The blue arrowheads mark  the positions of \nBloch lines, where the chirality of a domain wall is reversed.  \n"    },    {        "title": "2005.00716v1.Field_temperature_phase_diagram_of_magnetic_bubbles_spanning_charge___orbital_ordered_and_metallic_phases_in_La___1_x__Sr__x_MnO__3____x___0_125__.pdf",        "content": "1 \n Field-temperature phase diagram  of magnetic bubbles  spanning \ncharge/orb ital ordered and metallic phases in La1−xSrxMnO 3 (x = \n0.125)  \n \nA. Kotani1, H. Nakajima1, K. Harada1,2, Y. Ishii1 and S. Mori1,* \n  \n1Department of Materials Science, Osaka Prefecture University, Sakai, Osaka 599 -\n8531, Japan.  \n2Center for Emergent Mat ter Science, the Institute of Physical and Chemical Research \n(RIKEN), Hatoyama, Saitama 350 -0395, Japan.  \n \n \nAbstract  \nWe re port formation of magnetic textures in the ferromagnetic (FM) phase  of \nLa1−xSrxMnO 3 for x = 0.125; these textures  are magnetic bubbles, magnetic stripe \ndomains, and forced FM states.  In situ  Lorentz microscopy (LM) observations show that \nmagnetic bubbles exist in the FM insulating phase accompanying the formation of the \ncharge/orbital ordering (CO/OO). Furthermore, stable magnetic bubbles still exist in an \nintermediate temperature region betw een the C O/OO ( TCO=155 K) and FM ( Tc =190 K) \ntransition temperatures. These magnetic bubbles are believed to originate from the \nmagnet ocrystalline  anisotropy and the dipole –dipole interaction in  the FM phase . Based \non in situ  LM observations as a function of  both temperature and the strength of the \nexternal magnetic field applied, a magneti c field –temperature phase diagram is \nconstructed,  exhibiting  the stabilizing regions of the magnetic bubbles  in the FM phase \nof La 0.875Sr0.125MnO 3. \n \nP.A.C.S numbers: 75.70Kw, 75.47.Lx, 75.78.Fq, 68.37.Lp  \n \n  2 \n Introduction  \nNumerous  nanoscale  magnetic textures have recently been investig ated with \nrespect  to appl ying them to  high -performance spin devices ; magnetic skyrmions, for \nexample, have been  found to possess vortex -like nanometric spin structures, and their \nspin swirling is defined b y an integer winding number [1 -5]. Studies have shown that \nmagnetic skyrmions are induced by the Dzyaloshinskii –Moriya (DM) interaction in \nchiral magnets that hav e a B20-type cubic structure without inversion symmetry; such \nmagnets include MnSi and FeGe [6 -10]. Magnetic bubbles in uniaxial ferromagnet s that \nhave inversion symmetry have  also been found to have  the same  spin structure as that \nof the skyrmions  [11]. Magnetic bubbles are induced by applying  external magnetic \nfields parallel to a magnetic easy axis , and they are stabilized by magnetic aniso tropy  \n[12-14]. Additionally , magnetic dipole –dipole interaction s stabilize the swirling -spin \nconfiguration in a magnetic bubble’s domain wall.  These magnetic bubbles  are called \ntype -I magnetic bubbles , and they  have the same winding number as the skyrmi ons. \nTherefore, the type -I magnetic bubbles have topologically non -trivial spin structure \nwith a finite topological  charge. On the other hand, the magnetic bubble s called type -II \nmagnetic bubble s without spin rotation have  topologically trivial structures.  A large \nnumber of magnetic skyrmions in metallic  magnets have been studied [4-10,15 -18], but \nfew studies have been  conducted  on magnetic skyrmion s in the insulating phase ; \nexamples of materials studied include Cu2OSeO 3 and GaV 4S8 [19,20, 21].  \nHole-doped La1−xSrxMnO 3 has been known to exhibit  various magnetic phases  \nand crystal structures [22-24]. In addition, many types of magnetic textures  such as \nmagnetic bubbles  and magnetic stripe domains  have been found in the ferromagnetic  \n(FM)  phase of La1−xSrxMnO 3 [25-28]. When x = 0.125, La1−xSrxMnO 3 exhibits two \nsuccessive structural phase transitions; one is a transition from an orthorhombic \nstructure that has a Pbnm symmetry ( O, c/√2 < b ≈ a) to a monoclinic structure that \nhas a  P121/c1 symmetry ( M′, a/√2 < b ≈ c) in the paramagnetic insulating ( PM-I) phase \naccompanying the Jahn –Teller (JT) distortion of  the Mn3+ ion. The other is  a transition \nfrom the monoclinic structure ( M′) in the ferromagnetic metallic  (FM -M) state below \nTC ≈ 190 K  to the monoclinic structure  (M′′) with a charge /orbital -ordered (CO/OO) \nstructure in the  ferromagnetic insulating  (FM -I) state at TCO ≈ 150 K [29]. The CO/OO \nstructure here is characterized as a modulated structure with q = <1/2 1/4  1/4> [30]. Li \net al ., however, found that the spin orienta tion in  the FM-I phase  at 5 K  is almost parallel \nto the [110] direction , where , for simplicity,  the notations are based on the high -3 \n temperature cubic structure with a 𝑃𝑚3̅𝑚 space group (SG).  Thus, i n the M′′ structure , \nthe magnetic easy axis keeps  almost parallel to the [110] direction in the FM-I phase  \n[29]. \nIn a previous study , we investigated magnetic bubbles in  the FM-M phase  of \nLa1−xSrxMnO 3 for x = 0.175 , which  had an orthorhombic  structure with  inversion \nsymmetry ; we used in situ Lorentz microscopy ( LM) observations combined with small -\nangle electron diffraction experiments  [26,27] . We found  that magnetic textures such as \nmagnetic bubbles  and magnetic stripe domains appear ed in La 0.825Sr0.175MnO 3, and they \ndepend ed significantly on  the magnetocrystalline anisotropy.  Recently, Nagai et al . \nexamined the magnetic domain structure s in the CO/OO FM-I phase of \nLa0.875Sr0.125MnO 3 using  LM observations , and they  found that magnetic stripe domains \ntransform  into elliptically shaped magnetic nanodomains  when an  external magnetic \nfield  is applied perpendicular to a thin film  [31]. Notably, in the LM observations \nconducted by Nagai et al. , the orientation of the magnetization was not parallel to the \nmagnetic easy axis  and consequently  the elliptical magnetic domain s were  formed by \napplicatio n of magnetic field along the [1 11] direction. From the research currently \navailable, it is unclear whether magnetic bubbles form owing to the application of an \nexternal magnetic field parallel to the magnetic easy axis in the FM -I phase of \nLa0.875Sr0.125MnO 3. \nIn this present work, we investigate d the magnetic textures in the FM phase of \nLa0.875Sr0.125MnO 3 as both a function of temperature  and the strength of external \nmagnetic fields ; this was done through the use of  in situ LM observations  and small -\nangle electron diffraction experiments.  In particular, we investigate d different  magnetic \ntextures  by applying an external magnetic field along the magnetic easy axis in the FM  \nphase of La0.875Sr0.125MnO 3. Our results show  that in the FM -I phase that has a CO/OO \nstructure,  type-I magnetic bubbles with clockwise (CW) and counter clockwise (CCW)  \nspin rotations  exist  and they are  stabl e. Using LM and a phase reconstruction technique \nbased on a transport -of-intensit y equation (TIE)  [32, 33], we found  the spin \nconfiguration  of the type -I magnetic bubbles . Because of  our experimental results , we \nwere able to  construct a magneti c field –temperature phase diagram in the FM  phase of \nLa0.875Sr0.125MnO 3.  \n \nExperimental method  \nSingle crystalline samples of La 0.875Sr0.125MnO 3 were prepared using a  4 \n conventional solid -state reaction  and the floating zone method  [22]. The crystal \nstructures were determined  by powder X -ray diffraction . Note that the magnetic easy \naxis of  La0.875Sr0.125MnO 3 is parallel to the [110]  direction.  We characterized the \ncrystalline orientations in the obtained single crystals  and determined  them  by back -\nplate Laue -type X -ray diffraction experiments. The electrical resistivity was measured \nusing a  Physical Property Measurement System (PPMS) (Quantum Design, Inc.) that \nused a  conventional four -terminal method . The magnetization was measured using a \nvibrating sample magnetometer (VSM) equipped with PPMS. The thin films for the LM \nobservation s were prepared  by Ar+-ion s puttering at room temperature. Our in situ LM \nobservations  were conducted by a  TEM (JEM -2100F ) that used a double -tilted cooling \nholder in the temperature region of 80 –298 K.  External magnetic field s were  applied to \nthe thin films by exciting the objective lens . For simplicity, all indices of the crystal \nplanes and orientation s were indexed  based  on the cubic perovskite structure \nwith  𝑃𝑚3̅𝑚 SG.  \n \nExperimental r esult s \nFirst, to confirm the structural  and physical properties  of the single crystals  of \nLa0.875Sr0.125MnO 3 that we re obtained, we investigated  the electron diffraction patterns  \nand the crystals’  electric al and magnetic properties. In the temperature ( T) dependence \nof the electric al resistivity ( ), as shown in Fig. 1(a),  there is a  kink  at approximately \n260 K ; this indicat es that there is  a structural  phase transition from the orthorhombic  to \nthe monoclinic structure . The increase in  at 190 K can be ascribed to a ferromagnetic \ntransition , and the decrease at 155 K  corresponds to the metal –insulator transition that \noccurs when  the CO /OO structure  forms . These results are consistent with those \nobtained by a previous study [3 4]. To clarify  both  the structural phase transitions and \nthe formation of the CO/OO structure , we investigated how the electron diffraction \npatterns varied with temperature between 80 and 298 K.  In the high -temperature \nparamagnetic phase, fundamental reflection spots due to the o rthorhombic structure  \nwith Pbnm  SG can be observed , as shown in the top panel of Fig.  1(b). A s the \ntemperature fell below 170  K, superlattice reflection spots appeared at the h, k,  and l + \n1/2 reciprocal  positions  in the electron diffraction pattern with the [110] incidence ; this \nimpl ies that a structural phase transition to the monoclinic structure with P121/c1 SG  \ntook place  (note that h, k,  and l are all integers ). As the  temperature decreased to below \nTCO = 155 K, superlattice reflection  spots  were observe d at the h, k, and l + 1/4 reciprocal 5 \n position s, as indicated by  the yellow a rrows in the lower panel of Fig.  1(b); these \nindicate the formation of the CO /OO structure . These results are consistent with those \nof a previous study that used electron  diffraction  measurement techniques  [30,35]. In \nthis earlier  study , the M′ structure in the intermediate temperature region ( Ts > T > TCO) \nwas characterized by the presence of a large JT distortion , while the M′′ structure in the \nlower temperature region ( i.e., below TCO) wa s characterized by the formation of the \nCO/OO structure . Revealing the magnetic anisotropy in the two monoclinic structures  \n(M′ and M′′) is essential to understand the conditions wherein  magnetic bubbles  are \nformed . Because of this , we examined different  magnetic anisotropy constants ( Ku) as \na function of temperature  in the FM  phase of La 0.875Sr0.125MnO 3. Figure  1(c) shows the \nchanges in the magnetization curves as a function of temperature in the FM  phase of \nLa0.875Sr0.125MnO 3. Magnetic fields were applied along both the [110] axis and the \ncrystal plane perpendicular to the [110] axis. The v alues of the saturated magnetization \nat T = 170, 120 , and 80 K were approximately 3.12, 3. 55, and 3.75 B, respectively. The \nanisotropic magnetic field ( Hk) in the FM phase was obtained f rom the magnetization \ncurves at T = 170, 120 , and 80 K. Hk is defined  as being the critical value of the magnetic \nfield above which the difference in magnetization between H // c and H // ab is less than \n2%. The values of Hk at T = 170, 120, and 80 K are 0.95, 0.77, and 0.76 T, respectively.  \nThe Ku values obtained from  the anisotropic magnetic field s of T = 170, 120 , and 80 K \nwere 2.31, 2.13 , and 2.22 × 105 J/m3, respectively.  These values satisfied the formation \nconditions of the magnetic bubbles  [12,13 ]. This suggests  that magnetic bubbles can be \nformed in the FM  phase of La 0.875Sr0.125MnO 3 when  an external magnetic field is applied \nperpendicular to  the thin films  in a manner that would make them  parallel to the \nmagnetic easy axis along the [110] direction.   \nWe investigated the magnetic textures  in the FM phase of La0.875Sr0.125MnO 3 by \nusing in situ LM observation s. We found that m agnetic stripe domains appear in the \nCO/OO FM-I phase at 90  K, as shown in Fig. 2(a). The residual magnetic field in the \nobject ive lens of  the TEM  was observed  to be  less than 20 mT. The s patial configuration \nof the spins in the magnetic stripe domains was observed  to be  the same as  that in the \nFM phase of La 0.825Sr0.175MnO 3 [26]. A significant feature observed  in the domains was \nthe formation of zigzag walls , as shown in the inset of Fig. 2(a). External magnetic \nfields were applied perpendicular to the thin films along the [110] magnetic easy axis \nby exciting the objective lens of the TEM.  As the strength of  the external magnetic \nfields increased to 340 mT,  the magnetic stripe domains changed into type -I magnetic 6 \n bubbles with  a diameter of 150 nm (Fig.  2(b)). Type -I and type -II magnetic bubbles  \nwere found to co-exist  in a random manner  at 420 mT , and this can be clearly seen in \nFig. 2(c). The s patial configurations of the spins inside the type -I and type -II magnetic \nbubbles are schematically described in Fig.  2(f) [27]. The presence of these two types \nof magnetic bubbles is entirely distinct from the magnetic nanodomains found by Nagai \net al . [31]; for the circ ular-shaped magnetic bubble found in this work, spins rotates in \nthe magnetic domain wall and are oriented along the [110] direction, which is clearly \ncontrary to spins simply directed along the [11 1̅] direction in the elliptical magnetic \ndomain in the previous work by Nagai et al .. This implies that appl ying external \nmagnetic fields along the [110] magnetic easy axis is essential for induc ing magnetic \nbubbles. As the strength of the external magnetic fields was increased  further , the size \nof the bubbles began to decrease, shrinking  from 150 to 100 nm at 420 mT  before \nbecom ing approximately 80 nm at 470 mT  (Fig.  2(d)). At 530 mT , the magnetic bubbles \ndisappear ed completely  and the forced ferromagnetic state  was realized , as shown in \nFig. 2(e).  \nTo clarify  the spatial conf iguration of the spins in the type -I magnetic bubbles in \nFig. 2(c), we analyzed the spin configuration s of the bubbles indicated by  the white \nsquares A and B in Fig.  3(a). A phase retrieval technique based on the transport -of-\nintensity equation (TIE)  was used in this analysis . The spatial distribution of the type -I \nmagnetic bubbles that had dark and bright contrasts at their centers stemmed from the \nclockwise (CW) and counter  clockwise (CCW) spin rotations at their respective domain \nwalls.  We analyzed the LM image s of the magnetic bubbles (Figs. 3 (b) and 3 (c)) using \nthe TIE analysis method, and with this we were able to reproduce the orientation and \nmagnitude of the i n-plane magnetization. We found that the distinctive feature of the \nbubbles in Fig.  3(c) was  that they had no bright or dark point s at the ir core, unlike the \ntype -I magnetic bubbles found in La0.825Sr0.175MnO 3 [26-28]. For comparison, the \nmagnetic bubble formed in La0.825Sr0.175MnO 3 is shown in Fig. 3(d), [27]. In Fig.  3(e), \na schematic illustration of the transition process from the magne tic stripe domains to \nthe type -I magnetic bubbles caused by the appl ication of  externa l magnetic fields  is \ngiven , and we found that it  is similar to the formation process of the type -I magnetic \nbubbles of La 0.825Sr0.175MnO 3 [26,27] .  \nAs shown in  Fig. 3, we can see  that type -I magnetic bubbles a re formed in the \nCO/OO FM-I phase.  We can see that La0.875Sr0.125MnO 3 exhibits an insulating -to-\nmetallic transition in the FM phase at approximately 150 K; as such, we focused on 7 \n whether magnetic bubbles are formed in the FM -M phase between 150 –195 K.  Figures \n4(a)–(c) show  a series of magnetic textures obtained at 170 K as a function of the \nstrength of the external magnetic fields ; magnetic stripe domains can be observed at 20 \nmT. The m agnetic stripe domain s in Fig. 4 (a) form elliptical domains  at 215 mT, as \nindicated by  the white broken lines  in Fig.  4(b) and the contour highlighted by the \nyellow broken lines in the inset  of Fig. 4(b) ; meanwhile, no magne tic contrast is \nobserved at  250 mT , as shown in Fig.  4 (c). Figures 4 (d)–(f), however,  show that the  \nmagnetic bubbles vary as a function of temperature in a constant  350 mT  magnetic field ; \ntype-I magnetic bubbles are formed at 100  K and 350 mT, as shown in Fig.  4(d). As the \ntemperature was increased in th is constant magnet ic field, the magnetic bubbles shrank \nfrom 150 nm to 100 nm at 130 K and  disappear ed altogether  at 160 K. These results \nsuggest that magnetic bubbles appear in the narrow range of 200–250 mT.  \nWe also investigated how the  magnetic stripe domains vary as a function of \ntemperature in a constant magnetic field  of 200 mT . The m agnetic s tripe domains at 130 \nK in Fig. 5 (a) start to transform into magnetic bubbles at 150 K  (Fig. 5(b)) , and the \ndomains  and bubbles co -exist at 170 K  (Fig. 5(c)) . No magnetic con trast can be observed \nabove the FM  transition temperature of TC =195 K. Our findings imply that magnetic \nbubbles  exist  and are  stabl e in the FM -M phase below TC = 195 K.  \nBased on  the in situ LM observations we made , we constructed a T–B phase \ndiagram with  respect to  the stable magnetic bubbles, as shown in Fig. 6. The circles in \nthe phase diagram show the experimental points determined  by the in situ LM \nobservation s. A significant feature in the phase diagram is that the existence of the  \nCO/OO structure has some  influence on the formation of  the magnetic bubbles , as \ndiscussed below . In addition, in the temperature range between 155–195 K  in the FM -\nM phase , the magn etic bubbles exist stably in the magnetic field region below 200 mT , \nand they  exist stably  in a magnetic field of 100 mT just below the FM transition \ntemperature of TC =195  K.  \n \nDiscussion  \nIn the inset of Fig. 2(a), we can see zigzag magnetic stripe domains in the walls \nof the CO/OO FM -I phase of La0.875Sr0.125MnO 3. In our previous work on the phase -\nseparated manganite, La 0.25Pr0.375Ca 0.375MnO 3 [36], FM domains with zigzag domain \nwalls were found to be  formed in the phase -separated state consisting of the CO /OO and \nFM-M state s. In the case of La 0.25Pr0.375Ca 0.375MnO 3, the nucleation of the FM -M 8 \n domains inside the PM -CO/OO phase occurred  as the temperature decrease d below TC \n= 50 K and zigzag magnetic domain walls separate d the FM domains  and the CO -OO \ndomains [36]. We believe  that the pinning force originating from the structural \ndifference s between the crystal structures in the FM CO /OO phase s had a pronounced \ninfluence on the phase boundary. However , the magnetic stripe domain wall s in \nLa0.875Sr0.125MnO 3 are characterized by zigzag magnetic domain wall s, as shown in the \ninset of Fig.  2(a). As reported by Asaka et al ., a CO/OO structure with a nanometric  \ncoherent length of approximately 10 nm was superimposed on  the FM state  [35]. It is \nsuggest ed that the JT distortion in  the CO/OO structure had some significant influence \non the formation of the magnetic stripe domains, which resulted in the zigzag magnetic \ndomain walls.  As such, a l ocal strain effect due to the JT distortion should originate in \nthe zigzag mag netic stripe domain walls.  \nThe formation of type -I magnetic bubbles without bright or dark points at their \ncenter is found in La0.875Sr0.125MnO 3, as shown in Fig. 3(a).  As we have previously \nreported , type -I magnetic bubble s possess a bright point at their core  in the FM-M phase \nof La0.825Sr0.175MnO 3 at 90 K  [26-28] in Fig. 3( d). The bright point originate s from the \nconvergence of the transmission electron beam due to the rotation of the magnetization \naround the core of a magnetic bubble. However , the magnetic bubble s in the FM-I phase \nof La0.875Sr0.125MnO 3 have no bright point ; this implies that the spatial configuration of  \nthe spins inside the magnetic bubbles of La0.875Sr0.125MnO 3 and La0.825Sr0.175MnO 3 are \ndifferent.  Consequently , we considered the magnet ocrystalline  anisotropy in the FM  \nphase of  La0.875Sr0.125MnO 3 based on the anisotropic field ( Hk) obtained in Fig. 1(a). \nThe values of the anisotropic field, Hk [T], and saturation magnetization, Ms [B / f.u.],  \nobtained  by magnetization at 8 T were 0.95  and 3.12 (at 170 K ), 0.77  and 3.55 (at 120 \nK), and 0.76  and 3.75 (at 80 K ). The values of the magnet ocrystalline  anisotropy \nconstant s (Ku) calculated  by using the equation Ku = (1/2) Hk × Ms at each temperature  \nin La 0.875Sr0.125MnO 3 were 2.22 (at 80 K ), 2.13 (at 120 K), and 2.31 × 105 J/m3 (at 170 \nK). The magnet ocrystalline  anisotropy in the FM phase has a constant value of \napproximately 2.2 × 105 J/m3, as while Hk increases with increasing temperature, Ms \ndecreases;  the effects of this are relatively greater than when 1.52  × 105 J/m3 obtained \nin the FM phase of La 0.825Sr0.175MnO 3 at 90 K [27] . Note that  in the LM observation s \nwe made for the  magnetic easy axis  along the [110] direction , the external magnetic \nfield  to the thin films is applied parallel to the orientation of the magnet ocrystalline \nanisotropy . This situation implies that the magnetization  in the sample is  oriented  along 9 \n the perpendicular direction to the  thin film s. The FM phase of La0.875Sr0.125MnO 3 \nexhibits greater magnet ocrystalline  anisotropy than that of La0.82 5Sr0.17 5MnO 3and, as a \nresult,  the spins inside the magnetic bubble s are oriented  so as to be  perpendicular to \nthe thin film ; this gives rise to  no bright/dark point s at the core of the magnetic bubbles . \nIn conclusion , the spatial configuration of the spins inside a magnetic bubble  depends \nstrongly on  the magnetocrystalline anisotropy .  \nIn addition, m agnetic bubbles  are stabilized by the  shape anisotropy interacting \nwith  the demagnetiz ing effect  except for the uni-axial magnetic anisotropy including \nthe magnetocrystalline anisotropy along the direction perpendicular to a thin film. \nTherefore,  magnetic bubbles are formed in the thin film  as the TEM samples  we \nobserved with the thickness less than  200 nm . In bulk s amples, the shape anisotropy is  \nreduced so that the magnetic bubble state is not stable. Skyrmions induced by DM \ninteractions also requires  the shape anisotropy. Our observation revealed  that magnetic  \nbubbles were  formed in the thin film of La0.875Sr0.125MnO 3. However,  it has been \nunclear whether they are  formed in the bulk  sample . \n \nSummary  \nMagnetic textures , including magnetic bubbles , in the FM phase of \nLa0.875Sr0.125MnO 3 were investigated by means of in situ LM observations as function s \nof both  temperature  and the strength of external magnetic fields. We found  that magnetic \nstripe domain s transformed into magnetic bubbles  in the CO/OO FM-I phase. In the \nmagnetic bubbles , the  spins rotat e at the domain wall s and are oriented perpendicular ly \nto the thin film s inside the magnetic bubbles . The s patial configuration of the spins \ninside the magnetic bubbles depends significantly on  both  the strength of the \nmagnet ocrystalline  anisotropy and the dipole –dipole interaction in  the FM phase . Based \non our in situ LM observations, w e constructed a magnetic field –temperature  phase \ndiagram for the FM phase of La 0.875Sr0.125MnO 3. \nThis is the first observation of magnetic bubbles for a  FM-I phase that has  a \nCO/OO structure . Stable magnetic bubbles co -exist with the CO/OO structure  in the \nspecific region of the phase diagram. Magnetic bubbles  with  100–200 nm in diameter \nin the insulating phase exhibit great potential for possible future a pplications of \nmagnetic bubbles, e.g., low -energy spintronic devices with unique magnetoelectric \ncouplings could be developed using them.  Moreover, La0.875Sr0.125MnO 3 exhibits the \nhomogeneous CO/OO state in the FM phase. On the other hand, it has been recognized 10 \n that perovskite manganese oxides such as (La,Pr) 5/8Ca 3/8MnO 3 exhibit the electronic \nphase separation between the FM -M and the CO/OO -I phases [36]. It is antici pated that \nour experimental results in the present work will help to understand the formation \nmechanism of magnetic bubbles in a variety of electronic phases in manganese oxides, \nwhich were produced by strong interaction s among the spin, orbital and charge  degrees \nof freedom of electrons. In addition, magnetic textures including magnetic bubbles and \nmagnetic skyrmions in functional materials should play some crucial roles to give rise \nto new functionalities such as topological Hall effects.  \n \nACKNOWLEDGEMENT S \nThis work was partially supported by Grant -in-Aid (No. 16H03833 and No. 15K13306) from \nthe Ministry of Education, Culture, Sports, Science  and Technology (MEXT), Japan.  \n  11 \n References  \n* mori@mtr.osakafu -u.ac.jp  \n[1] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer A. Rosch, A. Neubauer, R. Georgii, \nand P. Böni, Science 323, 915 (2009).  \n[2] M. Onoda, G. Tatara, and N. Nagaosa, J. Phys. Soc. Jpn, 73, 2624 (2004).  \n[3] N. 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Murakami, H. Kasai, J. J. Kim, S. Mamishin, D. Shindo, S. Mori, and A. \nTonomura, Nat. Nanotech. 5, 37 (2010).  \n  14 \n Figure captions;  \nFigure 1 (color online).  (a) Variation of  resistivity as a function of temperature in \nLa0.875Sr0.125MnO 3. ‘PM-I’, ‘FM-M’, and ‘FM-CO/OO -I’ represent  the magnetic phase, \nsuch as,  the paramagnetic insulating, ferromagnetic metallic, and ferromagnetic charge -\norbital ordered insulating phases, respec tively. (b) [110] -zone axis electron diffraction \npatterns at  (upper) 293 K, (middle) 170 K, an d (lower) 120 K , showing the appearance \nof superlattice reflection spots at the h, k, and l + 1/2 (white arrows) and h, k, and  l + \n1/4 (yellow arrow heads ) reciprocal positions. (c) Variation of the M–H hysteresis curves \nas a function of temperature at 170, 120, and 80 K. In the figure, ‘O’, ‘M′’, and ‘M′′’ \nrepresent  the crystal structure, as  the orthorhombic, m onoclinic without CO/OO , and \nmonoclinic st ructure with CO/OO . \n \nFigure 2 (color online). Variation of magntic textures at 90 K in the (110)  plane as \na function of the strength of the external magnetic field between 20 –530 mT. The \nstrength of the magnetic fields are (a) 20, (b) 340, (c) 420, (d) 470, and (e) 530 mT, \nrespectively. The inset of (a) shows a magnified image of the zigzag magnetic s tripe \ndomain walls, indicated by the white broken squares in (a). The bar in the inset of \n(a) indicates 200 nm. (f) Schematic description of type -I and type -II magnetic \nbubbles. In (c), type -I and type -II magnetic bubbles are indicated by the broken  and \nsolid squares, respectively.  \n \nFigure 3  (color online) . Spatial configuration s of the sp ins inside the magnetic \nbubbles . (a) A Lorentz micro graph  exhibiting the presence of magnetic bubbles with \nCW and CCW spins obtained at 90 K and B = 350 mT. (b) and (c) Magnified images \nof two magnetic bubbles highlighted by broken  squares in (a). Color representations \nof the in -plane magnetization are calculated with the aid of the TIE analysis method. 15 \n The directions of the in -plane magnetization are type -I with (b)  CWW  spin rotation \nand (c) CW spin rotation . (d) A Lorentz micrograph exhibiting the magnetic bubble \nof La0.825Sr0.175MnO 3, obtained at 90 K and B = 570 mT , and the inset showing the \nin-plane magnetization mapping calculated from TIE  [27]. (e) Schematic illustration \nof a transition process from the magnetic stripe domains to the magnetic bubbles. \nMagnetization is depicted by the red arrows.  \n \nFigure 4  (color online) . (a) –(c) Variation in the magnetic textures at 170 K as a \nfunction of the strength of the a pplied magnetic fields in La0.875Sr0.125MnO 3. The \nexternal magnetic fields are (a) B = 20 mT, (b) 215 mT, and (c) 250 mT, respectively. \nThe inset of (b) shows an enlarged image of the area enclosed by broken lines in (b) \nin which the scale bar indicates 250 nm. The y ellow broken line represents  the \ncontour of the elliptical  domain.  (d)–(f) Variation in the magnetic textures at B = \n350 mT a s a function of temperature in La0.875Sr0.125MnO 3. The Lorentz micrographs  \nare obtained at (d) 100 K, (e) 130 K, and (f) 160 K, respectively.  \n \nFigure 5 (color online).  Variation  of the magnetic domains as a function of \ntemperature under a  constant magnetic field of 200 mT . The images  were obtained \nin the ( 001) crystal plane at (a) 130, (b) 150, and (c) 170 K , respectively.  The \nuniform magnetic stripe domains in (a) change to the bubble structure in (c).  In \nthe inset of (c), the magnified image of t he magnetic bubble, indicated by the yellow \narrow, is shown.  \n \nFigure 6 (color online). The T–B phase diagram  of La0.875Sr0.125MnO 3 constructed using \nthe experimental results. The circles show the experimental points. The blue area shows \nthe region in which the magnetic bubble domains were formed. The green and white \nareas show the regions in which the magnetic stripe domains and the forced FM state 16 \n were realized, respectively.  \n \n  17 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 1, A. Kotani et al. , Physical Review B  \n18 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 2, A. Kotani et al. , Physical Review B  \n19 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3, A. Kotani et al. , Physical Review B  \n \n  \n20 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 4, A. Kotani et al. , Physical Review B  \n \n  \n21 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 5, A. Kotani et al. , Physical Review B  \n \n  \n22 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 6, A. Kotani et al. , Physical Review B  \n"    },    {        "title": "2005.00720v1.Lorentz_microscopy_and_small_angle_electron_diffraction_study_of_magnetic_textures_in_La___1_x__Sr__x_MnO__3___0_15____x____0_30___the_role_of_magnetic_anisotropy.pdf",        "content": "1 \n Lorentz microscopy and small -angle electron diffraction study of \nmagnetic textures in La 1-xSrxMnO 3 (0.15 < x < 0.30): the role of \nmagnetic anisotropy  \n \nA. Kotani1, H. Nakajima1, K. Harada1,2, Y. Ishii1 and S. Mori1,* \n  \n1Department of Materials Science, Osaka Prefecture University, Sakai, Osaka 599 -\n8531, Japan.  \n2Center for Emergent Material Science, the Institute of Physical and Chemical \nResearch (RIKEN), Hatoyama, Saitama 350 -0395, Japan.  \n \n      \n \nMagnetic textures in the ferromagnetic phases of La 1-xSrxMnO 3 for 0 .15 < x < 0.30 \nhave been investigated by Lorentz microscopy combined  with  small -angle electron \ndiffraction experiments. Various types of magneti c textures characterized by stripe, \nplate -shaped , and cylindrical (magnetic bubble ) domains were found. Two distinct types \nof magnetic stripe domains appeared in the orthorhombic structure  with an inversion  \nsymmetry of La 0.825Sr0.175MnO 3, depending significantly on magnet ocrystalline  \nanisotr opy. Based on in-situ observations as functions of  temperature and the strength \nof the external magnetic field , a magneti c field -temperature phase diagram was \nconstructed,  showing the stabilization of  magnetic bubbles  in the ferromagnetic phase \nof La 0.825Sr0.175MnO 3. \n \nP.A.C.S numbers: 75.70Kw, 75.47.Lx, 75.78.Fq, 68.37.Lp  \n \n \n  2 \n Introduction  \nA number of intricate magnetic textures such as magnetic bubbles and periodic \nmagnetic stripes, which are induced by magnetic anisotropy and the uniform \nferromagne tic order due to the long -range  magnetic dipole interaction, have been \nstudied in magnetic materials [1 -5]. For example, magnetic bu bbles were  generated by \napplying a perpendicular magnetic field to periodic magnetic stripe domains in thin \nfilms such as hexago nal barium ferrite s and cobalt thin films [ 1,6,7,8]. In addition , when \na magne tic field was applied normal to thin film s of some manganese oxides such as \nLa0.875Sr0.125MnO 3 and Ru -doped La 1.2Sr1.8Mn2O7, periodic magnetic stripe domains in \nthe ferromagnetic phase turned into magnetic bubbles, which were  observed by Lorentz \nmicroscopy (LM)  [9,10,11].  \nManganese oxides with the perovskite structure exhibit ed a large number of ground \nstates and some anomalous phase transitions such as the magnetic -field -induced \ninsulator -to-metal transition  [12]. As shown in Fig. 1(a), La1-xSrxMnO 3 exhibited  a \nvariety of ground states  in the electronic and magnetic phase diagr am, which depended  \nstrongly upon the Sr concentration ( x) [13, 14 ]. The phase diagram was  mainly divided \ninto two distinct ground states,  the ferromagnetic insulating  phase with a  monoclinic \nstructure (space group  (SG) : P21/c) in x < 0.15 and the orthorhombic ferromagnetic \nmetallic  (FMM)  phase (SG: Pbnm)  in x > 0.15 . Note that the two SGs ( P21/c and Pbnm) \nhave an inversion symmetry. Our previous work on LM observation of the magnetic \ndomain structures revealed that the orthorhombic ferromagnetic phase of La1-xSrxMnO 3 \nfor x=0.175 was characterized by  nanoscale magnetic stripe domains with antiparallel \nmagnetic moments in adjacent stripe domains [15]. Conversely , in the rhombohedral \nferromagnetic phase  (SG;  R-3c), the macroscopic m agnetic plate -shaped domains were  \nstabilized. Note that La 0.825Sr0.175MnO 3 exhibited a  rhombohedral -to-orthorhombic \ntransition  at approximately 1 85 K during  cooling, accompanying a drastic change in  the \norientation of the magnetic easy axis from the [111] to the [001] directions.  \nRecently we found that magnetic domains in La 0.825Sr0.175MnO 3 depend significantly \non the relationship between the magnetic easy axis and the direction of the applied \nmagnetic field . Moreover, La0.825Sr0.175MnO 3 with  an inversion symmetry  showed  \nmagnetic cylindrical domains (magnetic bubbles) under the application of  the magnetic \nfields normal to the thin films [16]. Although a large number of magnetic textures such \nas magnetic skyrmions in magnets  without  inversion symmetry through the relativistic \nDzyaloshinski -Moriya (DM) inte raction have been studied to date  [17-27], few studies 3 \n of magnetic textures in magnets with inversion symmetry h ave been performed  [10]. In \nthis work, we systematically investigate  magnetic textures and their changes as \nfunctions of temperature and the strength of the external magne tic fields perpendicular \nto thin films in the ferromagnetic  metallic  phase of La 1-xSrxMnO 3 for 0 .15 < x < 0.30  \nwith inversion symmetry  using  LM and small -angle electron diffraction (SmAED) \nexperiments.  \n \nExperimental methods  \nSingle crystals of La 1-xSrxMnO 3 for x = 0.15, 0.175 , 0.20 , 0.25 and 0.30  were grown \nby the floating zone method. The cry stal structures and orientations in the obtained \nsingle crystals were examined by powder X-ray diffraction and back -plate Laue -type X-\nray diffraction experiments, r espectively. Thin films for LM observation were obtained \nby grinding with alumina powder and , subsequently , by Ar+-ion sputtering at room \ntemperature. In-situ LM experiments were conducted in order to clarify magnetic \ntextures and their changes by applying magnetic fields perpendicular to the thin films. \nNote that an external ma gnetic field was applied to the thin films by exciting the \nobjective lens of the conventional TEM (JEM -2100F and JEM -2010). The SmAED \nexperiments were performed at an  angular resolution of 10  -4 ~ 10  -6 rad [28,29,30 ]. For \nsimplicity, the indices  of the crystal planes and electron diffra ction spots were \nrepresented  on the basis of the cubic perovskite structure with  𝑃𝑚3̅𝑚. The in -plane \nmagnetizations of the magnetic  textures were analyzed with the aid of  a phase retrieval \ntechnique based on the transport -of-intensity equation (TIE) [31, 32]. Magnetization \ncurves were measured by using vibrating sample magnetometer (VSM) equipped with \nQuantum Design Physical Property Measurement System (PPMS).  \n \nResults  \nMagnetic textures in the ferromagnetic metallic  phase of La 1-xSrxMnO 3were  \ninvestig ated by obtaining LM (Fresnel) images and  SmAED patterns. Note that  LM \nimage s are  obtained by  imaging the intensity distribution at a distance f below or above \nthe specimen by under - or over -focusing , respectively  [33]. Figure 1(b) is a n LM image \n(f = -1000  nm) of La0.8Sr0.2MnO 3 at 297 K in the (111)  crystal plane of the \nrhombohedral  𝑅3̅c structure, in which the magnetic easy axis  is almost parallel to the \n<111> direction . A magnetic domain structure characterized as a macroscopic magnetic \ndomain  with a width of approximately  5 μm appears . As understood  in the SmAED 4 \n pattern of the inset of Fig. 1(b), the primary beam splits into two spots with an angular \nseparation of ±and there is a diffuse streak between the two split spots. Note that the \nmagnitude of corresponds to approximately rad. This implies  that the \nmacroscopic magnetic domain structure  in Fig. 1(b) comprises  180 degree  magnetic \ndomains with  Bloch -type magnetic domain walls , in which the direction s of the \nmagnetic moment s of adjacent domains are antiparallel to one another . Note that one -\nto-one correspondence between the twin structures due to the rhombohedral distortion \nand the magnetic domain wall can be seen. Conversely , magnetic textures in vari ous \ncrystal planes of the ferromagnetic metallic  phase with an orthorhombic structure  \n( Pbnm ) of La 0.825Sr0.175MnO 3 were examined  at 100 K . Note that the magnetic easy \naxis is parallel to the [001] direction in the orthorhombic structure. Figure s 1(c) and  (d) \nexhibit  LM images with a  defocused value of  Δf = -300 nm , exhibiting magnetic textures \nobtained in the (111) and  (001) crystal planes at 100 K , respectively.  Magnetic  stripe \ndomains comprise  regions separated by domain walls that appear as an alternating \narrangement of straight lines with bright and dark contrast  in the (111) crystal plane , as \nshown in  Fig. 1(c). In the SmAED pattern, t here are two split spots indicated by \n±’and there is a diffuse streak between them.  From the analysis of the SmAED \npattern, magnetic stripe domains  are characterized as the 180 degree  magnetic domain \nstructure  having the Bloch -type magnetic domain walls . Conversely , as shown in Fig. \n1(d), magnetic stripe domains were found in th e (001) crystal planes, which are  \nperpendicular  to the magnetic easy axis along the [001] direction.  In the LM image, \nalternative br ight and dark lines can be seen . Unlike the SmAED patterns of Figs. 1(b) \nand (c), the SmAED  pattern in Fig. 1(d) shows  the presence of the primary spot  in the \nreciprocal space , as indi cated by arrow O , in addition to two split spots ( +α and -α) due \nto magnetic deflection by the magnetic moment in the sample. The presence of the \nprimary  spot impl ies th at there is a magnetic component  perpendicular to the (001) \ncrystal plane. Based on the analysis of  both SmAED patterns and LM images , the spatial \ndistribution s of magnetization in each domain  are illustrated schematically in Figs. 1(e)-\n(g). The significant point of difference  between the  three distinct types of magnetic \ndomain structures  in Figs. 1(b) -(d) is that there exists magnetic moment  component s \nperpendicular to thin film s only in the magnetic domain structure of the (001) crystal \nplane of La 0.825Sr0.175MnO 3. \nChanges in  the magnetic domains due to the application of an  external magnetic \nfield were examined by in-situ LM observation. Figure 2 shows the magnetic field -5 \n dependence of a typical 180 degree  ferromagnetic domain structure in the (111) crystal \nplane of the rhombohedral structure  in La0.8Sr0.2MnO 3. When an external  magnetic field \nis applied  perpendicular to the (111) plane , magnetization in the magnetic domain s is \ngradually  saturated by the Zeeman effect , and magnetic contrasts almost disappear in \nthe field at  80 mT.  In contrast, it is found that magnetic stripe domains in the (111) \ncrystal plane of the orthorhombic structure of La 0.825Sr0.175MnO 3 changed in a different \nmanner from  those  of La 0.8Sr0.2MnO 3. Changes of magnetic stripe domains in the \northorhombic structure of La 0.825Sr0.175MnO 3 as function s of the strengths of  external \nmagnetic fields perpendicular to the thin films  are shown in Fig. 3 . At B =34 mT, the \nmagnetic stripe domains were  pinched off and elliptical isolated magnetic domains \nappeared at the ir edge s, as shown in the square region of Fig. 3(b). Note that the \nmagnified LM image and the schematic illustration of the elliptical isolated magnetic \ndomain are displayed in Fig. 3(d) , and it is found that this domain is similar to the ones  \nobserved in  the ferromagnetic phase of  La0.875Sr0.125MnO 3 [9]. As the strength  of the \nmagnetic fields increas ed, the magnetic stripe and elliptical isolated magnetic domains \nfinally disappeared at B =100 mT  in Fig. 3(c) . As the str ength of the magnetic field \ndecreased from B = 100 mT, the magnetic stripe domains appeared reversibly around  B \n= 50 mT with thermal hysteresis.  \nAs shown in Figs. 1 (c) and (d ), two distinct types of magnetic stripe domains \nappear ed in the (111) and (001) planes of the orthorhombic structure  of \nLa0.825Sr0.175MnO 3. The first were the 180 degree magnetic domains with in-plane \nmagnetization s antiparallel to those of  adjacent domains , and the second were the \nmagnetic stripe domai ns with magnetization s perpendicular to the (001) crystal planes. \nThus, we applied magnetic fields perpendicular to two distinct types of magnetic stripe \ndomains and investigated their changes as function s of the strength of the external \nmagnetic fields. Unlike the  changes observed in the  magnetic stripe domains in the \n(111) plane under the application of an  external magnetic  field,  as shown in Fig. 3, w e \nfound the appearance s of type -I and -II magnetic bubbles clockwise (CW)  and \ncounterclockwise (CCW)  spin rotations by applying magnetic field s perpendicular to \nthe magnetic stripe domains in the (001) crystal planes  [16]. Note that only the stripe \ndomains in the (001) plane changed into the bubble -type magnetic domains by applying \nexternal magenta field  perpendicular to the thin film. The transition process from the \nmagnetic stripe domains to type -I and -II magnetic bubbles ha ve been reported in our \nrecent work  [16]. Note that the type -I and -II magnetic bubbles are in accordance with 6 \n the definition in Ref. 3, and they are  illustrated in Fig. 4(a). In order to confirm the spin \nconfiguration s of type -I and -II magnetic bubbles found in the ferromagnetic phase of \nLa0.825Sr0.175MnO 3, we analyzed the LM  images showing type -I and -II magnetic bubbles \nwith the aid of the so-called phase retrieval TIE method. Figure 4 represents the LM \nimages taken at 80 K in the orthorhombic structure of La 0.825Sr0.175MnO 3 under a \nperpendicular magnetic fiel d of approximately 570 mT . The images show three kinds  of \nmagnetic domain patterns. Under a  zero field , the magnetic stripe domains were \nobserved. As the magnetic field was applied perpendicular to the thin film, the transition \nto the magnetic bubbles took place in a weak magnetic field of 400 mT . Three kinds  of \nmagnetic bubbles appeared at approximately 570 mT, as shown in Fig. 4(b). Note that \nthe LM image in Fig. 4(b) was  obtained at a defocused value of Δf = -0.2 mm. The \nspatial distribution of type -I magnetic bubbles with dark and bright contrast at the center \nstem med from the  CW and CCW  spin rotations of their domain walls. This randomness \nof the chirality of  the magnetic bubbles contrast s with  the single -chirality in DM \nhelimagnets [ 18,19]. Type -II magnetic bubbles also exhibit random chirality  with CW \nand CCW spin rotations . The LM image in Fig. 4(c) is obtained at the defocused value \nof Δf = +0.2 mm. The dark and bright contrast of the magnetic bubbles was completely \nreversed, as understood by comparing Figs. 4(b) and 4 (c). We analyzed the LM image \nshowing  the magnetic bubbles (Figs. 4( b) and 4( c)) using the TIE analysis method, \nwhich rep roduced the orientation and magnitude of the in -plane magnetization [32 -36]. \nNote that the color mapping in the inset of Fig. 4( e) exhibited  the direction of the in -\nplane mag netization. The directions of the in -plane magnetization s of type -I and -II \nmagnetic bubbles are visualized  in Fig. 4(d). The b lack color at the center s of type -I \nmagnetic bubbles shows t hat no in -plane magnetizat ion was  present , thus magnetization \nis assumed to be vertical to the thin film with magnetization being  parallel to the [001] \ndirection. The magnified images of type -I magnetic bubbles with left - and right -handed \nspin rotations and those of type -II magnet ic bubbles are  shown in Figs. 4( e), (f) and (g), \nrespectively, together with color representations of the in -plane magnetization s \ncalculated using  the TIE analysis method.   \nTo clarify the stabilizing r egion of  the magnetic  bubbles , we investigated the \ndisappearance of  magnetic bubbles as a function of the temperature and the strength of \nthe external magnetic fields.  Figure 5 shows  a series of changes in the  magnetic bubbles \non warming from  90 K at a constant magnetic field of 500 mT. The average diameter  of \nthe magnetic bubbles was approximately 300 nm at 90K. Upon  increasing the 7 \n temperature from 90 K,  magnetic bubbles shrunk gradually from 300  to 200 nm at 165 \nK and disappeared above  185 K, as shown in Fig . 5(d). Note that a  struc tural phase \ntransition from rhombohedral to orthorhombic structure occurred at approximately 1 85 \nK, during  which the magnetic easy axis changed from the [001] direction to the [111] \ndirection. Conversely , when the temperature  decreased below  200 K , magnetic bubbles \nappear ed reversibly around 185  K and the diameter s of the magnetic bubbles increased \nto 300 nm at 90  K. These reversible change s with respect to temperature at a constant \nstrength of B= 500 mT indica te that magnetic bubbles exist ed as a stable state in  the \ndefinite range of  the B-T phase diagram.  \nThe stability of the magnetic bubb les as a function of the strength of external \nmagnetic fields was investigated at constant temperatures of 90, 120  and 165  K in the \northorhombic phase. The average diameter  (r) of the magnetic bubbles was \napproximately 300 nm  at T = 90 K and B = 400 mT. When the strength s of the external \nmagnetic fields increased, t he diameter s (r) of the magnetic bubbles decreased and until \nthe magnetic bubbles disappeared at B = 720 mT. Similar tendencies were exhibited at \nT = 120 and 165 K , as shown in Fig. 6(a) . Based on these experimental results, the B-T \nphase diagram that shows  the stability of the magnetic bubbles is presented in Fig. 6(b) . \nNote that the specimen thickness  in the LM observation  is approximately  100 ~ 150 nm. \nThe range in which magnetic bubbles exist stably in the B-T phase diagram depends \nstrongly upon the specimen thickness , as in the B-T phase diagram of the magnetic \nskyr mion in some helical magnets [18 ].  \n \nDiscussion  \nHere  let us briefly discuss the magnetic bubbles found in the ferromagnetic metallic  \nphase of La1-xSrxMnO 3. Formation of magnetic bubbles depends strongly upon magnetic \nanisotropy. The m agnetic easy axis is parallel to the [ 001] direction in the orthorhombic \nphase and  to the <111> direction  in the rhombohedral phase . The magnetic 180 degree  \ndomain structure with  in-plane magnetization is formed in the (111) cr ystal plane of the \nrhombohedral phase , as shown in Fig. 2, although the magnetic easy axis is parallel to \nthe incidence  direction of the electron beam. This clearly implies that the \nmagnetocrystalline anisotropy in the rhombohedral structure is relatively low and that \nthe shap e anisotropy is more dominant . As a result, magnetic bubbles are not formed by \nthe application o f external magnetic field s perpendicular  to thin films  in the \nrhombohedral structure . Conversely , Nagai et al . obtained  a magnetic anisotropy 8 \n constant of Ku = 2.1 × 105 J/m3 in the ferromagnetic  phase of La 0.875Sr0.125MnO 3 at 100 \nK [9], which is larger than Ku ~ 1.0 × 104 J/m3 for the  ferromagnetic phase of \nLa0.7Sr0.3MnO 3 at 300 K. Moreover, it is believe d that this difference arises from the \nfact that magnetic anisot ropy in the ferromagnetic  phase is enhanced by orbital o rdering .   \nWe examined variations of magnetic anisotropy constants ( Ku) as a function of \ntemperature in the ferromagnetic phase with the orthorhombic structure of \nLa0.825Sr0.175MnO 3. Figure 7 shows changes of magnetization curves as a function of \ntemperature in the ferromagnetic state of La 0.825Sr0.175MnO 3 with the orthorhombic \nstructure. Magnetic fields were applied along the c axis and th e ab plane perpendicular \nto the c axis. Values of saturated magnetization at T= 90 K, 120 K and 165 K are \napproximately 3.7, 3.6 and 3.4 B, respectively. From magnetization curves at T= 90 K, \n120 K and 165 K, we obtained the anisotropic  magnetic  field (  Hk ) in the ferromagnetic \nphase with the orthorhombic structure. Note that H k is defined as the critical value of \nthe magnetic field, above which the difference in magnetization between H // c and H// ab \nis less than 2 %. Values of H k at T= 90 K, 120 K and 165 K are 5.3, 5.2 and 4.8 kOe, \nrespectively. The Ku values obtained based on  the anisotropic  magnetic field T= 90 K, \n120 K and 165 K are, respectively,  1.52 × 105 J/m3, 1.47 × 105 and 1.30 × 105 J/m3 , \nwhich are  larger than that of rhombohedral structure s such as La 0.7Sr0.3MnO 3. Based on \nthese considerations, i t is concluded that the magnetic stripe domains in the \northorhombic structure with magnetic moments perpendicular to  the thin film are \ntransformed into magnetic bubble s by the application of external magnetic field s \nperpendicular to the thin film.  \n \nConclusion  \nWe revealed magnetic textures in the ferromagnetic phase s with the inversion \nsymmetry  of La 1-xSrxMnO 3 for 0 .15 < x < 0.30 . These textures were  characterized as \nstripe s, plate -shapes  and magnetic bubbles. By applying external magnetic fields \nperpendicular to the (001) crystal plane of the ferromagnetic metallic phase in \nLa0.825Sr0.175MnO 3, the magnetic stripe doma ins changed into type -I and -II magnetic \nbubbles. Magnetic  states with type -I and -II magnetic bubbles in the ferromagnetic \nmetallic  phase exist sta bly in the temperature -magnetic -field phase diagram.  \n \nACKNOWLEDGEMENTS  \nThis work was partially supported by a Grant -in-Aid for Scientific Research from the 9 \n Ministry of Education, Culture, Sports, Science and Technology of Japan. The authors \nwould like to thank Dr. Y. Togawa and Dr. T. Koyama for useful discussion .  \n \nREFERENCES  \n* Corresponding author: mori@mtr.osakafu -u.ac.jp  \n[1] P. J. Grundy and S. R. Herd, P hys. Status Solidi (a) 20, 295 (1973) . \n[2] T. Garel and S. Doniach, Phys.  Rev. B 26, 325 (1982) . \n[3] T. Suzuki, et al ., J. Magn . Magn . 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Matsui , and Y. Tokura , Science 311, 359 (2006).  11 \n Figure captions  \nFigure 1.  (color online)  \n(a) Electronic phase diagram of La 1-xSrxMnO 3, in which O, R and M denote the \northorhombic, rhombohedral and monoclinic crystal systems and PM, CA and FM \ndenote the paramagnetic, canted antiferromagnetic , and ferromagnetic phases, \nrespectively [13]. (b) Ferromagnetic domain structures in the (111)  crystal plane of the \nrhombohedral  structure of La 0.8Sr0.2MnO 3 at room temperature . (c), (d) Fresnel images \nshowing magnetic stripe domains in the (c) (111) and (d) the  (001)  plane s of the \northorhombic  structure of La 0.825Sr0.175MnO 3 at 100 K. In the inset s of (b) -(d), SmAED \npatterns are shown. (e)-(f) Schematic depictions  of the magnetic domains with the in -\nplane antiparallel magnetization in the (111) plane of (e) the rhombohedral and (f) the \northorhombic structures , respectively.  (g) Schematic description of the magnetic \ndomains with magnetic moments normal to the (001) plane  in the orthorhombic  sturcture . \nRed arrows represent  the directions of magnetization in each magnetic domain . The \ndefocused values in ( b), (c) and ( d) are Δf = -1000 nm , -300 nm, -300 nm, respectively .  \n \nFigure 2.  (color online)  \nChanges of magnetic stripe domains in the (111) crystal plane as a function of the \nstrength of magnetic field perpendicular to the thin film of La 0.8Sr0.2MnO 3. The \nstrengths of the magnetic field were (a) 20 mT, (b) 50 mT and (c) 80 mT . Red arrows \nrepresent the directions of the magnetic moments in each magnetic domain.  \n \nFigure 3.  (color online)  \nChanges of magnetic stripe domains in the (111) crystal plane as a function of the \nstrength of the magnetic field perpendicular to the thin film of La 0.825Sr0.175MnO 3. The \nstrengths of the magnetic field were (a) 20 mT, (b) 34  mT and (c) 100 mT . (d) Magnif ied \nellipsoidal magnetic domain  obtained from the square region of (c) and schematic \nillustration of the ellipsoidal magnetic domain . The r ed arrow shows the direction of \nthe magnetic moment inside the ellipsoidal magnetic bubble  and the magnetic moment \nis perpendicular to thin film  outside . \n \nFigure 4 . (color online)  \n(a) Schematic illustration s of type -I and -II magnetic bubble s. Red arrows represent the \ndirection of the magnetic moment. (b), (c)  Magnetic bubbles generated by applying 12 \n external magnetic field s perpendicular ly to a thin film  of La0.825Sr0.175MnO 3 in the (001) \ncrystal plane . The strength  of the applied magnetic field is approximately 570 mT . The \nimages were taken at the defocused values of (b) Δf= -0.2 mm and (c) Δf=+0.2 mm. In \n(c), the type -I magnetic bubbles with left -handed and right -handed spin rotation s, and \nthe type -II magnetic bubbles are  indicated by capital letters A, B , and C, respectively. \n(d)-(g) Color representations of the in -plane magnetization calculated with the aid of \nthe TIE analysis method. The directions of the magnetization are (e) type -I with left -\nhanded spin rotation, (f) type -I with right -handed spin rotation , and (g) type -II magnetic  \nbubbles , which were  indicated by color mapping and vectorial representations by white \narrows.  \n \nFigure 5. (color online)   \nTemperature variations  of the magnetic bubble  domains in the ( 001) crystal plane under \na constant magnetic field of 500 mT normal to the thin film of La 0.825Sr0.175MnO 3. The \nimages  were obtained at (a) 90 K, (b) 140 K (c) 165  K and (c) 200 K, respectively.  \n \nFigure 6 (color online)  \n(a) Variations of the average diameters (r) of magnetic bubbles as function s of strength \nof the applied magnetic fields at 90, 120,  and 165  K, respectively . (b) The B-T phase \ndiagram  constructed on  the basis of the experimental results in this study , in which \ncircles indicate exper imentally -measured  point s. \n \nFigure 7 (color online)  \nMagnetization curves of La0.825Sr0.175MnO3 at 90 K, 120 K  and 165 K. Magnetic field \n(H) was applied parallel  (red circles)  or perpe ndicular  (blue circles)  to the c axis. Each \nsolid triangle shows the anisotropic field.  \n \n 13 \n  \n \nPhysical Review B  \nFigure 1, A. Kotani et al . \n14 \n  \n \nPhysical Review B  \nFigure 2, A. Kotani et al . \n \n15 \n  \n \n \nPhysical Review B  \nFigure 3, A. Kotani et al . \n \n16 \n  \n \nPhysical Review B  \nFigure 4, A. Kotani et al . \n \n \n \n17 \n  \n \n \nPhysical Review B  \nFigure 5, A. Kotani et al . \n \n \n  \n18 \n  \n \n \n \nPhysical Review B  \nFigure 6, A. Kotani et al . \n  \n19 \n  \n \n \n \n \n \nPhysical Review B  \nFigure 7, A. Kotani et al . \n \n"    }]